Fixed point theorems in partially ordered spaces and aplications

Fixed Point Theorems in Partially Ordered Spaces and Applications Tesis Doctoral Jackie Harjani Sauco Las Palmas de Gran Canaria Mayo de 2014 Fixed Point Theorems in Partially Ordered Spaces and Applications Jackie Harjani Sauco Departamento de Física P´agina reservada para los servicios administrativos de la Universidad de Las Palmas de Gran Canaria SALVADOR GALVÁN HERRERA, SECRETARIO DEL DEPARTAMENTO DE FÍSICA DE LA UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA, CERTIFICA, Que la Comisión de Doctores del Departamento en su sesión de fecha 19 de mayo de 2014 tomó el acuerdo de dar el consentimiento para su tramitación, a la tesis doctoral titulada “Fixed Point Theorems in Partially Ordered Spaces and Applications“, presentada por el doctorando D. Jackie Harjani Sauco, dirigida por el Dr. D. Kishin Sadarangani y codirigida por la Dra. Dª Belén López Brito. Y para que así conste, y a efectos de lo previsto en el Artº 73.2 del Reglamento de Estudios de Doctorado de esta Universidad, firmo la presente en Las Palmas de Gran Canaria, a diecinueve de mayo de dos mil catorce. Tesis Doctoral Fixed Point Theorems in Partially Ordered Spaces and Applications Jackie Harjani Sauco Doctorando Kishin Sadarangani Director Bel´ en L´ opez Brito Codirectora Departamento de F´ ısica Programa de Doctorado: F´ ısica, Matem´ aticas, Geolog´ ıa y Clima Las Palmas de Gran Canaria a Diecis´ eis de Mayo de 2014 Tesis presentada por Jackie Harjani Sauco para aspirar al grado de Doctor por la Universidad de Las Palmas de Gran Canaria, con la aprobaci´on y el visto bueno del Dr. Kishin Sadarangani y la Dra. Bel´ en L´ opez Brito, junto con los informes favorables del Dr. J¨ urgen Appell y el Dr. Adrian Petrus¸el. En Las Palmas de Gran Canaria a Diecis´eis de Mayo de 2014 ´ Indice general 1. Resumen de la Tesis / Summary of the thesis in Spanish 1 1.1. Teoremas del punto fijo en espacios m´etricos ordenados . . . . 9 1.1.1. Fixed point theorems for weakly contractive mappings in partially ordered sets . . . . . . . . . . . . . . . . . 11 1.1.2. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations 15 1.1.3. Contractive-like mapping principles in ordered metric spaces and applications to ordinary differential equations 19 1.1.4. Fixed point theorems for mappings satisfying a condition of integral type in partially ordered sets . . . . . . 23 1.1.5. A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially orderedmetricspace...................... 27 1.1.6. Fixed point theorems for weakly C-contractive mappings in ordered metric spaces . . . . . . . . . . . . . . 29 1.1.7. Fixed point theorems for mixed monotone operators and application to integral equations . . . . . . . . . . 33 1.1.8. A fixed point theorem for Meir-Keeler contractions in ordered metric spaces . . . . . . . . . . . . . . . . . . . 39 1.2. Teor´ıa de existencia y unicidad para las soluciones de un bvp . 41 1.2.1. Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem . . . . 43 IX 1.2.2. On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets............................. 47 1.2.3. Uniqueness of positive solutions for a class of fourthorder boundary value problems . . . . . . . . . . . . . 49 1.3. Fractional boundary value problem . . . . . . . . . . . . . . . 53 1.3.1. Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . . . . . . 53 1.3.2. Positive solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . . . . 57 1.3.3. Existence and uniqueness of positive solution for a boundary value problem of fractional order . . . . . . . 61 1.3.4. On existence and uniqueness of positive solutions to a class of fractional boundary value problems . . . . . . . 65 1.3.5. Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations . . . . . . . . . . . . . . . . . . . . 67 2. A short history approach 69 3. Fixed point theorems in partially ordered metric spaces 75 3.1. Fixed point theorems for weakly contractive mappings ... . . . 79 3.2. Generalized contractions in partially ordered metric spaces ... . 91 3.3. Contractive-like mapping principles in ordered metric spaces ... 105 3.4. Fixed point theorems for mappings satisfying a condition of ... 123 3.5. A fixed point theorem for mappings satisfying a contractive ... 141 3.6. Fixed point theorems for weakly C-contractive mappings in ... 151 3.7. Fixed point theorems for mixed monotone operators and ... . . 161 3.8. A fixed point theorem for Meir-Keeler contractions in ... . . . 179 4. Theory of existence and uniqueness of solution for boundary value problems 189 4.1. Classical boundary value problem . . . . . . . . . . . . . . . . 193 4.1.1. Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem . . . . 193 4.1.2. On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets.............................207 4.1.3. Uniqueness of positive solutions for a class of fourthorder boundary value problems . . . . . . . . . . . . . 219 4.2. Fractional boundary value problem . . . . . . . . . . . . . . . 237 4.2.1. Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . . . . . . 237 4.2.2. Positive solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . . . . 251 4.2.3. Existence and uniqueness of positive solution for a boundary value problem of fractional order . . . . . . . 263 4.2.4. On existence and uniqueness of positive solutions to a class of fractional boundary value problems . . . . . . . 279 4.2.5. Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations . . . . . . . . . . . . . . . . . . . . 291 5. Future lines of research 301 5.1. Operators of cyclical type . . . . . . . . . . . . . . . . . . . . 303 5.2. Best proximity point: approximation and optimization . . . . 305 5.3. Fixed points of decreasing operators and applications . . . . . 307 Bibliography 309 Cap´ıtulo 1 Resumen de la Tesis / Summary of the thesis in Spanish 1 2 1.1. Teoremas del punto fijo en espacios m´etricos ordenados ........................ 9 1.1.1. Fixed point theorems for weakly contractive mappings in partially ordered sets . . . . . . . . . . . . 11 1.1.2. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations ....................... 15 1.1.3. Contractive-like mapping principles in ordered metric spaces and applications to ordinary differential equations . . . . . . . . . . . . . . . . . . . 19 1.1.4. Fixed point theorems for mappings satisfying a condition of integral type in partially ordered sets 23 1.1.5. A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space . . . . . . . . . . . . . . 27 1.1.6. Fixed point theorems for weakly C-contractive mappings in ordered metric spaces . . . . . . . . . 29 1.1.7. Fixed point theorems for mixed monotone operators and application to integral equations . . . . . 33 1.1.8. A fixed point theorem for Meir-Keeler contractions in ordered metric spaces . . . . . . . . . . . . . . . 39 1.2. Teor´ıa de existencia y unicidad para las solucionesdeunbvp..................... 41 1.2.1. Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem . 43 1.2.2. On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem inorderedsets .................... 47 1.2.3. Uniqueness of positive solutions for a class of fourth-order boundary value problems . . . . . . . 49 1. Resumen de la Tesis / Summary of the thesis in Spanish 3 1.3. Fractional boundary value problem . . . . . . . . 53 1.3.1. Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . 53 1.3.2. Positive solutions for a class of singular fractional boundary value problems . . . . . . . . . . . . . . 57 1.3.3. Existence and uniqueness of positive solution for a boundary value problem of fractional order . . . 61 1.3.4. On existence and uniqueness of positive solutions to a class of fractional boundary value problems . 65 1.3.5. Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations . . . . . . . . . . . . . . . . . 67 1. Resumen de la Tesis / Summary of the thesis in Spanish 5 Recientemente, en la Teor´ıa del punto fijo, han aparecido muchos resultados que obtienen condiciones suficientes para la existencia de un punto fijo si trabajamos con aplicaciones en un conjunto dotado de un orden parcial. Generalmente, estos resultados combinan dos teoremas del punto fijo fundamentales: el Teorema de la contracci´on de Banach y el Teorema de Knaster-Tarski. El Teorema de la contracci´on de Banach fue demostrado en 1922 y su enunciado es el siguiente. Teorema 1 (Teorema de la contracci´on de Banach).Sea (X, d)un espacio m´etrico completo y T:X→Xuna aplicaci´on tal que, existe λ∈[0,1) y se verifica d(Tx, Ty)≤λ d(x, y)para todo x, y ∈X. Entonces Ttiene un ´unico punto fijo ¯x∈X(i.e., T¯x= ¯x). Adem´as, para cada x∈Xla sucesi´on {Tnx}converge a ¯x. Se han llevado a cabo un gran n´umero de generalizaciones de este principio, donde la condici´on contractiva que aparece en el Teorema 1 es reemplazada por otras condiciones (ver, B. E. Rhoades, A comparison of various definitions of contractive mappings. Transactions of the Amer. Math. Soc. 226, (1977), 257–290). El Teorema de Knaster-Tarski fue probado en 1955 y su formulaci´on viene recogida en el siguiente teorema. Teorema 2 (Teorema de Knaster-Tarski).Sea (A, ≤)un lattice completo (esto significa que cada subconjunto Atiene ´ınfimo y supremo en A) y T:A→Auna aplicaci´on que preserva el orden. Entonces Ttiene un punto fijo. Los teoremas del punto fijo en los espacios m´etricos ordenados est´an ´ıntimamente ligados con la monoton´ıa (tanto si preservan el orden como si no) de las aplicaciones, donde la condici´on contractiva es solamente satisfecha por loa elementos que son comparables. 12 Teoremas del punto fijo en espacios m´etricos ordenados contexto de los espacios m´etricos parcialmente ordenados. La principal contribuci´on puede ser resumida en el siguiente teorema. Teorema 6. Sea (X, ≤)un espacio parcialmente ordenado y supongamos que existen una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on continua y creciente tal que d(Tx, Ty)≤d(x, y)−ψd(x, y),para x, y ∈Xcon x≥y , donde ψ: [0,∞)→[0,∞)es una funci´on continua y creciente tal que es positiva en (0,∞)yψ(0) = 0. Si existe x0∈Xtal que x0≤Tx0entonces Ttiene un punto fijo. Demostramos que la condici´on de que Tsea continua es innecesaria si asumimos que en Xse cumple si (xn) es una sucesi´on creciente en Xtal que xn→x entonces xn≤x, para todo n∈N.(1.1) Esta condici´on fue usada por J. Nieto y R. Rodr´ıguez-L´opez en [118]. Con mayor precisi´on, demostramos el siguiente resultado. Teorema 7. Si en el Teorema 6 reemplazamos la continuidad de Tpor la condici´on (1.1) entonces obtendr´ıamos la misma conclusi´on. Para analizar la unicidad del punto fijo, aportamos un ejemplo en donde se comprueba que los Teoremas 6 y 7 no garantizan la unicidad. Por este motivo, presentamos una condici´on suficiente para la unicidad (que fue usada en [118]). Su enunciado es: Para x, y ∈Xexiste z∈Xque es comparable con xyy. (1.2) Teorema 8. A˜nadiendo la condici´on (1.2) a las hip´otesis del Teorema 6 (respectivamente Teorema 7), obtenemos la unicidad del punto fijo. Finalmente, aplicamos los resultados obtenidos para demostrar la exis- 1. Resumen de la Tesis / Summary of the thesis in Spanish 13 tencia de la soluci´on del siguiente problema peri´odico de primer orden (u0(t) = ft, u(t), t ∈[0, T], u(0) = u(T),(1.3) bajo la hip´otesis de la existencia de una subsoluci´on para (1.3), es decir, una funci´on α∈ C[0, T] tal que α0(t)≤ft, α(t),para t∈[0, T], α(0) ≤α(T). Es decir, obtenemos el siguiente resultado. Teorema 9. Supongamos que f: [0, T]×R→Res continua y existe λ > 0 tal que para x, y ∈Rcon y≥x, 0≤f(t, y) + λy −[f(t, x) + λx]≤λln(y−x+ 1) . Entonces, la existencia de una subsoluci´on para (1.3) nos garantiza la existencia de una ´unica soluci´on de (1.3). Este art´ıculo se inspira, fundamentalmente, en los resultados obtenidos por J.J. Nieto y R. Rodr´ıguez-L´opez en Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, (2005), 223–239.. 1. Resumen de la Tesis / Summary of the thesis in Spanish 15 1.1.2. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations En este art´ıculo hacemos uso de las funciones que alteran la distancia, que fueron introducidas por Khan, Swalesh y Sessa en Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30, (1984), 1–9. Diremos que una funci´on ϕ: [0,∞)→[0,∞) es una funci´on que altera la distancia si satisface: (a) ϕes continua y creciente. (b) ϕ(t) = 0 si y s´olo si t= 0. (Notar que estas funciones son las mismas que aparecen en la definici´on de aplicaci´on d´ebilmente contractivas en el art´ıculo anterior). En [161], los autores prueban este resultado: Teorema 10. Sea (X, d)un espacio m´etrico completo, ϕuna funci´on que altera la distancia y T:X→Xuna aplicaci´on tal que ϕd(Tx, Ty)≤c·ϕd(x, y), para cada x, y ∈X, donde c∈(0,1). Entonces Ttiene un ´unico punto fijo. En 2008, Dutta y Choudhury generalizaron el Teorema 5, obtenido por Geraghty en [136], de la siguiente manera. Teorema 11. Sea (X, d)un espacio m´etrico completo y T:X→Xuna aplicaci´on que satisface ϕd(Tx, Ty)≤ϕd(x, y)−φd(x, y),para cada x, y ∈X , donde ϕyφson funciones que alteran la distancia. Entonces Ttiene un ´unico punto fijo. 16 Teoremas del punto fijo en espacios m´etricos ordenados El principal objetivo del art´ıculo es presentar la versi´on del Teorema 11 en el contexto de espacios m´etricos parcialmente ordenados. Nuestra contribuci´on m´as importante puede ser resumida en los siguientes teoremas. Teorema 12. Sea (X, ≤)un espacio parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on continua y creciente tal que ψd(Tx, Ty)≤ψd(x, y)−φd(x, y)para cada x, y ∈Xcon x≥y , donde ψyφson funciones que alteran la distancia. Si existe x0∈Xy x0≤Tx0, entonces Ttiene un punto fijo. Teorema 13. Si en el Teorema 12 reemplazamos la condici´on de continuidad de Tpor si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤xpara todo n∈N, entonces llegar´ıamos a la misma conclusi´on. Teorema 14. A˜nadiendo la condici´on: Para x, y ∈Xexiste z∈Xque es comparable con xyy , a las hip´otesis del Teorema 12 (respectivamente Teorema 13) obtenemos la unicidad del punto fijo. Los principales resultados de nuestros art´ıculos anteriores, se obtienen como casos particulares del obtenido en este art´ıculo. Finalmente, presentamos dos ejemplos sobre problemas con valores en la frontera donde se pueden aplicar nuestros resultados. En el primer ejemplo, estudiamos la existencia de soluciones para este pro- 1. Resumen de la Tesis / Summary of the thesis in Spanish 17 blema peri´odico de primer orden (u0(t) = ft, u(t), t ∈[0, T], u(0) = u(T).(1.4) Nuestro resultado ser´ıa: Teorema 15. Bajo la hip´otesis de que f: [0, T]×R→Res continua y suponiendo que existen dos n´umeros reales positivos λ, α > 0que satisfacen α≤2λ(eλT −1) T(eλT + 1) 1 2 , y tales que, para x, y ∈Rcon y≥x 0≤f(t, y) + λy −f(t, x) + λx≤αqln (y−x)2+ 1. Entonces la existencia de una subsoluci´on en (1.4) (ver los comentarios del art´ıculo anterior) nos garantiza la existencia de una ´unica soluci´on para (1.4). El segundo ejemplo estudia la existencia de una soluci´on para la siguiente ecuaci´on diferencial de segundo orden con valores en la frontera para dos puntos.    −d2x dt2=f(t, x), t ∈[0,1] , x(0) = x(1) = 0 . (1.5) Obteniendo este resultado. Teorema 16. Bajo la hip´otesis de que f: [0,1] ×R→Res continua y creciente con respecto a la segunda variable, y tal que, para cada x, y ∈R con y≥x, f(t, y)−f(t, x)≤αpln[(y−x)2+ 1] , donde 0< α ≤8, entonces el Problema (1.5) tiene una ´unica soluci´on positiva. Adem´as, si f(t, x)6= 0 para t∈(0,1), la soluci´on de (1.5) es positiva 18 Teoremas del punto fijo en espacios m´etricos ordenados (esto significa que 0< x(t)para t∈(0,1)). 1. Resumen de la Tesis / Summary of the thesis in Spanish 19 1.1.3. Contractive-like mapping principles in ordered metric spaces and applications to ordinary differential equations En[58], Geraghty present´o una generalizaci´on del principio de la contracci´on de Banach usando la clase de funciones Sdadas por β: [0,∞)→[0,1) que satisfacen β(tn)→1⇒tn→0. El resultado m´as importante de [58] lo mostramos a continuaci´on. Teorema 17. Sea (X, d)un espacio m´etrico completo y T:X→Xuna aplicaci´on satisfaciendo d(Tx, Ty)≤βd(x, y)·d(x, y)para cada x, y ∈X , donde β∈ S. Entonces Ttiene un ´unico punto fijo. En 2010, Amini-Harandi y Emami demostraron una version del Teorema 17 en el contexto de espacios parcialmente ordenados (ver, [9]). La principal aportaci´on de [9] se resume en este teorema. Teorema 18. Sea (X, ≤)un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on creciente tal que d(Tx, Ty)≤βd(x, y)·d(x, y)para cada x, y ∈Xcon x≤y , donde β∈ S. Asumimos que Tes continua o Xsatisface la siguiente condici´on: si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤x, para todo n∈N. Adem´as, supongamos que para cada x, y ∈Xexiste z∈Xcomparable con x 20 Teoremas del punto fijo en espacios m´etricos ordenados ey. Si existe x0∈Xcon x0≤Tx0entonces Ttiene un ´unico punto fijo. Nuestro prop´osito en este art´ıculo era extender el Teorema 18 haciendo uso de las funciones que alteran la distancia (consultar los art´ıculos previos). La principal aportaci´on del art´ıculo puede ser resumida en el siguiente teorema. Teorema 19. Sea (X, ≤)un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on creciente tal que ψd(Tx, Ty)≤βd(x, y)·ψd(x, y)para cada x, y ∈Xcon x≤y, donde ψes una funci´on con distancia alterada y β∈ S. Asumamos tambi´en que Tes continua o Xsatisface la siguiente condici´on si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤x, para todo n∈N. Si existe x0∈Xcon x0≤Tx0entonces Ttiene un punto fijo. En el art´ıculo damos un ejemplo para ver que las hip´otesis del Teorema 19 no garantizan la unicidad del punto fijo. El siguiente resultado aporta una condici´on suficiente para la unicidad del punto fijo. Teorema 20. A˜nadiendo al condici´on: Para cada x, y ∈Xexiste z∈Xque es comparable con xyy , a las hip´otesis del Teorema 19, obtenemos la unicidad del punto fijo. Para ilustrar la aplicabilidad de los resultados obtenidos, damos un teorema que garantiza la existencia de soluci´on del problema peri´odico de primer orden (u0(t) = ft, u(t), t ∈[0, T], u(0) = u(T).(1.6) 1. Resumen de la Tesis / Summary of the thesis in Spanish 21 Necesitamos introducir la clase Ade funciones φ: [0,∞)→[0,∞) que satisfacen (a) φes creciente. (b) φ(x)< x, si x > 0. (c) β(x) = φ(x) x∈ S, El resultado ser´ıa: Teorema 21. Supongamos que f:I×R−→ Res continua y existe λ, α > 0 que satisfacen α≤ 2λ(eλT −1) T(eλT + 1) !1 2 , tal que para x, y ∈Rcon x≤y 0≤f(t, y) + λy −f(t, x) + λx≤αp(y−x)φ(y−x), donde φ∈ A. Entonces, la existencia de una subsoluci´on de (1.6) garantizar´ıa la existencia de una ´unica soluci´on de (1.6). Adem´as, en el art´ıculo se demuestra que si φ∈ A entonces la funci´on ϕ(x) = pxφ(x) satisface que ϕ∈ A Esta idea es fundamental a la hora de presentar un ejemplo del Problema (1.6) que puede ser tratado con los resultados del art´ıculo pero no puede estudiarse con los teoremas del art´ıculo [9]. 28 Teoremas del punto fijo en espacios m´etricos ordenados es m´as fuerte que la utilizada en los art´ıculos anteriores: si (xn) es una sucesi´on creciente en Xtal que xn→x entonces xn≤xpara todo n∈N. Para poder obtener la unicidad del punto fijo, usamos la misma condici´on que en los art´ıculos previos y demostramos este resultado. Teorema 28. A˜nadiendo la condici´on para x, y ∈Xexiste z∈Xque es comparable con xyy , a las hip´otesis del Teorema 27, obtenemos la unicidad del punto fijo. Finalmente, presentamos un ejemplo en donde se puede aplicar el Teorema 27 mientras que no puede ser abordado por el Teorema 26. 1. Resumen de la Tesis / Summary of the thesis in Spanish 29 1.1.6. Fixed point theorems for weakly C-contractive mappings in ordered metric spaces Choudhry en [B. S. Choudhury, Unique fixed point theorem for weak Ccontractive mappings, Katmandu University Journal of Science, Engineering and Technology, vol. 5, (1), (2009) 6–13], introdujo esta definici´on. Definition 1. Una aplicaci´on T:X→X, donde X, des un espacio m´etrico, se dice que es d´ebilmente C-contractiva (o d´ebil C-contracci´on) si para todo x, y ∈X, dTx, Ty≤1 2d(x, Ty) + d(y, Tx)−ϕd(x, Ty), d(y, Tx), donde ϕ: [0,∞)2→[0,∞)es una funci´on continua verificando que ϕ(x, y) = 0si y s´olo si x=y= 0. El autor tambi´en demuestra este resultado. Teorema 29. Supongamos que X, des un espacio m´etrico completo y T:X→Xes una aplicaci´on d´ebilmente C-contractiva, entonces Ttiene un ´unico punto fijo. El principal prop´osito de nuestro art´ıculo es dar una versi´on del Teorema 29 en el contexto de los espacios m´etricos parcialmente ordenado. El primer resultado que obtuvimos fue: Teorema 30. Sea X, ≤un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que X, des un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on creciente tal que dTx, Ty≤1 2dx, Ty+dy, Tx−ϕdx, Ty, dy, Tx, para cada x, y ∈Xcon x≥y, donde ϕ: [0,∞)2→[0,∞)es una funci´on continua cumpliendo que ϕ(x, y) = 0 si y s´olo si x=y= 0. 30 Teoremas del punto fijo en espacios m´etricos ordenados Asumamos adem´as que Tes continua o xsatisface la condici´on si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤xpara todo n∈N. Si existe x0∈Xtal que x0≤Tx0entonces Ttiene un punto fijo. Damos un ejemplo para ilustrar que las hip´otesis del Teorema 30 no garantizan la unicidad del punto fijo. Nuestro siguiente resultado fue dar una condici´on suficiente que nos garantizase la unicidad del punto fijo. Teorema 31. A˜nadiendo la condici´on para cada x, y ∈Xexiste z∈Xque es comparable con xyy , a las hip´otesis del Teorema 30, obtenemos la unicidad del punto fijo. Adem´as, en el art´ıculo se analiza qu´e ocurre si el operador Tes decreciente. Teorema 32. Sea X, ≤un conjunto parcialmente ordenado tal que para cada x, y ∈Xexiste z∈Xque es comparable con xyy. Supongamos que existe una m´etrica den Xtal que X, des un espacio m´etrico completo y sea T:X→Xuna aplicaci´on decreciente satisfaciendo dTx, Ty≤1 2dx, Ty+dy, Tx−ϕdx, Ty, dy, Tx, para cada x, y ∈Xcon x≥y, donde ϕ: [0,∞)2→[0,∞)es una funci´on continua tal que ϕ(x, y)=0si y s´olo si x=y= 0. Entonces (i) Si existe x0∈Xcomparable con Tx0entonces inf{d(x, Tx): x∈X}= 0. (ii) Si, adem´as, Xes compacto y Tes continua, entonces Ttiene un ´unico punto fijo. 1. Resumen de la Tesis / Summary of the thesis in Spanish 31 Para finalizar, damos un ejemplo que puede ser estudiado con el Teorema 30 y no puede ser tratado con el Teorema 29. 