Fixed Poin Theo ems in Pa ially
O de ed Spaces and Applica ions
Tesis Doc o al
Jackie Ha jani Sauco
Las Palmas de G an Cana ia
Mayo de 2014
Fixed Poin Theo ems in Pa ially O de ed Spaces and Applica ions
Jackie Ha jani Sauco
Depa amen o de Física
P´agina ese ada pa a los se icios adminis a i os de la Uni e sidad de Las
Palmas de G an Cana ia
SALVADOR GALVÁN HERRERA, SECRETARIO DEL
DEPARTAMENTO DE FÍSICA DE LA UNIVERSIDAD DE
LAS PALMAS DE GRAN CANARIA,
CERTIFICA,
Que la Comisión de Doc o es del Depa amen o en su
sesión de echa 19 de mayo de 2014 omó el acue do de
da el consen imien o pa a su ami ación, a la esis
doc o al i ulada “Fixed Poin Theo ems in Pa ially
O de ed Spaces and Applica ions“, p esen ada po el
doc o ando D. Jackie Ha jani Sauco, di igida po el D . D.
Kishin Sada angani y codi igida po la D a. Dª Belén
López B i o.
Y pa a que así cons e, y a e ec os de lo p e is o en
el A º 73.2 del Reglamen o de Es udios de Doc o ado de
es a Uni e sidad, i mo la p esen e en Las Palmas de
G an Cana ia, a diecinue e de mayo de dos mil ca o ce.

Tesis Doc o al
Fixed Poin Theo ems in Pa ially
O de ed Spaces and Applica ions
Jackie Ha jani Sauco
Doc o ando
Kishin Sada angani
Di ec o
Bel´
en L´
opez B i o
Codi ec o a
Depa amen o de F´
ısica
P og ama de Doc o ado: F´
ısica, Ma em´
a icas, Geolog´
ıa y Clima
Las Palmas de G an Cana ia a Diecis´
eis de Mayo de 2014
Tesis p esen ada po Jackie Ha jani Sauco pa a aspi a al g ado de
Doc o po la Uni e sidad de Las Palmas de G an Cana ia, con la ap o-
baci´on y el is o bueno del D . Kishin Sada angani y la D a. Bel´
en
L´
opez B i o, jun o con los in o mes a o ables del D . J¨
u gen Appell
y el D . Ad ian Pe us¸el.
En Las Palmas de G an Cana ia a Diecis´eis de Mayo de 2014

´
Indice gene al
1. Resumen de la Tesis / Summa y o he hesis in Spanish 1
1.1. Teo emas del pun o ijo en espacios m´e icos o denados . . . . 9
1.1.1. Fixed poin heo ems o weakly con ac i e mappings
in pa ially o de ed se s . . . . . . . . . . . . . . . . . 11
1.1.2. Gene alized con ac ions in pa ially o de ed me ic
spaces and applica ions o o dina y di e en ial equa ions 15
1.1.3. Con ac i e-like mapping p inciples in o de ed me ic
spaces and applica ions o o dina y di e en ial equa ions 19
1.1.4. Fixed poin heo ems o mappings sa is ying a condi-
ion o in eg al ype in pa ially o de ed se s . . . . . . 23
1.1.5. A ixed poin heo em o mappings sa is ying a con-
ac i e condi ion o a ional ype on a pa ially o de-
edme icspace...................... 27
1.1.6. Fixed poin heo ems o weakly C-con ac i e map-
pings in o de ed me ic spaces . . . . . . . . . . . . . . 29
1.1.7. Fixed poin heo ems o mixed mono one ope a o s
and applica ion o in eg al equa ions . . . . . . . . . . 33
1.1.8. A ixed poin heo em o Mei -Keele con ac ions in
o de ed me ic spaces . . . . . . . . . . . . . . . . . . . 39
1.2. Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p . 41
1.2.1. Exis ence and uniqueness o posi i e solu ions o a
nonlinea ou h-o de bounda y alue p oblem . . . . 43
IX
1.2.2. On posi i e solu ions o a nonlinea ou h o de boun-
da y alue p oblem ia a ixed poin heo em in o de ed
se s............................. 47
1.2.3. Uniqueness o posi i e solu ions o a class o ou h-
o de bounda y alue p oblems . . . . . . . . . . . . . 49
1.3. F ac ional bounda y alue p oblem . . . . . . . . . . . . . . . 53
1.3.1. Exis ence and uniqueness o posi i e and nondec ea-
sing solu ions o a class o singula ac ional boun-
da y alue p oblems . . . . . . . . . . . . . . . . . . . 53
1.3.2. Posi i e solu ions o a class o singula ac ional
bounda y alue p oblems . . . . . . . . . . . . . . . . . 57
1.3.3. Exis ence and uniqueness o posi i e solu ion o a
bounda y alue p oblem o ac ional o de . . . . . . . 61
1.3.4. On exis ence and uniqueness o posi i e solu ions o a
class o ac ional bounda y alue p oblems . . . . . . . 65
1.3.5. Posi i e and nondec easing solu ions o a singula
bounda y alue p oblem o nonlinea ac ional di -
e en ial equa ions . . . . . . . . . . . . . . . . . . . . 67
2. A sho his o y app oach 69
3. Fixed poin heo ems in pa ially o de ed me ic spaces 75
3.1. Fixed poin heo ems o weakly con ac i e mappings ... . . . 79
3.2. Gene alized con ac ions in pa ially o de ed me ic spaces ... . 91
3.3. Con ac i e-like mapping p inciples in o de ed me ic spaces ... 105
3.4. Fixed poin heo ems o mappings sa is ying a condi ion o ... 123
3.5. A ixed poin heo em o mappings sa is ying a con ac i e ... 141
3.6. Fixed poin heo ems o weakly C-con ac i e mappings in ... 151
3.7. Fixed poin heo ems o mixed mono one ope a o s and ... . . 161
3.8. A ixed poin heo em o Mei -Keele con ac ions in ... . . . 179
4. Theo y o exis ence and uniqueness o solu ion o bounda y
alue p oblems 189
4.1. Classical bounda y alue p oblem . . . . . . . . . . . . . . . . 193
4.1.1. Exis ence and uniqueness o posi i e solu ions o a
nonlinea ou h-o de bounda y alue p oblem . . . . 193
4.1.2. On posi i e solu ions o a nonlinea ou h o de boun-
da y alue p oblem ia a ixed poin heo em in o de ed
se s.............................207
4.1.3. Uniqueness o posi i e solu ions o a class o ou h-
o de bounda y alue p oblems . . . . . . . . . . . . . 219
4.2. F ac ional bounda y alue p oblem . . . . . . . . . . . . . . . 237
4.2.1. Exis ence and uniqueness o posi i e and nondec ea-
sing solu ions o a class o singula ac ional boun-
da y alue p oblems . . . . . . . . . . . . . . . . . . . 237
4.2.2. Posi i e solu ions o a class o singula ac ional
bounda y alue p oblems . . . . . . . . . . . . . . . . . 251
4.2.3. Exis ence and uniqueness o posi i e solu ion o a
bounda y alue p oblem o ac ional o de . . . . . . . 263
4.2.4. On exis ence and uniqueness o posi i e solu ions o a
class o ac ional bounda y alue p oblems . . . . . . . 279
4.2.5. Posi i e and nondec easing solu ions o a singula
bounda y alue p oblem o nonlinea ac ional di -
e en ial equa ions . . . . . . . . . . . . . . . . . . . . 291
5. Fu u e lines o esea ch 301
5.1. Ope a o s o cyclical ype . . . . . . . . . . . . . . . . . . . . 303
5.2. Bes p oximi y poin : app oxima ion and op imiza ion . . . . 305
5.3. Fixed poin s o dec easing ope a o s and applica ions . . . . . 307
Bibliog aphy 309
Cap´ı ulo 1
Resumen de la Tesis /
Summa y o he hesis in
Spanish
1

2
1.1. Teo emas del pun o ijo en espacios m´e icos o -
denados ........................ 9
1.1.1. Fixed poin heo ems o weakly con ac i e map-
pings in pa ially o de ed se s . . . . . . . . . . . . 11
1.1.2. Gene alized con ac ions in pa ially o de ed me-
ic spaces and applica ions o o dina y di e en ial
equa ions ....................... 15
1.1.3. Con ac i e-like mapping p inciples in o de ed
me ic spaces and applica ions o o dina y di e-
en ial equa ions . . . . . . . . . . . . . . . . . . . 19
1.1.4. Fixed poin heo ems o mappings sa is ying a
condi ion o in eg al ype in pa ially o de ed se s 23
1.1.5. A ixed poin heo em o mappings sa is ying a
con ac i e condi ion o a ional ype on a pa -
ially o de ed me ic space . . . . . . . . . . . . . . 27
1.1.6. Fixed poin heo ems o weakly C-con ac i e
mappings in o de ed me ic spaces . . . . . . . . . 29
1.1.7. Fixed poin heo ems o mixed mono one ope a-
o s and applica ion o in eg al equa ions . . . . . 33
1.1.8. A ixed poin heo em o Mei -Keele con ac ions
in o de ed me ic spaces . . . . . . . . . . . . . . . 39
1.2. Teo ´ıa de exis encia y unicidad pa a las solucio-
nesdeunb p..................... 41
1.2.1. Exis ence and uniqueness o posi i e solu ions o
a nonlinea ou h-o de bounda y alue p oblem . 43
1.2.2. On posi i e solu ions o a nonlinea ou h o de
bounda y alue p oblem ia a ixed poin heo em
ino de edse s .................... 47
1.2.3. Uniqueness o posi i e solu ions o a class o
ou h-o de bounda y alue p oblems . . . . . . . 49
1. Resumen de la Tesis / Summa y o he hesis in Spanish 3
1.3. F ac ional bounda y alue p oblem . . . . . . . . 53
1.3.1. Exis ence and uniqueness o posi i e and nonde-
c easing solu ions o a class o singula ac ional
bounda y alue p oblems . . . . . . . . . . . . . . 53
1.3.2. Posi i e solu ions o a class o singula ac ional
bounda y alue p oblems . . . . . . . . . . . . . . 57
1.3.3. Exis ence and uniqueness o posi i e solu ion o
a bounda y alue p oblem o ac ional o de . . . 61
1.3.4. On exis ence and uniqueness o posi i e solu ions
o a class o ac ional bounda y alue p oblems . 65
1.3.5. Posi i e and nondec easing solu ions o a singula
bounda y alue p oblem o nonlinea ac ional
di e en ial equa ions . . . . . . . . . . . . . . . . . 67
1. Resumen de la Tesis / Summa y o he hesis in Spanish 5
Recien emen e, en la Teo ´ıa del pun o ijo, han apa ecido muchos esul-
ados que ob ienen condiciones su icien es pa a la exis encia de un pun o
ijo si abajamos con aplicaciones en un conjun o do ado de un o den pa -
cial. Gene almen e, es os esul ados combinan dos eo emas del pun o ijo
undamen ales: el Teo ema de la con acci´on de Banach y el Teo ema de
Knas e -Ta ski.
El Teo ema de la con acci´on de Banach ue demos ado en 1922 y su enun-
ciado es el siguien e.
Teo ema 1 (Teo ema de la con acci´on de Banach).Sea (X, d)un espacio
m´e ico comple o y T:X→Xuna aplicaci´on al que, exis e λ∈[0,1) y se
e i ica
d(Tx, Ty)≤λ d(x, y)pa a odo x, y ∈X.
En onces T iene un ´unico pun o ijo ¯x∈X(i.e., T¯x= ¯x).
Adem´as, pa a cada x∈Xla sucesi´on {Tnx}con e ge a ¯x.
Se han lle ado a cabo un g an n´ume o de gene alizaciones de es e p in-
cipio, donde la condici´on con ac i a que apa ece en el Teo ema 1 es eem-
plazada po o as condiciones ( e , B. E. Rhoades, A compa ison o a ious
de ini ions o con ac i e mappings. T ansac ions o he Ame . Ma h. Soc.
226, (1977), 257–290).
El Teo ema de Knas e -Ta ski ue p obado en 1955 y su o mulaci´on iene
ecogida en el siguien e eo ema.
Teo ema 2 (Teo ema de Knas e -Ta ski).Sea (A, ≤)un la ice comple-
o (es o signi ica que cada subconjun o A iene ´ın imo y sup emo en A) y
T:A→Auna aplicaci´on que p ese a el o den. En onces T iene un pun o
ijo.
Los eo emas del pun o ijo en los espacios m´e icos o denados es ´an ´ın i-
mamen e ligados con la mono on´ıa ( an o si p ese an el o den como si no)
de las aplicaciones, donde la condici´on con ac i a es solamen e sa is echa
po loa elemen os que son compa ables.
12 Teo emas del pun o ijo en espacios m´e icos o denados
con ex o de los espacios m´e icos pa cialmen e o denados.
La p incipal con ibuci´on puede se esumida en el siguien e eo ema.
Teo ema 6. Sea (X, ≤)un espacio pa cialmen e o denado y supongamos que
exis en una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on con inua y c ecien e al que
d(Tx, Ty)≤d(x, y)−ψd(x, y),pa a x, y ∈Xcon x≥y ,
donde ψ: [0,∞)→[0,∞)es una unci´on con inua y c ecien e al que es
posi i a en (0,∞)yψ(0) = 0.
Si exis e x0∈X al que x0≤Tx0en onces T iene un pun o ijo.
Demos amos que la condici´on de que Tsea con inua es innecesa ia si
asumimos que en Xse cumple
si (xn) es una sucesi´on c ecien e en X al que xn→x
en onces xn≤x, pa a odo n∈N.(1.1)
Es a condici´on ue usada po J. Nie o y R. Rod ´ıguez-L´opez en [118]. Con
mayo p ecisi´on, demos amos el siguien e esul ado.
Teo ema 7. Si en el Teo ema 6 eemplazamos la con inuidad de Tpo la
condici´on (1.1) en onces ob end ´ıamos la misma conclusi´on.
Pa a analiza la unicidad del pun o ijo, apo amos un ejemplo en donde
se comp ueba que los Teo emas 6 y 7 no ga an izan la unicidad.
Po es e mo i o, p esen amos una condici´on su icien e pa a la unicidad (que
ue usada en [118]). Su enunciado es:
Pa a x, y ∈Xexis e z∈Xque es compa able con xyy. (1.2)
Teo ema 8. A˜nadiendo la condici´on (1.2) a las hip´o esis del Teo ema 6
( espec i amen e Teo ema 7), ob enemos la unicidad del pun o ijo.
Finalmen e, aplicamos los esul ados ob enidos pa a demos a la exis-

1. Resumen de la Tesis / Summa y o he hesis in Spanish 13
encia de la soluci´on del siguien e p oblema pe i´odico de p ime o den
(u0( ) =  , u( ), ∈[0, T],
u(0) = u(T),(1.3)
bajo la hip´o esis de la exis encia de una subsoluci´on pa a (1.3), es deci , una
unci´on α∈ C[0, T] al que
α0( )≤  , α( ),pa a ∈[0, T],
α(0) ≤α(T).
Es deci , ob enemos el siguien e esul ado.
Teo ema 9. Supongamos que : [0, T]×R→Res con inua y exis e λ > 0
al que pa a x, y ∈Rcon y≥x,
0≤ ( , y) + λy −[ ( , x) + λx]≤λln(y−x+ 1) .
En onces, la exis encia de una subsoluci´on pa a (1.3) nos ga an iza la exis-
encia de una ´unica soluci´on de (1.3).
Es e a ´ıculo se inspi a, undamen almen e, en los esul ados ob enidos
po J.J. Nie o y R. Rod ´ıguez-L´opez en Con ac i e mapping heo ems in pa -
ially o de ed se s and applica ions o o dina y di e en ial equa ions, O de
22, (2005), 223–239..
1. Resumen de la Tesis / Summa y o he hesis in Spanish 15
1.1.2. Gene alized con ac ions in pa ially o de ed
me ic spaces and applica ions o o dina y
di e en ial equa ions
En es e a ´ıculo hacemos uso de las unciones que al e an la dis ancia,
que ue on in oducidas po Khan, Swalesh y Sessa en Fixed poin heo ems
by al e ing dis ances be ween he poin s, Bull. Aus al. Ma h. Soc. 30, (1984),
1–9.
Di emos que una unci´on ϕ: [0,∞)→[0,∞) es una unci´on que al e a la
dis ancia si sa is ace:
(a) ϕes con inua y c ecien e.
(b) ϕ( ) = 0 si y s´olo si = 0.
(No a que es as unciones son las mismas que apa ecen en la de inici´on de
aplicaci´on d´ebilmen e con ac i as en el a ´ıculo an e io ).
En [161], los au o es p ueban es e esul ado:
Teo ema 10. Sea (X, d)un espacio m´e ico comple o, ϕuna unci´on que
al e a la dis ancia y T:X→Xuna aplicaci´on al que
ϕd(Tx, Ty)≤c·ϕd(x, y),
pa a cada x, y ∈X, donde c∈(0,1).
En onces T iene un ´unico pun o ijo.
En 2008, Du a y Choudhu y gene aliza on el Teo ema 5, ob enido po
Ge agh y en [136], de la siguien e mane a.
Teo ema 11. Sea (X, d)un espacio m´e ico comple o y T:X→Xuna
aplicaci´on que sa is ace
ϕd(Tx, Ty)≤ϕd(x, y)−φd(x, y),pa a cada x, y ∈X ,
donde ϕyφson unciones que al e an la dis ancia.
En onces T iene un ´unico pun o ijo.
16 Teo emas del pun o ijo en espacios m´e icos o denados
El p incipal obje i o del a ´ıculo es p esen a la e si´on del Teo ema 11
en el con ex o de espacios m´e icos pa cialmen e o denados.
Nues a con ibuci´on m´as impo an e puede se esumida en los siguien es
eo emas.
Teo ema 12. Sea (X, ≤)un espacio pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on con inua y c ecien e al que
ψd(Tx, Ty)≤ψd(x, y)−φd(x, y)pa a cada x, y ∈Xcon x≥y ,
donde ψyφson unciones que al e an la dis ancia. Si exis e x0∈Xy
x0≤Tx0, en onces T iene un pun o ijo.
Teo ema 13. Si en el Teo ema 12 eemplazamos la condici´on de con inuidad
de Tpo
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N,
en onces llega ´ıamos a la misma conclusi´on.
Teo ema 14. A˜nadiendo la condici´on:
Pa a x, y ∈Xexis e z∈Xque es compa able con xyy ,
a las hip´o esis del Teo ema 12 ( espec i amen e Teo ema 13) ob enemos la
unicidad del pun o ijo.
Los p incipales esul ados de nues os a ´ıculos an e io es, se ob ienen
como casos pa icula es del ob enido en es e a ´ıculo.
Finalmen e, p esen amos dos ejemplos sob e p oblemas con alo es en la
on e a donde se pueden aplica nues os esul ados.
En el p ime ejemplo, es udiamos la exis encia de soluciones pa a es e p o-
1. Resumen de la Tesis / Summa y o he hesis in Spanish 17
blema pe i´odico de p ime o den
(u0( ) =  , u( ), ∈[0, T],
u(0) = u(T).(1.4)
Nues o esul ado se ´ıa:
Teo ema 15. Bajo la hip´o esis de que : [0, T]×R→Res con inua y
suponiendo que exis en dos n´ume os eales posi i os λ, α > 0que sa is acen
α≤2λ(eλT −1)
T(eλT + 1) 1
2
,
y ales que, pa a x, y ∈Rcon y≥x
0≤ ( , y) + λy − ( , x) + λx≤αqln (y−x)2+ 1.
En onces la exis encia de una subsoluci´on en (1.4) ( e los comen a ios del
a ´ıculo an e io ) nos ga an iza la exis encia de una ´unica soluci´on pa a
(1.4).
El segundo ejemplo es udia la exis encia de una soluci´on pa a la siguien e
ecuaci´on di e encial de segundo o den con alo es en la on e a pa a dos
pun os.



−d2x
d 2= ( , x), ∈[0,1] ,
x(0) = x(1) = 0 .
(1.5)
Ob eniendo es e esul ado.
Teo ema 16. Bajo la hip´o esis de que : [0,1] ×R→Res con inua y
c ecien e con espec o a la segunda a iable, y al que, pa a cada x, y ∈R
con y≥x,
( , y)− ( , x)≤αpln[(y−x)2+ 1] ,
donde 0< α ≤8, en onces el P oblema (1.5) iene una ´unica soluci´on posi-
i a. Adem´as, si ( , x)6= 0 pa a ∈(0,1), la soluci´on de (1.5) es posi i a

18 Teo emas del pun o ijo en espacios m´e icos o denados
(es o signi ica que 0< x( )pa a ∈(0,1)).
1. Resumen de la Tesis / Summa y o he hesis in Spanish 19
1.1.3. Con ac i e-like mapping p inciples in o de ed
me ic spaces and applica ions o o dina y
di e en ial equa ions
En[58], Ge agh y p esen ´o una gene alizaci´on del p incipio de la con ac-
ci´on de Banach usando la clase de unciones Sdadas po β: [0,∞)→[0,1)
que sa is acen
β( n)→1⇒ n→0.
El esul ado m´as impo an e de [58] lo mos amos a con inuaci´on.
Teo ema 17. Sea (X, d)un espacio m´e ico comple o y T:X→Xuna
aplicaci´on sa is aciendo
d(Tx, Ty)≤βd(x, y)·d(x, y)pa a cada x, y ∈X ,
donde β∈ S. En onces T iene un ´unico pun o ijo.
En 2010, Amini-Ha andi y Emami demos a on una e sion del Teo e-
ma 17 en el con ex o de espacios pa cialmen e o denados ( e , [9]).
La p incipal apo aci´on de [9] se esume en es e eo ema.
Teo ema 18. Sea (X, ≤)un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on c ecien e al que
d(Tx, Ty)≤βd(x, y)·d(x, y)pa a cada x, y ∈Xcon x≤y ,
donde β∈ S. Asumimos que Tes con inua o Xsa is ace la siguien e condi-
ci´on:
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤x, pa a odo n∈N.
Adem´as, supongamos que pa a cada x, y ∈Xexis e z∈Xcompa able con x
20 Teo emas del pun o ijo en espacios m´e icos o denados
ey. Si exis e x0∈Xcon x0≤Tx0en onces T iene un ´unico pun o ijo.
Nues o p op´osi o en es e a ´ıculo e a ex ende el Teo ema 18 haciendo
uso de las unciones que al e an la dis ancia (consul a los a ´ıculos p e ios).
La p incipal apo aci´on del a ´ıculo puede se esumida en el siguien e eo-
ema.
Teo ema 19. Sea (X, ≤)un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on c ecien e al que
ψd(Tx, Ty)≤βd(x, y)·ψd(x, y)pa a cada x, y ∈Xcon x≤y,
donde ψes una unci´on con dis ancia al e ada y β∈ S.
Asumamos ambi´en que Tes con inua o Xsa is ace la siguien e condici´on
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤x, pa a odo n∈N.
Si exis e x0∈Xcon x0≤Tx0en onces T iene un pun o ijo.
En el a ´ıculo damos un ejemplo pa a e que las hip´o esis del Teo ema 19
no ga an izan la unicidad del pun o ijo.
El siguien e esul ado apo a una condici´on su icien e pa a la unicidad del
pun o ijo.
Teo ema 20. A˜nadiendo al condici´on:
Pa a cada x, y ∈Xexis e z∈Xque es compa able con xyy ,
a las hip´o esis del Teo ema 19, ob enemos la unicidad del pun o ijo.
Pa a ilus a la aplicabilidad de los esul ados ob enidos, damos un eo e-
ma que ga an iza la exis encia de soluci´on del p oblema pe i´odico de p ime
o den (u0( ) =  , u( ), ∈[0, T],
u(0) = u(T).(1.6)
1. Resumen de la Tesis / Summa y o he hesis in Spanish 21
Necesi amos in oduci la clase Ade unciones φ: [0,∞)→[0,∞) que sa is-
acen
(a) φes c ecien e.
(b) φ(x)< x, si x > 0.
(c) β(x) = φ(x)
x∈ S,
El esul ado se ´ıa:
Teo ema 21. Supongamos que :I×R−→ Res con inua y exis e λ, α > 0
que sa is acen
α≤ 2λ(eλT −1)
T(eλT + 1) !1
2
,
al que pa a x, y ∈Rcon x≤y
0≤ ( , y) + λy − ( , x) + λx≤αp(y−x)φ(y−x),
donde φ∈ A. En onces, la exis encia de una subsoluci´on de (1.6) ga an iza ´ıa
la exis encia de una ´unica soluci´on de (1.6).
Adem´as, en el a ´ıculo se demues a que si φ∈ A en onces la unci´on
ϕ(x) = pxφ(x) sa is ace que ϕ∈ A
Es a idea es undamen al a la ho a de p esen a un ejemplo del P oblema (1.6)
que puede se a ado con los esul ados del a ´ıculo pe o no puede es udia se
con los eo emas del a ´ıculo [9].
28 Teo emas del pun o ijo en espacios m´e icos o denados
es m´as ue e que la u ilizada en los a ´ıculos an e io es:
si (xn) es una sucesi´on c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N.
Pa a pode ob ene la unicidad del pun o ijo, usamos la misma condici´on
que en los a ´ıculos p e ios y demos amos es e esul ado.
Teo ema 28. A˜nadiendo la condici´on
pa a x, y ∈Xexis e z∈Xque es compa able con xyy ,
a las hip´o esis del Teo ema 27, ob enemos la unicidad del pun o ijo.
Finalmen e, p esen amos un ejemplo en donde se puede aplica el Teo e-
ma 27 mien as que no puede se abo dado po el Teo ema 26.

1. Resumen de la Tesis / Summa y o he hesis in Spanish 29
1.1.6. Fixed poin heo ems o weakly C-con ac i e
mappings in o de ed me ic spaces
Choudh y en [B. S. Choudhu y, Unique ixed poin heo em o weak C-
con ac i e mappings, Ka mandu Uni e si y Jou nal o Science, Enginee ing
and Technology, ol. 5, (1), (2009) 6–13], in odujo es a de inici´on.
De ini ion 1. Una aplicaci´on T:X→X, donde X, des un espacio m´e i-
co, se dice que es d´ebilmen e C-con ac i a (o d´ebil C-con acci´on) si pa a
odo 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 unci´on con inua e i icando que ϕ(x, y) =
0si y s´olo si x=y= 0.
El au o ambi´en demues a es e esul ado.
Teo ema 29. Supongamos que X, des un espacio m´e ico comple o y
T:X→Xes una aplicaci´on d´ebilmen e C-con ac i a, en onces T iene
un ´unico pun o ijo.
El p incipal p op´osi o de nues o a ´ıculo es da una e si´on del Teo e-
ma 29 en el con ex o de los espacios m´e icos pa cialmen e o denado.
El p ime esul ado que ob u imos ue:
Teo ema 30. Sea X, ≤un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que X, des un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on c ecien e al que
dTx, Ty≤1
2dx, Ty+dy, Tx−ϕdx, Ty, dy, Tx,
pa a cada x, y ∈Xcon x≥y, donde ϕ: [0,∞)2→[0,∞)es una unci´on
con inua cumpliendo que ϕ(x, y) = 0 si y s´olo si x=y= 0.
30 Teo emas del pun o ijo en espacios m´e icos o denados
Asumamos adem´as que Tes con inua o xsa is ace la condici´on
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N.
Si exis e x0∈X al que x0≤Tx0en onces T iene un pun o ijo.
Damos un ejemplo pa a ilus a que las hip´o esis del Teo ema 30 no ga-
an izan la unicidad del pun o ijo.
Nues o siguien e esul ado ue da una condici´on su icien e que nos ga an-
izase la unicidad del pun o ijo.
Teo ema 31. A˜nadiendo la condici´on
pa a cada x, y ∈Xexis e z∈Xque es compa able con xyy ,
a las hip´o esis del Teo ema 30, ob enemos la unicidad del pun o ijo.
Adem´as, en el a ´ıculo se analiza qu´e ocu e si el ope ado Tes dec e-
cien e.
Teo ema 32. Sea X, ≤un conjun o pa cialmen e o denado al que pa a
cada x, y ∈Xexis e z∈Xque es compa able con xyy. Supongamos que
exis e una m´e ica den X al que X, des un espacio m´e ico comple o y
sea T:X→Xuna aplicaci´on dec ecien e sa is aciendo
dTx, Ty≤1
2dx, Ty+dy, Tx−ϕdx, Ty, dy, Tx,
pa a cada x, y ∈Xcon x≥y, donde ϕ: [0,∞)2→[0,∞)es una unci´on
con inua al que ϕ(x, y)=0si y s´olo si x=y= 0.
En onces
(i) Si exis e x0∈Xcompa able con Tx0en onces in {d(x, Tx): x∈X}=
0.
(ii) Si, adem´as, Xes compac o y Tes con inua, en onces T iene un ´unico
pun o ijo.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 31
Pa a inaliza , damos un ejemplo que puede se es udiado con el Teo e-
ma 30 y no puede se a ado con el Teo ema 29.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 33
1.1.7. Fixed poin heo ems o mixed mono one
ope a o s and applica ion o in eg al equa ions
Los ope ado es mon´o onos mix os ue on in oducidos po Guo y Laksh-
mikan ham en [63]. Su de inici´on es la siguien e.
De ini ion 2. Sea (X, ≤)un conjun o pa cialmen e o denado y F:X×X→
Xuna aplicaci´on. Se dice que F iene la p opiedad mon´o ona mix a si F(x, y)
es c ecien e en xy es dec ecien e en y, es o es, pa a 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).
De ini ion 3. Sea F:X×X→Xuna aplicaci´on. Un pa (x, y)∈X×X
se dice que es un pun o ijo acoplado del ope ado Fsi
F(x, y) = xyF(y, x) = y .
En [18] Bhaska y Lakshmikan ham p oba on el siguien e eo ema del
pun o ijo.
Teo ema 33. Sea X, ≤un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Sea F:X×X→Xuna aplicaci´on que iene la p opiedad mon´o ona mix a
y supongamos que exis e k∈[0,1) al que
dF(x, y), F(u, )≤k
2[d(x, u) + d(y, )],pa a cada x≥u, y ≤ .
Si exis e x0, y0∈X al que
x0≤F(x0, y0)yy0≥F(y0, x0),
y supongamos ambi´en que Fes con inua o Xsa is ace:
si (xn)es una sucesi´on c ecien e en Xcon xn→x
en onces xn≤xpa a odo n∈N,