1. Resumen de la Tesis / Summary of the thesis in Spanish 33 1.1.7. Fixed point theorems for mixed monotone operators and application to integral equations Los operadores mon´otonos mixtos fueron introducidos por Guo y Lakshmikantham en [63]. Su definici´on es la siguiente. Definition 2. Sea (X, ≤)un conjunto parcialmente ordenado y F:X×X→ Xuna aplicaci´on. Se dice que Ftiene la propiedad mon´otona mixta si F(x, y) es creciente en xy es decreciente en y, esto es, para cada x, y ∈X, x1, x2∈X, x1≤x2⇒F(x1, y)≤F(x2, y), y1, y2∈X, y1≤y2⇒F(x, y1)≥F(x, y2). Definition 3. Sea F:X×X→Xuna aplicaci´on. Un par (x, y)∈X×X se dice que es un punto fijo acoplado del operador Fsi F(x, y) = xyF(y, x) = y . En [18] Bhaskar y Lakshmikantham probaron el siguiente teorema del punto fijo. Teorema 33. Sea X, ≤un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Sea F:X×X→Xuna aplicaci´on que tiene la propiedad mon´otona mixta y supongamos que existe k∈[0,1) tal que dF(x, y), F(u, v)≤k 2[d(x, u) + d(y, v)],para cada x≥u, y ≤v. Si existe x0, y0∈Xtal que x0≤F(x0, y0)yy0≥F(y0, x0), y supongamos tambi´en que Fes continua o Xsatisface: si (xn)es una sucesi´on creciente en Xcon xn→x entonces xn≤xpara todo n∈N, 34 Teoremas del punto fijo en espacios m´etricos ordenados y si (yn)es una sucesi´on decreciente en Xcon yn→x entonces y≤ynpara todo n∈N. Entonces Ftiene un punto fijo acoplado. El prop´osito de este art´ıculo es generalizar el Teorema 33 usando las funciones que alteran las distancias, esto es, funciones ϕ: [0,∞)→[0,∞) tales que ϕes continua y creciente con ϕ(t) = 0 si y s´olo si t= 0. El primer resultado obtenido en el art´ıculo es el siguiente. Teorema 34. Sea X, ≤un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que X, des un espacio m´etrico completo. Sea F:X×X→Xuna aplicaci´on que tiene la propiedad mon´otona mixta, continua y satisfaciendo ϕdF(x, y),F(u, v) ≤ϕmax d(x, u), d(y, v)−φmax d(x, u), d(y, v), para cada x, y, u, v ∈Xcon x≥uey≤v, donde ϕyφson funciones con distancias alteradas. Si existe x0, y0∈Xcon x0≤F(x0, y0)ey0≥F(y0, x0)entonces Ftiene un punto fijo acoplado. En el pr´oximo resultado, reemplazamos la continuidad de Fpor esta condici´on: si (xn) es una sucesi´on creciente en Xcon xn→x entonces xn≤xpara todo n∈N, y si (yn) es una sucesi´on decreciente en Xcon yn→x entonces y≤ynpara todo n∈N, (1.9) 1. Resumen de la Tesis / Summary of the thesis in Spanish 35 y obtenemos el siguiente teorema. Teorema 35. Si en el Teorema 34 reemplazamos la continuidad de Fpor la condici´on mencionada anteriormente, obtenemos la misma conclusi´on. Seguidamente, en el art´ıculo, presentamos un ejemplo que demuestra que las hip´otesis dadas en los Teoremas 34 y 35 no garantizan la unicidad del punto fijo acoplado. El siguiente resultado nos da una condici´on suficiente para obtener la unicidad del punto fijo acoplado. Teorema 36. Asumamos que para (x, y),(u, v)∈X×Xexiste (z, t)∈X×X que es comparable con (x, y)y(u, v), y consideramos en X×Xel orden parcial definido por (x, y)≤(u, v)si y s´olo si x≤uyy≥v . Bajo las hip´otesis del Teorema 34 (resp. Teorema 35) obtenemos la unicidad del punto fijo acoplado. Despu´es, presentamos algunas consecuencias de los resultados obtenidos. En particular, el principal resultado de [18] puede ser deducido de nuestros Teoremas 34 y 35. Adem´as, damos un teorema del punto fijo acoplado para aplicaciones que tienen la propiedad mon´otona mixta con una condici´on de tipo integral, que presentamos a continuaci´on. Teorema 37. Sea X, ≤un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que X, des un espacio m´etrico completo. Sea F:X×X→Xuna aplicaci´on con la propiedad mon´otona mixta y supongamos que existe k∈[0,1) tal que ZdF(x,y),F (u,v) 0 ρ(t)dt ≤KZmax d(x,u),d(y,v) 0 ρ(t)dt , 36 Teoremas del punto fijo en espacios m´etricos ordenados para todo x, y, u, v ∈Xcon x≥uyy≤v, donde ρ:R+→R+es una funci´on medible Lebesgue con integral finita sobre cada compacto de R+y tal que Rε 0ρ(t)dt > 0para ε > 0. Supongamos adem´as, que Fes continua o Xsatisface la condici´on (1.9). Si existe x0, y0∈Xcon x0≤F(x0, y0)ey0≥F(y0, x0)entonces Ftiene un punto fijo acoplado. Para concluir el art´ıculo, damos una aplicaci´on de nuestros resultados a la teor´ıa de existencia de soluciones en ecuaciones integrales no lineales. Con mayor precisi´on, consideramos la ecuaci´on integral x(t) = Z1 0k1(t, s) + k2(t, s)fs, x(s)+gs, x(s)ds +a(t), con t∈[0,1] , (1.10) y suponemos que se verificas las hip´otesis: (i) ki: [0,1]×[0,1] →R(i= 1,2) son continuas con k1(t, s)≥0 y k2(t, s)≤ 0. (ii) a∈ C[0,1]. (iii) f, g: [0,1] ×R→Rson funciones continuas. (iv) Existen constantes λ, µ > 0 tales que para cada x, y ∈Rcon x≥y 0≤f(t, x)−f(t, y)≤λpln[(y−x)2+ 1] y −µpln[(y−x)2+ 1] ≤g(t, x)−g(t, y)≤0. 1. Resumen de la Tesis / Summary of the thesis in Spanish 37 (v) Existe α, β ∈ C[0,1] tal que α(t)≤Z1 0 k1(t, s)(f(s, α(s)) + g(s, β(s)))ds +Z1 0 k2(t, s)(f(s, β(s)) + g(s, α(s)))ds +a(t) ≤Z1 0 k1(t, s)(f(s, β(s)) + g(s, α(s)))ds +Z1 0 k2(t, s)(f(s, α(s)) + g(s, β(s)))ds +a(t)≤β(t). (vi) 2 ·max(λ, µ)kk1−k2k∞≤1, donde kk1−k2k∞= sup n(k1(t, s)−k2(t, s)): t, s ∈[0,1]o. Entonces, la ecuaci´on integral (1.10) tiene una ´unica soluci´on en C[0,1]. 44 Teor´ıa de existencia y unicidad para las soluciones de un bvp donde ψes una funci´on que altera las distancias. Si existe x0∈Xtal que x0≤Tx0entonces Ttiene un punto fijo. Adem´as, si para cada x, y ∈Xexiste z∈Xcomparable con xeyentonces el punto fijo es ´unico. El principal resultado obtenido en el art´ıculo es el teorema que aparece a continuaci´on. Teorema 42. Consideremos el Problema (1.11) bajo las siguientes hip´otesis: (i) f: [0,1] ×[0,∞)→[0,∞)es una funci´on continua. (ii) fes creciente con respecto a la segunda variable y supongamos que existe 0< α ≤384 5tal que, para cada x, y ∈[0,∞)con y≥x f(t, y)−f(t, x)≤αln(y−x+ 1) . Entonces Problema (1.11) tiene una ´unica soluci´on no negativa. En el art´ıculo, se menciona que el Teorema 42 sigue siendo v´alido si reemplazamos la inecuaci´on de (ii) por f(t, y)−f(t, x)≤α ϕ(y−x) donde ϕ: [0,∞)→[0,∞) es continua y φ(x) = x−ϕ(x) cumple estas condiciones: (i) φ: [0,∞)→[0,∞) es creciente. (ii) φ(0) = 0. (iii) φes positiva en (0,∞). El pr´oximo teorema nos da una condici´on suficiente para la existencia y unicidad de una soluci´on positiva para Problema (1.11) (por soluci´on positiva entendemos que x(t)>0 para t∈(0,1)). 1. Resumen de la Tesis / Summary of the thesis in Spanish 45 Teorema 43. Bajo las hip´otesis del Teorema 42 y suponiendo que f(t, 0) 6= 0 para t∈A⊂[0,1] con µ(A)>0, donde µdenota la medida de Lebesgue, el Problema (1.11) tiene una ´unica soluci´on positiva. Queremos destacar que la condici´on que apareces en el Teorema 43 se satisface autom´aticamente cuando f: [0,1] ×[0,∞)→[0,∞) es continua y f(t0,0) 6= 0 para cierto t0∈[0,1]. En [J. A. Cid, D. Franco, F. Minh´os, Positive fixed points and fourth-order equations, Bull. London Math. Soc. 41 (2009), 72–78] los autores estudiaron este problema: (u(iv)(t) = λ h(t)f(u), t ∈(0,1), λ > 0, u(0) = u(1) = 0 = u00(0) = u00(1) ,(1.12) El principal resultado que obtuvieron es el siguiente teorema. Teorema 44. Supongamos que h: [0,1] →[0,∞)es continua y no id´enticamente nula en 1 4,3 4,f:R→[0,∞)continua y tal que lim s→∞ f(s) s= +∞y existe B∈[0,∞)tal que fes creciente en [0, B). Si 0< λ < sup s∈(0,B) s γ∗f(s), donde γ∗= max t∈[0,1] Z1 0 G(t, s)h(s)dsyG(t, s)es la funci´on de Green asociada a(1.12), dada por G(t, s) =            1 6s(1 −t)(2t−s2−t2),0≤s≤t, 1 6t(1 −s)(2s−t2−s2),0≤t≤s. Entonces el Problema (1.12) tiene al menos una soluci´on positiva. En el art´ıculo, comparamos nuestros resultados con los obtenidos en [38]. 46 Teor´ıa de existencia y unicidad para las soluciones de un bvp Consideramos el problema (u(iv)(t) = λ h(t) ln(u+ 2), t ∈(0,1), λ > 0 u(0) = u(1) = 0 = u00(0) = u00(1) ,(1.13) donde h: [0,1] →[0,∞) es continua y no id´enticamente nula en 1 4,3 4. Comprobamos que el Problema (1.13) cumple las condiciones del Teorema 44 y consecuentemente, obtenemos la existencia de una soluci´on positiva para este problema para cada λ > 0. Pero el Problema (1.13) tambi´en verifica las condiciones del nuestros Teoremas 42 y 43 obteniendo la existencia y unicidad de una soluci´on positiva para el mismo cuando λ≤384 5khk. Nuestra principal contribuci´on es este ejemplo es la unicidad de la soluci´on cuando λ≤384 5khk. 1. Resumen de la Tesis / Summary of the thesis in Spanish 47 1.2.2. On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets En este art´ıculo, estamos interesados en obtener la existencia y unicidad de una soluci´on positiva para el problema con valores en la frontera: (u(iv)(t) = f(t, u), t ∈(0,1) u(0) = u(1) = 0 = u00(0) = u00(1) .(1.14) Utilizaremos en nuestro estudio el teorema en espacios m´etricos parcialmente ordenados obtenido en [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72, (2010), 1188–1197]. Teorema 13. Sea (X, ≤)un espacio parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Asumamos que Xsatisface la condici´on si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤xpara todo n∈N. Sea T:X→Xuna aplicaci´on creciente tal que ψd(Tx, Ty)≤ψd(x, y)−φd(x, y)para cada x, y ∈Xcon x≥y , donde ψyφson funciones que alteran las distancias. Si existe x0∈Xcon x0≤Tx0, entonces Ttiene un punto fijo. Adem´as, si para cada x, y ∈Xexiste z∈Xque sea comparable con xey, entonces el punto fijo es ´unico. El principal resultado del art´ıculo puede ser resumido en este teorema. Teorema 45. Consideremos el Problema (1.14) bajo estas hip´otesis: (i) f: [0,1] ×[0,∞)→[0,∞)es una funci´on continua. 48 Teor´ıa de existencia y unicidad para las soluciones de un bvp (ii) Existe 0< α ≤q80640 17 tal que, para cada x, y ∈[0,∞)con y≥x 0≤f(t, y)−f(t, x)≤αpln[(y−x)2+ 1] . Entonces nuestro Problema (1.14) tiene una ´unica soluci´on no negativa. Adem´as, si f(t0,0) 6= 0 para cierto t0∈[0,1] entonces el Problema (1.14) tiene una ´unica soluci´on positiva. Para finalizar, comparamos los resultados obtenidos con otros precedentes que aparecen en [38]. Nuestra principal contribuci´on es que obtenemos la unicidad de la soluci´on. 1. Resumen de la Tesis / Summary of the thesis in Spanish 49 1.2.3. Uniqueness of positive solutions for a class of fourth-order boundary value problems El prop´osito de este art´ıculo es estudiar la existencia y unicidad de una soluci´on positiva y sim´etrica para este problema con valores en la frontera: (y(iv)(t) = ft, y(t), t ∈[0,1], y(0) = y(1) = y0(0) = y0(1) = 0 ,(1.15) Para ello, utilizamos el teorema del punto fijo en espacios m´etricos ordenados que citamos a continuaci´on, obtenido por A. Amini-Harandi y H. Emami [A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72, (2010), 2238–2242]. Teorema 46. Sea X, ≤un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que X, des un espacio m´etrico completo. Sea T:X→Xuna aplicaci´on creciente tal que existe un elemento x0∈X con x0≤Tx0yTsatisface d(Tx, Ty)≤βd(x, y)·d(x, y),para cada x, y ∈Xcon x≥y , donde β: [0,∞)→[0,1] tal que β(tn)→1implica tn→0. Asumamos tambi´en que Tes continua o Xes tal que if (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤xpara todo n∈N. Supongamos que para cada x, y ∈Xexiste z∈Xque es comparable con xey . Entonces Ttiene un ´unico punto fijo. En el art´ıculo se usa la clase de funciones A, definidas por las funciones φ: [0,∞)→[0,∞) que satisfaces estas condiciones: 50 Teor´ıa de existencia y unicidad para las soluciones de un bvp (i) φes creciente. (ii) Para cada x > 0, φ(x)< x. (iii) β(x) = φ(x) xes tal que β(tn)→1⇒tn→0. El resultado de mayor importancia obtenido es el siguiente teorema. Teorema 47. Consideremos el Problema (1.15) bajo las hip´otesis: (i) f: [0,1] ×[0,∞)→[0,∞)es una funci´on continua. (ii) Existe 0< α ≤384 tal que, para x, y ∈[0,∞)con y≥x 0≤f(t, y)−f(t, x)≤α φ(y−x),donde φ∈ A. (iii) f(t0,0) 6= 0, para cierto t0∈[0,1]. (iv) f(t, y) = f(1 −t, y), para (t, y)∈[0,1] ×[0,∞). Entonces el Problema (1.15) tiene una ´unica soluci´on positiva y sim´etrica (una soluci´on y(t)es sim´etrica si y(t) = y(1 −t)para cada t∈[0,1]). Este problema fue tratado por M. Pei y S. K. Chang, [M. Pei, S. K. Chang, Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Modelling, 51, (2010), 1260–1267]. El principal resultado que obtuvieron fue: Teorema 48. Supongamos que: (a) f: [0,1] ×[0,∞)→[0,∞)es una funci´on continua. (b) f(t, y)es creciente en y, para cada t∈[0,1]. (c) f(t, y) = f(1 −t, y), para cada (t, y)∈[0,1] ×[0,∞). Adem´as, supongamos que existen n´umeros positivos a>btales que max 0≤t≤1f(t, a)≤a A min 1 4≤t≤3 4 f(t, b 16)≥b B , 1. Resumen de la Tesis / Summary of the thesis in Spanish 51 donde A=max 0≤t≤1Z1 0 G(t, s)ds−1 yB= max 0≤t≤1Z3 4 1 4 G(t, s)ds!−1 , siendo G(t, s)la funci´on de Green asociada al Problema (1.15), que viene dada por G(t, s) = 1 6(t2(1 −s)2[(s−t) + 2(1 −t)s],0≤t≤s, s2(1 −t)2[(t−s) + 2(1 −s)t],0≤s≤t. Entonces el Problema (1.15) tiene al menos una soluci´on sim´etrica positiva. Para poder comparar nuestros resultados con los obtenidos por ellos en [126] consideramos este problema: (y(iv)(t) = c+λsin(πt) arctan y(t), t ∈(0,1), c, λ > 0 y(0) = y(1) = y0(0) = y0(1) = 0 (1.16) y probamos que para 0 < λ ≤384 el Problema (1.16) puede estudiarse tanto bajo nuestra perspectiva como por los resultados obtenidos en [126]. Sin embargo, nuestra principal contribuci´on es la unicidad de la soluci´on. Seguidamente, consideramos el problema (y(iv)(t) = c(t) + λsin(πt) arctan y(t), t ∈(0,1), c, λ > 0 y(0) = y(1) = y0(0) = y0(1) = 0 (1.17) donde c(t) es la funci´on dada por c(t) =              1−4t , 0≤t≤1 4 0,1 4≤t≤3 4 4t−3,3 4≤t≤1, y demostramos, usando nuestro resultado, que para 0 < λ ≤384, el Problema (1.17) tiene una ´unica soluci´on sim´etrica y positiva. 52 Teor´ıa de existencia y unicidad para las soluciones de un bvp Por otro lado, vemos que el Problema (1.17) no satisface las hip´otesis del Teorema 48 y, entonces, el Problema (1.17) no puede ser estudiado por los resultados obtenidos en [126]. 1. Resumen de la Tesis / Summary of the thesis in Spanish 53 1.3. Fractional boundary value problem 1.3.1. Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems El art´ıculo trata sobre la existencia y unicidad de una soluci´on positiva y creciente para el problema fraccionario con valores en la frontera (Dα 0+u(t) + ft, u(t)= 0 ,0< t < 1, u(0) = u0(1) = u00(0) = 0 ,(1.18) donde 2 < α ≤3, Dα 0+es la deriva de Caputo y f: (0,1] ×[0,∞)→[0,∞) con lim t→0+f(t, −) = ∞, es decir, fes singular en t= 0. En nuestro estudio, usamos el siguiente teorema del punto fijo en espacios m´etricos ordenados, que es el principal resultado obtenido en [69]. Teorema 7. Sea (X, ≤)un conjunto parcialmente ordenado y supongamos que existe una m´etrica den Xtal que (X, d)es un espacio m´etrico completo. Asumamos que Xsatisface la condici´on: si (xn)es una sucesi´on creciente en Xtal que xn→x entonces xn≤x, para todo n∈N. Sea T:X→Xuna aplicaci´on creciente tal que d(Tx, Ty)≤d(x, y)−ψd(x, y),para cada x, y ∈Xcon x≥y , donde ψes una funci´on que altera la distancia. Si existe x0∈Xcon x0≤Tx0entonces Ttiene un punto fijo. Adem´as, si para cada x, y ∈Xexiste z∈Xcomparable con xey, entonces el punto fijo es ´unico. El primer resultado obtenido en el art´ıculo es: 1. Resumen de la Tesis / Summary of the thesis in Spanish 61 1.3.3. Existence and uniqueness of positive solution for a boundary value problem of fractional order En el a˜no 2009, S. Liang y J. Zhang presentaron un art´ıculo [Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Analysis, 71, (2009), 5545–5550] donde se estudiaba la existencia de soluciones para el problema fraccionario con valores en la frontera: (Dα 0+u(t) + ft, u(t)= 0,0< t < 1,3< α ≤4, u(0) = u0(0) = u00(0) = u00(1) = 0.(1.20) La principal aportaci´on del art´ıculo fue este teorema. Teorema 53. El Problema (1.20) tiene una soluci´on positiva si las siguientes condiciones son satisfechas: (a) f(t, u)∈ C[0,1] ×[0,∞),R+. (b) f(t, u)es creciente en u. (c) ft, ρ(t)6= 0 para t∈(0,1), donde ρ(t) = Z1 0 G(t, s)ds=1 Γ(α)tα−1 α−2−1 αtα. (d) Existe una constante positiva µ < 1tal que kµf(t, u)≤f(t, k u),for any k∈[0,1]. A ra´ız de este art´ıculo, nos planteamos analizar la existencia y unicidad de ese problema. Nuestro estudio se basa en el teorema del punto fijo sobre conjuntos parcialmente ordenados presentado en [69]. Teorema 7. Sea (X, ≤)un conjunto parcialmente ordenado y supongamos que existe den Xtal que (X, d)es un espacio m´etrico completo. Asumamos 62 Fractional boundary value problem que Xverifica la condici´on: si (xn)es creciente en Xtal que xn→x entonces xn≤xpara todo n∈N. Sea T:X→Xuna aplicaci´on creciente tal que d(Tx, Ty)≤d(x, y)−ψd(x, y),para cada x, y ∈Xcon x≥y, donde ψes una funci´on que altera la distancia. Si existe x0∈Xcon x0≤ T(x0)entonces Ttiene un punto fijo. Adem´as, si para cada x, y ∈Xexiste z∈Xcomparable con xeyentonces el punto fijo es ´unico. Antes de presentar el principal resultado que obtuvimos, necesitamos introducir la clase de funciones A. La clase Aest´a definida por aquellas funciones ϕ: [0,∞)→[0,∞) continuas y crecientes tales que ψ(x) = x−ϕ(x) para x∈[0,∞) satisface: (a) ψ: [0,∞)→[0,∞). (b) ψes continua y creciente. (c) ψ(0) = 0. (d) ψ(t)>0 para t > 0. Teorema 54. Bajo las hip´otesis: (1) f: [0,1] ×[0,∞)→[0,∞)es continua y creciente con respecto a la segunda variable. (2) Existe t0∈[0,1] tal que f(t0,0) >0. (3) Existe 0< λ ≤(α−2)Γ(α+ 1) 2tal que, para cada x, y ∈[0,∞)con y≥xy cada t∈[0,1], f(t, y)−f(t, x)≤λ·ψ(y−x), 1. Resumen de la Tesis / Summary of the thesis in Spanish 63 donde ψ∈ A. El Problema (1.20) tiene una ´unica soluci´on positiva. Con el prop´osito de poder comparar nuestros resultados con los obtenidos en [106] presentamos este problema fraccionario con valores en la frontera: (D7/2 0+u(t)+(t2+ 1)ρu(t) + c= 0,0< t < 1, u(0) = u0(0) = u00(0) = u00(1) = 0 ,(1.21) con c > 0 y 0 <ρ<1. En el trabajo probamos que el Problema (1.21) puede ser tratado por nuestros resultados pero no puede ser estudiado por el Teorema 53. 1. Resumen de la Tesis / Summary of the thesis in Spanish 65 1.3.4. On existence and uniqueness of positive solutions to a class of fractional boundary value problems El art´ıculo estudia la existencia y unicidad de una soluci´on positiva para el problema fraccionario con valores en la frontera: (Dα 0+u(t) + ft, u(t)= 0,0<t<1, u(0) = u(1) = u0(0) = 0,(1.22) donde 2 < α ≤3, que representa la versi´on no mon´otona de (Dα 0+u(t) + λfu(t)= 0,0< t < 1, u(0) = u(1) = u0(0) = 0,(1.23) donde 2 < α ≤3 y λ > 0. Este ´ultimo problema fue analizado recientemente en [Y. Zhao, S. Sun, Z. Han, Q. Li, Positive solutions to boundary value problems of nonlinear fractional differential equations. Abs Appl Anal. 2011, (2011), Article ID 390543]. En nuestro trabajo, utilizamos el mismo teorema del punto fijo en espacios m´etricos parcialmente ordenados que en el art´ıculo anterior. Los resultados obtenidos pueden ser resumidos en pr´oximo teorema. Teorema 55. El Problema (1.22) tiene una ´unica soluci´on no negativa si las condiciones que aparecen a continuaci´on son satisfechas: (1) f: [0,1] ×[0,∞)→[0,∞)es continua y creciente con respecto al segundo argumento. (2) Existe 0< λ ≤1 Atal que, para cada x, y ∈[0,∞)con y≥xy cada t∈[0,1] f(t, y)−f(t, x)≤λϕ(y−x), donde ϕ∈ A, siendo Ala clase de funciones tratada en el art´ıculo anterior y A=1 Γ(α+1) hα−1 αα−1−α−1 ααi. 66 Fractional boundary value problem Adem´as, si f(t0,0) 6= 0 para cierto t0∈[0,1] entonces la soluci´on ´unica es positiva. En [174], usando un teorema del punto fijo en conos, los autores probaron este resultado. Teorema 56. Supongamos que existe l∈(0,1) tal que q(l)c2f0> F∞c1 entonces, para cada λ∈(q(l)c2f0)−1,(F∞c1)−1el Problema (1.23) tiene al menos una soluci´on positiva, donde: F∞= lim u→∞ sup f(u) u, q(t) = tα−1(1 −t), k(s) = s(1 −s)α−1, c1=1 Γ(α)Z1 0 (α−1)k(s)ds , c2=1 α−1Z1 0 1 α−1q(s)k(s)ds. En el art´ıculo, damos un ejemplo que puede ser tratado por los Teoremas 55 y 56, pero nuestra principal contribuci´on es la obtenci´on de unicidad en la soluci´on si 0 < λ ≤17,8682. Para finalizar, presentamos este problema con valores en la frontera, que no puede ser estudiado por el Teorema 56 y puede ser abordado por nuestros resultados. (D5/2 0+u(t) + λt+ arctan u(t)= 0,0<t<1, λ > 0, u(0) = u(1) = u0(0) = 0. 1. Resumen de la Tesis / Summary of the thesis in Spanish 67 1.3.5. Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations En este trabajo se analiza la existencia y unicidad de una soluci´on positiva y creciente para el siguiente problema fraccionario con valores en la frontera (Dα 0+u(t) + ft, u(t)= 0,0<t<1, u(0) = u0(1) = u00(0) = 0,(1.24) con 2 < α ≤3, y lim t→0+f(t, ·) = ∞, esto es fes singular en t= 0. Necesitamos la clase Ade aquellas funciones φ: [0,∞)→[0,∞) que verifican las condiciones: (i) φes creciente. (ii) φ(x)< x, para cada x > 0. (iii) β(x) = φ(x) xes tal que si b(tn)→1 implica tn→0. Nuestro principal resultado se puede resumir en el pr´oximo teorema. Teorema 57. Supongamos que 0< σ < 1y2< α ≤3. Bajo estas hip´otesis: (1) f: (0,1] ×[0,∞)→[0,∞)es una funci´on continua satisfaciendo lim t→0+f(t, −) = ∞. (2) tσf(t, y)es una funci´on continua en [0,1] ×[0,∞). (3) Existe 0< λ ≤Γ(α−σ) Γ(1−σ)and φ∈ A tal que 0≤tσ(f(t, y)−f(t, x)) ≤λφ(y−x), para cada x, y ∈[0,∞)con y≥xy cada t∈[0,1], El Problema (1.24) tiene una ´unica soluci´on no negativa. Adem´as, esta soluci´on es estrictamente creciente. 68 Fractional boundary value problem Este mismo problema fue abordado por T. Qiu y Z. Bai en Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 146, (2008), 1–9, donde aplicaron este teorema: Teorema 58. Sea 0< σ < 1,2< α ≤3,f: (0,1] ×[0,+∞)→[0,+∞)es continua y lim t→0+f(t, ·) = +∞,tσf(t, y)es una funci´on continua en [0,1] × [0,+∞). Asumamos que existen dos constantes positivas ρ, µ (ρ > µ)tales que (H1) tσf(t, ω)≤ρΓ(α−σ) Γ(1−σ), para (t, ω)∈[0,1] ×[0, ρ]; (H2) tσf(t, ω)≥µΓ(α−σ) Γ(1−σ), para (t, ω)∈[0,1] ×[0, µ]. Entonces el Problema (1.24) tiene al menos una soluci´on positiva. Nuestros resultados generalizan los obtenidos por estos autores, ya que la unicidad y la monoton´ıa de la soluci´on no se pueden deducir de sus resultados. Ilustramos nuestros resultados con un ejemplo que puede ser resuelto por el Teorema 57 y no puede ser estudiado por los resultados de [25]. Cap´ıtulo 2 A short history approach 69 76 3.1. Fixed point theorems for weakly contractive mappings......................... 79 3.2. Generalized contractions in partially ordered metric spaces ... . . . . . . . . . . . . . . . . . . . 91 3.3. Contractive-like mapping principles in ordered metric spaces ... . . . . . . . . . . . . . . . . . . . 105 3.4. Fixed point theorems for mappings satisfying a condition of ... . . . . . . . . . . . . . . . . . . . . 123 3.5. A fixed point theorem for mappings satisfying a contractive ... . . . . . . . . . . . . . . . . . . . . . 