34 Teo emas del pun o ijo en espacios m´e icos o denados
y
si (yn)es una sucesi´on dec ecien e en Xcon yn→x
en onces y≤ynpa a odo n∈N.
En onces F iene un pun o ijo acoplado.
El p op´osi o de es e a ´ıculo es gene aliza el Teo ema 33 usando las
unciones que al e an las dis ancias, es o es, unciones ϕ: [0,∞)→[0,∞)
ales que ϕes con inua y c ecien e con ϕ( ) = 0 si y s´olo si = 0.
El p ime esul ado ob enido en el a ´ıculo es el siguien e.
Teo ema 34. Sea X, ≤un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que X, des un espacio m´e ico comple o.
Sea F:X×X→Xuna aplicaci´on que iene la p opiedad mon´o ona mix a,
con inua y sa is aciendo
ϕdF(x, y),F(u, )
≤ϕmax d(x, u), d(y, )−φmax d(x, u), d(y, ),
pa a cada x, y, u, ∈Xcon x≥uey≤ , donde ϕyφson unciones con
dis ancias al e adas.
Si exis e x0, y0∈Xcon x0≤F(x0, y0)ey0≥F(y0, x0)en onces F iene un
pun o ijo acoplado.
En el p ´oximo esul ado, eemplazamos la con inuidad de Fpo es a
condici´on:
si (xn) es una sucesi´on c ecien e en Xcon xn→x
en onces xn≤xpa a odo n∈N,
y
si (yn) es una sucesi´on dec ecien e en Xcon yn→x
en onces y≤ynpa a odo n∈N,
(1.9)
1. Resumen de la Tesis / Summa y o he hesis in Spanish 35
y ob enemos el siguien e eo ema.
Teo ema 35. Si en el Teo ema 34 eemplazamos la con inuidad de Fpo la
condici´on mencionada an e io men e, ob enemos la misma conclusi´on.
Seguidamen e, en el a ´ıculo, p esen amos un ejemplo que demues a que
las hip´o esis dadas en los Teo emas 34 y 35 no ga an izan la unicidad del
pun o ijo acoplado.
El siguien e esul ado nos da una condici´on su icien e pa a ob ene la unici-
dad del pun o ijo acoplado.
Teo ema 36. Asumamos que
pa a (x, y),(u, )∈X×Xexis e (z, )∈X×X
que es compa able con (x, y)y(u, ),
y conside amos en X×Xel o den pa cial de inido po
(x, y)≤(u, )si y s´olo si x≤uyy≥ .
Bajo las hip´o esis del Teo ema 34 ( esp. Teo ema 35) ob enemos la unicidad
del pun o ijo acoplado.
Despu´es, p esen amos algunas consecuencias de los esul ados ob enidos.
En pa icula , el p incipal esul ado de [18] puede se deducido de nues os
Teo emas 34 y 35. Adem´as, damos un eo ema del pun o ijo acoplado pa a
aplicaciones que ienen la p opiedad mon´o ona mix a con una condici´on de
ipo in eg al, que p esen amos a con inuaci´on.
Teo ema 37. Sea X, ≤un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que X, des un espacio m´e ico comple o.
Sea F:X×X→Xuna aplicaci´on con la p opiedad mon´o ona mix a y
supongamos que exis e k∈[0,1) al que
ZdF(x,y),F (u, )
0
ρ( )d ≤KZmax d(x,u),d(y, )
0
ρ( )d ,
36 Teo emas del pun o ijo en espacios m´e icos o denados
pa a odo x, y, u, ∈Xcon x≥uyy≤ , donde ρ:R+→R+es una
unci´on medible Lebesgue con in eg al ini a sob e cada compac o de R+y al
que Rε
0ρ( )d > 0pa a ε > 0.
Supongamos adem´as, que Fes con inua o Xsa is ace la condici´on (1.9).
Si exis e x0, y0∈Xcon x0≤F(x0, y0)ey0≥F(y0, x0)en onces F iene un
pun o ijo acoplado.
Pa a conclui el a ´ıculo, damos una aplicaci´on de nues os esul ados a
la eo ´ıa de exis encia de soluciones en ecuaciones in eg ales no lineales.
Con mayo p ecisi´on, conside amos la ecuaci´on in eg al
x( ) = Z1
0k1( , s) + k2( , s) s, x(s)+gs, x(s)ds +a( ),
con ∈[0,1] ,
(1.10)
y suponemos que se e i icas las hip´o esis:
(i) ki: [0,1]×[0,1] →R(i= 1,2) son con inuas con k1( , s)≥0 y k2( , s)≤
0.
(ii) a∈ C[0,1].
(iii) , g: [0,1] ×R→Rson unciones con inuas.
(i ) Exis en cons an es λ, µ > 0 ales que pa a cada x, y ∈Rcon x≥y
0≤ ( , x)− ( , y)≤λpln[(y−x)2+ 1]
y
−µpln[(y−x)2+ 1] ≤g( , x)−g( , y)≤0.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 37
( ) Exis e α, β ∈ C[0,1] al que
α( )≤Z1
0
k1( , s)( (s, α(s)) + g(s, β(s)))ds
+Z1
0
k2( , s)( (s, β(s)) + g(s, α(s)))ds +a( )
≤Z1
0
k1( , s)( (s, β(s)) + g(s, α(s)))ds
+Z1
0
k2( , s)( (s, α(s)) + g(s, β(s)))ds +a( )≤β( ).
( i) 2 ·max(λ, µ)kk1−k2k∞≤1, donde
kk1−k2k∞= sup n(k1( , s)−k2( , s)): , s ∈[0,1]o.
En onces, la ecuaci´on in eg al (1.10) iene una ´unica soluci´on en C[0,1].
44 Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p
donde ψes una unci´on que al e a las dis ancias.
Si exis e x0∈X al que x0≤Tx0en onces T iene un pun o ijo.
Adem´as, si pa a cada x, y ∈Xexis e z∈Xcompa able con xeyen onces
el pun o ijo es ´unico.
El p incipal esul ado ob enido en el a ´ıculo es el eo ema que apa ece a
con inuaci´on.
Teo ema 42. Conside emos el P oblema (1.11) bajo las siguien es hip´o esis:
(i) : [0,1] ×[0,∞)→[0,∞)es una unci´on con inua.
(ii) es c ecien e con espec o a la segunda a iable y supongamos que
exis e 0< α ≤384
5 al que, pa a cada x, y ∈[0,∞)con y≥x
( , y)− ( , x)≤αln(y−x+ 1) .
En onces P oblema (1.11) iene una ´unica soluci´on no nega i a.
En el a ´ıculo, se menciona que el Teo ema 42 sigue siendo ´alido si
eemplazamos la inecuaci´on de (ii) po
( , y)− ( , x)≤α ϕ(y−x)
donde ϕ: [0,∞)→[0,∞) es con inua y φ(x) = x−ϕ(x) cumple es as
condiciones:
(i) φ: [0,∞)→[0,∞) es c ecien e.
(ii) φ(0) = 0.
(iii) φes posi i a en (0,∞).
El p ´oximo eo ema nos da una condici´on su icien e pa a la exis encia y
unicidad de una soluci´on posi i a pa a P oblema (1.11) (po soluci´on posi i a
en endemos que x( )>0 pa a ∈(0,1)).

1. Resumen de la Tesis / Summa y o he hesis in Spanish 45
Teo ema 43. Bajo las hip´o esis del Teo ema 42 y suponiendo que ( , 0) 6= 0
pa a ∈A⊂[0,1] con µ(A)>0, donde µdeno a la medida de Lebesgue, el
P oblema (1.11) iene una ´unica soluci´on posi i a.
Que emos des aca que la condici´on que apa eces en el Teo ema 43 se
sa is ace au om´a icamen e cuando : [0,1] ×[0,∞)→[0,∞) es con inua y
( 0,0) 6= 0 pa a cie o 0∈[0,1].
En [J. A. Cid, D. F anco, F. Minh´os, Posi i e ixed poin s and ou h-o de
equa ions, Bull. London Ma h. Soc. 41 (2009), 72–78] los au o es es udia on
es e p oblema:
(u(i )( ) = λ h( ) (u), ∈(0,1), λ > 0,
u(0) = u(1) = 0 = u00(0) = u00(1) ,(1.12)
El p incipal esul ado que ob u ie on es el siguien e eo ema.
Teo ema 44. Supongamos que h: [0,1] →[0,∞)es con inua y no id´en ica-
men e nula en 1
4,3
4, :R→[0,∞)con inua y al que lim
s→∞
(s)
s= +∞y
exis e B∈[0,∞) al que es c ecien e en [0, B). Si
0< λ < sup
s∈(0,B)
s
γ∗ (s),
donde γ∗= max
∈[0,1] Z1
0
G( , s)h(s)dsyG( , s)es la unci´on de G een asociada
a(1.12), dada po
G( , s) = 










1
6s(1 − )(2 −s2− 2),0≤s≤ ,
1
6 (1 −s)(2s− 2−s2),0≤ ≤s.
En onces el P oblema (1.12) iene al menos una soluci´on posi i a.
En el a ´ıculo, compa amos nues os esul ados con los ob enidos en [38].
46 Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p
Conside amos el p oblema
(u(i )( ) = λ h( ) ln(u+ 2), ∈(0,1), λ > 0
u(0) = u(1) = 0 = u00(0) = u00(1) ,(1.13)
donde h: [0,1] →[0,∞) es con inua y no id´en icamen e nula en 1
4,3
4.
Comp obamos que el P oblema (1.13) cumple las condiciones del Teo ema 44
y consecuen emen e, ob enemos la exis encia de una soluci´on posi i a pa a
es e p oblema pa a cada λ > 0.
Pe o el P oblema (1.13) ambi´en e i ica las condiciones del nues os Teo-
emas 42 y 43 ob eniendo la exis encia y unicidad de una soluci´on posi i a
pa a el mismo cuando λ≤384
5khk.
Nues a p incipal con ibuci´on es es e ejemplo es la unicidad de la soluci´on
cuando λ≤384
5khk.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 47
1.2.2. On posi i e solu ions o a nonlinea ou h
o de bounda y alue p oblem ia a ixed poin
heo em in o de ed se s
En es e a ´ıculo, es amos in e esados en ob ene la exis encia y unicidad
de una soluci´on posi i a pa a el p oblema con alo es en la on e a:
(u(i )( ) = ( , u), ∈(0,1)
u(0) = u(1) = 0 = u00(0) = u00(1) .(1.14)
U iliza emos en nues o es udio el eo ema en espacios m´e icos pa cialmen e
o denados ob enido en [J. Ha jani, K. Sada angani, Gene alized con ac ions
in pa ially o de ed me ic spaces and applica ions o o dina y di e en ial
equa ions, Nonlinea Anal. 72, (2010), 1188–1197].
Teo ema 13. Sea (X, ≤)un espacio pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Asumamos que Xsa is ace la condici´on
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N.
Sea T:X→Xuna aplicaci´on c ecien e al que
ψd(Tx, Ty)≤ψd(x, y)−φd(x, y)pa a cada x, y ∈Xcon x≥y ,
donde ψyφson unciones que al e an las dis ancias. Si exis e x0∈Xcon
x0≤Tx0, en onces T iene un pun o ijo.
Adem´as, si pa a cada x, y ∈Xexis e z∈Xque sea compa able con xey,
en onces el pun o ijo es ´unico.
El p incipal esul ado del a ´ıculo puede se esumido en es e eo ema.
Teo ema 45. Conside emos el P oblema (1.14) bajo es as hip´o esis:
(i) : [0,1] ×[0,∞)→[0,∞)es una unci´on con inua.
48 Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p
(ii) Exis e 0< α ≤q80640
17 al que, pa a cada x, y ∈[0,∞)con y≥x
0≤ ( , y)− ( , x)≤αpln[(y−x)2+ 1] .
En onces nues o P oblema (1.14) iene una ´unica soluci´on no nega i a.
Adem´as, si ( 0,0) 6= 0 pa a cie o 0∈[0,1] en onces el P oblema (1.14)
iene una ´unica soluci´on posi i a.
Pa a inaliza , compa amos los esul ados ob enidos con o os p eceden es
que apa ecen en [38].
Nues a p incipal con ibuci´on es que ob enemos la unicidad de la soluci´on.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 49
1.2.3. Uniqueness o posi i e solu ions o a class o
ou h-o de bounda y alue p oblems
El p op´osi o de es e a ´ıculo es es udia la exis encia y unicidad de una
soluci´on posi i a y sim´e ica pa a es e p oblema con alo es en la on e a:
(y(i )( ) =  , y( ), ∈[0,1],
y(0) = y(1) = y0(0) = y0(1) = 0 ,(1.15)
Pa a ello, u ilizamos el eo ema del pun o ijo en espacios m´e icos o denados
que ci amos a con inuaci´on, ob enido po A. Amini-Ha andi y H. Emami
[A ixed poin heo em o con ac ion ype maps in pa ially o de ed me ic
spaces and applica ion o o dina y di e en ial equa ions, Nonlinea Anal., 72,
(2010), 2238–2242].
Teo ema 46. Sea X, ≤un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que X, des un espacio m´e ico comple o.
Sea T:X→Xuna aplicaci´on c ecien e al que exis e un elemen o x0∈X
con x0≤Tx0yTsa is ace
d(Tx, Ty)≤βd(x, y)·d(x, y),pa a cada x, y ∈Xcon x≥y ,
donde β: [0,∞)→[0,1] al que β( n)→1implica n→0.
Asumamos ambi´en que Tes con inua o Xes al que
i (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N.
Supongamos que
pa a cada x, y ∈Xexis e z∈Xque es compa able con xey .
En onces T iene un ´unico pun o ijo.
En el a ´ıculo se usa la clase de unciones A, de inidas po las unciones
φ: [0,∞)→[0,∞) que sa is aces es as condiciones:

50 Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p
(i) φes c ecien e.
(ii) Pa a cada x > 0, φ(x)< x.
(iii) β(x) = φ(x)
xes al que β( n)→1⇒ n→0.
El esul ado de mayo impo ancia ob enido es el siguien e eo ema.
Teo ema 47. Conside emos el P oblema (1.15) bajo las hip´o esis:
(i) : [0,1] ×[0,∞)→[0,∞)es una unci´on con inua.
(ii) Exis e 0< α ≤384 al que, pa a x, y ∈[0,∞)con y≥x
0≤ ( , y)− ( , x)≤α φ(y−x),donde φ∈ A.
(iii) ( 0,0) 6= 0, pa a cie o 0∈[0,1].
(i ) ( , y) = (1 − , y), pa a ( , y)∈[0,1] ×[0,∞).
En onces el P oblema (1.15) iene una ´unica soluci´on posi i a y sim´e ica
(una soluci´on y( )es sim´e ica si y( ) = y(1 − )pa a cada ∈[0,1]).
Es e p oblema ue a ado po M. Pei y S. K. Chang, [M. Pei, S. K.
Chang, Mono one i e a i e echnique and symme ic posi i e solu ions o a
ou h-o de bounda y alue p oblem, Ma h. Compu . Modelling, 51, (2010),
1260–1267]. El p incipal esul ado que ob u ie on ue:
Teo ema 48. Supongamos que:
(a) : [0,1] ×[0,∞)→[0,∞)es una unci´on con inua.
(b) ( , y)es c ecien e en y, pa a cada ∈[0,1].
(c) ( , y) = (1 − , y), pa a cada ( , y)∈[0,1] ×[0,∞).
Adem´as, supongamos que exis en n´ume os posi i os a>b ales que
max
0≤ ≤1 ( , a)≤a A min
1
4≤ ≤3
4
( , b
16)≥b B ,
1. Resumen de la Tesis / Summa y o he hesis in Spanish 51
donde
A=max
0≤ ≤1Z1
0
G( , s)ds−1
yB= max
0≤ ≤1Z3
4
1
4
G( , s)ds!−1
,
siendo G( , s)la unci´on de G een asociada al P oblema (1.15), que iene
dada po
G( , s) = 1
6( 2(1 −s)2[(s− ) + 2(1 − )s],0≤ ≤s,
s2(1 − )2[( −s) + 2(1 −s) ],0≤s≤ .
En onces el P oblema (1.15) iene al menos una soluci´on sim´e ica posi i a.
Pa a pode compa a nues os esul ados con los ob enidos po ellos en
[126] conside amos es e p oblema:
(y(i )( ) = c+λsin(π ) a c an y( ), ∈(0,1), c, λ > 0
y(0) = y(1) = y0(0) = y0(1) = 0 (1.16)
y p obamos que pa a 0 < λ ≤384 el P oblema (1.16) puede es udia se
an o bajo nues a pe spec i a como po los esul ados ob enidos en [126].
Sin emba go, nues a p incipal con ibuci´on es la unicidad de la soluci´on.
Seguidamen e, conside amos el p oblema
(y(i )( ) = c( ) + λsin(π ) a c an y( ), ∈(0,1), c, λ > 0
y(0) = y(1) = y0(0) = y0(1) = 0 (1.17)
donde c( ) es la unci´on dada po
c( ) = 












1−4 , 0≤ ≤1
4
0,1
4≤ ≤3
4
4 −3,3
4≤ ≤1,
y demos amos, usando nues o esul ado, que pa a 0 < λ ≤384, el P oble-
ma (1.17) iene una ´unica soluci´on sim´e ica y posi i a.
52 Teo ´ıa de exis encia y unicidad pa a las soluciones de un b p
Po o o lado, emos que el P oblema (1.17) no sa is ace las hip´o esis del
Teo ema 48 y, en onces, el P oblema (1.17) no puede se es udiado po los
esul ados ob enidos en [126].
1. Resumen de la Tesis / Summa y o he hesis in Spanish 53
1.3. F ac ional bounda y alue p oblem
1.3.1. Exis ence and uniqueness o posi i e and non-
dec easing solu ions o a class o singula
ac ional bounda y alue p oblems
El a ´ıculo a a sob e la exis encia y unicidad de una soluci´on posi i a y
c ecien e pa a el p oblema acciona io con alo es en la on e a
(Dα
0+u( ) +  , u( )= 0 ,0< < 1,
u(0) = u0(1) = u00(0) = 0 ,(1.18)
donde 2 < α ≤3, Dα
0+es la de i a de Capu o y : (0,1] ×[0,∞)→[0,∞)
con lim
→0+ ( , −) = ∞, es deci , es singula en = 0. En nues o es udio,
usamos el siguien e eo ema del pun o ijo en espacios m´e icos o denados,
que es el p incipal esul ado ob enido en [69].
Teo ema 7. Sea (X, ≤)un conjun o pa cialmen e o denado y supongamos
que exis e una m´e ica den X al que (X, d)es un espacio m´e ico comple o.
Asumamos que Xsa is ace la condici´on:
si (xn)es una sucesi´on c ecien e en X al que xn→x
en onces xn≤x, pa a odo n∈N.
Sea T:X→Xuna aplicaci´on c ecien e al que
d(Tx, Ty)≤d(x, y)−ψd(x, y),pa a cada x, y ∈Xcon x≥y ,
donde ψes una unci´on que al e a la dis ancia.
Si exis e x0∈Xcon x0≤Tx0en onces T iene un pun o ijo.
Adem´as, si pa a cada x, y ∈Xexis e z∈Xcompa able con xey, en onces
el pun o ijo es ´unico.
El p ime esul ado ob enido en el a ´ıculo es:

1. Resumen de la Tesis / Summa y o he hesis in Spanish 61
1.3.3. Exis ence and uniqueness o posi i e solu ion o
a bounda y alue p oblem o ac ional o de
En el a˜no 2009, S. Liang y J. Zhang p esen a on un a ´ıculo [Posi i-
e solu ions o bounda y alue p oblems o nonlinea ac ional di e en ial
equa ion, Nonlinea Analysis, 71, (2009), 5545–5550] donde se es udiaba la
exis encia de soluciones pa a el p oblema acciona io con alo es en la on-
e a:
(Dα
0+u( ) +  , u( )= 0,0< < 1,3< α ≤4,
u(0) = u0(0) = u00(0) = u00(1) = 0.(1.20)
La p incipal apo aci´on del a ´ıculo ue es e eo ema.
Teo ema 53. El P oblema (1.20) iene una soluci´on posi i a si las siguien es
condiciones son sa is echas:
(a) ( , u)∈ C[0,1] ×[0,∞),R+.
(b) ( , u)es c ecien e en u.
(c)  , ρ( )6= 0 pa a ∈(0,1), donde
ρ( ) = Z1
0
G( , s)ds=1
Γ(α) α−1
α−2−1
α α.
(d) Exis e una cons an e posi i a µ < 1 al que
kµ ( , u)≤ ( , k u), o any k∈[0,1].
A a´ız de es e a ´ıculo, nos plan eamos analiza la exis encia y unicidad
de ese p oblema. Nues o es udio se basa en el eo ema del pun o ijo sob e
conjun os pa cialmen e o denados p esen ado en [69].
Teo ema 7. Sea (X, ≤)un conjun o pa cialmen e o denado y supongamos
que exis e den X al que (X, d)es un espacio m´e ico comple o. Asumamos
62 F ac ional bounda y alue p oblem
que X e i ica la condici´on:
si (xn)es c ecien e en X al que xn→x
en onces xn≤xpa a odo n∈N.
Sea T:X→Xuna aplicaci´on c ecien e al que
d(Tx, Ty)≤d(x, y)−ψd(x, y),pa a cada x, y ∈Xcon x≥y,
donde ψes una unci´on que al e a la dis ancia. Si exis e x0∈Xcon x0≤
T(x0)en onces T iene un pun o ijo.
Adem´as, si pa a cada x, y ∈Xexis e z∈Xcompa able con xeyen onces
el pun o ijo es ´unico.
An es de p esen a el p incipal esul ado que ob u imos, necesi amos in-
oduci la clase de unciones A. La clase Aes ´a de inida po aquellas un-
ciones ϕ: [0,∞)→[0,∞) con inuas y c ecien es ales que ψ(x) = x−ϕ(x)
pa a x∈[0,∞) sa is ace:
(a) ψ: [0,∞)→[0,∞).
(b) ψes con inua y c ecien e.
(c) ψ(0) = 0.
(d) ψ( )>0 pa a > 0.
Teo ema 54. Bajo las hip´o esis:
(1) : [0,1] ×[0,∞)→[0,∞)es con inua y c ecien e con espec o a la
segunda a iable.
(2) Exis e 0∈[0,1] al que ( 0,0) >0.
(3) Exis e 0< λ ≤(α−2)Γ(α+ 1)
2 al que, pa a cada x, y ∈[0,∞)con
y≥xy cada ∈[0,1],
( , y)− ( , x)≤λ·ψ(y−x),
1. Resumen de la Tesis / Summa y o he hesis in Spanish 63
donde ψ∈ A.
El P oblema (1.20) iene una ´unica soluci´on posi i a.
Con el p op´osi o de pode compa a nues os esul ados con los ob enidos
en [106] p esen amos es e p oblema acciona io con alo es en la on e a:
(D7/2
0+u( )+( 2+ 1)ρu( ) + c= 0,0< < 1,
u(0) = u0(0) = u00(0) = u00(1) = 0 ,(1.21)
con c > 0 y 0 <ρ<1.
En el abajo p obamos que el P oblema (1.21) puede se a ado po nues os
esul ados pe o no puede se es udiado po el Teo ema 53.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 65
1.3.4. On exis ence and uniqueness o posi i e solu-
ions o a class o ac ional bounda y alue p o-
blems
El a ´ıculo es udia la exis encia y unicidad de una soluci´on posi i a pa a
el p oblema acciona io con alo es en la on e a:
(Dα
0+u( ) +  , u( )= 0,0< <1,
u(0) = u(1) = u0(0) = 0,(1.22)
donde 2 < α ≤3, que ep esen a la e si´on no mon´o ona de
(Dα
0+u( ) + λ u( )= 0,0< < 1,
u(0) = u(1) = u0(0) = 0,(1.23)
donde 2 < α ≤3 y λ > 0. Es e ´ul imo p oblema ue analizado ecien emen e
en [Y. Zhao, S. Sun, Z. Han, Q. Li, Posi i e solu ions o bounda y alue
p oblems o nonlinea ac ional di e en ial equa ions. Abs Appl Anal. 2011,
(2011), A icle ID 390543].
En nues o abajo, u ilizamos el mismo eo ema del pun o ijo en espacios
m´e icos pa cialmen e o denados que en el a ´ıculo an e io .
Los esul ados ob enidos pueden se esumidos en p ´oximo eo ema.
Teo ema 55. El P oblema (1.22) iene una ´unica soluci´on no nega i a si
las condiciones que apa ecen a con inuaci´on son sa is echas:
(1) : [0,1] ×[0,∞)→[0,∞)es con inua y c ecien e con espec o al se-
gundo a gumen o.
(2) Exis e 0< λ ≤1
A al que, pa a cada x, y ∈[0,∞)con y≥xy cada
∈[0,1]
( , y)− ( , x)≤λϕ(y−x),
donde ϕ∈ A, siendo Ala clase de unciones a ada en el a ´ıculo
an e io y A=1
Γ(α+1) hα−1
αα−1−α−1
ααi.

66 F ac ional bounda y alue p oblem
Adem´as, si ( 0,0) 6= 0 pa a cie o 0∈[0,1] en onces la soluci´on ´unica es
posi i a.
En [174], usando un eo ema del pun o ijo en conos, los au o es p oba on
es e esul ado.
Teo ema 56. Supongamos que exis e l∈(0,1) al que q(l)c2 0> F∞c1
en onces, pa a cada λ∈(q(l)c2 0)−1,(F∞c1)−1el P oblema (1.23) iene al
menos una soluci´on posi i a, donde:
F∞= lim
u→∞ sup (u)
u,
q( ) = α−1(1 − ),
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 a ´ıculo, damos un ejemplo que puede se a ado po los Teo e-
mas 55 y 56, pe o nues a p incipal con ibuci´on es la ob enci´on de unicidad
en la soluci´on si 0 < λ ≤17,8682.
Pa a inaliza , p esen amos es e p oblema con alo es en la on e a, que no
puede se es udiado po el Teo ema 56 y puede se abo dado po nues os
esul ados.
(D5/2
0+u( ) + λ + a c an u( )= 0,0< <1, λ > 0,
u(0) = u(1) = u0(0) = 0.
1. Resumen de la Tesis / Summa y o he hesis in Spanish 67
1.3.5. Posi i e and nondec easing solu ions o a singu-
la bounda y alue p oblem o nonlinea ac-
ional di e en ial equa ions
En es e abajo se analiza la exis encia y unicidad de una soluci´on posi i a
y c ecien e pa a el siguien e p oblema acciona io con alo es en la on e a
(Dα
0+u( ) +  , u( )= 0,0< <1,
u(0) = u0(1) = u00(0) = 0,(1.24)
con 2 < α ≤3, y lim
→0+ ( , ·) = ∞, es o es es singula en = 0.
Necesi amos la clase Ade aquellas unciones φ: [0,∞)→[0,∞) que e i ican
las condiciones:
(i) φes c ecien e.
(ii) φ(x)< x, pa a cada x > 0.
(iii) β(x) = φ(x)
xes al que si b( n)→1 implica n→0.
Nues o p incipal esul ado se puede esumi en el p ´oximo eo ema.
Teo ema 57. Supongamos que 0< σ < 1y2< α ≤3. Bajo es as hip´o esis:
(1) : (0,1] ×[0,∞)→[0,∞)es una unci´on con inua sa is aciendo
lim
→0+ ( , −) = ∞.
(2) σ ( , y)es una unci´on con inua en [0,1] ×[0,∞).
(3) Exis e 0< λ ≤Γ(α−σ)
Γ(1−σ)and φ∈ A al que
0≤ σ( ( , y)− ( , x)) ≤λφ(y−x),
pa a cada x, y ∈[0,∞)con y≥xy cada ∈[0,1],
El P oblema (1.24) iene una ´unica soluci´on no nega i a.
Adem´as, es a soluci´on es es ic amen e c ecien e.
68 F ac ional bounda y alue p oblem
Es e mismo p oblema ue abo dado po T. Qiu y Z. Bai en Exis ence
o posi i e solu ions o singula ac ional di e en ial equa ions, Elec onic
Jou nal o Di e en ial Equa ions, 146, (2008), 1–9, donde aplica on es e eo-
ema:
Teo ema 58. Sea 0< σ < 1,2< α ≤3, : (0,1] ×[0,+∞)→[0,+∞)es
con inua y lim
→0+ ( , ·) = +∞, σ ( , y)es una unci´on con inua en [0,1] ×
[0,+∞). Asumamos que exis en dos cons an es posi i as ρ, µ (ρ > µ) ales
que
(H1) σ ( , ω)≤ρΓ(α−σ)
Γ(1−σ), pa a ( , ω)∈[0,1] ×[0, ρ];
(H2) σ ( , ω)≥µΓ(α−σ)
Γ(1−σ), pa a ( , ω)∈[0,1] ×[0, µ].
En onces el P oblema (1.24) iene al menos una soluci´on posi i a.
Nues os esul ados gene alizan los ob enidos po es os au o es, ya que la
unicidad y la mono on´ıa de la soluci´on no se pueden deduci de sus esul ados.
Ilus amos nues os esul ados con un ejemplo que puede se esuel o po el
Teo ema 57 y no puede se es udiado po los esul ados de [25].
Cap´ı ulo 2
A sho his o y app oach
69
76
3.1. Fixed poin heo ems o weakly con ac i e
mappings......................... 79
3.2. Gene alized con ac ions in pa ially o de ed
me ic spaces ... . . . . . . . . . . . . . . . . . . . 91
3.3. Con ac i e-like mapping p inciples in o de ed
me ic spaces ... . . . . . . . . . . . . . . . . . . . 105
3.4. Fixed poin heo ems o mappings sa is ying a
condi ion o ... . . . . . . . . . . . . . . . . . . . . 123
3.5. A ixed poin heo em o mappings sa is ying a
con ac i e ... . . . . . . . . . . . . . . . . . . . . . 141
3.6. Fixed poin heo ems o weakly C-con ac i e
mappings in ... . . . . . . . . . . . . . . . . . . . . 151
3.7. Fixed poin heo ems o mixed mono one ope-
a o sand... ..................... 161
3.8. A ixed poin heo em o Mei -Keele con ac-
ionsin.......................... 179