141 3.6. Fixed point theorems for weakly C-contractive mappings in ... . . . . . . . . . . . . . . . . . . . . 151 3.7. Fixed point theorems for mixed monotone operatorsand... ..................... 161 3.8. A fixed point theorem for Meir-Keeler contractionsin.......................... 179 3. Fixed point theorems in partially ordered metric spaces 77 In this chapter, we present some fixed point theorems in partially ordered metric spaces obtained in our research. The main objective of our study was to extend, to improve and to generalize some classical fixed point theorems in the context of partially ordered metric spaces. For a better readability, we present the papers in connection with this section and, then, briefly we discuss each article. We will show these papers: a) J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71, (2009), 3403– 3410. b) J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72, (2010), 1188–1197. c) J. Caballero, J. Harjani, K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory and Applications, vol. 2010, Article ID916064, 14 pages. d) J. Harjani, K. Sadarangani, Fixed point theorems for mappings satisfying a condition of integral type in partially ordered sets, Journal of Convex Analysis 17, (2010), 597–609. e) J. Harjani, B. L´opez, K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstract and Applied Analysis vol 2010, Article ID190701, 8 pages. f) J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for weakly C-contractive mappings in ordered metric spaces, Computer and Mathematics with Applications 61, (2011), 790–796. g) J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for mixed monotone operators and application to integral equations, Nonlinear Anal. 74, (2011), 1749–1760. 78 h) J. Harjani, B. L´opez, K. Sadarangani, A fixed point theorem for MeirKeeler contractions in ordered metric spaces, Fixed Point Theory and Applications, vol. 2011, doi: 10.1186/1687-1812-2011-83. 3. Fixed point theorems in partially ordered metric spaces 79 3.1. Fixed point theorems for weakly contractive mappings in partially ordered sets In this paper, we present some fixed point theorems for weakly contractive mappings in partially ordered metric spaces. The weakly contractive mappings were defined by Alber and Guerre-Delabriere in [6], in the context of Banach spaces, as a generalization of the classical contractive mappings. More precisely, let (X, k k) be a Banach space and Ta self mapping on X, we say that Tis weakly contractive if, for any x, y ∈X, kTx −Tyk ≤ kx−yk−ψkx−yk, where ψ: [0,∞)→[0,∞) is a continuous and nondecreasing mapping such that ψis positive on (0,∞), ψ(0) = 0 and lim t→∞ ψ(t) = ∞. Alber and Guerre-Delabriere proved that any weakly contractive self mapping defined in a Hilbert space has a fixed point. In [136] Rhoades extended the definition of weakly contractive mapping to the context of metric spaces and proved a fixed point theorem for these mappings. The main result of [136] is the following. Theorem 5. Let (X, d)be a complete metric space and T:X→Xa mapping satisfying d(Tx, Ty)≤d(x, y)−ψd(x, y), for any x, y ∈X, where ψ: [0,∞)→[0,∞)is a continuous and nondecreasing function such that ψis positive on (0,∞)and ψ(0) = 0. Then Thas a unique fixed point. Notice that the condition lim t→∞ ψ(t) = ∞, used by Alber and GuerreDelabriere, is not necessary for our objective as it is proved in Theorem 5. 80 Fixed point theorems for weakly contractive mappings ... Our purpose in the paper is to present a version of Theorem 5 in the context of partially ordered metric spaces. Our main results can be summarized in the following theorems. Theorem 6. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let T:X→Xbe a continuous and nondecreasing mapping such that d(Tx, Ty)≤d(x, y)−ψd(x, y),for x, y ∈Xwith x≥y , where ψ: [0,∞)→[0,∞)is a continuous and nondecreasing function such that it is positive on (0,∞)and ψ(0) = 0. If there exists x0∈Xsuch that x0≤Tx0then Thas a fixed point. We prove that the condition Tcontinuous is unnecessary, assuming the following assumption in X if (xn) is a nondecreasing sequence in Xsuch that xn→x then xn≤x, for all n∈N.(3.1) This condition was used by J. Nieto and R. Rodr´ıguez-L´opez in [118]. More precisely, we prove the following result. Theorem 7. If in Theorem 6 we replace the continuity of Tby assumption (3.1) then we obtain the same conclusion. Respect to the uniqueness of the fixed point, we present an example which shows that Theorems 6 and 7 do not guarantee this uniqueness. Next, we give a sufficient condition for the uniqueness (which was used in [118]). This condition says: For x, y ∈Xthere exists z∈Xwhich is comparable to xand y. (3.2) Theorem 8. Adding condition (3.2) to the assumptions of Theorem 6 (resp. Theorem 7), we obtain the uniqueness of the fixed point. 3. Fixed point theorems in partially ordered metric spaces 81 Finally, we apply our result to the existence of solution for the following first-order periodic problem (u0(t) = ft, u(t), t ∈[0, T], u(0) = u(T),(3.3) under assumption about the existence of a lower solution for (3.3), i.e., a function α∈ C[0, T] such that α0(t)≤ft, α(t),for t∈[0, T], α(0) ≤α(T). More precisely, we obtain the following result. Theorem 9. Suppose that f: [0, T]×R→Ris continuous and there exists λ > 0such that for x, y ∈Rwith y≥x, 0≤f(t, y) + λy −[f(t, x) + λx]≤λln(y−x+ 1) . Then, the existence of a lower solution for (3.3) provides the existence of a unique solution of (3.3). This paper is strongly inspired by [118]. Nonlinear Analysis 71 (2009) 3403–3410 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Fixed point theorems for weakly contractive mappings in partially ordered sets J. Harjani, K. Sadarangani∗ Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain article info Article history: Received 10 July 2008 Accepted 30 January 2009 MSC: 47H10 Keywords: Fixed point Weakly contractive map Partially ordered set abstract The purpose of this paper is to present some fixed point theorems for weakly contractive maps in a complete metric space endowed with a partial order. ©2009 Elsevier Ltd. All rights reserved. 1. Introduction Alber and Guerre-Delabriere in [1] define weakly contractive maps. In this paper they confine their theorems to Hilbert spaces, but acknowledge that their results are true, at least for uniformly smooth and uniformly convex Banach spaces. In [2] Rhoades extends some results appearing in [1] to arbitrary Banach spaces. If Xis an arbitrary Banach space, then a selfmap Tof Xsatisfies the Banach contraction principle if there exists a constant ksatisfying 0 ≤k<1 such that, for x,y∈X kTx −Tyk ≤ kkx−yk.(1) As noted in the introduction of [1], inequality (1) can be written in the form kTx −Tyk≤kx−yk − qkx−yk(2) where k=1−qwith q∈(0,1]. The extension of (2) in the context of Banach spaces to what are called weakly contractive maps is a natural one. A selfmap Tof Xis weakly contractive if, for every x,y∈X, kTx −Tyk≤kx−yk − ψ(kx−yk)(3) where ψ: [0,∞)−→ [0,∞)is continuous and nondecreasing such that ψis positive on (0,∞),ψ(0)=0 and limt→∞ ψ(t)= ∞ (an example of such function ψis ψ(t)=ln(t+1)). Now, let (X,d)be a metric space and T:X−→ X.Tis said to be weakly contractive if for x,y∈X d(Tx,Ty)≤d(x,y)−ψ(d(x,y)) ∗Corresponding author. Fax: +34 928 45 88 11. E-mail address: [email protected] (K. Sadarangani). 0362-546X/$ – see front matter ©2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.240 3. Fixed point theorems in partially ordered metric spaces 83 3404 J. Harjani, K. Sadarangani / Nonlinear Analysis 71 (2009) 3403–3410 where ψ: [0,∞)−→ [0,∞)satisfies the above mentioned conditions. Notice that to be weakly contractive implies continuity. The purpose of this paper is to present some fixed point theorems for weakly contractive operators in the context of ordered metric spaces which are extensions of those in [2]. Existence of fixed point in partially ordered sets has been considered recently in [3–16]. Tarski’s theorem is used in [8] to show the existence of solutions for fuzzy equations and in [10] to prove existence theorems for fuzzy differential equations. In [15,9,12] some applications to matrix equations and to ordinary differential equations are presented, respectively. In [4–6, 16] it is proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems [6, 16]. Some other references on the topic are [17,18]. The usual contraction condition is weakened but at the expense that the operator is monotone. The main idea in [9,15] involve combining the ideas in the contraction principle with those in the monotone iterative technique [19]. 2. Fixed point theorems Definition 1. If (X,≤)is a partially ordered set and f:X−→ X, we say that fis monotone nondecreasing if x,y∈X, x≤y⇒f(x)≤f(y). This definition coincides with the notion of a nondecreasing function in the case where X=Rand ≤represents the usual total order in R. In [2], the following theorem is proved. Theorem 1. Let (X,d)be a complete metric space and f :X−→ X is a weakly contractive map. Then f has a unique fixed point in X. In what follows we prove the following theorem which is a version of Theorem 1 in the context of ordered metric spaces. Theorem 2. Let (X,≤)be a partially ordered set and suppose that there exists a metric d in X such that (X,d)is a complete metric space. Let f :X−→ X be a continuous and nondecreasing mapping such that d(f(x), f(y)) ≤d(x,y)−ψ(d(x,y)) for x ≥y(4) where ψ: [0,∞)−→ [0,∞)is a continuous and nondecreasing function such that it is positive in (0,∞),ψ(0)=0and limt→∞ ψ(t)= ∞. If there exists x0∈X with x0≤f(x0), then f has a fixed point. Proof. If f(x0)=x0then the proof is finished. Suppose that x0<f(x0). Since x0<f(x0)and fis a nondecreasing function, we obtain by induction that x0<f(x0)≤f2(x0)≤f3(x0)≤ · · · ≤ fn(x0)≤fn+1(x0)≤ · · · Put xn+1=f(xn). Then for each integer n≥1, from (4) and, as the elements xnand xn+1are comparable, we get d(xn+1,xn)=d(f(xn), f(xn−1)) ≤d(xn,xn−1)−ψ(d(xn,xn−1)). If there exists n0∈Nsuch that d(xn0,xn0−1)=0 then xn0=f(xn0−1)=xn0−1and xn0−1is a fixed point and the proof is finished. In other case, suppose that d(xn+1,xn)6= 0 for all n∈N. Then taking into account (4) and our assumptions about ψ d(xn+1,xn)≤d(xn,xn−1)−ψ(d(xn,xn−1)) < d(xn,xn−1). Put ρn=d(xn+1,xn). Then we have ρn≤ρn−1−ψ(ρn−1)<ρn−1.(5) Therefore {ρn}is a nonnegative nonincreasing sequence and hence possesses a limit ρ∗. From (5), taking limit when n→ ∞, we get ρ∗≤ρ∗−ψ(ρ∗)≤ρ∗ and, consequently, ψ(ρ∗)=0. By our assumptions about ψ,ρ∗=0. In what follows we will show that {xn}is a Cauchy sequence. Fix ε > 0. As ρn=d(xn+1,xn)→0, there exists n0∈Nsuch that d(xn0+1,xn0)≤min nε 2, ψ ε 2o.(6) We claim that fB(xn0, ε) ∩ {y∈X:y≥xn0}⊂B(xn0, ε). Let z∈B(xn0, ε) ∩ {y∈X:y≥xn0}. Then there are two cases: 84 Fixed point theorems for weakly contractive mappings ... J. Harjani, K. Sadarangani / Nonlinear Analysis 71 (2009) 3403–3410 3405 Case 1. d(z,xn0)≤ε 2. In this case, as zand xn0are comparable, we have d(f(z), xn0)≤d(f(z), f(xn0)) +d(f(xn0), xn0) =d(f(z), f(xn0)) +d(xn0+1,xn0) ≤d(z,xn0)−ψ(d(z,xn0)) +d(xn0+1,xn0) ≤d(z,xn0)+d(xn0+1,xn0)≤ε 2+ε 2=ε. Case 2. ε 2<d(z,xn0)≤ε. In this case, as ψis a nondecreasing function, ψ(d(z,xn0)) ≥ψ( ε 2). Therefore from (6) we get d(f(z), xn0)≤d(f(z), f(xn0)) +d(f(xn0), xn0) =d(f(z), f(xn0)) +d(xn0+1,xn0) ≤d(z,xn0)−ψ(d(z,xn0)) +d(xn0+1,xn0) ≤d(z,xn0)−ψε 2+d(xn0+1,xn0) ≤d(z,xn0)−ψε 2+ψε 2≤d(z,xn0)≤ε. This proves the claim. As xn0+1∈B(xn0, ε) ∩ {y∈X:y≥xn0}, the claim gives us that f(xn0+1)=xn0+2∈B(xn0, ε) ∩ {y∈X:y≥xn0}. Repeating this process it follows that xn∈B(xn0, ε) for n≥n0. Since εis arbitrary, {xn}is a Cauchy sequence. Since Xis a complete metric space there exists z∈Xsuch that limn→∞ xn=z. The continuity of fimplies that zis a fixed point. Thus, the proof is complete.  In what follows we prove that Theorem 1 is still valid for fnot necessarily continuous, assuming the following hypothesis in X(which appears in Theorem 1 of [9]) if {xn}is a nondecreasing sequence in Xsuch that xn→xthen xn≤xfor all n∈N.(7) Theorem 3. Let (X,≤)be a partially ordered set and suppose that there exists a metric d in X such that (X,d)is a complete metric space. Assume that X satisfies (7). Let f :X−→ X be a nondecreasing mapping such that d(f(x), f(y)) ≤d(x,y)−ψ(d(x,y)) for x ≥y where ψ: [0,∞)−→ [0,∞)is continuous and nondecreasing function such that ψis positive in (0,∞),ψ(0)=0and limt→∞ ψ(t)= ∞. If there exists x0∈X with x0≤f(x0), then f has a fixed point. Proof. Following the proof of Theorem 2 we only have to check that f(z)=z. In fact, d(f(z), z)≤d(f(z), f(xn)) +d(f(xn), z) ≤d(z,xn)−ψ(d(z,xn)) +d(xn+1,z) and taking limit as n→ ∞,d(f(z), z)≤0 and this proves that d(f(z), z)=0 and, consequently, f(z)=z. Now, we present an example where it can be appreciated that hypotheses in Theorems 2 and 3do not guarantee uniqueness of the fixed point. This example appears in [9]. Let X= {(1,0), (0,1)} ⊂ R2and consider the usual order (x,y)≤(z,t)⇔x≤zand y≤t. Thus, (X,≤)is a partially ordered set, whose different elements are not comparable. Besides, (X,d2)is a complete metric space considering d2the euclidean distance. The identity map f(x,y)=(x,y)is trivially continuous and nondecreasing and condition (4) of Theorem 2 is satisfied since elements in Xare only comparable to themselves. Moreover, (1,0)≤f(1,0)= (1,0)and fhas two fixed points in X. In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2 and 3. This condition is for x,y∈Xthere exists a lower bound or an upper bound. (8) In [9] it is proved that condition (8) is equivalent to for x,y∈Xthere exists z∈Xwhich is comparable to xand y.(9) Theorem 4. Adding condition (9) to the hypotheses of Theorem 2(resp. Theorem 3) we obtain uniqueness of the fixed point of f . 3. Fixed point theorems in partially ordered metric spaces 85 92 Generalized contractions in partially ordered metric spaces ... The main objective in this paper is to present a version of Theorem 11 in the context of partially ordered metric spaces. Our main results can be summarized in the following theorems. Theorem 12. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let T:X→Xbe a continuos and nondecreasing mapping such that ψd(Tx, Ty)≤ψd(x, y)−φd(x, y)for any x, yıXwith x≥y , where ψand φare altering distance functions. If there exist x0∈Xwith x0≤Tx0, then Thas a fixed point. Theorem 13. If in Theorem 12 we replace the condition of continuity of T by if (xn)is a nondecreasing sequence in Xsuch that xn→x then xn≤xfor all n∈N, then we obtain the same conclusion. Theorem 14. Adding the condition: For x, y ∈Xthere exists z∈Xwhich is comparable to xand y , to the hypotheses of Theorem 12 (resp. Theorem 13) we obtain uniqueness of the fixed point. The main results of our previous paper are particular cases of the ones obtained in this paper. Finally, we present two examples about boundary value problems where our results can be applied. In the first example, we study the existence of solutions for the following first-order periodic problem (u0(t) = ft, u(t), t ∈[0, T], u(0) = u(T).(3.4) 3. Fixed point theorems in partially ordered metric spaces 93 Our result is the following. Theorem 15. Under assumption f: [0, T]×R→Ris continuous and suppose that there exists two positive real numbers λ, α > 0satisfying α≤2λ(eλT −1) T(eλT + 1) 1 2 , and such that, for x, y ∈Rwith y≥x 0≤f(t, y) + λy −f(t, x) + λx≤αqln (y−x)2+ 1. Then the existence of a lower solution for (3.4) (see, comments about previous paper) provides the existence of an unique solution of (3.4). The second example studies the existence of solution for the following two-point boundary value problem of the second order differential equation    −d2x dt2=f(t, x), t ∈[0,1] , x(0) = x(1) = 0 . (3.5) We get the following result. Theorem 16. Under assumption f: [0,1] ×R→Ris continuous and nondecreasing with respect to the second variable, and such that, for any x, y ∈R with y≥x, f(t, y)−f(t, x)≤αpln[(y−x)2+ 1] , when 0< α ≤8, then Problem (3.5) has a unique nonnegative solution. Moreover, if f(t, x)6= 0 for t∈(0,1), the solution of (3.5) is positive (this means that 0< x(t)for t∈(0,1)). Nonlinear Analysis 72 (2010) 1188–1197 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations J. Harjani, K. Sadarangani∗ Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain article info Article history: Received 28 May 2009 Accepted 3 August 2009 MSC: 47H10 Keywords: Fixed point Altering distance function Partially ordered set abstract The purpose of this paper is to present some fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions. We also present some applications to first and second order ordinary differential equations. ©2009 Elsevier Ltd. All rights reserved. 1. Introduction The Banach contraction mapping principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics. Generalization of the above principle has been a heavily investigated branch of research. In particular, there has been a number of works involving altering distance functions. There are control functions which alter the distance between two points in a metric space. Such functions were introduced by Khan et al. in [1], where they present some fixed point theorems with the help of such functions. Previously, we recall the definition of altering distance function. Definition 1.1. An altering distance function is a function ψ: [0,∞)→ [0,∞)which satisfies (a) ψis continuous and non-decreasing. (b) ψ(t)=0 if and only if t=0. In [1], the authors prove the following result. Theorem 1.1 ([1]). Let (X,d)be a complete metric space, ψan altering distance function and T :X→X satisfying ψ(d(Tx,Ty))≤c·ψ(d(x,y)), for x,y∈X and 0<c<1. Then T has an unique fixed point theory. Altering distance has been used in metric fixed point theory in recent papers (see, for example, [2–4]). On the other hand, Alber and Guerre-Delabriere in [5] define weakly contractive maps and they confine their theorems to Hilbert spaces. Rhoades [6] proved that those results are also valid in complete metric spaces. ∗Corresponding author. E-mail addresses: [email protected] (J. Harjani), [email protected] (K. Sadarangani). 0362-546X/$ – see front matter ©2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.08.003 3. Fixed point theorems in partially ordered metric spaces 95 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 1189 Theorem 1.2 ([6]). Let (X,d)be a complete metric space, ψan altering distance function and T :X→X satisfying d(Tx,Ty)≤d(x,y)−ψ(d(x,y)) for x,y∈X. Then T has a unique fixed point. In fact, Alber and Guerre-Delabriere assumed an additional assumption on ψwhich is limt→∞ ψ(t)= ∞but Rhoades proved Theorem 1.2 without this particular condition on ψ. Dutta and Choudhury in [7] present a generalization of Theorems 1.1 and 1.2 proving the following result. Theorem 1.3 ([7]). Let (X,d)be a complete metric space, T :X→X satisfying ψ(d(Tx,Ty))≤ψ(d(x,y))−φ(d(x,y)),for x,y∈X, where ψand φare altering distance functions. Then T has an unique fixed point. The purpose of this paper is to present some fixed point theorems involving altering distance functions in the context of ordered metric spaces. Existence of fixed point in partially ordered sets has been considered recently in [8–22]. Tarski’s theorem is used in [14] to show the existence of solutions for fuzzy equations and in [16] to prove existence theorems for fuzzy differential equations. In [15,21] some applications to ordinary differential equations and to matrix equations are presented, respectively. In [9–11, 22] some fixed point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems. In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone. The main idea in [15,21] involve combining the ideas in the contraction principle with those in the monotone iterative technique [23]. 2. Fixed point theorems Definition 2.1. If (X,≤)is a partially ordered set and f:X→X, we say that fis monotone nondecreasing if x,y∈X, x≤y⇒f(x)≤f(y). This definition coincides with the notion of a nondecreasing function in the case where X=Rand ≤represents the usual total order in R. In what follows, we prove the following theorem which is a version of Theorem 1.3 in the context of ordered metric spaces. Theorem 2.1. Let (X,≤)be a partially ordered set and suppose that there exists a metric d in X such that (X,d)is a complete metric space. Let f :X→X be a continuous and nondecreasing mapping such that ψ(d(f(x), f(y)))≤ψ(d(x,y))−φ(d(x,y)),for x ≥y(1) where ψand φare altering distance functions. If there exists x0∈X with x0≤f(x0)then f has a fixed point. Proof. If f(x0)=x0then the proof is finished. Suppose that x0<f(x0). Since x0<f(x0)and fis a nondecreasing function, we obtain by induction that x0<f(x0)≤f2(x0)≤f3(x0)≤ ··· ≤ fn(x0)≤fn+1(x0)≤ ··· Put xn+1=f(xn). Then, for each integer n≥1, from (1) and, as the elements xnand xn+1are comparable, we get ψ(d(xn+1,xn))=ψ(d(f(xn), f(xn−1))) ≤ψ(d(xn,xn−1))−φ(d(xn,xn−1)) ≤ψ(d(xn,xn−1)).(2) Using the fact that ψis nondecreasing we have d(xn+1,xn)≤d(xn,xn−1). (3) If there exists n0∈Nsuch that d(xn0,xn0−1)=0 then xn0=f(xn0−1)=xn0−1and xn0−1is a fixed point and the proof is finished. In other case, suppose that d(xn+1,xn)6= 0 for all n∈N. Then, taking into account (3), the sequence {d(xn+1,xn)} is decreasing and, consequently, there exists r≥0 such that d(xn+1,xn)−→ ras n→ ∞. Letting n→ ∞in (2) we get ψ(r)≤ψ(r)−φ(r)≤ψ(r) 96 Generalized contractions in partially ordered metric spaces ... 1190 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 and this implies φ(r)=0. As φis an altering distance function, r=0, and, hence, lim n→∞d(xn+1,xn)=0.(4) In what follows, we will show that {xn}is a Cauchy sequence. Suppose that {xn}is not a Cauchy sequence. Then, there exists  > 0 for which we can find subsequences {xm(k)}and {xn(k)}of {xn}with n(k) > m(k) > ksuch that dxn(k),xm(k)≥. (5) Further, corresponding to m(k)we can choose n(k)in such a way that it is the smallest integer with n(k) > m(k)and satisfying (5). Then dxn(k)−1,xm(k)< . (6) Using (5),(6) and the triangular inequality, we have ≤dxn(k),xm(k) ≤dxn(k),xn(k)−1+dxn(k)−1,xm(k) <dxn(k),xn(k)−1+. Letting k→ ∞and using (4) lim k→∞dxn(k),xm(k)=. (7) Again, the triangular inequality gives us dxn(k),xm(k)≤dxn(k),xn(k)−1+dxn(k)−1,xm(k)−1+dxm(k)−1,xm(k), dxn(k)−1,xm(k)−1≤dxn(k)−1,xn(k)+dxn(k),xm(k)+dxm(k),xm(k)−1. Letting k→ ∞in the above two inequalities and using (4) and (7), we have lim k→∞dxn(k)−1,xm(k)−1=. (8) As n(k) > m(k)and xn(k)−1and xm(k)−1are comparable (in fact, xm(k)−1≤xn(k)−1), setting x=xn(k)−1and y=xm(k)−1in (1), we obtain ψd(xn(k),xm(k))≤ψd(xn(k)−1,xm(k)−1)−φd(xn(k)−1,xm(k)−1). Letting k→ ∞and taking into account (7) and (8), we have ψ() ≤ψ() −φ(). As ψis an altering distance function, the last inequality gives us φ() =0 and, consequently, =0 which is a contradiction. This shows that {xn}is a Cauchy sequence and, since Xis a complete metric space, there exists z∈Xsuch that limn→∞ xn=z. Moreover, the continuity of fimplies that z=lim n→∞f(xn)=lim n→∞xn+1=f(z) and this proves that zis a fixed point.  