3. Fixed poin heo ems in pa ially o de ed me ic spaces 77
In his chap e , we p esen some ixed poin heo ems in pa ially o de ed
me ic spaces ob ained in ou esea ch.
The main objec i e o ou s udy was o ex end, o imp o e and o gene alize
some classical ixed poin heo ems in he con ex o pa ially o de ed me ic
spaces.
Fo a be e eadabili y, we p esen he pape s in connec ion wi h his sec ion
and, hen, b ie ly we discuss each a icle. We will show hese pape s:
a) J. Ha jani, K. Sada angani, Fixed poin heo ems o weakly con ac i e
mappings in pa ially o de ed se s, Nonlinea Anal. 71, (2009), 3403–
3410.
b) J. Ha jani, K. Sada angani, Gene alized con ac ions in pa ially o -
de ed me ic spaces and applica ions o o dina y di e en ial equa ions,
Nonlinea Anal. 72, (2010), 1188–1197.
c) J. Caballe o, J. Ha jani, K. Sada angani, Con ac i e-like mapping
p inciples in o de ed me ic spaces and applica ions o o dina y di -
e en ial equa ions, Fixed Poin Theo y and Applica ions, ol. 2010,
A icle ID916064, 14 pages.
d) J. Ha jani, K. Sada angani, Fixed poin heo ems o mappings sa is -
ying a condi ion o in eg al ype in pa ially o de ed se s, Jou nal o
Con ex Analysis 17, (2010), 597–609.
e) J. Ha jani, B. L´opez, K. Sada angani, A ixed poin heo em o map-
pings sa is ying a con ac i e condi ion o a ional ype on a pa ially
o de ed me ic space, Abs ac and Applied Analysis ol 2010, A icle
ID190701, 8 pages.
) J. Ha jani, B. L´opez, K. Sada angani, Fixed poin heo ems o weakly
C-con ac i e mappings in o de ed me ic spaces, Compu e and Ma -
hema ics wi h Applica ions 61, (2011), 790–796.
g) J. Ha jani, B. L´opez, K. Sada angani, Fixed poin heo ems o mixed
mono one ope a o s and applica ion o in eg al equa ions, Nonlinea
Anal. 74, (2011), 1749–1760.
78
h) J. Ha jani, B. L´opez, K. Sada angani, A ixed poin heo em o Mei -
Keele con ac ions in o de ed me ic spaces, Fixed Poin Theo y and
Applica ions, ol. 2011, doi: 10.1186/1687-1812-2011-83.
3. Fixed poin heo ems in pa ially o de ed me ic spaces 79
3.1. Fixed poin heo ems o weakly con-
ac i e mappings in pa ially o de ed
se s
In his pape , we p esen some ixed poin heo ems o weakly con ac i e
mappings in pa ially o de ed me ic spaces.
The weakly con ac i e mappings we e de ined by Albe and Gue e-Dela-
b ie e in [6], in he con ex o Banach spaces, as a gene aliza ion o he
classical con ac i e mappings. Mo e p ecisely, le (X, k k) be a Banach space
and Ta sel mapping on X, we say ha Tis weakly con ac i e i , o any
x, y ∈X,
kTx −Tyk ≤ kx−yk−ψkx−yk,
whe e ψ: [0,∞)→[0,∞) is a con inuous and nondec easing mapping such
ha ψis posi i e on (0,∞), ψ(0) = 0 and lim
→∞ ψ( ) = ∞.
Albe and Gue e-Delab ie e p o ed ha any weakly con ac i e sel mapping
de ined in a Hilbe space has a ixed poin .
In [136] Rhoades ex ended he de ini ion o weakly con ac i e mapping o
he con ex o me ic spaces and p o ed a ixed poin heo em o hese
mappings.
The main esul o [136] is he ollowing.
Theo em 5. Le (X, d)be a comple e me ic space and T:X→Xa map-
ping sa is ying
d(Tx, Ty)≤d(x, y)−ψd(x, y),
o any x, y ∈X, whe e ψ: [0,∞)→[0,∞)is a con inuous and nondec ea-
sing unc ion such ha ψis posi i e on (0,∞)and ψ(0) = 0.
Then Thas a unique ixed poin .
No ice ha he condi ion lim
→∞ ψ( ) = ∞, used by Albe and Gue e-
Delab ie e, is no necessa y o ou objec i e as i is p o ed in Theo em 5.
80 Fixed poin heo ems o weakly con ac i e mappings ...
Ou pu pose in he pape is o p esen a e sion o Theo em 5 in he con ex
o pa ially o de ed me ic spaces.
Ou main esul s can be summa ized in he ollowing heo ems.
Theo em 6. Le (X, ≤)be a pa ially o de ed se and suppose ha he-
e exis s a me ic din Xsuch ha (X, d)is a comple e me ic space. Le
T:X→Xbe a con inuous and nondec easing mapping such ha
d(Tx, Ty)≤d(x, y)−ψd(x, y), o x, y ∈Xwi h x≥y ,
whe e ψ: [0,∞)→[0,∞)is a con inuous and nondec easing unc ion such
ha i is posi i e on (0,∞)and ψ(0) = 0.
I he e exis s x0∈Xsuch ha x0≤Tx0 hen Thas a ixed poin .
We p o e ha he condi ion Tcon inuous is unnecessa y, assuming he
ollowing assump ion in X
i (xn) is a nondec easing sequence in Xsuch ha xn→x
hen xn≤x, o all n∈N.(3.1)
This condi ion was used by J. Nie o and R. Rod ´ıguez-L´opez in [118]. Mo e
p ecisely, we p o e he ollowing esul .
Theo em 7. I in Theo em 6 we eplace he con inui y o Tby assum-
p ion (3.1) hen we ob ain he same conclusion.
Respec o he uniqueness o he ixed poin , we p esen an example which
shows ha Theo ems 6 and 7 do no gua an ee his uniqueness.
Nex , we gi e a su icien condi ion o he uniqueness (which was used in
[118]). This condi ion says:
Fo x, y ∈X he e exis s z∈Xwhich is compa able o xand y. (3.2)
Theo em 8. Adding condi ion (3.2) o he assump ions o Theo em 6 ( esp.
Theo em 7), we ob ain he uniqueness o he ixed poin .
3. Fixed poin heo ems in pa ially o de ed me ic spaces 81
Finally, we apply ou esul o he exis ence o solu ion o he ollowing
i s -o de pe iodic p oblem
(u0( ) =  , u( ), ∈[0, T],
u(0) = u(T),(3.3)
unde assump ion abou he exis ence o a lowe solu ion o (3.3), i.e., a
unc ion α∈ C[0, T] such ha
α0( )≤  , α( ), o ∈[0, T],
α(0) ≤α(T).
Mo e p ecisely, we ob ain he ollowing esul .
Theo em 9. Suppose ha : [0, T]×R→Ris con inuous and he e exis s
λ > 0such ha o x, y ∈Rwi h y≥x,
0≤ ( , y) + λy −[ ( , x) + λx]≤λln(y−x+ 1) .
Then, he exis ence o a lowe solu ion o (3.3) p o ides he exis ence o a
unique solu ion o (3.3).
This pape is s ongly inspi ed by [118].

Nonlinea Analysis 71 (2009) 3403–3410
Con en s lis s a ailable a ScienceDi ec
Nonlinea Analysis
jou nal homepage: www.else ie .com/loca e/na
Fixed poin heo ems o weakly con ac i e mappings in pa ially
o de ed se s
J. Ha jani, K. Sada angani∗
Depa amen o de Ma emá icas, Uni e sidad de Las Palmas de G an Cana ia, Campus de Ta i a Baja, 35017 Las Palmas de G an Cana ia, Spain
a icle in o
A icle his o y:
Recei ed 10 July 2008
Accep ed 30 Janua y 2009
MSC:
47H10
Keywo ds:
Fixed poin
Weakly con ac i e map
Pa ially o de ed se
abs ac
The pu pose o his pape is o p esen some ixed poin heo ems o weakly con ac i e
maps in a comple e me ic space endowed wi h a pa ial o de .
©2009 Else ie L d. All igh s ese ed.
1. In oduc ion
Albe and Gue e-Delab ie e in [1] de ine weakly con ac i e maps. In his pape hey con ine hei heo ems o Hilbe
spaces, bu acknowledge ha hei esul s a e ue, a leas o uni o mly smoo h and uni o mly con ex Banach spaces. In [2]
Rhoades ex ends some esul s appea ing in [1] o a bi a y Banach spaces.
I Xis an a bi a y Banach space, hen a sel map To Xsa is ies he Banach con ac ion p inciple i he e exis s a cons an
ksa is ying 0 ≤k<1 such ha , o x,y∈X
kTx −Tyk ≤ kkx−yk.(1)
As no ed in he in oduc ion o [1], inequali y (1) can be w i en in he o m
kTx −Tyk≤kx−yk − qkx−yk(2)
whe e k=1−qwi h q∈(0,1].
The ex ension o (2) in he con ex o Banach spaces o wha a e called weakly con ac i e maps is a na u al one. A sel map
To Xis weakly con ac i e i , o e e y x,y∈X,
kTx −Tyk≤kx−yk − ψ(kx−yk)(3)
whe e ψ: [0,∞)−→ [0,∞)is con inuous and nondec easing such ha ψis posi i e on (0,∞),ψ(0)=0 and
lim →∞ ψ( )= ∞ (an example o such unc ion ψis ψ( )=ln( +1)).
Now, le (X,d)be a me ic space and T:X−→ X.Tis said o be weakly con ac i e i o x,y∈X
d(Tx,Ty)≤d(x,y)−ψ(d(x,y))
∗Co esponding au ho . Fax: +34 928 45 88 11.
E-mail add ess: [email p o ec ed] (K. Sada angani).
0362-546X/$ – see on ma e ©2009 Else ie L d. All igh s ese ed.
doi:10.1016/j.na.2009.01.240
3. Fixed poin heo ems in pa ially o de ed me ic spaces 83
3404 J. Ha jani, K. Sada angani / Nonlinea Analysis 71 (2009) 3403–3410
whe e ψ: [0,∞)−→ [0,∞)sa is ies he abo e men ioned condi ions. No ice ha o be weakly con ac i e implies
con inui y.
The pu pose o his pape is o p esen some ixed poin heo ems o weakly con ac i e ope a o s in he con ex o
o de ed me ic spaces which a e ex ensions o hose in [2].
Exis ence o ixed poin in pa ially o de ed se s has been conside ed ecen ly in [3–16]. Ta ski’s heo em is used in [8] o
show he exis ence o solu ions o uzzy equa ions and in [10] o p o e exis ence heo ems o uzzy di e en ial equa ions.
In [15,9,12] some applica ions o ma ix equa ions and o o dina y di e en ial equa ions a e p esen ed, espec i ely. In [4–6,
16] i is p o ed some ixed poin heo ems o a mixed mono one mapping in a me ic space endowed wi h pa ial o de and
he au ho s apply hei esul s o p oblems o exis ence and uniqueness o solu ions o some bounda y alue p oblems [6,
16]. Some o he e e ences on he opic a e [17,18].
The usual con ac ion condi ion is weakened bu a he expense ha he ope a o is mono one. The main idea in [9,15]
in ol e combining he ideas in he con ac ion p inciple wi h hose in he mono one i e a i e echnique [19].
2. Fixed poin heo ems
De ini ion 1. I (X,≤)is a pa ially o de ed se and :X−→ X, we say ha is mono one nondec easing i x,y∈X,
x≤y⇒ (x)≤ (y).
This de ini ion coincides wi h he no ion o a nondec easing unc ion in he case whe e X=Rand ≤ ep esen s he
usual o al o de in R.
In [2], he ollowing heo em is p o ed.
Theo em 1. Le (X,d)be a comple e me ic space and :X−→ X is a weakly con ac i e map. Then has a unique ixed poin
in X.
In wha ollows we p o e he ollowing heo em which is a e sion o Theo em 1 in he con ex o o de ed me ic spaces.
Theo em 2. Le (X,≤)be a pa ially o de ed se and suppose ha he e exis s a me ic d in X such ha (X,d)is a comple e
me ic space. Le :X−→ X be a con inuous and nondec easing mapping such ha
d( (x), (y)) ≤d(x,y)−ψ(d(x,y)) o x ≥y(4)
whe e ψ: [0,∞)−→ [0,∞)is a con inuous and nondec easing unc ion such ha i is posi i e in (0,∞),ψ(0)=0and
lim →∞ ψ( )= ∞. I he e exis s x0∈X wi h x0≤ (x0), hen has a ixed poin .
P oo . I (x0)=x0 hen he p oo is inished. Suppose ha x0< (x0). Since x0< (x0)and is a nondec easing unc ion,
we ob ain by induc ion ha
x0< (x0)≤ 2(x0)≤ 3(x0)≤ · · · ≤ n(x0)≤ n+1(x0)≤ · · ·
Pu xn+1= (xn). Then o each in ege n≥1, om (4) and, as he elemen s xnand xn+1a e compa able, we ge
d(xn+1,xn)=d( (xn), (xn−1)) ≤d(xn,xn−1)−ψ(d(xn,xn−1)).
I he e exis s n0∈Nsuch ha d(xn0,xn0−1)=0 hen xn0= (xn0−1)=xn0−1and xn0−1is a ixed poin and he p oo is
inished.
In o he case, suppose ha d(xn+1,xn)6= 0 o all n∈N. Then aking in o accoun (4) and ou assump ions abou ψ
d(xn+1,xn)≤d(xn,xn−1)−ψ(d(xn,xn−1)) < d(xn,xn−1).
Pu ρn=d(xn+1,xn). Then we ha e
ρn≤ρn−1−ψ(ρn−1)<ρn−1.(5)
The e o e {ρn}is a nonnega i e noninc easing sequence and hence possesses a limi ρ∗. F om (5), aking limi when n→ ∞,
we ge
ρ∗≤ρ∗−ψ(ρ∗)≤ρ∗
and, consequen ly, ψ(ρ∗)=0. By ou assump ions abou ψ,ρ∗=0.
In wha ollows we will show ha {xn}is a Cauchy sequence.
Fix ε > 0. As ρn=d(xn+1,xn)→0, he e exis s n0∈Nsuch ha
d(xn0+1,xn0)≤min nε
2, ψ ε
2o.(6)
We claim ha B(xn0, ε) ∩ {y∈X:y≥xn0}⊂B(xn0, ε).
Le z∈B(xn0, ε) ∩ {y∈X:y≥xn0}. Then he e a e wo cases:
84 Fixed poin heo ems o weakly con ac i e mappings ...
J. Ha jani, K. Sada angani / Nonlinea Analysis 71 (2009) 3403–3410 3405
Case 1. d(z,xn0)≤ε
2.
In his case, as zand xn0a e compa able, we ha e
d( (z), xn0)≤d( (z), (xn0)) +d( (xn0), xn0)
=d( (z), (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 his case, as ψis a nondec easing unc ion, ψ(d(z,xn0)) ≥ψ( ε
2). The e o e om (6) we ge
d( (z), xn0)≤d( (z), (xn0)) +d( (xn0), xn0)
=d( (z), (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 p o es he claim.
As xn0+1∈B(xn0, ε) ∩ {y∈X:y≥xn0}, he claim gi es us ha (xn0+1)=xn0+2∈B(xn0, ε) ∩ {y∈X:y≥xn0}.
Repea ing his p ocess i ollows ha xn∈B(xn0, ε) o n≥n0. Since εis a bi a y, {xn}is a Cauchy sequence.
Since Xis a comple e me ic space he e exis s z∈Xsuch ha limn→∞ xn=z.
The con inui y o implies ha zis a ixed poin . Thus, he p oo is comple e. 
In wha ollows we p o e ha Theo em 1 is s ill alid o no necessa ily con inuous, assuming he ollowing hypo hesis
in X(which appea s in Theo em 1 o [9])
i {xn}is a nondec easing sequence in Xsuch ha xn→x hen xn≤x o all n∈N.(7)
Theo em 3. Le (X,≤)be a pa ially o de ed se and suppose ha he e exis s a me ic d in X such ha (X,d)is a comple e
me ic space. Assume ha X sa is ies (7). Le :X−→ X be a nondec easing mapping such ha
d( (x), (y)) ≤d(x,y)−ψ(d(x,y)) o x ≥y
whe e ψ: [0,∞)−→ [0,∞)is con inuous and nondec easing unc ion such ha ψis posi i e in (0,∞),ψ(0)=0and
lim →∞ ψ( )= ∞. I he e exis s x0∈X wi h x0≤ (x0), hen has a ixed poin .
P oo . Following he p oo o Theo em 2 we only ha e o check ha (z)=z. In ac ,
d( (z), z)≤d( (z), (xn)) +d( (xn), z)
≤d(z,xn)−ψ(d(z,xn)) +d(xn+1,z)
and aking limi as n→ ∞,d( (z), z)≤0 and his p o es ha d( (z), z)=0 and, consequen ly, (z)=z.
Now, we p esen an example whe e i can be app ecia ed ha hypo heses in Theo ems 2 and 3do no gua an ee
uniqueness o he ixed poin . This example appea s in [9].
Le X= {(1,0), (0,1)} ⊂ R2and conside he usual o de
(x,y)≤(z, )⇔x≤zand y≤ .
Thus, (X,≤)is a pa ially o de ed se , whose di e en elemen s a e no compa able. Besides, (X,d2)is a comple e me ic
space conside ing d2 he euclidean dis ance. The iden i y map (x,y)=(x,y)is i ially con inuous and nondec easing and
condi ion (4) o Theo em 2 is sa is ied since elemen s in Xa e only compa able o hemsel es. Mo eo e , (1,0)≤ (1,0)=
(1,0)and has wo ixed poin s in X.
In wha ollows, we gi e a su icien condi ion o he uniqueness o he ixed poin in Theo ems 2 and 3. This condi ion
is
o x,y∈X he e exis s a lowe bound o an uppe bound. (8)
In [9] i is p o ed ha condi ion (8) is equi alen o
o x,y∈X he e exis s z∈Xwhich is compa able o xand y.(9)
Theo em 4. Adding condi ion (9) o he hypo heses o Theo em 2( esp. Theo em 3) we ob ain uniqueness o he ixed poin o .
3. Fixed poin heo ems in pa ially o de ed me ic spaces 85
92 Gene alized con ac ions in pa ially o de ed me ic spaces ...
The main objec i e in his pape is o p esen a e sion o Theo em 11 in
he con ex o pa ially o de ed me ic spaces.
Ou main esul s can be summa ized in he ollowing heo ems.
Theo em 12. Le (X, ≤)be a pa ially o de ed se and suppose ha he-
e exis s a me ic din Xsuch ha (X, d)is a comple e me ic space. Le
T:X→Xbe a con inuos and nondec easing mapping such ha
ψd(Tx, Ty)≤ψd(x, y)−φd(x, y) o any x, yıXwi h x≥y ,
whe e ψand φa e al e ing dis ance unc ions. I he e exis x0∈Xwi h
x0≤Tx0, hen Thas a ixed poin .
Theo em 13. I in Theo em 12 we eplace he condi ion o con inui y o T
by
i (xn)is a nondec easing sequence in Xsuch ha xn→x
hen xn≤x o all n∈N,
hen we ob ain he same conclusion.
Theo em 14. Adding he condi ion:
Fo x, y ∈X he e exis s z∈Xwhich is compa able o xand y ,
o he hypo heses o Theo em 12 ( esp. Theo em 13) we ob ain uniqueness o
he ixed poin .
The main esul s o ou p e ious pape a e pa icula cases o he ones
ob ained in his pape .
Finally, we p esen wo examples abou bounda y alue p oblems whe e ou
esul s can be applied.
In he i s example, we s udy he exis ence o solu ions o he ollowing
i s -o de pe iodic p oblem
(u0( ) =  , u( ), ∈[0, T],
u(0) = u(T).(3.4)

3. Fixed poin heo ems in pa ially o de ed me ic spaces 93
Ou esul is he ollowing.
Theo em 15. Unde assump ion : [0, T]×R→Ris con inuous and sup-
pose ha he e exis s wo posi i e eal numbe s λ, α > 0sa is ying
α≤2λ(eλT −1)
T(eλT + 1) 1
2
,
and such ha , o x, y ∈Rwi h y≥x
0≤ ( , y) + λy − ( , x) + λx≤αqln (y−x)2+ 1.
Then he exis ence o a lowe solu ion o (3.4) (see, commen s abou p e ious
pape ) p o ides he exis ence o an unique solu ion o (3.4).
The second example s udies he exis ence o solu ion o he ollowing
wo-poin bounda y alue p oblem o he second o de di e en ial equa ion



−d2x
d 2= ( , x), ∈[0,1] ,
x(0) = x(1) = 0 .
(3.5)
We ge he ollowing esul .
Theo em 16. Unde assump ion : [0,1] ×R→Ris con inuous and non-
dec easing wi h espec o he second a iable, and such ha , o any x, y ∈R
wi h y≥x,
( , y)− ( , x)≤αpln[(y−x)2+ 1] ,
when 0< α ≤8, hen P oblem (3.5) has a unique nonnega i e solu ion.
Mo eo e , i ( , x)6= 0 o ∈(0,1), he solu ion o (3.5) is posi i e ( his
means ha 0< x( ) o ∈(0,1)).
Nonlinea Analysis 72 (2010) 1188–1197
Con en s lis s a ailable a ScienceDi ec
Nonlinea Analysis
jou nal homepage: www.else ie .com/loca e/na
Gene alized con ac ions in pa ially o de ed me ic spaces and
applica ions o o dina y di e en ial equa ions
J. Ha jani, K. Sada angani∗
Depa amen o de Ma emá icas, Uni e sidad de Las Palmas de G an Cana ia, Campus de Ta i a Baja, 35017 Las Palmas de G an Cana ia, Spain
a icle in o
A icle his o y:
Recei ed 28 May 2009
Accep ed 3 Augus 2009
MSC:
47H10
Keywo ds:
Fixed poin
Al e ing dis ance unc ion
Pa ially o de ed se
abs ac
The pu pose o his pape is o p esen some ixed poin heo ems in a comple e me ic
space endowed wi h a pa ial o de by using al e ing dis ance unc ions. We also p esen
some applica ions o i s and second o de o dina y di e en ial equa ions.
©2009 Else ie L d. All igh s ese ed.
1. In oduc ion
The Banach con ac ion mapping p inciple is one o he pi o al esul s o analysis. I is widely conside ed as he sou ce
o me ic ixed poin heo y. Also i s signi icance lies in i s as applicabili y in a numbe o b anches o ma hema ics.
Gene aliza ion o he abo e p inciple has been a hea ily in es iga ed b anch o esea ch. In pa icula , he e has been a
numbe o wo ks in ol ing al e ing dis ance unc ions. The e a e con ol unc ions which al e he dis ance be ween wo
poin s in a me ic space. Such unc ions we e in oduced by Khan e al. in [1], whe e hey p esen some ixed poin heo ems
wi h he help o such unc ions.
P e iously, we ecall he de ini ion o al e ing dis ance unc ion.
De ini ion 1.1. An al e ing dis ance unc ion is a unc ion ψ: [0,∞)→ [0,∞)which sa is ies
(a) ψis con inuous and non-dec easing.
(b) ψ( )=0 i and only i =0.
In [1], he au ho s p o e he ollowing esul .
Theo em 1.1 ([1]). Le (X,d)be a comple e me ic space, ψan al e ing dis ance unc ion and T :X→X sa is ying
ψ(d(Tx,Ty))≤c·ψ(d(x,y)),
o x,y∈X and 0<c<1. Then T has an unique ixed poin heo y.
Al e ing dis ance has been used in me ic ixed poin heo y in ecen pape s (see, o example, [2–4]).
On he o he hand, Albe and Gue e-Delab ie e in [5] de ine weakly con ac i e maps and hey con ine hei heo ems
o Hilbe spaces. Rhoades [6] p o ed ha hose esul s a e also alid in comple e me ic spaces.
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (J. Ha jani), [email p o ec ed] (K. Sada angani).
0362-546X/$ – see on ma e ©2009 Else ie L d. All igh s ese ed.
doi:10.1016/j.na.2009.08.003
3. Fixed poin heo ems in pa ially o de ed me ic spaces 95
J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197 1189
Theo em 1.2 ([6]). Le (X,d)be a comple e me ic space, ψan al e ing dis ance unc ion and T :X→X sa is ying
d(Tx,Ty)≤d(x,y)−ψ(d(x,y))
o x,y∈X. Then T has a unique ixed poin .
In ac , Albe and Gue e-Delab ie e assumed an addi ional assump ion on ψwhich is lim →∞ ψ( )= ∞bu Rhoades
p o ed Theo em 1.2 wi hou his pa icula condi ion on ψ.
Du a and Choudhu y in [7] p esen a gene aliza ion o Theo ems 1.1 and 1.2 p o ing he ollowing esul .
Theo em 1.3 ([7]). Le (X,d)be a comple e me ic space, T :X→X sa is ying
ψ(d(Tx,Ty))≤ψ(d(x,y))−φ(d(x,y)), o x,y∈X,
whe e ψand φa e al e ing dis ance unc ions. Then T has an unique ixed poin .
The pu pose o his pape is o p esen some ixed poin heo ems in ol ing al e ing dis ance unc ions in he con ex o
o de ed me ic spaces.
Exis ence o ixed poin in pa ially o de ed se s has been conside ed ecen ly in [8–22]. Ta ski’s heo em is used in [14] o
show he exis ence o solu ions o uzzy equa ions and in [16] o p o e exis ence heo ems o uzzy di e en ial equa ions.
In [15,21] some applica ions o o dina y di e en ial equa ions and o ma ix equa ions a e p esen ed, espec i ely. In [9–11,
22] some ixed poin heo ems a e p o ed o a mixed mono one mapping in a me ic space endowed wi h pa ial o de and
he au ho s apply hei esul s o p oblems o exis ence and uniqueness o solu ions o some bounda y alue p oblems.
In he con ex o o de ed me ic spaces, he usual con ac ion is weakened bu a he expense ha he ope a o is
mono one. The main idea in [15,21] in ol e combining he ideas in he con ac ion p inciple wi h hose in he mono one
i e a i e echnique [23].
2. Fixed poin heo ems
De ini ion 2.1. I (X,≤)is a pa ially o de ed se and :X→X, we say ha is mono one nondec easing i x,y∈X,
x≤y⇒ (x)≤ (y).
This de ini ion coincides wi h he no ion o a nondec easing unc ion in he case whe e X=Rand ≤ ep esen s he
usual o al o de in R.
In wha ollows, we p o e he ollowing heo em which is a e sion o Theo em 1.3 in he con ex o o de ed me ic
spaces.
Theo em 2.1. Le (X,≤)be a pa ially o de ed se and suppose ha he e exis s a me ic d in X such ha (X,d)is a comple e
me ic space. Le :X→X be a con inuous and nondec easing mapping such ha
ψ(d( (x), (y)))≤ψ(d(x,y))−φ(d(x,y)), o x ≥y(1)
whe e ψand φa e al e ing dis ance unc ions. I he e exis s x0∈X wi h x0≤ (x0) hen has a ixed poin .
P oo . I (x0)=x0 hen he p oo is inished. Suppose ha x0< (x0). Since x0< (x0)and is a nondec easing unc ion,
we ob ain by induc ion ha
x0< (x0)≤ 2(x0)≤ 3(x0)≤ ··· ≤ n(x0)≤ n+1(x0)≤ ···
Pu xn+1= (xn). Then, o each in ege n≥1, om (1) and, as he elemen s xnand xn+1a e compa able, we ge
ψ(d(xn+1,xn))=ψ(d( (xn), (xn−1)))
≤ψ(d(xn,xn−1))−φ(d(xn,xn−1))
≤ψ(d(xn,xn−1)).(2)
Using he ac ha ψis nondec easing we ha e
d(xn+1,xn)≤d(xn,xn−1). (3)
I he e exis s n0∈Nsuch ha d(xn0,xn0−1)=0 hen xn0= (xn0−1)=xn0−1and xn0−1is a ixed poin and he p oo is
inished. In o he case, suppose ha d(xn+1,xn)6= 0 o all n∈N. Then, aking in o accoun (3), he sequence {d(xn+1,xn)}
is dec easing and, consequen ly, he e exis s ≥0 such ha
d(xn+1,xn)−→ as n→ ∞.
Le ing n→ ∞in (2) we ge
ψ( )≤ψ( )−φ( )≤ψ( )
96 Gene alized con ac ions in pa ially o de ed me ic spaces ...
1190 J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197
and his implies φ( )=0. As φis an al e ing dis ance unc ion, =0, and, hence,
lim
n→∞d(xn+1,xn)=0.(4)
In wha ollows, we will show ha {xn}is a Cauchy sequence.
Suppose ha {xn}is no a Cauchy sequence. Then, he e exis s  > 0 o which we can ind subsequences {xm(k)}and
{xn(k)}o {xn}wi h n(k) > m(k) > ksuch ha
dxn(k),xm(k)≥. (5)
Fu he , co esponding o m(k)we can choose n(k)in such a way ha i is he smalles in ege wi h n(k) > m(k)and
sa is ying (5). Then
dxn(k)−1,xm(k)< . (6)
Using (5),(6) and he iangula inequali y, we ha e
≤dxn(k),xm(k)
≤dxn(k),xn(k)−1+dxn(k)−1,xm(k)
<dxn(k),xn(k)−1+.
Le ing k→ ∞and using (4)
lim
k→∞dxn(k),xm(k)=. (7)
Again, he iangula inequali y gi es 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.
Le ing k→ ∞in he abo e wo inequali ies and using (4) and (7), we ha e
lim
k→∞dxn(k)−1,xm(k)−1=. (8)
As n(k) > m(k)and xn(k)−1and xm(k)−1a e compa able (in ac , xm(k)−1≤xn(k)−1), se ing x=xn(k)−1and y=xm(k)−1in (1),
we ob ain
ψd(xn(k),xm(k))≤ψd(xn(k)−1,xm(k)−1)−φd(xn(k)−1,xm(k)−1).
Le ing k→ ∞and aking in o accoun (7) and (8), we ha e
ψ() ≤ψ() −φ().
As ψis an al e ing dis ance unc ion, he las inequali y gi es us φ() =0 and, consequen ly, =0 which is a con adic ion.
This shows ha {xn}is a Cauchy sequence and, since Xis a comple e me ic space, he e exis s z∈Xsuch ha
limn→∞ xn=z.
Mo eo e , he con inui y o implies ha
z=lim
n→∞ (xn)=lim
n→∞xn+1= (z)
and his p o es ha zis a ixed poin . 
In wha ollows, we p o e ha Theo em 2.1 is s ill alid o no necessa ily con inuous, assuming he ollowing
hypo hesis in X(which appea s in Theo em 1 o [15])
i (xn)is a nondec easing sequence in Xsuch ha xn→x hen xn≤x o all n∈N.(9)
Theo em 2.2. Le (X,≤)be a pa ially o de ed se and suppose ha he e exis s a me ic d in X such ha (X,d)is a comple e
me ic space. Assume ha X sa is ies (9). Le :X→X be a nondec easing mapping such ha
ψ(d( (x), (y)))≤ψ(d(x,y))−φ(d(x,y)), o x ≥y,
whe e ψand φa e al e ing dis ance unc ions. I he e exis s x0∈X wi h x0≤ (x0) hen has a ixed poin .
P oo . Following he p oo o Theo em 2.1 we only ha e o check ha (z)=z. As (xn)is a nondec easing sequence in X
and limn→∞ xn=z hen he condi ion (9) gi es us ha xn≤z o e e y n∈Nand, consequen ly,
ψ(d(xn+1, (z)))=ψ(d( (xn), (z)))≤ψ(d(xn,z))−φ(d(xn,z)).
3. Fixed poin heo ems in pa ially o de ed me ic spaces 97