In what follows, we prove that Theorem 2.1 is still valid for fnot necessarily continuous, assuming the following hypothesis in X(which appears in Theorem 1 of [15]) if (xn)is a nondecreasing sequence in Xsuch that xn→xthen xn≤xfor all n∈N.(9) Theorem 2.2. Let (X,≤)be a partially ordered set and suppose that there exists a metric d in X such that (X,d)is a complete metric space. Assume that X satisfies (9). Let f :X→X be a nondecreasing mapping such that ψ(d(f(x), f(y)))≤ψ(d(x,y))−φ(d(x,y)),for x ≥y, where ψand φare altering distance functions. If there exists x0∈X with x0≤f(x0)then f has a fixed point. Proof. Following the proof of Theorem 2.1 we only have to check that f(z)=z. As (xn)is a nondecreasing sequence in X and limn→∞ xn=zthen the condition (9) gives us that xn≤zfor every n∈Nand, consequently, ψ(d(xn+1,f(z)))=ψ(d(f(xn), f(z)))≤ψ(d(xn,z))−φ(d(xn,z)). 3. Fixed point theorems in partially ordered metric spaces 97 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 1191 Letting n→ ∞and taking into account that ψand ψare altering distance functions, we have ψ(d(z,f(z)))≤ψ(0)−φ(0)=0. This implies ψ(d(z,f(z)))=0. Thus, d(z,f(z))=0, or equivalently, f(z)=z. Now, we present an example where it can be appreciated that hypotheses in Theorems 2.1 and 2.2 do not guarantee uniqueness of the fixed point. This example appears in [15]. Let X= {(1,0), (0,1)} ⊂ R2and consider the usual order (x,y)≤(z,t)⇔x≤zand y≤t. Thus, (X,≤)is a partially ordered set whose different elements are not comparable. Besides, (X,d2)is a complete metric space considering d2the Euclidean distance. The identity map f(x,y)=(x,y)is trivially continuous and nondecreasing and condition (1) of Theorem 2.2 is satisfied since elements in Xare only comparable to themselves. Moreover, (1,0)≤ f(1,0)=(1,0)and fhas two fixed points in X. In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.1 and 2.2. This condition is for x,y∈Xthere exists a lower bound or an upper bound. (10) In [15] it is proved that condition (10) is equivalent to for x,y∈Xthere exists z∈Xwhich is comparable to xand y.(11) Theorem 2.3. Adding condition (11) to the hypotheses of Theorem 2.1 (resp. Theorem 2.2) we obtain uniqueness of the fixed point of f . Proof. Suppose that there exist z,y∈Xwhich are fixed points. We distinguish two cases: Case1. If yis comparable to zthen fn(y)=yis comparable to fn(z)=zfor n=0,1,2, . . . and ψ(d(z,y))=ψd(fn(z), fn(y)) ≤ψd(fn−1(z), fn−1(y))−φd(fn−1(z), fn−1(y)) ≤ψ(d(z,y))−φ(d(z,y)). As ψand φare altering distance functions, the last inequality gives us φ(d(z,y))=0 and this implies z=y. Case2. If yis not comparable to zthen there exists x∈Xcomparable to yand z. Monotonicity of fimplies that fn(x)is comparable to fn(y)=yand to fn(z)=z, for n=0,1,2, . . .. Moreover, ψd(z,fn(x))=ψd(fn(z), fn(x)) ≤ψd(fn−1(z), fn−1(x))−φd(fn−1(z), fn−1(x)) ≤ψd(fn−1(z), fn−1(x)) =ψd(z,fn(x)).(12) Hence, the last inequality proves that {ψ(d(z,fn(x)))}is a nonnegative decreasing sequence. Monotonicity of ψ, gives us that {d(z,fn(x))}is also a nonnegative decreasing sequence and, consequently, there exists γsuch that lim n→∞dz,fn(x)=γ . Letting n→ ∞in (12) and, taking into account that ψand φare altering distance functions, we obtain ψ(γ ) ≤ψ(γ ) −φ(γ ) ≤ψ(γ ) and this implies φ(γ ) =0 and, consequently, γ=0. Analogously, it can be proved that lim n→∞dy,fn(x)=0. Finally, as lim n→∞dz,fn(x)=lim n→∞dy,fn(x)=0, the uniqueness of the limit gives us y=z. This finishes the proof.  98 Generalized contractions in partially ordered metric spaces ... 1192 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 Remark 2.1. Under the assumptions of Theorem 2.3, it can be proved that for every x∈X, limn→∞ fn(x)=z, where zis the fixed point (i.e. the operator fis Picard). In fact, if xis comparable to zthen using the same argument that in Theorem 2.3 can be proved that limn→∞ d(z,fn(x))= 0 and, consequently, limn→∞ fn(x)=z. If xis not comparable with z, we take y∈Xcomparable both with xand zand the same reasoning that Theorem 2.3 gives us lim n→∞dz,fn(y)=lim n→∞dfn(z), fn(y)=0 and lim n→∞dfn(x), fn(y)=0. Finally, using dz,fn(x)≤dz,fn(y)+dfn(y), fn(x) and taking limit as n→ ∞, we obtain limn→∞ d(z,fn(x))=0, or equivalently, limn→∞ fn(x)=z. Remark 2.2. Notice that if (X,≤)is totally ordered set, Theorem 2.3 gives us the uniqueness of the fixed point. Remark 2.3. By using Zermelo’s well ordering theorem, the set Xcan be well ordered and the condition (11) of our Theorem 2.3 is valid for each x,y∈X. Moreover, x0=min Xsatisfies x0≤f(x0)and Theorem 2.3 gives us Theorem 1.2 in [7] for the particular case that fis a nondecreasing function. Remark 2.4. Theorems 2 and 3 in [12] are particular cases of our Theorems 2.2 and 2.3 for ψthe identity function. 3. Application to ordinary differential equations In this section we present two examples where our Theorems 2.2 and 2.3 can be applied. The first example is inspired in [15]. We study the existence of solution for the following first-order periodic problem u0(t)=f(t,u(t)),t∈ [0,T] u(0)=u(T), (13) where T>0 and f:I×R−→ Ris a continuous function. Previously, we considered the space C(I)(I= [0,T]) of continuous functions defined on I. Obviously, this space with the metric given by d(x,y)=sup{|x(t)−y(t)| : t∈I},for x,y∈C(I), is a complete metric space. C(I)can also be equipped with a partial order given by x,y∈C(I), x≤y⇔x(t)≤y(t)for t∈I. Clearly, (C(I), ≤)satisfies condition (10), since for x,y∈C(I)the functions max{x,y}and min{x,y}are least upper and greatest lower bounds of xand y, respectively. Moreover, in [15] it is proved that (C(I), ≤)with the above mentioned metric satisfies condition (9). Now we give the following definition. Definition 3.1. A lower solution for (13) is a function α∈C1(I)such that α0(t)=f(t, α(t)),for t∈I α(0)≤α(T). Theorem 3.1. Consider problem (13) with f :I×R−→ Rcontinuous and suppose that there exist λ, α > 0with α≤2λ(eλT−1) T(eλT+1)1 2 , such that for x,y∈Rwith x ≥y 0≤f(t,y)+λy−[f(t,x)+λx]≤αqln (y−x)2+1. Then the existence of a lower solution for (13) provides the existence of an unique solution of (13). 3. Fixed point theorems in partially ordered metric spaces 99 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 1193 Proof. Problem (13) can be written as u0(t)+λu(t)=f(t,u(t))+λu(t), for t∈I= [0,T] u(0)=u(T). This problem is equivalent to the integral equation u(t)=ZT 0 G(t,s)[f(s,u(s)) +λu(s)]ds, where G(t,s)is the Green function given by G(t,s)=       eλ(T+s−t) eλT−1,0≤s<t≤T eλ(s−t) eλT−1,0≤t<s≤T. Define F:C(I)−→ C(I)by (Fu)(t)=ZT 0 G(t,s)[f(s,u(s)) +λu(s)]ds. Note that if u∈C(I)is a fixed point of Fthen u∈C1(I)is a solution of (11). In what follows, we check that hypotheses in Theorems 2.2 and 2.3 are satisfied. The mapping Fis nondecreasing, since for u≥v, and using our assumption, we can obtain f(t,u)+λu≥f(t, v) +λv, which implies, since G(t,s) > 0, that for t∈I, (Fu)(t)=ZT 0 G(t,s)[f(s,u(s)) +λu(s)]ds≥ZT 0 G(t,s)[f(s, v(s)) +λv(s)]ds=(Fv)(t). Besides, for u≥v, we have d(Fu,Fv) =sup t∈I|(Fu)(t)−(Fv)(t)| =sup t∈I ((Fu)(t)−(Fv)(t)) =sup t∈IZT 0 G(t,s)[f(s,u(s)) +λu(s)−f(s, v(s)) −λv(s)]ds ≤sup t∈IZT 0 G(t,s)α qln (u(s)−v(s))2+1ds.(14) Using the Cauchy–Schwarz inequality in the last integral we get ZT 0 G(t,s)α qln (u(s)−v(s))2+1ds≤ZT 0 G(t,s)2ds 1 2ZT 0 α2ln (u(s)−v(s))2+1ds 1 2 .(15) The first integral gives us ZT 0 G(t,s)2ds=Zt 0 G(t,s)2ds+ZT t G(t,s)2ds =Zt 0 e2λ(T+s−t) eλT−12ds+ZT t e2λ(s−t) eλT−12ds =1 2λeλT−12e2λT−1 =eλT+1 2λeλT−1.(16) 100 Generalized contractions in partially ordered metric spaces ... 1194 J. Harjani, K. Sadarangani / Nonlinear Analysis 72 (2010) 1188–1197 The second integral in (15) gives us the following estimate ZT 0 α2ln (u(s)−v(s))2+1ds≤α2ln ku−vk2+1·T =α2ln d(u, v)2+1·T.(17) Taking into account (14)–(17) we have d(Fu,Fv) ≤sup t∈I eλT+1 2λeλT−1!1 2 ·α2ln d(u, v)2+1·T1 2 = eλT+1 2λeλT−1!1 2 ·α·√T·ln d(u, v)2+11 2 and from the last inequality we obtain d(Fu,Fv)2≤eλT+1 2λeλT−1·α2·T·ln d(u, v)2+1 or, equivalently, 2λeλT−1d(Fu,Fv)2≤eλT+1·α2·T·ln d(u, v)2+1. By our assumption, as α≤2λ(eλT−1) T(eλT+1)1 2 , the last inequality gives us 2λeλT−1d(Fu,Fv)2≤2λeλT−1·ln d(u, v)2+1 and, hence, d(Fu,Fv)2≤ln d(u, v)2+1 =d(u, v)2−d(u, v)2−ln d(u, v)2+1.(18) Put ψ(x)=x2and φ(x)=x2−ln(x2+1). Obviously, ψand φare altering distance functions. From (18), we obtain for u≥v ψ(d(Fu,Fv))≤ψ(d(u, v))−φ(d(u, v)). Finally, let α(t)be a lower solution for (13), we claim that α≤F(α). In fact, α0(t)+λα(t)≤f(t, α(t))+λα(t), for t∈I. Multiplying by eλt α(t)eλt0≤[f(t, α(t))+λα(t)]eλt,for t∈I, and this gives us α(t)eλt≤α(0)+Zt 0 [f(s, α(s))+λα(s)]eλsds,for t∈I.(19) As α(0)≤α(T), the last inequality gives us α(0)eλT≤α(T)eλT≤α(0)+ZT 0 [f(s, α(s))+λα(s)]eλsds and so α(0)≤ZT 0 eλs eλT−1[f(s, α(s))+λα(s)]ds. This and (19) give us 3. Fixed point theorems in partially ordered metric spaces 101 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 916064, 14 pages doi:10.1155/2010/916064 Research Article Contractive-Like Mapping Principles in Ordered Metric Spaces and Application to Ordinary Differential Equations J. Caballero, J. Harjani, and K. Sadarangani Departamento de Matem´ aticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain Correspondence should be addressed to K. Sadarangani, [email protected] Received 25 November 2009; Revised 10 March 2010; Accepted 30 March 2010 Academic Editor: Tomonari Suzuki Copyright q2010 J. Caballero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations. 1. Introduction Existence of fixed point in partially ordered sets has been considered recently in 1–17. Tarski’s theorem is used in 9to show the existence of solutions for fuzzy equations and in 11to prove existence theorems for fuzzy differential equations. In 2,6,7,10,13some applications to ordinary differential equations and to matrix equations are presented. In 3– 5,17some fixed point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems. In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone. The main tool in the proof of the results in this context combines the ideas in the contraction principle with those in the monotone iterative technique 18. Let Sdenote the class of the class of the functions β:0,∞→0,1which satisfies the condition βtn−→ 1⇒tn−→ 0.1.1 In 19the following generalization of Banach’s contraction principle appears. 3. Fixed point theorems in partially ordered metric spaces 109 2 Fixed Point Theory and Applications Theorem 1.1. Let X, dbe a complete metric space and let T:X→Xbe a mapping satisfying dTx,Ty≤βdx, y·dx, y,for x, y ∈X, 1.2 where β∈S.ThenThas a unique fixed point z∈Xand {Tnx}converges to zfor each x∈X. Recently, in 2the authors prove a version of Theorem 1.1 in the context of ordered complete metric spaces. More precisely, they prove the following result. Theorem 1.2. Let X, ≤be a partially ordered set and suppose that there exists a metric din Xsuch that X, dis a complete metric space. Let T:X→Xbe a nondecreasing mapping such that dTx,Ty≤βdx, y·dx, y,for x, y ∈Xwith x≤y, 1.3 where β∈S. Assume that either Tis continuous or Xsatisfies the following condition: if {xn}is a nondecreasing sequence in Xsuch that xn−→ x, then xn≤x∀n∈N.1.4 Besides, suppose that for each x,y ∈Xthere exists z∈Xwhich is comparable to xand y.Ifthere exists x0∈Xwith x0≤Tx0,thenThas a unique fixed point. The purpose of this paper is to generalize Theorem 1.2 with the help of the altering functions. We recall the definition of such functions. Definition 1.3. An altering function is a function ψ:0,∞→0,∞which satisfies the following. aψis continuous and nondecreasing. bψt0 if and only if t0. Altering functions have been used in metric fixed point theory in recent papers 20– 22. In 7the authors use these functions and they prove some fixed point theorems in ordered metric spaces. 2. Fixed Point Theorems Definition 2.1. If X, ≤is a partially ordered set and T:X→X, we say that Tis monotone nondecreasing if for x,y ∈X, x≤y⇒Tx≤Ty.2.1 This definition coincides with the notion of a nondecreasing function in the case XR and ≤represents the usual total order in R. In the sequel, we prove the main result of the paper. 110 Contractive-like mapping principles in ordered metric spaces ... Fixed Point Theory and Applications 3 Theorem 2.2. Let X, ≤be a partially ordered set and suppose that there exists a metric din Xsuch that X, dis a complete metric space. Let T:X→Xbe a continuous and nondecreasing mapping such that ψdTx,Ty≤βdx,y·ψdx, y,for x≥y, 2.2 where ψis an altering function and β∈S. If there exist x0∈Xwith x0≤Tx0,thenThas a fixed point. Proof. If Tx0x0, then the proof is finished. Suppose that x0<Tx0. Since x0<Tx0and Tis a nondecreasing mapping, we obtain by induction that x0<T x0≤T2x0≤T3x0≤···≤Tnx0≤Tn1x0≤···.2.3 Put xn1Txn. Taking into account that β∈Sand since xn≤xn1for each n∈N,then, by 2.2,weget ψdxn1,x n ψdTxn,Txn−1 ≤βdxn,x n−1 ·ψdxn,x n−1 ≤ψdxn,x n−1. 2.4 Using the fact that ψis nondecreasing, we have dxn1,x n≤dxn,x n−1.2.5 If there exists n0∈Nsuch that dxn0,x n0−10, then xn0Txn0−1xn0−1and xn0−1is a fixed point and the proof is finished. In another case, suppose that dxn1,x n/ 0 for all n∈N. Then, taking into account 2.5, the sequence {dxn1,x n}is decreasing and bounded below, so lim n→∞dxn1,x nr≥02.6 Assume that r>0. Then, from 2.4, we have ψdxn1,x n ψdxn,x n−1 ≤βdxn,x n−1 <1.2.7 Letting n→∞in the last inequality and by the fact that ψis an altering function, we get 1≤lim n→∞βdxn,x n−1 ≤12.8 3. Fixed point theorems in partially ordered metric spaces 111 4 Fixed Point Theory and Applications and, consequently, limn→∞βdxn,x n−1  1.Since β∈Sthis implies that limn→∞dxn1, xn0 and this contradicts our assumption that r>0.Hence, lim n→∞dxn1,x n 0.2.9 In what follows, we will show that {xn}is a Cauchy sequence. Suppose that {xn}is not a Cauchy sequence. Then, there exists >0 for which we can find subsequences {xmk}and {xnk}of {xn}with nk>mk>ksuch that dxnk,x mk≥. 2.10 Further, corresponding to mk, we can choose nkin such a way that it is the smallest integer with nk>mkand satisfying 2.10, then dxnk−1,x mk<. 2.11 Using 2.10,2.11, and the triangular inequality, we have ≤dxnk,x mk ≤dxnk,x nk−1dxnk−1,x mk <d xnk,x nk−1. 2.12 Letting k→∞and using 2.9,weget lim k→∞dxnk,x mk. 2.13 Again, the triangular inequality gives us dxnk,x mk≤dxnk,x nk−1dxnk−1,x mk−1dxmk−1,x mk, dxnk−1,x mk−1≤dxnk−1,x nkdxnk,x mkdxmk,x mk−1. 2.14 Letting k→∞in the above two inequalities and using 2.9and 2.13, we have lim k→∞dxnk−1,x mk−1. 2.15 As nk>mkand xnk−1≥xmk−1,by2.2,weobtain ψdxnk,x mkψdTxnk−1,Tx mk−1 ≤βdxnk−1,x mk−1·ψdxnk−1,x mk−1 ≤ψdxnk−1,x mk−1. 2.16 112 Contractive-like mapping principles in ordered metric spaces ... Fixed Point Theory and Applications 5 Taking into account 2.13and 2.15and the fact that ψis continuous and letting k→∞in 2.16,weget ψ≤lim k→∞βdxnk−1,x mk−1·ψ≤ψ.2.17 As ψis an altering function, ψ>0, the last inequality gives us lim k→∞βdxnk−1,x mk−11.2.18 Since β∈S, this means that lim k→∞dxnk−1,x mk−10.2.19 This fact and 2.15give us 0 which is a contradiction. This shows that {xn}is a Cauchy sequence. Since X, dis a complete metric space, there exists z∈Xsuch that limn→∞xnz. Moreover, the continuity of Timplies that zlim n→∞Txnlim n→∞xn1Tz, 2.20 and this proves that zis a fixed point. In what follows, we prove that Theorem 2.2 is still valid for Tnot necessarily continuous, assuming the following hypothesis in Xwhich appears in 10, Theorem 1: if xnis a nondecreasing sequence in Xsuch that xn−→ x,then xn≤x∀n∈N.2.21 Theorem 2.3. Let X, ≤be a partially ordered set and suppose that there exists a metric din X such that X, dis a complete metric space. Assume that Xsatisfies 2.21.LetT:X→Xbe a nondecreasing mapping such that ψdTx,Ty≤βdx,y·ψdx, y,for x≥y, 2.22 where ψis an altering function and β∈S. If there exists x0∈Xwith x0≤Tx0,thenThas a fixed point. Proof. Following the proof of Theorem 2.2, we only have to check that Tzz.Asxnis a nondecreasing sequence in Xand limn→∞xnzthen, by 2.21, we have xn≤zfor all n∈N, and, consequently, ψdxn1,fzψdTxn,Tz ≤βdxn,z  ·ψdxn,z  ≤ψdxn,z .2.23 3. Fixed point theorems in partially ordered metric spaces 113 6 Fixed Point Theory and Applications Letting n→∞and using the continuity of ψ, we have 0≤ψdz, Tz ≤ψ00,2.24 or, equivalently, ψdz, Tz 0.2.25 As ψis an altering function, this gives us dz, Tz  0 and, thus, Tzz. Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.2 and 2.3 do not guarantee uniqueness of the fixed point. This example appears in 10. Let X{1,0,0,1}⊂R2and consider the usual order x,y≤z, t⇐⇒ x≤z, y ≤t. 2.26 X, ≤is a partially ordered set whose different elements are not comparable. Besides, X, d2 is a complete metric space considering d2as the Euclidean distance. The identity map Tx,yx,yis trivially continuous and nondecreasing and condition 2.2of Theorem 2.2 is satisfied since elements in Xare only comparable to themselves. Moreover, 1,0≤T1,0 1,0and Thas two fixed points in X. In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.2 and 2.3. This condition appears in 16and says that for x,y ∈X, there exists a lower bound or an upper bound.2.27 In 10it is proved that condition 2.27is equivalent to for x,y ∈X, there exists z∈Xwhich is comparable to xand y. 2.28 Theorem 2.4. Adding condition 2.28to the hypotheses of Theorem 2.2 (resp., Theorem 2.3), we obtain uniqueness of the fixed point of f. Proof. Suppose that there exist y,z ∈Xwhich are fixed points of Tand y/ z. We distinguish two cases. Case 1. If yand zare comparable, then Tnyyand Tnzzare comparable for n 0,1,2,.... Using the contractive condition appearing in Theorem 2.2 or Theorem 2.3and the fact that β∈S,weget ψdy,zψdTny,Tnz ≤βdTn−1y,Tn−1z·ψdTn−1y,Tn−1z ≤βdy,z·ψdy,z <ψ dy,z, 2.29 which is a contradiction. 114 Contractive-like mapping principles in ordered metric spaces ... Fixed Point Theory and Applications 7 Case 2. Using condition 2.28, there exists x∈Xcomparable to yand z. Monotonicity of T implies that Tnxis comparable to Tnyyand to Tnzz,forn0,1,2,....Moreover, as β∈S,weget ψdz, Tnx ψdTnz,Tnx ≤βdTn−1z,Tn−1x·ψdTn−1z,Tn−1x βdz, Tn−1x·ψdz, Tn−1x ≤ψdz, Tn−1x. 2.30 Since ψis nondecreasing the above inequality gives us dz, Tnx ≤dz, Tn−1x.2.31 Thus, limn→∞dz, Tnxγ≥0. Assume that γ>0. Taking into account that ψis an altering function and letting n→∞in 2.30,we obtain ψγ≤lim n→∞βdz, Tn−1x·ψγ≤ψγ,2.32 and this implies that limn→∞βdz, Tn−1x  1. Since β∈Sthen we get lim n→∞dz, Tn−1x0,2.33 and, consequently, γ0, which is a contradiction. Hence, limn→∞dz, Tnx0. Analogously, it can be proved that lim n→∞dy,Tnx0.2.34 Finally, as dz, y≤dz, Tnx dTnx,y2.35 and taking limit, we obtain dz, y0. This finishes the proof. Remark 2.5. Under the assumptions of Theorem 2.4, it can be proved that for every x∈X, limn→∞Tnxz, where zis the fixed point i.e., the operator Tis Picard. 3. Fixed point theorems in partially ordered metric spaces 115 8 Fixed Point Theory and Applications In fact, for x∈Xand xcomparable to zthen using the same argument that is in Case 1of Theorem 2.4 can prove that limn→∞dz, Tnx0 and, hence, limn→∞Tnxz. If xis not comparable to z, we take that y∈Xis comparable to xand z. Using a similar argument that is in Case 2of Theorem 2.4,weobtain lim n→∞dz, Tny0,lim n→∞dTnx,Tny0.2.36 Finally, dz, Tnx ≤dz, TnydTny,Tnx,2.37 and taking limit as n→∞, we obtain limn→∞dz, Tnx0 or, equivalently, limn→∞Tnx z. Remark 2.6. Notice that if X, ≤is totally ordered, condition 2.28is obviously satisfied. Remark 2.7. Considering ψthe identity mapping in Theorem 2.4,weobtainTheorem 1.2, being the main result of 2. 3. Application to Ordinary Differential Equations In this section we present an example where our results can be applied. This example is inspired by 10. We study the existence of solution for the following first-order periodic problem utft, ut,t∈0,T, u0uT, 3.1 where T>0andf:I×R→Ris a continuous function. Previously, we considered the space CII0,T of continuous functions defined on I. Obviously, this space with the metric given by dx,ysupxt−yt:t∈I,for x,y ∈C I,3.2 is a complete metric space. CIcan also be equipped with a partial order given by x,y ∈C I,x≤y⇐⇒ xt≤yt,for t∈I. 3.3 Clearly, CI,≤satisfies condition 2.28,sinceforx, y ∈CIthe function max{x, y}∈CI. Moreover, in 10it is proved that CI,≤with the above-mentioned metric satisfies condition 2.21. 116 Contractive-like mapping principles in ordered metric spaces ... Fixed Point Theory and Applications 9 Now, let Adenote the class of functions φ:0,∞→0,∞satisfying the following. iφis nondecreasing. iiφx<x,forx>0. iiiβxφx/x ∈S, where Sis the class of functions defined in Section 1. Examples of such functions are φtμ·t,with0≤μ<1, φtt/1t,and φtln1t. Recall now the following definition Definition 3.1. A lower solution for 3.1is a function α∈C 1Isuch that αt≤ft, αt for t∈I, α0≤αT. 3.4 Now, we present the following theorem about the existence of solution for problem 3.1in presence of a lower solution. Theorem 3.2. Consider problem 3.1with f:I×R→Rcontinuous and suppose that there exist λ, α > 0with α≤2λeλT −1 TeλT 11/2 ,3.5 such that for x, y ∈Rwith x≤y 0≤ft, yλy −ft, xλx≤αy−xφy−x,3.6 where φ∈A. Then the existence of a lower solution for 3.1provides the existence of a unique solution of 3.1. Proof. Problem 3.1can be written as utλutft, ut λutfor t∈I0,T, u0uT. 3.7 This problem is equivalent to the integral equation utT 0 Gt, sfs, us λusds, 3.8 3. Fixed point theorems in partially ordered metric spaces 117 124 Fixed point theorems for mappings satisfying a condition of ... k∈[0,1) such that Zd(T(x),T(y)) 0 ϕ(t)dt ≤kZm(x,y) 0 ϕ(t)dt for x, y ∈Xwith x≥y, where ϕis a Lebesgue measurable function with finite integral on each compact subset of [0,∞), satisfying Zε 0 ϕ(t)dt > 0for ε > 0. Assume that either T is continuous or Xsatisfies the condition if (xn)is a nondecreasing sequence in Xsuch that xn→x then xn≤xfor all n∈N. If there exists x0∈Xwith x0≤T(x0)then Thas a fixed point. In this case, we have not been able to prove the uniqueness of the fixed point under classical assumption For any x, y ∈Xthere exists z∈Xcomparable to xand y . Our Theorem 23 can be considered as a version of the following result which appears in [B. E. Rhoades, Two fixed point theorems for mapping satisfying a general contractive condition of integral type, Int. J. Math. Sci., 63, (2003), 4007–4013]. Theorem 24. Let (X, d)be a complete metric space, k∈[0,1) and T:X→ Xa mapping such that, for each x, y ∈X, Zd(T(x),T(y)) 0 ϕ(t)dt ≤kZm(x,y) 0 ϕ(t)dt , where ϕ:R+→R+is a Lebesgue-measurable function with finite integral on each compact subset of R+, satisfying Zε 0 ϕ(t)dt > 0for ε > 0. Then Thas a fixed point. In the paper,we ask if in the contractive condition of Theorem th4.2 we 3. Fixed point theorems in partially ordered metric spaces 125 can replace m(x, y) by M(x, y), being M(x, y) = max{d(x, y), dx, T(x), dy, T(y), dx, T(y), dy, T(x)}, This quantity was previously used by ´ Ciri´c in [L. B. ´ Ciri´c, A generalization of Banach’s contraction principle, Proc. Amer Math. Soc. 45 (1974) 267–273]. We present an example that this is not possible. This example appears in [B. E. Rhoades, Two fixed point theorems for mapping satisfying a general contractive condition of integral type, Int. J. Math. Sci., 63, (2003), 4007–4013]. We answer affirmatively to our question with the addition of certain condition. More precisely, we prove the following result. Theorem 25. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let T:X→Xbe a nondecreasing mapping such that there exists k∈[0,1) with ZdT(x),T(y) 0 ϕ(t)dt ≤kZM(x,y) 0 ϕ(t)dt, for x≥y, (3.7) where ϕ:R+→R+is a Lebesgue-measurable function with finite integral on each compact subset of R+, such that Zε 0 ϕ(t)dt > 0for ε > 0. Assume that Tis continuous or Xsatisfies the condition if (xn)is a nondecreasing sequence in Xwith xn→x then xn≤xfor all n∈N. If there exists x0∈Xwith x0≤T(x0)and the orbit of x0is bounded, then Thas a fixed point. Note that our extra assumption is that the orbit of x0is bounded. Journal of Convex Analysis Volume 17 (2010), No. 2, 597–609 Fixed Point Theorems for Mappings Satisfying a Condition of Integral Type in Partially Ordered Sets∗ J. Harjani Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain K. Sadarangani Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain ksadar[email protected]gc.es Dedicated to Professor Jos´e Rodr´ıguez Exp´osito on the occasion of his 60th birthday. Received: July 14, 2008 Revised manuscript received: June 22, 2009 The purpose of this paper is to present some fixed point theorems for monotone operators in a metric space endowed with a partial order using a general contractive condition of integral type. Keywords: Fixed point, partially ordered metric spaces 1991 Mathematics Subject Classification: 47H10 1. Preliminaries Recently the Banach contraction principle [8] was discussed in a metric space endowed with a partial order where some applications to matrix equations [16] and to ordinary differential equations [11, 13] are presented. The usual contraction condition is weakened but at the expense that the operator is monotone. The main idea in [11, 16] involves combining the ideas in the contraction principle with those in the monotone iterative technique [2, 3]. This article presents new results for contractions satisfying a condition of integral type in ordered metric spaces and these results are slight extensions of those in [11, 16]. Existence of fixed point in partially ordered sets starts with Tarki’s theorem [18]. Recently, a lot of papers have treated this equation (see, for example [5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 19]). ∗Partially supported by Ministerio de Ciencia y Tecnolog´ıa, project MTM 2007-65706. ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 3. Fixed point theorems in partially ordered metric spaces 127 598 J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... 2. Fixed point theorems Suppose (X, ≤) is a partially ordered set and f:X−→ X. We say fis non-decreasing if x, y ∈X,x≤yimplies f(x)≤f(y). In a recent paper [1], R. Agarwal, M. El-Gebeily and D. O’Regan established the following theorem. Theorem 2.1. Let (X, ≤)be a partially ordered set and suppose that there is a metric don Xsuch that (X, d)is a complete metric space. Assume there is a non-decreasing function ψ: [0,∞)−→ [0,∞)with limn→∞ ψn(t) = 0 for each t > 0and also suppose F:X−→ Xis a nondecreasing mapping with d(F(x), F(y)) ≤ψmax{d(x, y), d(x, F(x)), d(y, F(y)),1 2[d(x, F(y)) + d(y, F(x))]}, for all x≥y. Also suppose either Fis continuous or if (xn)⊂Xis a nondecreasing sequence with xn→xin Xthen xn≤xfor all n∈N. If there exists x0∈Xwith x0≤F(x0), then Fhas a fixed point. Now, we present our main result in this paper. Previously, we define for F:X−→ X m(x, y) = max d(x, y), d(x, F(x)), d(y, F(y)),1 2[d(x, F(y)) + d(y, F(x))]. Theorem 2.2. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let F:X→Xbe a continuous and nondecreasing mapping such that there exists k∈[0,1) with Zd(F(x),F (y)) 0 ϕ(t)dt ≤kZm(x,y) 0 ϕ(t)dt for x≥y, (1) where ϕ:R+−→ R+is a Lebesgue-integrable mapping such that Rε 0ϕ(t)>0for ε > 0. If there exists x0∈Xwith x0≤F(x0)then Fhas a fixed point. Proof. If F(x0) = x0then the proof is finished. Suppose that x0< F(x0). Since x0< F(x0) and Fis nondecreasing, we obtain by induction that x0≤F(x0)≤F2(x0)≤ · · · ≤ Fn(x0)≤Fn+1(x0)≤.... Put xn+1 =Fn(x0). Then for each integer n≥1, from (1) and, as the elements xnand xn+1 are comparable, we get Zd(xn,xn+1) 0 ϕ(t)dt =Zd(F(xn−1),F (xn)) 0 ϕ(t)dt ≤kZm(xn−1,xn) 0 ϕ(t)dt. (2) 128 Fixed point theorems for mappings satisfying a condition of ... J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... 599 Taking into account that m(xn−1, xn) = max d(xn−1, xn), d(xn−1, F (xn−1)), d(xn, F(xn)),1 2[d(xn−1, F(xn)) + d(xn, F (xn−1))] = max d(xn−1, xn), d(xn−1, xn), d(xn, xn+1),1 2[d(xn−1, xn+1) + d(xn, xn)] = max d(xn−1, xn), d(xn, xn+1),1 2[d(xn−1, xn+1)], and, as d(xn−1, xn+1) 2≤d(xn−1, xn) + d(xn, xn+1) 2≤max{d(xn−1, xn), d(xn, xn+1)} we obtain m(xn−1, xn) = max{d(xn−1, xn), d(xn, xn+1)}. Substituting into (2) we obtain Zd(xn,xn+1) 0 ϕ(t)dt ≤kZmax{d(xn−1,xn),d(xn,xn+1)} 0 ϕ(t)dt =kmax (Zd(xn−1,xn) 0 ϕ(t)dt, Zd(xn,xn+1) 0 ϕ(t)dt).(3) If max nRd(xn−1,xn) 0ϕ(t)dt, Rd(xn,xn+1) 0ϕ(t)dto=Rd(xn,xn+1) 0ϕ(t), then, by (3), Zd(xn,xn+1) 0 ϕ(t)dt ≤kZd(xn,xn+1) 0 ϕ(t)dt. and, as k∈[0,1), we have that Rd(xn,xn+1) 0ϕ(t)dt = 0. By our hypothesis about ϕ, we get d(xn, xn+1) = 0, or, equivalently, xn=xn+1 =F(xn) and xnis a fixed point of F. If max nRd(xn−1,xn) 0ϕ(t)dt, Rd(xn,xn+1) 0ϕ(t)dto=Rd(xn−1,xn) 0ϕ(t) then, from (3), we get Zd(xn,xn+1) 0 ϕ(t)dt ≤kZd(xn−1,xn) 0 ϕ(t)dt. (4) Using induction we have Zd(xn,xn+1) 0 ϕ(t)dt ≤kZd(xn−1,xn) 0 ϕ(t)dt ≤ · · · ≤ knZd(x0,x1) 0 ϕ(t)dt. Taking limit as n→ ∞ lim n→∞ Zd(xn,xn+1) 0 ϕ(t)dt = 0.(5) 3. Fixed point theorems in partially ordered metric spaces 129 600 J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... On the other hand, by (4), as k∈[0,1), Zd(xn,xn+1) 0 ϕ(t)dt ≤kZd(xn−1,xn) 0 ϕ(t)dt < Zd(xn−1,xn) 0 ϕ(t)dt and, as ϕis a non-negative function, we obtain that {d(xn, xn+1)}is a non-negative and non-increasing sequence. We put limn→∞ d(xn, xn+1) = a. In what follows, we will prove that a= 0. Suppose that a > 0. As 0 < a ≤d(xn, xn+1) for all n, and, taking into account our assumption about ϕ, 0<Za 0 ϕ(t)dt ≤Zd(xn,xn+1) 0 ϕ(t)dt. Taking limit as n→ ∞ and, from (5), 0<Za 0 ϕ(t)dt ≤lim n→∞ Zd(xn,xn+1) 0 ϕ(t)dt = 0, which is a contradiction. Therefore, lim n→∞ d(xn, xn+1) = 0.(6) Now, we show that {xn}is a Cauchy sequence. Suppose that {xn}is not a Cauchy sequence there exists an ε > 0 and subsequences {m(p)}and {n(p)}such that m(p)< n(p)< m(p+ 1) with d(xm(p), xn(p))≥εand d(xm(p), xn(p)−1)< ε. (7) Then m(xm(p)−1, xn(p)−1) = max d(xm(p)−1, xn(p)−1), d(xm(p)−1, xm(p)), d(xn(p)−1, xn(p)), 1 2[d(xm(p)−1, xn(p)) + d(xm(p), xn(p)−1)]. By (5), we have lim p→∞ Zd(xm(p)−1,xm(p)) 0 ϕ(t)dt = lim p→∞ Zd(xn(p)−1,xn(p)) 0 ϕ(t)dt = 0.(8) By the triangular inequality and (7) d(xm(p)−1, xn(p)−1)≤d(xm(p)−1, xm(p)) + d(xm(p), xn(p−1))< d(xm(p)−1, xm(p)) + ε and, by (5), this implies lim p→∞ Zd(xm(p)−1,xn(p)−1) 0 ϕ(t)dt ≤Zε 0 ϕ(t)dt. (9) 130 Fixed point theorems for mappings satisfying a condition of ... J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... 601 Again, using the triangular inequality and (7), we get 1 2[d(xm(p)−1, xn(p)) + d(xm(p), xn(p)−1)] ≤1 2[d(xm(p)−1, xm(p)) + d(xm(p), xn(p)−1) + d(xn(p)−1, xn(p)) + d(xm(p), xn(p)−1)] =1 2[d(xm(p)−1, xm(p)) + 2d(xm(p), xn(p)−1) + d(xn(p)−1, xn(p))] =1 2[d(xm(p)−1, xm(p)) + d(xn(p)−1, xn(p))] + d(xm(p), xn(p)−1) <1 2[d(xm(p)−1, xm(p)) + d(xn(p)−1, xn(p))] + ε. Taking into account (6), we obtain lim p→∞ Z1 2[d(xm(p)−1,xn(p))+d(xm(p),xn(p)−1)] 0 ϕ(t)dt ≤Zε 0 ϕ(t)dt. (10) From (1) and (7), we can get Zε 0 ϕ(t)dt ≤Zd(xm(p),xn(p)) 0 ϕ(t)dt =Zd(F(xm(p)−1),F (xn(p)−1)) 0 ϕ(t)dt ≤kZm(xm(p)−1,xn(p)−1) 0 ϕ(t)dt =kmax Zd(xm(p)−1,xn(p)−1) 0 ϕ(t)dt, Zd(xm(p)−1,xm(p)) 0 ϕ(t)dt, Zd(xn(p)−1,xn(p)) 0 ϕ(t)dt, Z1 2[d(xm(p)−1,xn(p))+d(xm(p),xn(p)−1)] 0 ϕ(t)dt!, and, taking limit as p→ ∞, and taking into account (8), (9) and (10), we obtain Zε 0 ϕ(t)dt ≤kZε 0 ϕ(t)dt. As k∈[0,1), this implies Rε 0ϕ(t)dt = 0 which is a contradiction. Therefore, {xn}is a Cauchy sequence. Since Xis a complete metric space there exists z∈Xsuch that limn→∞ xn=z. Finally, we prove that z∈Xis a fixed point of F. As Fis a continuous mapping and limn→∞ xn=z, then z= lim n→∞ xn+1 = lim n→∞ F(xn) = F(z) and the proof is complete. In what follows, we prove that Theorem 2.2 is still valid for Fnot necessarily continuous, assuming the following hypothesis in X(which appears in Theorem 1 of [1]): if (xn)⊂Xis a nondecreasing sequence with xn→xthen xn≤xfor all n∈N.(11) 3. Fixed point theorems in partially ordered metric spaces 131 602 J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... Theorem 2.3. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let F:X−→ Xbe a nondecreasing mapping such that there exists k∈[0,1) with Zd(F(x),F (y)) 0 ϕ(t)dt ≤kZm(x,y) 0 ϕ(t)dt, for x≥y, where ϕ:R+−→ R+is a Lebesgue-integrable mapping such that Rε 0ϕ(t)dt > 0for ε > 0. Assume that Xsatisfies (11) and there exists x0∈Xwith x0≤F(x0), then F has a fixed point. Proof. Following the proof of Theorem 2.2, we only have to check that F(z) = z. From (2) and (11), we have Zd(F(z),xn+1) 0 ϕ(t)dt ≤kZm(z,xn) 0 ϕ(t)dt =kmax (Zd(z,xn) 0 ϕ(t)dt, Zd(z,F (z)) 0 ϕ(t)dt, Zd(xn+1,xn) 0 ϕ(t)dt, Z1 2[d(z,xn+1)+d(xn,F (z))] 0 ϕ(t)dt), and, taking limit as n→ ∞, and, by (5), we get Zd(F(z),z) 0 ϕ(t)dt ≤kZd(F(z),z) 0 ϕ(t)dt, which implies that Rd(F(z),z) 0ϕ(t)dt = 0. By our assumption about ϕ, this gives us d(F(z), z) = 0 and this proves that zis a fixed point of F. Remark 2.4. If we assume that ϕis a nonincreasing function in Theorem 2.2 its proof is less complicated. In fact, perhaps the more difficult part in Theorem 2.2 is to prove that {xn}is a Cauchy sequence. Under assumption that ϕis a nonincreasing function, for m > n we can get Zd(xm,xn) 0 ϕ(t)dt ≤Zd(xm,xm−1)+d(xm−1,xm−2)+···+d(xn+1,xn) 0 ϕ(t)dt =Zd(xn+1,xn) 0 ϕ(t)dt +Zd(xn+2,xn+1)+d(xn+1,xn) d(xn+1,xn) ϕ(t)dt +···+Zd(xn+1,xn)+···+d(xm−1,xm−2)+d(xm,xm−1) d(xn+1,xn)+···+d(xm−1,xm−2) ϕ(t)dt. Applying a simple change of variables, our integrals can be transformed in Zd(xm,xn) 0 ϕ(t)dt ≤ m X i=n+1 Zd(xi,xi−1) 0 ϕ s+ i−1 X j=n+1 d(xj, xj−1)!ds 132 Fixed point theorems for mappings satisfying a condition of ... J. Harjani, K. Sadarangani / Fixed Point Theorems for Mappings Satisfying ... 603 and, as ϕis a nonincreasing function, we can get Zd(xm,xn) 0 ϕ(t)dt ≤ m X i=n+1 Zd(xi,xi−1) 0 ϕ s+ i−1 X j=n+1 d(xj, xj−1)!ds ≤ m X i=n+1 Zd(xi,xi−1) 0 ϕ(s)ds. Taking into account (4) in the proof of Theorem 2.2, we obtain Zd(xm,xn) 0 ϕ(t)dt ≤ m X i=n+1 Zd(xi,xi−1) 0 ϕ(t)dt ≤ m X i=n+1 ki−1Zd(x0,x1) 0 ϕ(t)dt = Zd(x0,x1) 0 ϕ(t)dt!(kn+···+km−1) ≤ Zd(x0,x1) 0 ϕ(t)dt!kn 1−k. Taking limit as n→ ∞ we have lim m,n→∞ Zd(xm.xn) 0 ϕ(t)dt = 0.(12) Now, suppose that {xn}is not a Cauchy sequence. This means that there exists an ε > 0 such that for any p∈Nwe can find m(p), n(p)∈Nwith m(p), n(p)> p satisfying d(xm(p),xn(p))≥ε. Consequently, Zd(xm(p),xn(p)) 0 ϕ(t)dt ≥Zε 0 ϕ(t)dt > 0, and, taking limit as p→ ∞, we get lim p→∞ Zd(xm(p),xn(p)) 0 ϕ(t)dt ≥Zε 0 ϕ(t)dt > 0 and this contradicts to (12). Remark 2.5. If we put ϕ(t) = 1 in (1) of Theorem 2.2, we have d(F(x), F(y)) ≤k m(x, y) for x≥y and our Theorem 2.2 is a particular case of Theorem 2.2 of [1] for the function ψ(t) = kt with k∈[0,1). Remark 2.6. If we put ϕ(t) = 1 in (1) of Theorem 2.2 then the condition d(F(x), F(y)) ≤kd(x, y) for x≥yimplies d(F(x), F (y)) ≤k m(x, y) and Theorem 2.1 in [11] and Theorem 2.1 in [16] are particular cases of our Theorem 2.2. 3. Fixed point theorems in partially ordered metric spaces 133 Theory of existence and uniqueness of solution for bvp 237 4.2. Fractional boundary value problem 4.2.1. Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to the next fractional boundary value problem (Dα 0+u(t) + ft, u(t)= 0 ,0< t < 1, u(0) = u0(1) = u00(0) = 0 ,(4.8) where 2 < α ≤3, Dα 0+is the Caputo’s differentiation and f: (0,1]×[0,∞)→ [0,∞) with lim t→0+f(t, −) = ∞, i.e., fis singular at t= 0. Our main tool in this paper is the following fixed point theorem in partially ordered metric spaces, which is the main result of the first paper of Section 1. Theorem 7. Let (X, ≤)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Assume that Xsatisfies the following condition: if (xn)is a nondecreasing sequence in Xsuch that xn→x then xn≤x, for all n∈N. Let T:X→Xbe a nondecreasing mapping such that d(Tx, Ty)≤d(x, y)−ψd(x, y),for any x, y ∈Xwith x≥y , where ψ)is an altering distance function. If there exists x0∈Xwith x0≤Tx0then Thas a fixed point. Besides, if for any x, y ∈Xthere exists z∈Xwhich is comparable to xand y, then the fixed point is unique. The first result of the paper is: 238 Fractional boundary value problem Theorem 49. Consider Problem (4.8), where 2< α ≤3, under these assumptions: (i) f: (0,1]×[0,∞)→[0,∞)is continuous and satisfies lim t→0+f(t, −) = ∞. (ii) The function tσf(t, y)is a continuous function on [0,1]×[0,∞), where 0< σ < 1. (iii) There exists 0< λ ≤Γ(α−σ) Γ(1−σ)such that, for any x, y ∈[0,∞)with y≥x and t∈[0,1], 0≤tσf(t, y)−f(t, x)≤λ·ln(y−x+ 1) . Then Problem (4.8) has a unique nonnegative solution. We notice in the paper that Theorem 49 remains valid if we replace assumption (iii) by (iii)’ There exists 0 < λ ≤Γ(α−σ) Γ(1−σ)such that, for any x, y ∈[0,∞) with y≥x and t∈[0,1], 0≤tσf(t, y)−f(t, x)≤λ·ψ(y−x+ 1) , where ψ: [0,∞)→[0,∞) is a continuous function such that ϕ(x) = x−ψ(x) satisfies: (a) ϕ: [0,∞)→[0,∞) is nondecreasing. (b) ϕ(0) = 0. (c) ϕis positive in (0,∞). Next, we probe that the Green’s function G(t, s) associated to Problem (4.8) which is given by G(t, s) =        (α−1) t(1 −s)α−2−(t−s)α−1 Γ(α),0≤s≤t≤1, t(1 −s)α−2 Γ(α−1) ,0≤t≤s≤1, Theory of existence and uniqueness of solution for bvp 239 is strictly increasing in the first variable in (0,1). Our main result is the following. Theorem 50. Under assumption of Theorem 49, Problem (4.8) has a unique strictly increasing and positive solution. Our results can be compared with the ones obtained by T. Qiu, Z. Bai, Existence of positive solutions for singular fractional differential equations, Elect. J. Diff. Eq. vol. (2008), 146, 2008, 1–9, where the authors study the same problem, but their results do not give uniqueness of the solution and neither monotonic character of this solution. Our main contribution in this paper is the uniqueness of the solution and , moreover, the strictly increasing character of this solution. Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 421310, 10 pages doi:10.1155/2009/421310 Research Article Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems J. Caballero Mena, J. Harjani, and K. Sadarangani Departamento de Matem´ aticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain Correspondence should be addressed to K. Sadarangani, [email protected] Received 24 April 2009; Accepted 14 June 2009 Recommended by Juan Jos´ e Nieto We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets. Copyright q2009 J. Caballero Mena et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function see, e.g., 1,2 and to problems of analyticity in the complex domain 3. Moreover, Delbosco and Rodino 4considered the existence of a solution for the nonlinear fractional differential equation Dα 0uft, u, where 0 <α<1andf:0,a×R→R,0<a≤∞ is a given continuous function in 0,a×R. They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang 5 considered the existence of positive solution for equation Dα 0uft, u, where 0 <α<1 and f:0,1×0,∞→0,∞is a given continuous function by using the suband supersolution methods. In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation Dα 0utft, ut 0,0<t<1, u0u1u00, 1.1 Theory of existence and uniqueness of solution for bvp 241 2 Boundary Value Problems where 2 <α≤3, Dα 0is the Caputo’s differentiation and f:0,1×0,∞→0,∞with limt→0ft, −∞i.e., fis singular at t0. Note that this problem was considered in 6where the authors proved the existence of one positive solution for 1.1by using Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function f.In6the uniqueness of the solution is not treated. In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem 1.1by using a fixed point theorem in partially ordered sets. Existence of fixed point in partially ordered sets has been considered recently in 7–12. This work is inspired in the papers 6,8. For existence theorems for fractional differential equation and applications, we refer to the survey 13. Concerning the definitions and basic properties we refer the reader to 14. Recently, some existence results for fractional boundary value problem have appeared in the literature see, e.g., 15–17. 2. Preliminaries and Previous Results For the convenience of the reader, we present here some notations and lemmas that will be used in the proofs of our main results. Definition 2.1. The Riemman-Liouville fractional integral of order α>0ofafunction f: 0,∞→Ris given by Iα 0ft1 Γαt 0 t−sα−1fsds 2.1 provided that the right-hand side is pointwise defined on 0,∞. Definition 2.2. The Caputo fractional derivative of order α>0 of a continuous function f: 0,∞→Ris given by Dα 0ft1 Γn−αt 0 fns t−sα−n1ds, 2.2 where n−1<α≤n, provided that the right-hand side is pointwise defined on 0,∞. The following lemmas appear in 14. Lemma 2.3. Let n−1<α≤n,u∈Cn0,1.Then Iα 0Dα 0utut−c1−c2t−···−cntn−1,2.3 where ci∈R,i1,2,...,n. 242 Fractional boundary value problem Boundary Value Problems 3 Lemma 2.4. The relation Iα 0Iβ 0ϕIαβ 0ϕ2.4 is valid when Re β>0,Reαβ>0,ϕx∈L10,b. The following lemmas appear in 6. Lemma 2.5. Givenf∈C0,1and 2<α≤3, the unique solution of Dα 0utft0,0<t<1, u0u1u00, 2.5 is given by ut1 0 Gt, sfsds, 2.6 where Gt, s⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α−1t1−sα−2−t−sα−1 Γα,0≤s≤t≤1, t1−sα−2 Γα−1,0≤t≤s≤1. 2.7 Remark 2.6. Note that Gt, s>0fort/ 0andG0,s0see 6. Lemma 2.7. Let 0<σ<1,2<α≤3and F:0,1→Ris a continuous function with limt→0Ft∞. Suppose that tσFtis a continuous function on 0,1. Then the function defined by Ht1 0 Gt, sFsds 2.8 is continuous on [0,1], where Gt, sis the Green function defined in Lemma 2.5. Now, we present some results about the fixed point theorems which we will use later. These results appear in 8. Theorem 2.8. Let X, ≤be a partially ordered set and suppose that there exists a metric din Xsuch that X, dis a complete metric space. Assume that Xsatisfies the following condition: if {xn}is a non decreasing sequence in Xsuch that xn→xthen xn≤xfor all n∈N.LetT:X→Xbe a nondecreasing mapping such that dTx,Ty≤dx, y−ψdx, y,for x≥y, 2.9 Theory of existence and uniqueness of solution for bvp 243 4 Boundary Value Problems where ψ:0,∞→0,∞is continuous and nondecreasing function such that ψis positive in 0,∞,ψ00and limt→∞ψt∞. If there exists x0∈Xwith x0≤Tx0then Thas a fixed point. If we consider that X, ≤satisfies the following condition: for x,y ∈Xthere exists z∈Xwhich is comparable to xand y, 2.10 then we have the following theorem 8. Theorem 2.9. Adding condition 2.10to the hypotheses of Theorem 2.8 one obtains uniqueness of the fixed point of f. In our considerations, we will work in the Banach space C0,1{x:0,1→ R,continuous}with the standard norm xmax0≤t≤1|xt|. Note that this space can be equipped with a partial order given by x,y ∈C0,1,x≤y⇐⇒ xt≤yt,for t∈0,1.2.11 In 10it is proved that C0,1,≤with the classic metric given by dx,ymax 0≤t≤1xt−yt2.12 satisfies condition 2of Theorem 2.8. Moreover, for x, y ∈C0,1, as the function max{x,y} is continuous in 0,1,C0,1,≤satisfies condition 2.10. 3. Main Result Theorem 3.1. Let 0<σ<1,2<α≤3,f:0,1×0,∞→0,∞is continuous and limt→0ft, −∞,tσft, yis a continuous function on 0,1×0,∞. Assume that there exists 0<λ≤Γα−σ/Γ1−σsuch that for x, y ∈0,∞with y≥xand t∈0,1 0≤tσft, y−ft, x≤λ·lny−x13.1 Then one’s problem 1.1has an unique nonnegative solution. Proof. Consider the cone P{u∈C0,1:ut≥0}.3.2 Note that, as Pis a closed set of C0,1,Pis a complete metric space. 244 Fractional boundary value problem Boundary Value Problems 5 Now, for u∈Pwe define the operator Tby Tut1 0 Gt, sfs, usds. 3.3 By Lemma 2.7,Tu ∈C0,1. Moreover, taking into account Remark 2.6 and as tσft, y≥0 for t, y∈0,1×0,∞by hypothesis, we get Tut1 0 Gt, ss−σsσfs, usds ≥0.3.4 Hence, TP⊂P. In what follows we check that hypotheses in Theorems 2.8 and 2.9 are satisfied. Firstly, the operator Tis nondecreasing since, by hypothesis, for u≥v Tut1 0 Gt, sfs, usds 1 0 Gt, ss−σsσfs, usds ≥1 0 Gt, ss−σsσfs, vsds Tvt. 3.5 Besides, for u≥v dTu,Tvmax t∈0,1|Tut−Tvt| max t∈0,1Tut−Tvt max t∈0,11 0 Gt, sfs, us −fs, vsds max t∈0,11 0 Gt, ss−σsσfs, us −fs, vsds ≤max t∈0,11 0 Gt, ss−σλ·lnus−vs1ds 3.6 As the function ϕxlnx1is nondecreasing then, for u≥v, lnus−vs1≤lnu−v13.7 Theory of existence and uniqueness of solution for bvp 245 252 Fractional boundary value problem y, then Thas a unique fixed point. Before to present the main result of the paper, we prove some lemmas. The Green’s function associated to Problem (4.9) is given by G(t, s) =        tα−1(1 −s)α−1−(t−s)α−1 Γ(α),0≤s≤t≤1, tα−1(1 −s)α−1 Γ(α),0≤t≤s≤1, being Γ the gamma function. Lemma 3. Suppose that 0< σ < 1,1< α ≤2and F: (0,1] →Ris a continuous function such that lim t→0+F(t) = ∞. If tσF(t)is a continuous function on [0,1] then the function H(t) = Z1 0 G(t, s)F(s)ds , is continuous on [0,1]. Lemma 4. Assume that 0< σ < 1. Then, max 0≤t≤1Z1 0 G(t, s)s−σds=Aα−1−Aα−σ Γ(α)β(1 −σ, α), where A=α−1 α−σ1 1−σand βis the Euler beta function. By commodity, we denote by K=Aα−1−Aα−σ Γ(α)β(1 −σ, α). For our main result, we need the class of functions Agiven by φ∈ A if φ: [0,∞)→[0,∞) and it satisfies (i) φis nondecreasing. (ii) φ(x)< x, for any x > 0. (iii) β(x) = φ(x) xis such that β(tn)→1⇒tn→0. The main result of the paper is the next theorem. Theory of existence and uniqueness of solution for bvp 253 Theorem 52. Suppose that 0< σ < 1and 1< α ≤2. Under the following assumptions: (i) f: (0,1] ×[0,∞)→[0,∞)is a continuous function such that lim t→0+f(t, −) = ∞, (ii) tσf(t, y)is a continuous function on [0,1] ×[0,∞). (iii) There exist 0< λ ≤1 Ksuch that, for x, y ∈[0,∞)with y≥xand t∈[0,1], 0≤tσ(f(t, y)−f(t, x)) ≤λφ(y−x), where φ∈ A, Problem (4.9) has a unique positive solution. Finally, we present an example illustrating our results. Computers and Mathematics with Applications 62 (2011) 1325–1332 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Positive solutions for a class of singular fractional boundary value problems✩ J. Caballero∗, J. Harjani, K. Sadarangani Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain article info Keywords: Fractional boundary value problem Fixed point theorem Positive solution abstract In this paper, we investigate the existence and uniqueness of positive solutions for the following singular fractional boundary value problem Dα 0+u(t)+f(t,u(t)) =0,0<t<1, u(0)=u(1)=0, where 1 < α ≤2, Dα 0+is the standard Riemann–Liouville differentiation and f:(0,1] × [0,∞)−→ [0,∞)with limt→0+f(t,−)= ∞ (i.e., fis singular at t=0). Our analysis relies on a fixed point theorem in partially ordered sets. ©2011 Elsevier Ltd. All rights reserved. 