J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197 1191
Le ing n→ ∞and aking in o accoun ha ψand ψa e al e ing dis ance unc ions, we ha e
ψ(d(z, (z)))≤ψ(0)−φ(0)=0.
This implies ψ(d(z, (z)))=0. Thus, d(z, (z))=0, o equi alen ly, (z)=z.
Now, we p esen an example whe e i can be app ecia ed ha hypo heses in Theo ems 2.1 and 2.2 do no gua an ee
uniqueness o he ixed poin . This example appea s in [15].
Le X= {(1,0), (0,1)} ⊂ R2and conside he usual o de
(x,y)≤(z, )⇔x≤zand y≤ .
Thus, (X,≤)is a pa ially o de ed se whose di e en elemen s a e no compa able. Besides, (X,d2)is a comple e me ic
space conside ing d2 he Euclidean dis ance. The iden i y map (x,y)=(x,y)is i ially con inuous and nondec easing
and condi ion (1) o Theo em 2.2 is sa is ied since elemen s in Xa e only compa able o hemsel es. Mo eo e , (1,0)≤
(1,0)=(1,0)and has wo ixed poin s in X.
In wha ollows, we gi e a su icien condi ion o he uniqueness o he ixed poin in Theo ems 2.1 and 2.2. This condi-
ion is
o x,y∈X he e exis s a lowe bound o an uppe bound. (10)
In [15] i is p o ed ha condi ion (10) is equi alen o
o x,y∈X he e exis s z∈Xwhich is compa able o xand y.(11)
Theo em 2.3. Adding condi ion (11) o he hypo heses o Theo em 2.1 ( esp. Theo em 2.2) we ob ain uniqueness o he ixed
poin o .
P oo . Suppose ha he e exis z,y∈Xwhich a e ixed poin s. We dis inguish wo cases:
Case1. I yis compa able o z hen n(y)=yis compa able o n(z)=z o n=0,1,2, . . . and
ψ(d(z,y))=ψd( n(z), n(y))
≤ψd( n−1(z), n−1(y))−φd( n−1(z), n−1(y))
≤ψ(d(z,y))−φ(d(z,y)).
As ψand φa e al e ing dis ance unc ions, he las inequali y gi es us φ(d(z,y))=0 and his implies z=y.
Case2. I yis no compa able o z hen he e exis s x∈Xcompa able o yand z. Mono onici y o implies ha n(x)is
compa able o n(y)=yand o n(z)=z, o n=0,1,2, . . .. Mo eo e ,
ψd(z, n(x))=ψd( n(z), n(x))
≤ψd( n−1(z), n−1(x))−φd( n−1(z), n−1(x))
≤ψd( n−1(z), n−1(x))
=ψd(z, n(x)).(12)
Hence, he las inequali y p o es ha {ψ(d(z, n(x)))}is a nonnega i e dec easing sequence. Mono onici y o ψ,
gi es us ha {d(z, n(x))}is also a nonnega i e dec easing sequence and, consequen ly, he e exis s γsuch ha
lim
n→∞dz, n(x)=γ .
Le ing n→ ∞in (12) and, aking in o accoun ha ψand φa e al e ing dis ance unc ions, we ob ain
ψ(γ ) ≤ψ(γ ) −φ(γ ) ≤ψ(γ )
and his implies φ(γ ) =0 and, consequen ly, γ=0.
Analogously, i can be p o ed ha
lim
n→∞dy, n(x)=0.
Finally, as
lim
n→∞dz, n(x)=lim
n→∞dy, n(x)=0,
he uniqueness o he limi gi es us y=z. This inishes he p oo . 
98 Gene alized con ac ions in pa ially o de ed me ic spaces ...
1192 J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197
Rema k 2.1. Unde he assump ions o Theo em 2.3, i can be p o ed ha o e e y x∈X, limn→∞ n(x)=z, whe e zis
he ixed poin (i.e. he ope a o is Pica d).
In ac , i xis compa able o z hen using he same a gumen ha in Theo em 2.3 can be p o ed ha limn→∞ d(z, n(x))=
0 and, consequen ly, limn→∞ n(x)=z.
I xis no compa able wi h z, we ake y∈Xcompa able bo h wi h xand zand he same easoning ha Theo em 2.3 gi es
us
lim
n→∞dz, n(y)=lim
n→∞d n(z), n(y)=0 and lim
n→∞d n(x), n(y)=0.
Finally, using
dz, n(x)≤dz, n(y)+d n(y), n(x)
and aking limi as n→ ∞, we ob ain limn→∞ d(z, n(x))=0, o equi alen ly, limn→∞ n(x)=z.
Rema k 2.2. No ice ha i (X,≤)is o ally o de ed se , Theo em 2.3 gi es us he uniqueness o he ixed poin .
Rema k 2.3. By using Ze melo’s well o de ing heo em, he se Xcan be well o de ed and he condi ion (11) o ou
Theo em 2.3 is alid o each x,y∈X. Mo eo e , x0=min Xsa is ies x0≤ (x0)and Theo em 2.3 gi es us Theo em
1.2 in [7] o he pa icula case ha is a nondec easing unc ion.
Rema k 2.4. Theo ems 2 and 3 in [12] a e pa icula cases o ou Theo ems 2.2 and 2.3 o ψ he iden i y unc ion.
3. Applica ion o o dina y di e en ial equa ions
In his sec ion we p esen wo examples whe e ou Theo ems 2.2 and 2.3 can be applied. The i s example is inspi ed
in [15].
We s udy he exis ence o solu ion o he ollowing i s -o de pe iodic p oblem
u0( )= ( ,u( )), ∈ [0,T]
u(0)=u(T), (13)
whe e T>0 and :I×R−→ Ris a con inuous unc ion.
P e iously, we conside ed he space C(I)(I= [0,T]) o con inuous unc ions de ined on I. Ob iously, his space wi h
he me ic gi en by
d(x,y)=sup{|x( )−y( )| : ∈I}, o x,y∈C(I),
is a comple e me ic space. C(I)can also be equipped wi h a pa ial o de gi en by
x,y∈C(I), x≤y⇔x( )≤y( ) o ∈I.
Clea ly, (C(I), ≤)sa is ies condi ion (10), since o x,y∈C(I) he unc ions max{x,y}and min{x,y}a e leas uppe and
g ea es lowe bounds o xand y, espec i ely.
Mo eo e , in [15] i is p o ed ha (C(I), ≤)wi h he abo e men ioned me ic sa is ies condi ion (9).
Now we gi e he ollowing de ini ion.
De ini ion 3.1. A lowe solu ion o (13) is a unc ion α∈C1(I)such ha
α0( )= ( , α( )), o ∈I
α(0)≤α(T).
Theo em 3.1. Conside p oblem (13) wi h :I×R−→ Rcon inuous and suppose ha he e exis λ, α > 0wi h
α≤2λ(eλT−1)
T(eλT+1)1
2
,
such ha o x,y∈Rwi h x ≥y
0≤ ( ,y)+λy−[ ( ,x)+λx]≤αqln (y−x)2+1.
Then he exis ence o a lowe solu ion o (13) p o ides he exis ence o an unique solu ion o (13).
3. Fixed poin heo ems in pa ially o de ed me ic spaces 99
J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197 1193
P oo . P oblem (13) can be w i en as
u0( )+λu( )= ( ,u( ))+λu( ), o ∈I= [0,T]
u(0)=u(T).
This p oblem is equi alen o he in eg al equa ion
u( )=ZT
0
G( ,s)[ (s,u(s)) +λu(s)]ds,
whe e G( ,s)is he G een unc ion gi en by
G( ,s)=






eλ(T+s− )
eλT−1,0≤s< ≤T
eλ(s− )
eλT−1,0≤ <s≤T.
De ine F:C(I)−→ C(I)by
(Fu)( )=ZT
0
G( ,s)[ (s,u(s)) +λu(s)]ds.
No e ha i u∈C(I)is a ixed poin o F hen u∈C1(I)is a solu ion o (11).
In wha ollows, we check ha hypo heses in Theo ems 2.2 and 2.3 a e sa is ied.
The mapping Fis nondec easing, since o u≥ , and using ou assump ion, we can ob ain
( ,u)+λu≥ ( , ) +λ ,
which implies, since G( ,s) > 0, ha o ∈I,
(Fu)( )=ZT
0
G( ,s)[ (s,u(s)) +λu(s)]ds≥ZT
0
G( ,s)[ (s, (s)) +λ (s)]ds=(F )( ).
Besides, o u≥ , we ha e
d(Fu,F ) =sup
∈I|(Fu)( )−(F )( )|
=sup
∈I
((Fu)( )−(F )( ))
=sup
∈IZT
0
G( ,s)[ (s,u(s)) +λu(s)− (s, (s)) −λ (s)]ds
≤sup
∈IZT
0
G( ,s)α qln (u(s)− (s))2+1ds.(14)
Using he Cauchy–Schwa z inequali y in he las in eg al we ge
ZT
0
G( ,s)α qln (u(s)− (s))2+1ds≤ZT
0
G( ,s)2ds
1
2ZT
0
α2ln (u(s)− (s))2+1ds
1
2
.(15)
The i s in eg al gi es us
ZT
0
G( ,s)2ds=Z
0
G( ,s)2ds+ZT
G( ,s)2ds
=Z
0
e2λ(T+s− )
eλT−12ds+ZT
e2λ(s− )
eλT−12ds
=1
2λeλT−12e2λT−1
=eλT+1
2λeλT−1.(16)
100 Gene alized con ac ions in pa ially o de ed me ic spaces ...
1194 J. Ha jani, K. Sada angani / Nonlinea Analysis 72 (2010) 1188–1197
The second in eg al in (15) gi es us he ollowing es ima e
ZT
0
α2ln (u(s)− (s))2+1ds≤α2ln ku− k2+1·T
=α2ln d(u, )2+1·T.(17)
Taking in o accoun (14)–(17) we ha e
d(Fu,F ) ≤sup
∈I eλT+1
2λeλT−1!1
2
·α2ln d(u, )2+1·T1
2
= eλT+1
2λeλT−1!1
2
·α·√T·ln d(u, )2+11
2
and om he las inequali y we ob ain
d(Fu,F )2≤eλT+1
2λeλT−1·α2·T·ln d(u, )2+1
o , equi alen ly,
2λeλT−1d(Fu,F )2≤eλT+1·α2·T·ln d(u, )2+1.
By ou assump ion, as
α≤2λ(eλT−1)
T(eλT+1)1
2
,
he las inequali y gi es us
2λeλT−1d(Fu,F )2≤2λeλT−1·ln d(u, )2+1
and, hence,
d(Fu,F )2≤ln d(u, )2+1
=d(u, )2−d(u, )2−ln d(u, )2+1.(18)
Pu ψ(x)=x2and φ(x)=x2−ln(x2+1). Ob iously, ψand φa e al e ing dis ance unc ions.
F om (18), we ob ain o u≥
ψ(d(Fu,F ))≤ψ(d(u, ))−φ(d(u, )).
Finally, le α( )be a lowe solu ion o (13), we claim ha α≤F(α).
In ac ,
α0( )+λα( )≤ ( , α( ))+λα( ), o ∈I.
Mul iplying by eλ
α( )eλ 0≤[ ( , α( ))+λα( )]eλ , o ∈I,
and his gi es us
α( )eλ ≤α(0)+Z
0
[ (s, α(s))+λα(s)]eλsds, o ∈I.(19)
As α(0)≤α(T), he las inequali y gi es us
α(0)eλT≤α(T)eλT≤α(0)+ZT
0
[ (s, α(s))+λα(s)]eλsds
and so
α(0)≤ZT
0
eλs
eλT−1[ (s, α(s))+λα(s)]ds.
This and (19) gi e us
3. Fixed poin heo ems in pa ially o de ed me ic spaces 101