1. Introduction Many papers and books on fractional differential equations have appeared recently (see, for example, [1–12]). Most of them are devoted to the solvability of linear fractional equations in terms of a special function (see, e.g., [3,8]) and to problems of analyticity in the complex domain [7]. Moreover, Delbosco and Rodino [4] considered the existence of a solution for the nonlinear fractional differential equation Dα 0+u=f(t,u), where 0 < α < 1 and f:[0,a] × R−→ R,0<a≤ +∞ is a given continuous function in (0,a)×R. They obtained their results by using the Schauder fixed point theorem and the Banach contraction principle. Zhang [11] considered the existence of positive solution for the equation Dα 0+u=f(t,u), where 0 < α < 1 and f:[0,1]×[0,∞)−→ [0,∞), is a given continuous function by using the suband super-solution methods. Recently, Bai and Lü [1] have investigated the existence and multiplicity of positive solutions for the boundary value problem Dα 0+u(t)+f(t,u(t)) =0,0<t<1 u(0)=u(1)=0,(1) where 1 < α ≤2 and f:[0,1]×[0,∞)−→ [0,∞)is continuous, by using some fixed point theorems on cones. Motivated by [1], in this paper we discuss the existence and uniqueness of positive solutions for Problem (1) assuming that f:(0,1]×[0,∞)−→ [0,∞)is such that limt→0+f(t,−)= ∞(i.e., fis singular at t=0). Our study is based on a fixed point theorem in partially ordered sets. The existence of fixed points in partially ordered sets has been considered recently in [13–17]. This work is inspired by papers [13,2,9]. For existence theorems of fractional differential equations and applications, we refer to surveys [5,8]. Concerning the definitions and basic properties, we refer the reader to [10]. ✩This research was partially supported by ‘‘Ministerio de Educación y Ciencia’’ Project MTM 2007/65706. ∗Corresponding author. E-mail addresses: [email protected] (J. Caballero), [email protected] (J. Harjani), [email protected] (K. Sadarangani). 0898-1221/$ – see front matter ©2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.04.013 Theory of existence and uniqueness of solution for bvp 255 1326 J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 2. Preliminaries and basic facts For the convenience of the reader, we present here some notation and lemmas which will be used in the proofs of our results. Definition 1. The Riemann–Liouville fractional integral of order α > 0 of a function f:(0,∞)→Ris defined by Iα 0+f(t)=1 Γ(α) ∫t 0 (t−s)α−1f(s)ds, provided that the right-hand side is pointwise defined on (0,∞), and where Γ(α) denotes the classical gamma function. Definition 2. The Riemann–Liouville fractional derivative of order α > 0 of a function f:(0,∞)→Ris given by Dα 0+f(t)=1 Γ(n−α) d dtn∫t 0 f(s) (t−s)α−n+1ds, where n= [α]+1 and [α]denotes the integer part of α. The following two lemmas can be found in [10]. Lemma 1. Let α > 0and u ∈C(0,1)∩L1(0,1). Then the fractional differential equation Dα 0+u(t)=0 has u(t)=c1tα−1+c2tα−2+···+cntα−n, where ci∈R(i=1,2,...,n)and n = [α]+1as unique solution. Lemma 2. Assume that u ∈C(0,1)∩L1(0,1)with fractional derivative of order α > 0that belongs to C(0,1)∩L1(0,1). Then Iα 0+Dα 0+u(t)=u(t)+c1tα−1+c2tα−2+···+cntα−n, for some ci∈R(i=1,...,n)and n = [α]+1. Using Lemma 2, in [1] the following result is proved. Lemma 3. Given f ∈C[0,1]and 1< α ≤2, the unique solution of Dα 0+u(t)+f(t)=0,0<t<1, u(0)=u(1)=0, is u(t)=∫1 0 G(t,s)f(s)ds, where G(t,s)=       tα−1(1−s)α−1−(t−s)α−1 Γ(α) ,0≤s≤t≤1 tα−1(1−s)α−1 Γ(α) ,0≤t≤s≤1. Remark 1. It is easily checked that G(t,s)is a continuous function on [0,1]×[0,1]and it satisfies G(t,s) > 0, for t,s∈(0,1). In what follows, we present the fixed point theorem which we will use later. This result appears in [13]. By Jwe denote the class of those functions β:[0,∞)−→ [0,1)satisfying the following condition β(tn)→1 implies tn→0. Theorem 1 (Theorem 2.1 of [13]).Let (X,≤)be a partially ordered set and suppose that there exists a metric d in X such that (X,d)is a complete metric space. Let T:X−→ X be a nondecreasing mapping such that there exists an element x0∈X with x0≤Tx0. Suppose that there exists β∈Jsuch that d(Tx,Ty)≤β(d(x,y)) ·d(x,y), for x,y∈X with x ≥y. 256 Fractional boundary value problem J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 1327 Assume that either T is continuous or X is such that if {xn}is a nondecreasing sequence in X such that xn→x then xn≤x for all n ∈N.(2) Besides, if for each x,y∈X there exists z ∈X which is comparable to x and y,(3) then T has a unique fixed point. In our considerations, we will work in the Banach space C[0,1]={x: [0,1] → R,continuous}with the classical metric given by d(x,y)=sup0≤t≤1{|x(t)−y(t)|}. Notice that this space can be equipped with a partial order given by x,y∈C[0,1],x≤y⇔x(t)≤y(t), for t∈ [0,1]. In [15] it is proved that (C[0,1],≤)satisfies condition (2) of Theorem 1. Moreover, for x,y∈C[0,1], as the function max(x,y)∈C[0,1], (C[0,1],≤)satisfies condition (3). 3. Main result Our starting point of this section is the following lemma. Lemma 4. Let 0< σ < 1,1< α ≤2and F :(0,1] −→ Ris a continuous function with limt→0+F(t)= ∞. Suppose that tσF(t)is a continuous function on [0,1]. Then the function defined by H(t)=∫1 0 G(t,s)F(s)ds is continuous on [0,1], where G(t,s)is the Green function appearing in Lemma 3. Proof. We divide the proof into three cases. Case 1: t0=0. It is easily checked that H(0)=0. Since tσF(t)is continuous on [0,1], we can find a constant M>0 such that |tσF(t)| ≤ Mfor any t∈ [0,1]. Hence |H(t)−H(0)| = |H(t)| = ∫1 0 G(t,s)F(s)ds =∫1 0 G(t,s)s−σsσF(s)ds =∫t 0 tα−1(1−s)α−1−(t−s)α−1 Γ(α) s−σsσF(s)ds+∫1 t tα−1(1−s)α−1 Γ(α) s−σsσF(s)ds =∫1 0 tα−1(1−s)α−1 Γ(α) s−σsσF(s)ds−∫t 0 (t−s)α−1 Γ(α) s−σsσF(s)ds ≤∫1 0 tα−1(1−s)α−1 Γ(α) s−σsσF(s)ds+∫t 0 (t−s)α−1 Γ(α) s−σsσF(s)ds ≤M∫1 0 tα−1(1−s)α−1 Γ(α) s−σds+M∫t 0 (t−s)α−1 Γ(α) s−σds =Mtα−1 Γ(α) ∫1 0 (1−s)α−1s−σds+M Γ(α) ∫t 0 (t−s)α−1s−σds =Mtα−1 Γ(α) ∫1 0 (1−s)α−1s−σds+Mtα−1 Γ(α) ∫t 01−s tα−1 s−σds.(4) If in the integral t 01−s tα−1s−σdswe make the change of variables u=s tthen we obtain ∫t 01−s tα−1 s−σds=t1−σ∫1 0 (1−u)α−1u−σdu. Theory of existence and uniqueness of solution for bvp 257 1328 J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 By taking (4) into account, |H(t)| ≤ Mtα−1 Γ(α) ∫1 0 (1−s)α−1s−σds+Mtα−1 Γ(α) t1−σ∫1 0 (1−u)α−1u−σdu =Mtα−1 Γ(α) +Mtα−σ Γ(α) ·β(1−σ , α), where βdenotes the beta function. In the last expression, when t→0 we see that |H(t)| → 0 and this proves the continuity of Hat t0=0. Case 2: t0∈(0,1). We take tn→t0and we have to prove that H(tn)→H(t0). Without loss of generality, we consider tn>t0(the same argument works for tn<t0). In fact, |H(tn)−H(t0)| = ∫tn 0 tα−1 n(1−s)α−1−(tn−s)α−1 Γ(α) s−σsσF(s)ds +∫1 tn tα−1 n(1−s)α−1 Γ(α) s−σsσF(s)ds−∫1 t0 tα−1 0(1−s)α−1 Γ(α) s−σsσF(s)ds −∫t0 0 tα−1 0(1−s)α−1−(t0−s)α−1 Γ(α) s−σsσF(s)ds =∫1 0 tα−1 n(1−s)α−1 Γ(α) s−σsσF(s)ds−∫tn 0 (tn−s)α−1 Γ(α) s−σsσF(s)ds −∫1 0 tα−1 0(1−s)α−1 Γ(α) s−σsσF(s)ds+∫t0 0 (t0−s)α−1 Γ(α) s−σsσF(s)ds =∫1 0 (tα−1 n−tα−1 0)(1−s)α−1 Γ(α) s−σsσF(s)ds− −∫t0 0 (tn−s)α−1−(t0−s)α−1 Γ(α) s−σsσF(s)ds−∫tn t0 (tn−s)α−1 Γ(α) s−σsσF(s)ds ≤M·(tα−1 n−tα−1 0) Γ(α) ∫1 0 (1−s)α−1s−σds+M Γ(α) ∫t0 0 ((tn−s)α−1−(t0−s)α−1)s−σds +M Γ(α) ∫tn t0 (tn−s)α−1s−σds ≤M(tα−1 n−tα−1 0) Γ(α) β(1−σ , α) +M Γ(α)I1 n+M Γ(α)I2 n,(5) where I1 n=∫t0 0 ((tn−s)α−1−(t0−s)α−1)s−σds I2 n=∫tn t0 (tn−s)α−1s−σds. We claim that I1 n→0 when n→ ∞. In fact, as tn→t0, then ((tn−s)α−1−(t0−s)α−1)s−σ−→ 0,when n→ ∞. Moreover, ((tn−s)α−1−(t0−s)α−1)s−σ≤(|tn−s|α−1+|t0−s|α−1)s−σ≤2s−σ and, as ∫1 0 2s−σds=2s−σ+1 −σ+1]1 0=2 1−σ<∞, 258 Fractional boundary value problem J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 1329 we have that the sequence ((tn−s)α−1−(t0−s)α−1)s−σconverges pointwise to the zero function and |(tn−s)α−1− (t0−s)α−1|s−σis bounded by a function belonging to L1[0,1], then by Lebesgue’s dominated convergence theorem I1 n→0 when n→ ∞.(6) This proves the claim. Now, we prove that I2 n→0, when n→ ∞. In fact, as I2 n=∫tn t0 (tn−s)α−1s−σds ≤∫tn t0 s−σds=1 1−σ(t1−σ n−t1−σ 0) and, taking into account that tn→t0, from the last expression we get I2 n→0,when n→ ∞.(7) Finally, from (5)–(7) we obtain |H(tn)−H(t0)| −→ 0,when n→ ∞. Case 3: t0=1. It is easily checked that H(1)=0. Following the same lines in the proof of Case 1, we can prove the continuity of H at t0=1.  Lemma 5. Suppose that 0< σ < 1. Then, max 0≤t≤1∫1 0 G(t,s)s−σds=Aα−1−Aα−σ Γ(α) β(1−σ , α), where G(t,s)is the Green function appearing in Lemma 3and A =α−1 α−σ1 1−σ. Proof. In fact, taking into account Case 1 of Lemma 4, we get ∫1 0 G(t,s)s−σds=∫t 0 tα−1(1−s)α−1−(t−s)α−1 Γ(α) s−σds+∫1 t tα−1(1−s)α−1 Γ(α) s−σds =∫1 0 tα−1(1−s)α−1 Γ(α) s−σds−∫t 0 (t−s)α−1 Γ(α) s−σds =tα−1 Γ(α) ∫1 0 (1−s)α−1s−σds−1 Γ(α) ∫t 0 (t−s)α−1s−σds =tα−1 Γ(α)β(1−σ , α) −tα−σ Γ(α)β(1−σ , α) =tα−1−tα−σ Γ(α) ·β(1−σ , α). Now, using elemental calculus we can prove that the function f(t)=tα−1−tα−σhas a maximum at the point t0=A= α−1 α−σ1 1−σ. This says us that max 0≤t≤1∫1 0 G(t,s)s−σds=Aα−1−Aα−σ Γ(α) β(1−σ , α).  Now, we introduce the following class of functions. By Awe denote the class of functions φ:[0,∞)−→ [0,∞) satisfying: (i) φis nondecreasing, (ii) φ(x) < x, for any x>0, (iii) β(x)=φ(x) x∈J, where Jis the class of functions appearing in Theorem 1. Examples of functions φ∈Aare φ(x)=µx, with 0 ≤µ < 1, φ(x)=x 1+xand φ(x)=ln(1+x). Denote by Kthe constant appearing in Lemma 5, i.e., K=max 0≤t≤1∫1 0 G(t,s)s−σds=Aα−1−Aα−γ Γ(α) β(1−σ , α). In what follows, we present our main result. Theory of existence and uniqueness of solution for bvp 259 1330 J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 Theorem 2. Let 0< σ < 1,1< α ≤2,f:(0,1] × [0,∞)−→ [0,∞)is continuous and limt→0+f(t,·)= ∞,tσf(t,y) is a continuous function on [0,1] × [0,∞). Assume that there exists 0< λ ≤1 Ksuch that, for x,y∈ [0,∞)with y ≥x and t∈ [0,1], 0≤tσ(f(t,y)−f(t,x)) ≤λφ(y−x), where φ∈A. Then, Problem (1) has a unique positive solution (this means that x(t) > 0, for t ∈(0,1)). Proof. Consider the cone P= {u∈C[0,1]:u(t)≥0}. Notice that, as Pis a closed set of C[0,1],Pis a complete metric space. It is easily checked that Psatisfies conditions (2) and (3) of Theorem 1. Now, for u∈Pwe define the operator Tby (Tu)(t)=∫1 0 G(t,s)f(s,u(s))ds=∫1 0 G(t,s)s−σsσf(s,u(s))ds. By Lemma 4,Tu ∈C[0,1]. Moreover, in view of nonnegativeness of G(t,s)and tσf(t,y), for u∈Pwe have Tu ∈P. Hence, T:P−→ P. In what follows, we check that assumptions in Theorem 1 are satisfied. First, the operator Tis nondecreasing. In fact, taking into account our assumption, for u≥vwe have (Tu)(t)=∫1 0 G(t,s)f(s,u(s))ds=∫1 0 G(t,s)s−σsσf(s,u(s))ds ≥∫1 0 G(t,s)s−σsσf(s, v(s))ds=(Tv)(t). Besides, for u≥vand u= v d(Tu,Tv) =max t∈[0,1]|(Tu)(t)−(Tv)(t)| =max t∈[0,1]((Tu)(t)−(Tv)(t)) =max t∈[0,1][∫1 0 G(t,s)(f(s,u(s)) −f(s, v(s)))ds] =max t∈[0,1][∫1 0 G(t,s)s−σsσ(f(s,u(s)) −f(s, v(s)))ds] ≤max t∈[0,1][∫1 0 G(t,s)s−σλφ(u(s)−v(s))ds]. Taking into account that φis nondecreasing, from last inequality we get d(Tu,Tv) ≤max t∈[0,1][∫1 0 G(t,s)s−σλφ(u(s)−v(s))ds] ≤max t∈[0,1][∫1 0 G(t,s)s−σλφ(d(u, v))ds] =λφ(d(u, v)) max t∈[0,1]∫1 0 G(t,s)s−σds. Now, Lemma 5 and the fact that 0 < λ ≤Kgive us d(Tu,Tv) ≤λφ(d(u, v)) ·max t∈[0,1]∫1 0 G(t,s)s−σds≤φ(d(u, v)) =φ(d(u, v)) d(u, v) ·d(u, v) =β(d(u, v)) ·d(u, v). Obviously, the last inequality is satisfied for u=v. Now, taking into account that the zero function satisfies 0 ≤T0, Theorem 1 says us that the operator Thas a unique fixed point in K, or, equivalently, Problem (1) has a unique nonnegative solution x(t)∈C[0,1]. In what follows, we will prove that x(t)is positive solution. 260 Fractional boundary value problem J. Caballero et al. / Computers and Mathematics with Applications 62 (2011) 1325–1332 1331 In contrary case, there exists 0 <t∗<1 such that x(t∗)=0. As the nonnegative solution x(t)of Problem (1) is a fixed point of the operator T, this says us that x(t)=∫1 0 G(t,s)f(s,x(s))ds,for 0 <t<1, and, particularly, x(t∗)=∫1 0 G(t∗,s)f(s,x(s))ds=0. The nonnegative character of G(t,s)and f(s,u)and the last relation give G(t∗,s)·f(s,x(s)) =0 a.e. (s). (8) Taking into account limt→0+f(t,0)= ∞ means that for M>0, we can find δsuch that, for s∈ [0,1] ∩ (0, δ) we have f(s,0) > M. Observe that [0,1]∩(0, δ) ⊂ {s∈ [0,1] : f(s,x(s)) > M}and µ([0,1]∩(0, δ)) > 0, where µis the Lebesgue measure on [0,1]. This and (8) give us that G(t∗,s)=0 a.e. (s) and this is a contradiction because G(t∗,s)is a rational function in the variable s. Therefore, x(t) > 0, for t∈(0,1). This finishes the proof.  In what follows, we present an example which illustrates Theorem 2. Example 1. Consider the following singular fractional boundary value problem D 3 2 0+u(t)+λ(t2+1)arctan(u(t)) √t=0,0<t<1 and λ > 0, u(0)=u(1)=0.(9) In this case, f(t,u)=λ(t2+1)arctan u √t, for (t,u)∈(0,1]×[0,∞). Notice that fis continuous in (0,1]×[0,∞)and limt→0+f(t,·)= ∞. Moreover, σ=1 2and α=3 2. Now we prove that f(t,u)satisfies assumptions of Theorem 2. Previously, we prove that the function φ, defined by φ:[0,∞)−→ 0,π 2 φ(x)=arctan x, satisfies that, for u≥v, φ(u)−φ(v) ≤φ(u−v). In fact, as φ(x)=arctan xis a nondecreasing function because φ′(x)=1 1+x2>0 and, consequently, for u≥v, 0≤φ(u)−φ(v). Put φ(u)=arctan u=αand φ(v) =arctan v=β(notice that the nondecreasing character of φgives us α≥βfor u≥v). Taking into account the trigonometric formula tan(α −β) =tan α−tan β 1+tan αtan β and, as tan α=uand tan β=vbelong to [0,∞), we can obtain tan(α −β) ≤tan α−tan β. As φis nondecreasing, the last inequality gives us α−β≤arctan(tan α−tan β) or, equivalently, φ(u)−φ(v) =arctan u−arctan v≤arctan(u−v) =φ(u−v). This proves our claim. Now, we check that f(t,u)satisfies assumptions appearing in Theorem 2. Theory of existence and uniqueness of solution for bvp 261 2 Abstract and Applied Analysis In 17the authors studied the following two-point boundary value problem of fractional order: Dα 0utatft, ut 0,0<t<1,1<α≤2, u0u10,1.1 and they proved the existence of positive solutions by means of the Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem. In 18the author investigated the existence of solutions of cDα 0utft, ut,0<t<1,1<α≤2, u0ν/ 0,u 1ρ/ 0.1.2 Since boundary values are nonzero, the Riemann-Liouville fractional derivative Dα 0is not suitable and the author used the Caputo fractional derivative cDα 0. Motivated by these works, in this paper we discuss the existence and uniqueness of positive solutions for the following nonlinear boundary value problem of fractional order: Dα 0utft, ut 0,0<t<1,3<α≤4, u0u0u0u10.1.3 This problem was studied in 21, where the authors use lower and upper solution method and the Schauder fixed point theorem which cannot ensure the uniqueness of the solution. The practical relevance of 3 <α≤4 appears in problems related with other areas as physics and economics which can be modeled by these fractional boundary values problems. Particularly, these problems appear in the Hamiltonian formulation for the lagrangians depending on fractional derivatives of coordinates when the systems are nonconservative see, e.g., 7. Our main interest in this paper is to give an alternative answer to the main results of the paper 21. The main tool used in our study is a fixed point theorem in partially ordered sets which gives us uniqueness of the solution. 2. Preliminaries and Previous Results For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems. Definition 2.1. The Riemann-Liouville fractional integral of order α>0ofafunctionf: 0,∞→Ris given by Iα 0ft1 Γαt 0 t−sα−1fsds 2.1 provided that the right-hand side is pointwise defined on 0,∞and where Γαdenotes the gamma function. 268 Fractional boundary value problem Abstract and Applied Analysis 3 Definition 2.2. The Riemann-Liouville fractional derivative of order α>0ofafunctionf: 0,∞→Ris given by Dα 0ft1 Γn−αd dtnt 0 fs t−sα−n1ds, 2.2 where nα1andαdenotes the integer part of α. The following two lemmas can be found in 17,22. Lemma 2.3. Let α>0and u∈C0,1∩L10,1. Then fractional differential equation Dα 0ut02.3 has utc1tα−1c2tα−2···cntα−n2.4 for some ci∈R(i1,2,...n) and nα1as unique solution. Lemma 2.4. Assume that u∈C0,1∩L10,1with a fractional derivative of order α>0that belongs to C0,1∩L10,1.Then Iα 0Dα 0ututc1tα−1c2tα−2···cntα−n,2.5 for some ci∈Ri1,...,nand nα1. Using Lemma 2.4,in21the following result is proved. Lemma 2.5. Given f∈C0,1and ft≥0, the unique nonnegative solution for Dα 0utft0,0<t<1,3<α≤4, u0u0u0u10 2.6 is ut1 0 Gt, sfsds, 2.7 where Gt, s⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ tα−11−sα−3−t−sα−1 Γα,0≤s≤t≤1, tα−11−sα−3 Γα,0≤t≤s≤1. 2.8 Theory of existence and uniqueness of solution for bvp 269 4 Abstract and Applied Analysis In the sequel, we present the fixed-point theorems which we will use later. These results appear in 23. Theorem 2.6. Let X, ≤be a partially ordered set and suppose that there exists a metric din Xsuch that X, dis a complete metric space. Assume that Xsatisfies the following condition if {xn}is a nondecreasing sequence in Xsuch that xn→x, then xn≤x∀n∈N.2.9 Let T:X→Xbe a nondecreasing mapping such that dTx,Ty≤dx, y−ψdx, y,for x≥y, 2.10 where ψ:0,∞→0,∞is a continuous and nondecreasing function such that ψis positive in 0,∞,ψ00and limt→∞ψt∞. If there exists x0∈Xwith x0≤Tx0then Thas a fixed point. Moreover, if X, ≤satisfies the following condition: for x, y ∈Xthere exists z∈Xwhich is comparable to xand y, 2.11 which appears in 24, the following result is proved 23. Theorem 2.7. Adding condition 2.11to the hypotheses of Theorem 2.6 one obtains the uniqueness of the fixed point. Remark 2.8. In Theorems 2.6 and 2.7 the condition limt→∞ψt∞is redundant. In our considerations we will work in the Banach space C0,1{x:0,1→R, continuous}with the standard norm xsup{|xt|:t∈0,1}. Notice that this space can be equipped with a partial order given by x,y ∈C0,1,x≤y⇐⇒ xt≤yt,for t∈0,1.2.12 In 24it is proved that C0,1,≤with the classical metric given by dx,ysup 0≤t≤1xt−yt2.13 satisfies condition 2.9of Theorem 2.6. Moreover, for x, y ∈C0,1, as the function maxx, y∈C0,1,C0,1,≤satisfies condition 2.11. Finally, by Fwe denote the class of functions ψ:0,∞→0,∞continuous, nondecreasing, positive in 0,∞and ψ00. By Jwe denote the class of functions ϕ:0,∞→0,∞continuous, nondecreasing, satisfying that I−ϕ∈F, where Idenotes the identity mapping on 0,∞. 270 Fractional boundary value problem Abstract and Applied Analysis 5 3. Main Result The main result of the paper is the following. Theorem 3.1. Problem 1.3has a unique positive solution utif the following conditions are satisfied. H1f:0,1×0,∞→0,∞is continuous and nondecreasing with respect to the second argument. H2There exists t0∈0,1such that ft0,0>0. H3There exists 0<λ≤α−2Γα1/2such that, for x,y ∈0,∞with y≥xand t∈0,1, ft, y−ft, x≤λ·ψy−x,3.1 where ψ∈J. Before the proof of Theorem 3.1, we will need some properties of Green’s function appearing in Lemma 2.5. Lemma 3.2. Gt, s≥0, and Gis a continuous function on 0,1×0,1. Proof. The continuity of Gis easily checked. In order to prove the nonnegativness of Gt, s, for 0 ≤t≤s≤1, it is obvious that Gt, stα−11−sα−3 Γα≥0.3.2 In the case of 0 ≤s≤t≤1witht/ 0, we have Gt, s1 Γαtα−11−sα−3−t−sα−1 1 Γαtα−11−sα−3−1−s tα−1. 3.3 As s≤s/t, we have 1 −s≥1−s/tand, consequently, 1−sα−3≥1−s tα−3.3.4 Taking into account that the function gαtαwith α>0andt∈0,1is decreasing we have 1−sα−3≥1−s tα−3≥1−s tα−1.3.5 The last inequality and 3.3give us Gt, s≥0witht/ 0. Finally, notice that G0,s0, and this finishes the proof. Theory of existence and uniqueness of solution for bvp 271 6 Abstract and Applied Analysis Lemma 3.3. One has sup t∈0,11 0 Gt, sds 2 α−2Γα1.3.6 Proof. Since 1 0 Gt, sds t 0 Gt, sds 1 t Gt, sds 1 Γαt 0tα−11−sα−3−t−sα−1ds 1 Γα1 t tα−11−sα−3ds 1 Γαtα−1 α−2−1 αtα 3.7 and if we put ϕt1 0Gt, sds 1/Γαtα−1/α−2−1/αtα, then, as ϕt1 Γαα−1 α−2tα−2−tα−1>0,for t>0,3.8 we deduce that ϕt1 0Gt, sds is strictly increasing and, consequently, sup t∈0,11 0 Gt, sds 1 0 G1,s ds 1 Γα1 α−2−1 α 2 αα−2Γα2 α−2Γα1. 3.9 In the sequel, we give the proof of Theorem 3.1. Proof of Theorem 3.1.Consider the cone P{u∈C0,1:ut≥0}.3.10 Obviously, Pis a closed set of C0,1,and,thus,Pis a complete metric space with the distance given by du, vsupt∈0,1{|ut−vt|}.Pcan be equipped with a partial order defined by x, y ∈P, x ≤y⇐⇒ xt≤yt,for t∈0,1.3.11 Using a similar argument to that in 24, it can be proved that P, ≤satisfies condition 2.9 of Theorem 2.6. Moreover, as for x, y ∈Pthe function maxx, y∈P,P, ≤satisfies condition 2.11. 272 Fractional boundary value problem Abstract and Applied Analysis 7 Now, we consider the operator Tdefined on Pand given by Tut1 0 Gt, sfs, usds, for u∈P. 3.12 By H1and Lemma 3.2,Tapplies Pinto itself. In the sequel we check that Tsatisfies the assumptions of Theorem 2.6. Firstly, we prove that Tis a nondecreasing operator. In fact, by H1,foru, v ∈Pwith u≥vand t∈0,1, we have Tut1 0 Gt, sfs, usds ≥1 0 Gt, sfs, vsds Tvt.3.13 Now, we prove that Tsatisfies the contractive condition appearing in Theorem 2.6. In fact, for u, v ∈Pand u≥v, taking into account assumption H3,weget dTu,Tvsup t∈0,1{|Tut−Tvt|} sup t∈0,1 Tut−Tvt sup t∈0,11 0 Gt, sfs, us −fs, vsds ≤sup t∈0,11 0 Gt, sλ·ψus−vsds. 3.14 As ψ∈Jand, thus, ψis nondecreasing and by Lemma 3.3, from the last inequality we obtain dTu,Tv≤λψdu, v ·sup t∈0,11 0 Gt, sds λ·ψdu, v ·2 α−2Γα1. 3.15 Using the fact that λ≤2/α−2Γα1assumption H3, we have dTu,Tv≤ψdu, v du, v−du, v−ψdu, v.3.16 Put ϕxx−ψx,Asψ∈J, this means that ϕ∈F. The last inequality gives us dTu,Tv≤du, v−ϕdu, v.3.17 This proves that Tsatisfies the contractive condition of Theorem 2.6. Finally, as Gt, s≥0Lemma 3.2and f≥0assumption H1, we have T0t1 0 Gt, sfs, 0ds ≥0,3.18 where 0 denotes the zero function. Theory of existence and uniqueness of solution for bvp 273 8 Abstract and Applied Analysis Now, Theorem 2.6 shows that problem 1.3has at least one nonnegative solution. As P, ≤satisfies condition 2.11, we obtain the uniqueness of the solution. In what follows, we will prove that this solution is positive this means that xt>0, for t∈0,1. Finally, we will prove that the zero function is not the solution for problem 1.3.In fact, in contrary case, the zero function is a fixed point of Tand, thus, we have 01 0 G0,s fs, 0ds, for t∈0,1.3.19 The nonnegative character of the functions Gand fand the last expression give us Gt, s·fs, 00a.e.s,for t∈0,1.3.20 This and the fact that Gt, s/ 0a.e.sfor any t∈0,1because Gt, sis given by a polynomial implies fs, 00a.e.s.3.21 Taking into account assumption H2,ft0,0>0 for certain t0∈0,1. By the continuity of f we can find a set A⊂0,1with t0∈Aand μA>0, where μis the Lebesgue measure, such that ft, 0>0fort∈A. This contradicts 3.21. This proves that the zero function is not the solution for problem 1.3. Now, we will prove that the solution xis positive. In the contrary case, we find 0 <t ∗<1 such that xt∗0. As the solution xis a fixed point of the operator T, this means that xt∗1 0 Gt∗,s fs, xsds 0.3.22 Since x∈Pand, thus, x≥0 and by the fact that fis nondecreasing in the second variable and Gt, s≥0, we can get 0xt∗1 0 Gt∗,s fs, xsds ≥1 0 Gt∗,s fs, 0ds ≥0,3.23 and this inequality implies xt∗1 0 Gt∗,s fs, 0ds 0.3.24 Using a similar reasoning to the one above used we obtain a contradiction. Therefore, xt>0, for t∈0,1. This finishes the proof. 274 Fractional boundary value problem Abstract and Applied Analysis 9 Remark 3.4. In Theorem 3.1, condition H2seems to be a strong condition in order to obtain a positive solution for problem 1.3, but when there is uniqueness of solution one will see that this condition is a very adjusted one. More precisely, under the assumption that problem 1.3has a unique nonnegative solution xtone has ft0,0>0 for certain t0∈0,1if and only if xtis a positive solution.3.25 In fact, if ft0,0>0 for certain t0∈0,1the argument used in the proof of Theorem 3.1 give us that xtis a positive solution. For the other implication, suppose that ft, 00 for any t∈0,1. Under this assumption, our problem 1.3admits as solutions the function xtand the zero function and this contradicts the hypothesis about uniqueness of solution to problem 1.3. Therefore, ft0,0> 0 for certain t0∈0,1. Remark 3.5. Notice that the assumptions in Theorem 3.1 are invariant by additive perturbations. More precisely, if ft, 00 for any t∈0,1and fsatisfies H1and H3of Theorem 3.1, then gt, uatft, u,witha:0,1→0,∞a nondecreasing continuous function with at0/ 0 for certain t0∈0,1,satisfiesH1,H2,andH3of Theorem 3.1 and the following nonlinear boundary value problem of fractional order: Dα 0utgt, ut 0,0<t<1,3<α≤4, u0u0u0u10 3.26 has a unique positive solution by Theorem 3.1. In the sequel we present an example where the results can be applied. Example 3.6. Consider the fractional boundary value problem D7/2 0utt21ln2ut 0,0<t<1, u0u0u0u10. 3.27 In this case, ft, ut21ln2ufor t, u∈0,1×0,∞. Obviously, fis a continuous function and ft, 0t21ln 2 / 0fort∈0,1.As∂f/∂u t211/2u >0for u∈0,∞,fis nondecreasing with respect to the second variable. Besides, for u≥vand t∈0,1, we have t21ln2u−ln2vt21·ln2u 2v t21ln2vu−v 2vt21ln1u−v 2v ≤t21ln1u−v ≤2ln 1u−v. 3.28 A straightforward calculation gives us that ψxln1xsatisfies that ψ∈J. Theory of existence and uniqueness of solution for bvp 275 10 Abstract and Applied Analysis Moreover, in this case λ2, α7/2 and we have α−2Γα1 27/2−2Γ7/21 23 4Γ7 213 4·7 2·Γ7 2≈8.7228 >2λ. 3.29 Finally, Theorem 3.1 proves the existence and uniqueness of a positive solution for problem 3.27. 4. A Final Remark In connection with problem 1.3, the main result in 21is the following. Theorem 4.1 see 21, Theorem 3.1.Problem 1.3has a positive solution utif the following conditions are satisfied: Hfft, u∈C0,1×0,∞,Ris nondecreasing relative to u,ft, pt / 0for t∈0,1, where pt1 0Gt, sds 1/Γαtα−1/α−2−1/αtα, and there exists a positive constant μ<1such that kμft, u≤ft, ku,∀0≤k≤1.4.1 In the sequel, we present an example which can be treated by Theorem 3.1 and it cannot be covered by Theorem 4.1. Example 4.2. Consider the fractional boundary value problem D7/2 Outt21ρutc0,0≤t≤1, u0u0u0u10, 4.2 with c>0and0<ρ<1. In this case, ft, ut21ρuc,fort, u∈0,1×0,∞. Obviously, fis continuous and nondecreasing with respect to the second variable since ∂f/∂u ρt21>0. Besides, if u≥vand t∈0,1, we have ft, u−ft, vt21ρu c−ρv c t21ρu−v≤2ρu−v. 4.3 In this case, ψxρx and it is easily seen that ϕxx−ψx1−ρxbelongs to F. 276 Fractional boundary value problem Abstract and Applied Analysis 11 Moreover, in this case, λ2and,asα7/2, we have α−2Γα1 27/2−2Γ7/21 23 4Γ7 21 3 4·7 2·Γ7 2≈8.7228 >2λ. 4.4 As ft, 0ct21>0 for any t∈0,1,Theorem 3.1 gives us the existence and uniqueness of positive solution for problem 4.2. On the other hand, we will show that ft, ut21ρu cwith 0 <ρ<1andc>0 does not satisfy Hfof Theorem 4.1. In fact, suppose that there exists 0 <μ<1 such that kμft, u≤ft, ku,for any 0 ≤k≤1.4.5 This implies that kμ≤ft, ku ft, ut21ρku c t21ρu cρku c ρu c.4.6 Notice that limu→∞ρku c/ρu ck, and, consequently, taking limit as u→∞in the last inequality, we get kμ≤k, 4.7 this is false because 0 <μ<1 and the function hαkαis decreasing when 0 <k< 1. Therefore, problem 4.2can be covered by Theorem 3.1 and it cannot be studied by Theorem 4.1. 5. Conclusions Our main contribution in this paper is to prove under certain assumptions the existence and the uniqueness of positive solution for problem 1.3which was treated in 21.In21the question of uniqueness of solution was not considered. Moreover, we present an example which can be covered by the results of this paper and cannot be treated by the ones obtained in 21. Acknowledgment This paper was partially supported by Ministerio de Educaci´ on y Ciencia Project MTM 2007/65706. References 1K. Diethelm and A. D. Freed, “On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity,” in Scientifice Computing in Chemical Engineering II: Theory of existence and uniqueness of solution for bvp 277 In [22], it is proved that (C[0, 1], ≤) with the above-mentioned metric satisfies condition (4) of Theorem 1. Moreover, for x,yÎC[0, 1], as the function max(x,y)ÎC[0, 1], (C[0, 1], ≤) satisfies condition (5). By F , we denote the class of functions ψ:[0,∞)®[0, ∞) continuous, nondecreasing, positive in (0, ∞) and ψ(0) = 0, and by J the class of functions : [0, ∞)®[0, ∞) continuous, nondecreasing, and satisfying that I− ϕ ∈ F ,whereIdenotes the identity mapping on [0, ∞). 3 Main result Our starting point of this section is the following result about Green’s function appearing in Section 2. Lemma 4. maxt∈[0,1] 1  0 G(t,s)ds =1 (α+1) α−1 αα−1−α−1 α  Proof. In fact, 1  0 G(t,s)ds= t  0 G(t,s)ds+ 1 t G(t,s)ds = t  0 tα−1(1 −s)α−1−(t−s)α−1 (α)ds+ 1 t tα−1(1 −s)α−1 (α)d s = 1  0 tα−1(1 −s)α−1 (α)ds− t  0 (t−s)α−1 (α)ds =1  ( α ) tα−1 α−tα α  =1  ( α+1 ) (tα−1−tα) By an elemental calculation, it can be proved that the maximum of h(t)=1 0G(t,s)ds=1  ( α+1 ) (tα−1−tα ) is reached at t0=α−1 α , thus, max 0 ≤t≤1 1  0 G(t,s)ds=1 (α+1)α−1 αα−1 −α−1 αα  □ In the sequel, we present the main result of this paper. For convenience, we put A=1 (α+1)  α−1 αα−1−α−1 αα  . Theorem 3.Our Problem (2) has a unique nonnegative solution u(t)if the following conditions are satisfied: (H1) f: [0, 1] × [0, ∞)®[0, ∞)is continuous and nondecreasing respect to the second argument. (H2) There exists 0<λ≤ 1 A such that, for x,yÎ[0, ∞)with y ≥x and t Î[0, 1], f ( t,y ) −f ( t,x ) ≤λϕ ( y−x ), where ϕ ∈ J . Proof. Consider the cone P={u∈C[0, 1] : u ( t ) ≥0} . Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 4 of 9 284 Fractional boundary value problem Obviously, (P,d) with d(x,y) = sup{|x(t)-y(t)|: tÎ[0, 1]} is a complete metric space satisfying conditions (4) and (5). Consider the operator defined by (Tx)(t)= 1  0 G(t,s)f(s,x(s))ds,forx∈P , where G(t,s) is the Green’s function appearing in Section 2. Obviously, Tapplies P into itself since f(t,x) and G(t,s) are nonnegative continuous functions. In what follows we check that assumptions in Theorem 2 are satisfied. Firstly, the operator Tis nondecreasing. Indeed, by (H1), for u,vÎP,u≥v, and tÎ[0, 1], we have (Tu)(t)=1 0 G(t,s)f(s,u(s))ds≥1 0 G(t,s)f(s,v(s))ds=(Tv)(t) . Now, we prove that Tsatisfies the contractive condition appearing in Theorem 1. In fact, for u,vÎPand u≥vand, taking into account assumption (H2), we get d(Tu,Tv)= sup t∈[0,1]{|Tu(t)−Tv(t)|} =sup t∈[0,1]{(Tu(t)−Tv(t))} =sup t∈[0,1] 1  0 G(t,s)(f(s,u(s)) −f(s,v(s)))d s ≤sup t∈[0,1] 1  0 G(t,s)λϕ(u(s)−v(s))ds. As ϕ ∈ J ,is nondecreasing, and, taking into account (H2) and Lemma 4, we obtain d(Tu,Tv)≤λϕ(d(u,v)) ·sup t∈[0,1] 1  0 G(t,s)ds =λϕ ( d ( u,v )) ·A≤ϕ ( d ( u,v )) =d ( u,v ) − ( d ( u,v ) −ϕ ( d ( u,v ))). Put ψ(x)=x-(x). As ϕ ∈ J , this means that ψ ∈ F and from the last inequality d( Tu,Tv ) ≤ d( u,v ) −ψ (d( u,v )). This proves that Tsatisfies the contractive condition of Theorem 1. Finally, the nonnegative character of the function G(t,s) and f(t,x) [assumption (H1)] gives us (T0)(t)= 1  0 G(t,s)f(s,0)ds≥0 , where 0 denotes the zero function. Therefore, Theorem 2 says us that Problem (2) has a unique nonnegative solution. □ Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 5 of 9 Theory of existence and uniqueness of solution for bvp 285 In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means x(t)>0fortÎ(0,1)).The proof of this condition is similar to the proof of Theorem 2.3 of [23]. We present this proof for completeness. Theorem 4.Under assumptions of Theorem 3 and suppose that f(t 0 ,0)≠0for certain t 0 Î[0, 1]. Then, Problem (2) has a unique positive solution. Proof. Consider the nonnegative solution x(t) for Problem (2) whose existence is guaranteed by Theorem 3. In the sequel, we will prove that x(t) is a positive solution. Firstly, notice that x(t) is a fixed point of the operator (Tu)(t)=1 0 G(t,s)f(s,u(s))d s and, consequently, x(t)= 1  0 G(t,s)f(s,x(s))ds . Now, suppose that there exists 0 <t* < 1 such that x(t*) = 0. This means that x(t∗)= 1  0 G(t∗,s)f(s,x(s))ds=0 . Using that x(t) is a nonnegative function, f(t,y) is nondecreasing with respect to the second argument and the nonnegative character of G(t,s), we get 0=x(t∗)= 1  0 G(t∗,s)f(s,x(s))ds≥ 1  0 G(t∗,s)f(s,0)ds≥0 . This gives us x(t∗)=1 0 G(t∗,s)f(s,0)ds= 0 . As G(t,s)≥0 and f(s,0)≥0, the last expression implies G ( t∗,s ) f ( s,0 ) =0 a.e ( s ). As G(t*, s)≠0 a.e (s) (because G(t*, s) is given by a polynomial), we can obtain f ( s,0 ) =0 a.e ( s ). (6) On the other hand, as f(t 0 ,0)≠0 for certain t 0 Î[0, 1], the nonnegative character of f(t,y) gives us f(t 0 ,0)>0.Asf(t,y) is a continuous function, we can find a set A⊂[0, 1] with t 0 ÎA,μ(A)>0,whereμis the Lebesgue measure and f(t,0)>0foranytÎ A. This contradicts (6). Therefore, x(t) > 0 for tÎ(0, 1). This finishes the proof. □ Remark 3. In Theorem 4, the condition f(t 0 ,0)≠0 for certain t 0 Î[0, 1] seems to be a strong condition in order to obtain a positive solution for Problem (2), but when the solution is unique, we will see that this condition is very adjusted one. In fact, suppose that Problem (2) has a unique nonnegative solution x(t) then f ( t,0 ) =0foreacht∈[0, 1] if and only if x ( t ) ≡0 . In fact, if f(t, 0) = 0 for each tÎ[0, 1], it is easily seen that the zero function satisfies Problem (2) and the uniqueness of the solution gives us x(t) = 0. The reverse implication is obvious. Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 6 of 9 286 Fractional boundary value problem Remark 4. Notice that the hypotheses in Theorem 3 are invariant by continuous perturbation. More precisely, if f(t, 0) = 0 for any tÎ[0, 1] and fsatisfies (H1) and (H2) of Theorem 3 then g(t,x)=a(t)+f(t,x)witha: [0, 1] ®[0, ∞) continuous and a≠ 0, satisfies assumptions of Theorem 4, and this means that the following boundary value problem Dα 0+u(t)+g(t,u(t)) = 0, 0 <t<1 u(0) = u(1) = u(0) = 0  has a unique positive solution. Now, we present an example that illustrates our results. Example 1. Consider the boundary value problem D 5 2 0+u(t)+c+λ·arctg u(t)=0, 0<t<1, c,λ>0 u(0) = u(1) = u(0) = 0 ⎫ ⎪ ⎬ ⎪ ⎭ (7) In this case, α = 5 2 and f(t,u)=c+l·arctg u. It is easily seen that f(t,u)satisfies (H1) of Theorem 3. In the sequel, we prove that f(t,u) satisfies (H2) of Theorem 3. Previously, we consider the function j:[0,∞)®[0, ∞) given by j(u)=arctg u and we will see that jsatisfies φ ( u ) −φ ( v ) ≤φ ( u−v ) ,foru≥v . In fact, put j(u)=arctag u =aand j(v)=arctg v =b(notice that, as u≥vand jis nondecreasing, a≥b). Then, from tg(α−β)= tgα−tg β 1+t g α·t g β and, as α,β∈[0, π 2), then tga,tgbÎ[0, ∞), we can obtain tg ( α−β ) ≤tgα−tgβ . Applying jto the last inequality and taking into account the nondecreasing character of j, we obtain α −β≤arctg ( tgα−tgβ ), or, equivalently, φ ( u ) −φ ( v ) =arctg u −arctg v =α−β≤arctg ( u−v ) =φ ( u−v ). This proof our previous claim. Now, for u≥vand tÎ[0, 1], we have, f( t,u ) − f( t,v ) =λ ( arctg u −arctg v ) ≤λarctg ( u−v ). Now, we prove that j(u)=arctg u belongs to J .Obviously,j:[0,∞)®[0, ∞)isa continuous and nondecreasing function. Moreover, ψ(u)=u-j(u)=u-arctg u is also continuous and nondecreasing and satisfies ψ(u)>0foru> 0 and ψ(0) = 0. Consequently, φ ∈ J . Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 7 of 9 Theory of existence and uniqueness of solution for bvp 287 Finally, as f(t,0)=c+arctg 0=c> 0, by Theorem 4, Problem (7) has a unique positive solution for 0<λ≤  1 (52+1)  3 53/2 −3 55/2  −1 ≈17.8682 . 4 Some remarks In a recent paper [18], the authors study the existence of positive solutions of a particular case of Problem (2). More precisely, they study the following fractional autonomous boundary value problem Dα 0+u(t)+λf(u(t)) = 0, 0 <t< 1 u ( 0 ) =u ( 1 ) =u ( 0 ) =0, (8) where 2 <a≤3, lis a positive parameter and f:(0,∞)®(0, ∞) is continuous. The main tool used by the authors in this paper is Guo-Kranosel’skii fixed-point theorem on cones. In [18], the question about the uniqueness of solutions is not treated. One of the results of [18] is the following theorem. Theorem 5.[[18],Theorem3.2]If there exists l Î(0, 1) such that q(l)c 2 f 0 >F ∞ c 1 holds then, for each lÎ((q(l)c 2 f 0 ) -1 ,(F ∞ c 1 ) -1 ), the boundary value problem (8) has at least one positive solution. Here, we consider (q(l)c 2 f 0 ) -1 =0iff 0 =∞and (F ∞ c 1 ) -1 =∞if F ∞ =0,where F∞= limu→+∞sup f (u) u ,F∞= limu→+∞sup f (u) u ,q(t)=t a-1 (1 - t), k(s)=s(1 - s) a-1 , c1=1  ( α ) 1 0(α−1)k(s)d s , and c2=1  ( α ) 1 0 1 α−1q(s)k(s)d s . Now, we present the following example. Example 2. Consider the boundary value problem that is a variant of Example 1. D5/2 0+u(t)+λ(c+arctg u(t)) = 0, 0 <t<1, c,λ>0 , u ( 0 ) =u ( 1 ) =u ( 0 ) =0, (9) In this case, α =5 2 and f(u)=c+arctg u. Then, we have F ∞ =0andf 0 =∞.Moreover, c 1 = 0.129, c 2 = 0.0077, and q(12) = √2 8 = 0.176 8 [[18], Example 5.1]. Thus, q (1/2)c 2 f 0 >F ∞ c 1 holds. Theorem 5 gives us the existence of a positive solution for Problem (9) for each lÎ(0, ∞). The question of uniqueness cannot be treated by the results of [18]. On the other hand, following a similar reasoning that in Example 1, Theorem 4 gives us the existence of a unique positive solution for Problem (9) when 0<λ≤  1 (5/2+1)  3 53/2−3 55/2  −1 ≈17.868 2 . Our main contribution is the uniqueness of positive solution for Problem (9) when 0 <l≤17.8682. Now, we present an example that cannot be studied by the results of [18], and it can be treated by the ones obtained in this paper. Example 3. Consider the following boundary value problem D5/2 0+u(t)+λ(t+arctg u(t)) = 0, 0 <t<1, λ>0 , u ( 0 ) =u ( 1 ) =u ( 0 ) =0, (10) Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 8 of 9 288 Fractional boundary value problem In this case, the boundary value problem is nonautonomous, and thus, this problem cannot be studied by the results of [18]. On the other hand, using a similar argument that in example 1, and using Theorem 4, we obtain the existence of a unique positive solution for Problem (10) when 0 <l≤ 17.868. Acknowledgements This research was partially supported by “Ministerio de Educación y Ciencia”Project MTM 2007/65706. Authors’contributions We are part of the same research group and work together therefore, we can affirm that the contents of this paper has been prepared by all the authors: JC, JH, and KS. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 28 February 2011 Accepted: 18 September 2011 Published: 18 September 2011 References 1. Diethelm, K, Freed, AD: On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity. In: Keil F, Mackens W, Voss H, Werthers J (eds.) Scientific Computing in Chemical Engineering II– Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. pp. 217–224. Springer, Heidelberg (1999) 2. Gaul, L, Klein, P, Kempffe, S: Damping description involving fractional operators. Mech Syst Signal Process. 5,81–88 (1991). doi:10.1016/0888-3270(91)90016-X 3. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys J. 68,46–53 (1995). doi:10.1016/S0006-3495(95)80157-8 4. Mainardi, F: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri CA, Mainardi F (eds.) Fractal and Fractional Calculus in Continuum Mechanics. pp. 291–348. Springer, Wien (1997) 5. Metzler, F, Schick, W, Kilian, HG, Nonnenmache, TF: Relaxation in filled polymers: A fractional calculus approach. J Chem Phys. 103, 7180–7186 (1995). doi:10.1063/1.470346 6. Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974) 7. Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: Methods, results and problems-I. Appl Anal. 78, 153–192 (2001). doi:10.1080/00036810108840931 8. Samko, SG, Marichev, OI, Kilbas, AA: Fractional Integral and Derivative, Theory and Applications. Gordon and Breach, Yverdon, Switzerland (1993) 9. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 10. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) 11. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008). doi:10.1016/j.na.2007.08.042 12. Lakshmikantham, V: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337–3343 (2008). doi:10.1016/j.na.2007.09.025 13. Bai, C: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign. Nonlinear Anal. 64, 677–685 (2006). doi:10.1016/j.na.2005.04.047 14. Bai, Z, Ge, W: Existence of three positive solutions for some second-order boundary value problems. Comput Math Appl. 48, 699–707 (2004). doi:10.1016/j.camwa.2004.03.002 15. Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J Math Anal Appl. 311, 495–505 (2005). doi:10.1016/j.jmaa.2005.02.052 16. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916–924 (2010). doi:10.1016/j.na.2009.07.033 17. Zhang, S: Existence of solution for a boundary value problem of fractional order. Acta Math Sci. 26, 220–228 (2006) 18. Zhao, Y, Sun, S, Han, Z, Li, Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abs Appl Anal. 2011, Article ID390543 (2011) 19. Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009). doi:10.1016/j.na.2009.01.240 20. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies, vol. 204,Elsevier, Amsterdam (2006) 21. Yu, Y, Jiang, D: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Northeast Normal University, Changchun, Jilin Province, Republic of China PRC (2009) 22. Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22, 223–239 (2005). doi:10.1007/s11083-005-9018-5 23. Harjani, J, López, B, Sadarangani, K: On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets. Dyn Syst Appl. 19, 625–634 (2010) doi:10.1186/1687-2770-2011-25 Cite this article as: Caballero et al.: On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Boundary Value Problems 2011 2011:25. Caballero et al.Boundary Value Problems 2011, 2011:25 http://www.boundaryvalueproblems.com/content/2011/1/25 Page 9 of 9 Theory of existence and uniqueness of solution for bvp 289 Theory of existence and uniqueness of solution for bvp 291 4.2.5. Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution for the following fractional boundary value problem (Dα 0+u(t) + ft, u(t)= 0,0<t<1, u(0) = u0(1) = u00(0) = 0,(4.14) with 2 < α ≤3, and lim t→0+f(t, ·) = ∞, that is fis singular at t= 0. We need the class Aof those functions φ: [0,∞)→[0,∞) satisfying the conditions: (i) φis nondecreasing. (ii) φ(x)< x, for any x > 0. (iii) β(x) = φ(x) xis such that b(tn)→1 implies tn→0. Our main result is the next theorem. Theorem 57. Suppose that 0< σ < 1and 2< α ≤3. Under these assumptions: (1) f: (0,1] ×[0,∞)→[0,∞)is a continuous function satisfying lim t→0+f(t, −) = ∞, (2) tσf(t, y)is a continuous function on [0,1] ×[0,∞), (3) there exists 0< λ ≤Γ(α−σ) Γ(1−σ)and φ∈ A such that 0≤tσ(f(t, y)−f(t, x)) ≤λφ(y−x), for any x, y ∈[0,∞)with y≥xand any t∈[0,1], Problem (4.14) has a unique nonnegative solution. Moreover, this solution is strictly increasing. 292 Fractional boundary value problem The same problem was treated by T. Qiu, Z. Bai, in Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 146, (2008), 1–9, using the next theorem. Theorem 58. Let 0< σ < 1,2< α ≤3,f: (0,1] ×[0,+∞)→[0,+∞)is continuous and lim t→0+f(t, ·)=+∞,tσf(t, y)is continuous function on [0,1]× [0,+∞). Assume that there exist two distinct positive constant ρ, µ(ρ>µ) such that (H1) tσf(t, ω)≤ρΓ(α−σ) Γ(1−σ), for (t, ω)∈[0,1] ×[0, ρ]; (H2) tσf(t, ω)≥µΓ(α−σ) Γ(1−σ), for (t, ω)∈[0,1] ×[0, µ]. Then (1.1) has at least one positive solution. Our result improves the ones obtained by them, since the uniqueness and the monotonicity of the solution cannot be deduced by their paper. We present an example which can be treated by Theorem 57 and it cannot be studied by Theorem . Communications in Applied Analysis 15 (2011), no. 2, 3 and 4, 265–272 POSITIVE AND NONDECREASING SOLUTIONS TO A SINGULAR BOUNDARY VALUE PROBLEM FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS J. CABALLERO1, J. HARJANI1, AND K. SADARANGANI1 1Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017. Las Palmas de Gran Canaria, Spain. E-mail: jmen[email protected] E-mail: [email protected] E-mail: [email protected] This paper is dedicated to Professor Jeff Webb on the occasion of his retirement ABSTRACT. In this paper we establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equations. Our analysis relies on a fixed point theorem in partially ordered sets. AMS (MOS) Subject Classification. 45M99,47H09. 1. INTRODUCTION Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, for example [3, 12]) and to problems of analyticity in the complex domain [11]. Moreover, Delbosco and Rodino [7] considered the existence of a solution for the nonlinear fractional differential equation Dα 0+u=f(t, u), where 0 < α < 1 and f: [0, a]×R→R, 0 < a ≤+∞is a given continuous function in (0, a)×R. They obtained existence results by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang [19] considered the existence of a positive solution for the equation Dα 0+u=f(t, u), where 0 < α < 1 and f: [0,1] ×[0,∞)→[0,∞) is a given continuous function by using the sub and super-solution method. In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to the boundary value problem Dα 0+u(t) + f(t, u(t)) = 0,0< t < 1, u(0) = u′(1) = u′′(0) = 0,(1.1) Received September 21, 2010 1083-2564 $15.00 c Dynamic Publishers, Inc. Theory of existence and uniqueness of solution for bvp 293 272 J. CABALLERO, J. HARJANI, AND K. SADARANGANI [3] L.M.C.M. Campos, On the solution of some simple fractional differential equation, Int. J. Math. Sci. 13 (1990), 481-496. [4] M. Belmekki; J.J. Nieto; R. Rodr´ıguez-L´opez, Existence of Periodic Solutions for a Nonlinear Fractional Differential Equation, Boundary Value Problems, vol. 2009, Article ID 324561, (2009). [5] Y.K. Chang; J.J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Mathematical and Computer Modelling, 49 (2009), 605-609. [6] L. Ciri´c; N. Caki´c; M. Rajovi´c; J.S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory and Applications, vol. 2008 Article 131294. [7] D. Delbosco; L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625. [8] J. Harjani; K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009) 3403-3410. [9] J. Caballero, J. Harjani, K. Sadarangani, Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems, Boundary value problems vol. 2009, Article ID 421310 (2009). [10] A.A. Kilbas; J.J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Applicable Analysis 78 (2001), 153-192. [11] Y. Ling; S. Ding, A class of analytic functions defined by fractional derivative, J. Math. Anal. Appl. 186 (1994), 504-513. [12] K. S. Miller; B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York, 1993. [13] J.J. Nieto; R.L. Pouso; R. Rodr´ıguez-L´opez, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135 (2007), 2505-2517. [14] J.J. Nieto; R. Rodr´ıguez-L´opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, (2005), 223-239. [15] J.J. Nieto; R. Rodr´ıguez-L´opez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica 23 (2007), 22052212. [16] D. O’Regan; A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008), 1241-1252. [17] T. Qiu; Z. Bai, Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 146 (2008), 1-9. [18] S. G. Samko; A.A. Kilbas; O.I. Marichev, Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, 1993. [19] S.Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2000), 804-812. 300 Fractional boundary value problem Cap´ıtulo 5 Future lines of research 301 302 5.1. Operators of cyclical type . . . . . . . . . . . . . 303 5.2. Best proximity point: approximation and optimization........................ 305 5.3. Fixed points of decreasing operators and applications......................... 307 5. Future lines of research 303 5.1. Operators of cyclical type The operators of cyclical type were introduced by Kirk, Srinivasan and Veeramani in [W. A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, (1), (2003), 79-89] and they are defined of the following form. Let Xbe a nonempty set, ma positive integer and T:X→Xa mapping. X=∪m i=1Ai is said to be a cyclic representation of Xwith respect to Tif (i) Ai, i = 1,2, . . . , m are nonempty sets. (ii) T(A1)⊂A2, . . . , T(Am−1)⊂Am, T(Am)⊂A1. In [99] the authors prove some fixed point theorems for this type of operators using several contractive conditions. Our aim is to study some fixed point theorems for operators of cyclical type in the context of partially ordered metric spaces. We have achieved some results in this area as evidenced by the following papers: [HLS1] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for cyclic ϕ-contractions in ordered metric spaces, Fixed Point Theory (accepted). [HLS2] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for cyclic weak contractions in compact metric spaces, J. Nonlinear Sci. Appl. (accepted). [HSS1] J. Harjani, F. Sabetghadam, K. Sadarangani, Fixed point theorem for cyclic weak contractions in partially ordered sets endowed with a complete metric, Carpathian J. Math. (accepted). [HSS2] J. Harjani, F. Sabetghadam, K. Sadarangani, Fixed points for mappings of cyclical type in ordered metric spaces, Fixed Point Theory (accepted). 304 Operators of cyclical type [KS] E. Karapinar, K. Sadarangani, Fixed point theory for cyclic (ϕ−ψ)- contraction, Fixed Point Theory and Applications, vol. 2011, doi: 10.1186/1687-1812-2011-69. 5. Future lines of research 305 5.2. Best proximity point: approximation and optimization Let Aand Btwo nonempty subsets of a metric space (X, d) and let T:A→Bbe a mapping. Since Tis not a self mapping, the equation Tx =x is unlikely to have a solution. Therefore, is interesting question is to find an element x∈Athat in some sense is closest to Tx. Best approximation theorems and best proximity point theorems are relevant under this perspective. It is clear that d(x, Tx)≥d(A, B), and an absolute optimal approximate solution is an element xfor which the error d(x, Tx) assumes the least possible value which is d(A, B). The points x∈Asatisfying d(x, Tx) = d(A, B) are known as best proximity point of T. In this field, we have written the following preprint: J. Caballero, J. Harjani, K. Sadarangani, Best proximity point theorems for non-self order contractions of Geraghty-type in partially ordered metric spaces. Some references about this topic are: [AS] M. A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70, (10), (2009), 3665–3671. [A] A. Amini-Harandi, Best proximity points for proximal generalized contractions in metric spaces, Optim. Letter. doi: 10.1007./s11590-0120470-z. [EV] A. Anthony Eldred, P. Veerameni, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323, (2006), 1001–1006. [BSV] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic MeirKeeler contractions, Nonlinear Anal. 69, (11), (2008), 3790–3794. [KA] S. Karpagam, S. Agarwal, Best proximity point theorems for ρ-cyclic Meir-Keeler contractions, Fixed Point Theory Appl. (2009), Art. ID 197308, 9 pages. 306 Best proximity point: approximation and optimization [KRV] W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best porximitypair theorems, Numer. Funct. Anal. Optim. 24, (2003), 851– 862. [B1] S. Sadiq Basha, Best proximity points: global optimal approximate solutions, J. Glob. Optim. 49, (2011), 15–21. [B2] S. Sadiq Basha, Best proximity point theorems, J. Approx. Theory 163, (2011), 1772–1781. [B3] S. Sadiq Basha, Discrete optimization in partially ordered sets, J. Glob. Optim. doi: 10.1007./s10898-011-9774-2. [BSJ] S. Sadiq Basha, N. Shahzad, R. Jeyaraj, Best proximity points: approximation and opritmization, Optim. Lett. doi: 10.1007/s11590-0110404-1. [BV] S. Sadiq Basha, P. Veeramani, Best porximity pair theorems for multifunctions with open fibres, J. Approx. Theory 103, (2000), 119–129. 5. Future lines of research 307 5.3. Fixed points of decreasing operators and applications In the paper [J. J. Nieto, R. Rodr´ıguez-L´opez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica 23, 12, (2007), 2205–2212] the authors prove the following result. Theorem 59. Let (X, ≤)be a partially ordered set such that for each x, y ∈ Xthere exist z∈Xwhich is comparable to xand y. Suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let T:X→Xbe a nonincreasing mapping such that there exists k∈[0,1) satisfying d(Tx, Ty)≤k d(x, y)for x, y ∈Xwith x≥y. Suppose also that either Tis continuous or Xis such that if (xn)⊂Xwith xn→xand consecutive terms are comparable then there exists a subsequence xnkof xnsuch that every term is comparable to the limit x. (5.1) If there exists x0∈Xwith x0≤Tx0or x0≥Tx0then Thas a unique fixed point. Notice that a difficult question is to find partially ordered metric spaces satisfying (5.1) and, consequently, Theorem 59 is not useful when Tis not continuous. In the practice, when we want to investigate the existence and uniqueness of solutions for boundary value problems where the data function is decreasing appears the above mentioned question and Theorem 59 does not work. In the literature, there exist fixed point theorems for decreasing operators which can be applied to boundary value problems. Our aim is to study these fixed point theorems and to apply them to boundary 308 Fixed points of decreasing operators and applications value problems of ordinary or fractionary differential equations. In this area, we include these references: [C] J. A. Cid-Araujo, The uniqueness of fixed points for decreasing operators, Appl. Math. Lerr. 17, (2004), 861–866. [G] D. Guo, Existence and uniqueness of positive fixed points for noncompact decreasing operators, Indian J. Pure Appl. Math. 31, (2000), 551– 562. [GN] Q. Guo, P. Niu, Some theorems on existence and uniqueness of fixed points for decreasing operators, Comput. Math. Appl. 57, (2009), 1515– 1521. [LLX] K. Li, J. Liang, T. J.Xiao, A fixed point theorem for convex and decreasing operators, Nonlinear Anal. 63, (2005), 209–216. [S] V. Seda, Monotone-iterative technique for decreasing mappings, Nonlinear Anal. 40, (2000) 577–588. [ZC] Z. Zhao, X. Chen, Fixed points of decreasing operators in ordered Banach spaces and applications to nonlinear second order elliptic equations, Comput. Math. Appl. 58, (2009), 1223–1229. Bibliography 309 [71] J. Harjani, K. Sadarangani, Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem, Positivity 14 (2010) 849–858. [72] J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72, (2010), 1188–1197. [73] J. Harjani, B. L´opez, K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstract and Applied Analysis vol 2010, Article ID190701, 8 pages. [74] J. Harjani, B. L´opez and K. Sadarangani, On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets, Dynamic Systems and Applications 19, (2010), 625–634. [75] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for weakly Ccontractive mappings in ordered metric spaces, Comput. Math. Appl., 61, (2011), 790–796. [76] J. Harjani, B. L´opez, K. Sadarangani, A fixed point theorem for Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory and Applications, vol. 2011, doi: 10.1186/1687-1812-2011-83. [77] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for mixed monotone operators and application to integral equations, Nonlinear Anal. 74, (2011), 1749–1760. [78] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for cyclic ϕcontractions in ordered metric spaces, Fixed Point Theory (accepted). [79] J. Harjani, B. L´opez, K. Sadarangani, Fixed point theorems for cyclic weak contractions in compact metric spaces, J. Nonlinear Sci. Appl. (accepted). [80] J. Harjani, F. Sabetghadam, K. Sadarangani, Fixed point theorem for cyclic weak contractions in partially ordered sets endowed with a complete metric, Carpathian J. Math. (accepted). [81] J. Harjani, F. Sabetghadam, K. Sadarangani, Fixed points for mappings of cyclical type in ordered metric spaces, Fixed Point Theory (accepted). [82] G. E. Hern´andez, R. Manasevich, Existence and multiplicity of solutions of a fourth order equation, Appl. Anal., 54, (1994), 237–250. [83] D. Ilic, V. Rakocevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341, (2008), 876-882. [84] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136, (2008), 1359–1373. [85] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal., 74, (2011), 768–774. [86] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math., 8, (1977), 223–230. [87] A. J. Jerri, Introduction to integral equations with applications, 2nd ed. Wiley-Interscience, New York, (1999). [88] R. Kannan, Some results on fixed points-II, Amer. Math. Monthly, 76, (1969), 405–408. [89] E. R. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 3, 1–11, (2008). [90] E. Karapinar, Fixed point theory for cyclic weak φ-contraction, Appl. Math. Lett., 24, (2011), 822–825. [91] E. Karapinar, K. Sadarangani, Fixed point theory for cyclic (ϕ− ψ)-contraction, Fixed Point Theory and Applications, vol. 2011, doi: 10.1186/1687-1812-2011-69. [92] S. Kempfle, I. Sch¨afer, H. Beyer, Functional calculus and a link to fractional calculus, Fract. Calc. Appl. Anal, 5, (2002), 411–426. [93] S. Kempfle, I. Sch¨afer, H. Beyer, Fractional calculus via functional calculus: theory and applications., Nonlinear Dynam., 29, (2002), 99–127. [94] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30, (1), (1984), 1–9. [95] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69, (2008), 2942–2949. [96] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, (2006). [97] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Applicable Analysis, 78, (2001), 153–192. [98] W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24, (2003), 851–862. [99] W. A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, (1), (2003), 79–89. [100] P. Korman, Uniqueness and exact multiplicity of solutions for class of fourthorder semilinear problems, Proceedings of the Royal Society of Edinburgh, 134, (2004), 179–190. [101] N. Kosmatov, Countably many solutions of a fourth order boundary-value problem, Elect. J. Qualitative Th. Diff. Eq., 12, (2004), 1–15. [102] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis, 69, (2008), 3337–3343. [103] V. Lakshmikantham, L. ´ Ciri´c, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70, (2009), 4341–4349. [104] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis, 69, (2008), 2677–2682. [105] C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59, (2010), 1363–1375. [106] S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Analysis, 71, (2009), 5545– 5550. [107] X. L. Liu, W. T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl., 327, (2007), 362–375. [108] J. Liu, W. Xu, Positive solutions for some beam equation boundary value problems, Boundary Value Problems, Vol. 2009, (2009), Article ID 393259. [109] R. Ma, Existence and uniqueness theorems for some fourth-order nonlinear boundary-value problems, Internat. J. Math. Math. Sci., 23, (2003), 783–788. [110] R. Ma, Existence of positive solutions of a fourth-order boundary value problem, Appl. Math. Comput., 168, (2005), 1219–1231. [111] J. Markin, N. Shahzad, Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces, Nonlinear Anal., 70, (2009), 2435–2441. [112] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28, (1969), 326–329. [113] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York, (1993). [114] F. Minh´os, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem, Discrete Contin. Dyn. Syst. Suppl., (2005), 662–671. [115] S.V.R. Naidu, Some fixed point theorems in metric spaces by altering distances, Czechoslovak Math. J., 53, (2003), 205–212. [116] H. K. Nashine, B. Samet, C. Vetro, Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces, Math. Comput. Modelling 54, (2011), 712–720. [117] A. Nicolae, D. O’Regan, A. Petru¸sel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J., 18, (2011), 307–327. [118] J.J. Nieto, R. Rodr´ıguez-L´opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, (2005), 223–239. [119] J.J. Nieto, R. Rodr´ıguez-L´opez, Existence of extremal solutions for quadratic fuzzy equations, Fixed Point Theory and Applications, (2005), 321–342. [120] J. J. Nieto, R. Rodr´ıguez-L´opez, Applications of contractive - like mapping principles to fuzzy equations, Rev. Math. Complut., 19, (2006), 361–383. [121] J. J. Nieto, R. Rodr´ıguez-L´opez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica 23, 12, (2007), 2205–2212. [122] J.J. Nieto, R.L. Pouso, R. Rodr´ıguez-L´opez, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135, (2007), 2505–2517. [123] D. O’Regan, A. Petru¸sel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341, (2008), 1241–1252. [124] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, (1974). [125] M. Pacurar, I. A. Rus, Fixed point theory for cyclic ϕ-contactions, Nonlinear Anal., 72, (2010), 1181–1187. [126] M. Pei, S. K. Chang, Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Modelling, 51, (2010), 1260–1267. [127] M. Petric, Some results concerning cyclical contractive mappings, Gen. Math. 18, (2010), No4, 213–226. [128] M. Petric, Best proximity point theorems for weak cyclic Kannan contractions, Filomat, 25, (2011), No1, 145–154. [129] A. Petru¸sel, I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc., 134, 2, (2006), 411–418. [130] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, San Diego, Calif. USA, (1999). [131] T. Qiu, Z. Bai, Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 146, (2008), 1–9. [132] A. C. M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132, (2004), 1435–1443. [133] E. M. Rabei, K. I. Nawafleh, R. S. Hijjawi, S. I. Muslih, D. Baleanu, The Hamilton formalism with fractional derivatives, J. Math. Anal. Appl., 327, (2007), 891–897. [134] S. Radenovic, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl., 60, (2010), 1776–1783. [135] B. E. Rhoades, A comparison of various definitions of contractive mappings. Transactions of the Amer. Math. Soc. 226, (1977), 257–290 [136] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47, (2001), 2683–2693. [137] B. E. Rhoades, Two fixed point theorems for mapping satisfying a general contractive condition of integral type, Int. J. Math. Sci., 63, (2003), 4007– 4013. [138] I. A. Rus, Cyclic representation and fixed points, Ann. Tiberiu Popoviciu Seminar Funct. Eq., Approx. Convexity, 3, (2005), 171–178. [139] B. P. Rynne, Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Methods Nonlinear Anal., 19, (2002), 303–312. [140] R. Saadati, S.M. Vaezpour, Monotone generalized weak contractions in partially ordered metric spaces, Fixed Point Theory, 11, (2), (2010), 375–382. [141] S. Sadiq Basha, Best proximity points: global optimal approximate solutions, J. Glob. Optim., 49, (2011), 15–21. [142] S. Sadiq Basha, Best proximity point theorems, J. Approx. Theory, 163, (2011), 1772–1781. [143] S. Sadiq Basha, Discrete optimization in partially ordered sets, J. Glob. Optim., (2011), doi: 10.1007./s10898-011-9774-2. [144] S. Sadiq Basha, N. Shahzad, R. Jeyaraj, Best proximity points: approximation and opritmization, Optim. Lett., (2011), doi: 10.1007/s11590-011-04041. [145] S. Sadiq Basha, P. Veeramani, Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103, (2000), 119–129. [146] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72, 12, (2010), 4508–4517. [147] B. Samet, C. Vetro, Coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear Anal., 74, (2011), 4260–4268. [148] S. G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, (1993). [149] V. Sankar Raj, P. Veeramani, Best proximity pair theorems for relatively nonexpansive mappings, Appl. Gen. Topol., 10, (2009), 21–28. [150] V. Sankar Raj, A best proximity theorem for weakly contractive non-self mappings, Nonlinear Anal., 74, (2011), 4804–4808. [151] V. Sankar Raj, Banach’s contraction principle for non-self mappings, (preprint). [152] K.P.R. Sastry, G.V.R. Babu, Some fixed point theorems by altering distances between the points, Indian J. of Pure and Appl. Math., 30, (1999), 641–647. [153] W. Shatanawi, Fixed point theory for contractive mappings satisfying φ-maps in G-metric spaces, Fixed Point Theory and Applications, 2010, ID 181650. [154] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Computers and Mathematics with Applications, 60, (8), (2010), 2508–2515. [155] W. Shatanawi, Fixed point theorems for nonlinear weakly C-contractive mappings in metric spaces, Mathematical and Computer Modelling, 54, 2011, 2816–2826. [156] Z. Mustafa, W. Shatanawi, M. Bataineh, Existence of fixed point results in G-metric spaces, International Journal of Mathematics and Mathematical Sciences, 2009, ID 283028. [157] P.V. Subrahmanyam, Completeness and fixed points, Monatsh. Math., 80, (1975), 325–330. [158] Y. Sun, A fixed point theorem for mixed monotone operator with applications, J. Math. Anal. Appl., 156, (1991), 240–252. [159] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136, (2008), 1861–1869. [160] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340, (2008), 1088–1095. [161] M. Swaleh and S. Sessa Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30, (1984), 1–9. [162] A. Tarski, A lattice-theoretical fixed point theorem and its applications, Pacific J. Math., 5, (1955), 285–309. [163] S. P. Timoshenko, Theory of elastic stability, McGraw-Hill, New York, (1961). [164] J. R. L. Webb, G. Infante, D. Franco, Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions, Proc. Roy. Soc. Edimburgh Sect. A, 148, (2008), 427–446. [165] Y. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl., 341, (2008), 883–893. [166] B. Yang, Positive solutions for a fourth order boundary value problem, Elect. J. Qualitative Th. Diff. Eq., 3, (2005), 1–17. [167] B. Yang, Positive solutions for the beam equation under certain boundary conditions, Elect. J. Qualitative Th. Diff. Eq., 78, (2005), 1–8. [168] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23, (1972), 292–298. [169] M.Y. Zhang, Fixed point theorems of φconvexψconcave mixed monotone operators and applications, J. Math. Anal. Appl., 339, (2008), 970–981. [170] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252, (2000), 804–812. [171] S. Zhang, Existence of solution for a boundary value problem of fractional order, Acta Mathematica Scientia, 26, (2006), 220–228. 323 [172] S.S. Zhang, Y.H. Ma, Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solution for a class of functional equations arising in dynamic programming, J. Math. Anal. Appl., 160, (1991), 468–479. [173] Z. Zhang, New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl., 204, (1996), 307–319. [174] Y. Zhao, S. Sun, Z. Han, Q. Li, Positive solutions to boundary value problems of nonlinear fractional differential equations. Abs Appl Anal. 2011, (2011), Article ID 390543. [175] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Applications, 11, (2010), 4465–4475. Fixed Point Theorems in Partially Ordered Spaces and Applications Tesis Doctoral Jackie Harjani Sauco Las Palmas de Gran Canaria Mayo de 2014 Fixed Point Theorems in Partially Ordered Spaces and Applications Jackie Harjani Sauco Departamento de Física