Hindawi Publishing Co po a ion
Fixed Poin Theo y and Applica ions
Volume 2010, A icle ID 916064, 14 pages
doi:10.1155/2010/916064
Resea ch A icle
Con ac i e-Like Mapping P inciples in O de ed
Me ic Spaces and Applica ion o O dina y
Di e en ial Equa ions
J. Caballe o, J. Ha jani, and K. Sada angani
Depa amen o de Ma em´
a icas, Uni e sidad de Las Palmas de G an Cana ia, Campus de Ta i a Baja,
35017 Las Palmas de G an Cana ia, Spain
Co espondence should be add essed o K. Sada angani, [email p o ec ed]
Recei ed 25 No embe 2009; Re ised 10 Ma ch 2010; Accep ed 30 Ma ch 2010
Academic Edi o : Tomona i Suzuki
Copy igh q2010 J. Caballe o e al. This is an open access a icle dis ibu ed unde he C ea i e
Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in
any medium, p o ided he o iginal wo k is p ope ly ci ed.
The pu pose o his pape is o p esen a ixed poin heo em o gene alized con ac ions in
pa ially o de ed comple e me ic spaces. We also p esen an applica ion o i s -o de o dina y
diffe en ial equa ions.
1. In oduc ion
Exis ence o ixed poin in pa ially o de ed se s has been conside ed ecen ly in 1–17.
Ta ski’s heo em is used in 9 o show he exis ence o solu ions o uzzy equa ions and
in 11 o p o e exis ence heo ems o uzzy diffe en ial equa ions. In 2,6,7,10,13some
applica ions o o dina y diffe en ial equa ions and o ma ix equa ions a e p esen ed. In 3–
5,17some ixed poin heo ems a e p o ed o a mixed mono one mapping in a me ic space
endowed wi h pa ial o de and he au ho s apply hei esul s o p oblems o exis ence and
uniqueness o solu ions o some bounda y alue p oblems.
In he con ex o o de ed me ic spaces, he usual con ac ion is weakened bu a he
expense ha he ope a o is mono one. The main ool in he p oo o he esul s in his con ex
combines he ideas in he con ac ion p inciple wi h hose in he mono one i e a i e echnique
18.
Le Sdeno e he class o he class o he unc ions β:0,∞→0,1which sa is ies
he condi ion
β n−→ 1⇒ n−→ 0.1.1
In 19 he ollowing gene aliza ion o Banach’s con ac ion p inciple appea s.
3. Fixed poin heo ems in pa ially o de ed me ic spaces 109
2 Fixed Poin Theo y and Applica ions
Theo em 1.1. Le X, dbe a comple e me ic space and le T:X→Xbe a mapping sa is ying
dTx,Ty≤βdx, y·dx, y, o x, y ∈X, 1.2
whe e β∈S.ThenThas a unique ixed poin z∈Xand {Tnx}con e ges o z o each x∈X.
Recen ly, in 2 he au ho s p o e a e sion o Theo em 1.1 in he con ex o o de ed
comple e me ic spaces. Mo e p ecisely, hey p o e he ollowing esul .
Theo em 1.2. Le X, ≤be a pa ially o de ed se and suppose ha he e exis s a me ic din Xsuch
ha X, dis a comple e me ic space. Le T:X→Xbe a nondec easing mapping such ha
dTx,Ty≤βdx, y·dx, y, o x, y ∈Xwi h x≤y, 1.3
whe e β∈S. Assume ha ei he Tis con inuous o Xsa is ies he ollowing condi ion:
i {xn}is a nondec easing sequence in Xsuch ha xn−→ x, hen xn≤x∀n∈N.1.4
Besides, suppose ha o each x,y ∈X he e exis s z∈Xwhich is compa able o xand y.I he e
exis s x0∈Xwi h x0≤Tx0, henThas a unique ixed poin .
The pu pose o his pape is o gene alize Theo em 1.2 wi h he help o he al e ing
unc ions.
We ecall he de ini ion o such unc ions.
De ini ion 1.3. An al e ing unc ion is a unc ion ψ:0,∞→0,∞which sa is ies he
ollowing.
aψis con inuous and nondec easing.
bψ 0 i and only i 0.
Al e ing unc ions ha e been used in me ic ixed poin heo y in ecen pape s 20–
22.
In 7 he au ho s use hese unc ions and hey p o e some ixed poin heo ems in
o de ed me ic spaces.
2. Fixed Poin Theo ems
De ini ion 2.1. I X, ≤is a pa ially o de ed se and T:X→X, we say ha Tis mono one
nondec easing i o x,y ∈X,
x≤y⇒Tx≤Ty.2.1
This de ini ion coincides wi h he no ion o a nondec easing unc ion in he case XR
and ≤ ep esen s he usual o al o de in R.
In he sequel, we p o e he main esul o he pape .
110 Con ac i e-like mapping p inciples in o de ed me ic spaces ...
Fixed Poin Theo y and Applica ions 3
Theo em 2.2. Le X, ≤be a pa ially o de ed se and suppose ha he e exis s a me ic din Xsuch
ha X, dis a comple e me ic space. Le T:X→Xbe a con inuous and nondec easing mapping
such ha
ψdTx,Ty≤βdx,y·ψdx, y, o x≥y, 2.2
whe e ψis an al e ing unc ion and β∈S.
I he e exis x0∈Xwi h x0≤Tx0, henThas a ixed poin .
P oo . I Tx0x0, hen he p oo is inished. Suppose ha x0<Tx0. Since x0<Tx0and
Tis a nondec easing mapping, we ob ain by induc ion ha
x0<T
x0≤T2x0≤T3x0≤···≤Tnx0≤Tn1x0≤···.2.3
Pu xn1Txn. Taking in o accoun ha β∈Sand since xn≤xn1 o each n∈N, hen, by
2.2,wege
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 he ac ha ψis nondec easing, we ha e
dxn1,x
n≤dxn,x
n−1.2.5
I he e exis s n0∈Nsuch ha dxn0,x
n0−10, hen xn0Txn0−1xn0−1and xn0−1is a ixed
poin and he p oo is inished. In ano he case, suppose ha dxn1,x
n/
0 o all n∈N.
Then, aking in o accoun 2.5, he sequence {dxn1,x
n}is dec easing and bounded below,
so
lim
n→∞dxn1,x
n ≥02.6
Assume ha >0.
Then, om 2.4, we ha e
ψdxn1,x
n
ψdxn,x
n−1 ≤βdxn,x
n−1 <1.2.7
Le ing n→∞in he las inequali y and by he ac ha ψis an al e ing unc ion, we ge
1≤lim
n→∞βdxn,x
n−1 ≤12.8
3. Fixed poin heo ems in pa ially o de ed me ic spaces 111
4 Fixed Poin Theo y and Applica ions
and, consequen ly, limn→∞βdxn,x
n−1  1.Since β∈S his implies ha limn→∞dxn1,
xn0 and his con adic s ou assump ion ha >0.Hence,
lim
n→∞dxn1,x
n 0.2.9
In wha ollows, we will show ha {xn}is a Cauchy sequence.
Suppose ha {xn}is no a Cauchy sequence. Then, he e exis s >0 o which we can
ind subsequences {xmk}and {xnk}o {xn}wi h nk>mk>ksuch ha
dxnk,x
mk≥. 2.10
Fu he , co esponding o mk, we can choose nkin such a way ha i is he smalles
in ege wi h nk>mkand sa is ying 2.10, hen
dxnk−1,x
mk<. 2.11
Using 2.10,2.11, and he iangula inequali y, we ha e
≤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
Le ing k→∞and using 2.9,wege
lim
k→∞dxnk,x
mk. 2.13
Again, he iangula inequali y gi es 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
Le ing k→∞in he abo e wo inequali ies and using 2.9and 2.13, we ha e
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,weob ain
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 Con ac i e-like mapping p inciples in o de ed me ic spaces ...
Fixed Poin Theo y and Applica ions 5
Taking in o accoun 2.13and 2.15and he ac ha ψis con inuous and le ing
k→∞in 2.16,wege
ψ≤lim
k→∞βdxnk−1,x
mk−1·ψ≤ψ.2.17
As ψis an al e ing unc ion, ψ>0, he las inequali y gi es us
lim
k→∞βdxnk−1,x
mk−11.2.18
Since β∈S, his means ha
lim
k→∞dxnk−1,x
mk−10.2.19
This ac and 2.15gi e us 0 which is a con adic ion.
This shows ha {xn}is a Cauchy sequence.
Since X, dis a comple e me ic space, he e exis s z∈Xsuch ha limn→∞xnz.
Mo eo e , he con inui y o Timplies ha
zlim
n→∞Txnlim
n→∞xn1Tz, 2.20
and his p o es ha zis a ixed poin .
In wha ollows, we p o e ha Theo em 2.2 is s ill alid o Tno necessa ily
con inuous, assuming he ollowing hypo hesis in Xwhich appea s in 10, Theo em 1:
i xnis a nondec easing sequence in Xsuch ha xn−→ x, hen xn≤x∀n∈N.2.21
Theo em 2.3. Le X, ≤be a pa ially o de ed se and suppose ha he e exis s a me ic din X
such ha X, dis a comple e me ic space. Assume ha Xsa is ies 2.21.Le T:X→Xbe a
nondec easing mapping such ha
ψdTx,Ty≤βdx,y·ψdx, y, o x≥y, 2.22
whe e ψis an al e ing unc ion and β∈S. I he e exis s x0∈Xwi h x0≤Tx0, henThas a ixed
poin .
P oo . Following he p oo o Theo em 2.2, we only ha e o check ha Tzz.Asxnis a
nondec easing sequence in Xand limn→∞xnz hen, by 2.21, we ha e xn≤z o all n∈N,
and, consequen ly,
ψdxn1, zψdTxn,Tz ≤βdxn,z
 ·ψdxn,z
 ≤ψdxn,z
.2.23
3. Fixed poin heo ems in pa ially o de ed me ic spaces 113

6 Fixed Poin Theo y and Applica ions
Le ing n→∞and using he con inui y o ψ, we ha e
0≤ψdz, Tz ≤ψ00,2.24
o , equi alen ly,
ψdz, Tz 0.2.25
As ψis an al e ing unc ion, his gi es us dz, Tz  0 and, hus, Tzz.
Now, we p esen an example whe e i can be app ecia ed ha he hypo heses in
Theo ems 2.2 and 2.3 do no gua an ee uniqueness o he ixed poin . This example appea s
in 10.
Le X{1,0,0,1}⊂R2and conside he usual o de
x,y≤z, ⇐⇒ x≤z, y ≤ . 2.26
X, ≤is a pa ially o de ed se whose diffe en elemen s a e no compa able. Besides, X, d2
is a comple e me ic space conside ing d2as he Euclidean dis ance. The iden i y map
Tx,yx,yis i ially con inuous and nondec easing and condi ion 2.2o Theo em 2.2
is sa is ied since elemen s in Xa e only compa able o hemsel es. Mo eo e , 1,0≤T1,0
1,0and Thas wo ixed poin s in X.
In wha ollows, we gi e a sufficien condi ion o he uniqueness o he ixed poin in
Theo ems 2.2 and 2.3. This condi ion appea s in 16and says ha
o x,y ∈X, he e exis s a lowe bound o an uppe bound.2.27
In 10i is p o ed ha condi ion 2.27is equi alen o
o x,y ∈X, he e exis s z∈Xwhich is compa able o xand y. 2.28
Theo em 2.4. Adding condi ion 2.28 o he hypo heses o Theo em 2.2 ( esp., Theo em 2.3), we
ob ain uniqueness o he ixed poin o .
P oo . Suppose ha he e exis y,z ∈Xwhich a e ixed poin s o Tand y/
z. We dis inguish
wo cases.
Case 1. I yand za e compa able, hen Tnyyand Tnzza e compa able o n
0,1,2,.... Using he con ac i e condi ion appea ing in Theo em 2.2 o Theo em 2.3and
he ac ha β∈S,wege
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 con adic ion.
114 Con ac i e-like mapping p inciples in o de ed me ic spaces ...
Fixed Poin Theo y and Applica ions 7
Case 2. Using condi ion 2.28, he e exis s x∈Xcompa able o yand z. Mono onici y o T
implies ha Tnxis compa able o Tnyyand o Tnzz, o n0,1,2,....Mo eo e ,
as β∈S,wege
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 nondec easing he abo e inequali y gi es us
dz, Tnx ≤dz, Tn−1x.2.31
Thus, limn→∞dz, Tnxγ≥0.
Assume ha γ>0.
Taking in o accoun ha ψis an al e ing unc ion and le ing n→∞in 2.30,we
ob ain
ψγ≤lim
n→∞βdz, Tn−1x·ψγ≤ψγ,2.32
and his implies ha limn→∞βdz, Tn−1x  1.
Since β∈S hen we ge
lim
n→∞dz, Tn−1x0,2.33
and, consequen ly, γ0, which is a con adic ion.
Hence, limn→∞dz, Tnx0.
Analogously, i can be p o ed ha
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 aking limi , we ob ain dz, y0.
This inishes he p oo .
Rema k 2.5. Unde he assump ions o Theo em 2.4, i can be p o ed ha o e e y x∈X,
limn→∞Tnxz, whe e zis he ixed poin i.e., he ope a o Tis Pica d.
3. Fixed poin heo ems in pa ially o de ed me ic spaces 115
8 Fixed Poin Theo y and Applica ions
In ac , o x∈Xand xcompa able o z hen using he same a gumen ha is in Case
1o Theo em 2.4 can p o e ha limn→∞dz, Tnx0 and, hence, limn→∞Tnxz.
I xis no compa able o z, we ake ha y∈Xis compa able o xand z. Using a simila
a gumen ha is in Case 2o Theo em 2.4,weob ain
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 aking limi as n→∞, we ob ain limn→∞dz, Tnx0 o , equi alen ly, limn→∞Tnx
z.
Rema k 2.6. No ice ha i X, ≤is o ally o de ed, condi ion 2.28is ob iously sa is ied.
Rema k 2.7. Conside ing ψ he iden i y mapping in Theo em 2.4,weob ainTheo em 1.2,
being he main esul o 2.
3. Applica ion o O dina y Di e en ial Equa ions
In his sec ion we p esen an example whe e ou esul s can be applied.
This example is inspi ed by 10.
We s udy he exis ence o solu ion o he ollowing i s -o de pe iodic p oblem
u   , u , ∈0,T,
u0uT,
3.1
whe e T>0and :I×R→Ris a con inuous unc ion.
P e iously, we conside ed he space CII0,T o con inuous unc ions de ined
on I. Ob iously, his space wi h he me ic gi en by
dx,ysupx −y : ∈I, o x,y ∈C
I,3.2
is a comple e me ic space. CIcan also be equipped wi h a pa ial o de gi en by
x,y ∈C
I,x≤y⇐⇒ x ≤y , o ∈I. 3.3
Clea ly, CI,≤sa is ies condi ion 2.28,since o x, y ∈CI he unc ion max{x, y}∈CI.
Mo eo e , in 10i is p o ed ha CI,≤wi h he abo e-men ioned me ic sa is ies
condi ion 2.21.
116 Con ac i e-like mapping p inciples in o de ed me ic spaces ...
Fixed Poin Theo y and Applica ions 9
Now, le Adeno e he class o unc ions φ:0,∞→0,∞sa is ying he ollowing.
iφis nondec easing.
iiφx<x, o x>0.
iiiβxφx/x ∈S,
whe e Sis he class o unc ions de ined in Sec ion 1.
Examples o such unc ions a e φ μ· ,wi h0≤μ<1, φ  /1 ,and
φ ln1 .
Recall now he ollowing de ini ion
De ini ion 3.1. A lowe solu ion o 3.1is a unc ion α∈C
1Isuch ha
α ≤  , α  o ∈I,
α0≤αT.
3.4
Now, we p esen he ollowing heo em abou he exis ence o solu ion o p oblem
3.1in p esence o a lowe solu ion.
Theo em 3.2. Conside p oblem 3.1wi h :I×R→Rcon inuous and suppose ha he e exis
λ, α > 0wi h
α≤2λeλT −1
TeλT 11/2
,3.5
such ha o x, y ∈Rwi h x≤y
0≤  , yλy −  , xλx≤αy−xφy−x,3.6
whe e φ∈A. Then he exis ence o a lowe solu ion o 3.1p o ides he exis ence o a unique
solu ion o 3.1.
P oo . P oblem 3.1can be w i en as
u λu   , u  λu  o ∈I0,T,
u0uT.
3.7
This p oblem is equi alen o he in eg al equa ion
u T
0
G , s s, us λusds, 3.8
3. Fixed poin heo ems in pa ially o de ed me ic spaces 117
124 Fixed poin heo ems o mappings sa is ying a condi ion o ...
k∈[0,1) such ha
Zd(T(x),T(y))
0
ϕ( )d ≤kZm(x,y)
0
ϕ( )d o x, y ∈Xwi h x≥y,
whe e ϕis a Lebesgue measu able unc ion wi h ini e in eg al on each compac
subse o [0,∞), sa is ying Zε
0
ϕ( )d > 0 o ε > 0. Assume ha ei he T
is con inuous o Xsa is ies he condi ion
i (xn)is a nondec easing sequence in Xsuch ha xn→x
hen xn≤x o all n∈N.
I he e exis s x0∈Xwi h x0≤T(x0) hen Thas a ixed poin .
In his case, we ha e no been able o p o e he uniqueness o he ixed
poin unde classical assump ion
Fo any x, y ∈X he e exis s z∈Xcompa able o xand y .
Ou Theo em 23 can be conside ed as a e sion o he ollowing esul which
appea s in [B. E. Rhoades, Two ixed poin heo ems o mapping sa is ying
a gene al con ac i e condi ion o in eg al ype, In . J. Ma h. Sci., 63, (2003),
4007–4013].
Theo em 24. Le (X, d)be a comple e me ic space, k∈[0,1) and T:X→
Xa mapping such ha , o each x, y ∈X,
Zd(T(x),T(y))
0
ϕ( )d ≤kZm(x,y)
0
ϕ( )d ,
whe e ϕ:R+→R+is a Lebesgue-measu able unc ion wi h ini e in eg al on
each compac subse o R+, sa is ying Zε
0
ϕ( )d > 0 o ε > 0. Then Thas
a ixed poin .
In he pape ,we ask i in he con ac i e condi ion o Theo em h4.2 we

3. Fixed poin heo ems in pa ially o de ed me ic spaces 125
can eplace 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 quan i y was p e iously used by ´
Ci i´c in [L. B. ´
Ci i´c, A gene aliza ion o
Banach’s con ac ion p inciple, P oc. Ame Ma h. Soc. 45 (1974) 267–273].
We p esen an example ha his is no possible. This example appea s in [B.
E. Rhoades, Two ixed poin heo ems o mapping sa is ying a gene al con-
ac i e condi ion o in eg al ype, In . J. Ma h. Sci., 63, (2003), 4007–4013].
We answe a i ma i ely o ou ques ion wi h he addi ion o ce ain condi-
ion. Mo e p ecisely, we p o e he ollowing esul .
Theo em 25. Le (X, ≤)be a pa ially o de ed se and suppose ha he-
e exis s a me ic din Xsuch ha (X, d)is a comple e me ic space. Le
T:X→Xbe a nondec easing mapping such ha he e exis s k∈[0,1) wi h
ZdT(x),T(y)
0
ϕ( )d ≤kZM(x,y)
0
ϕ( )d , o x≥y, (3.7)
whe e ϕ:R+→R+is a Lebesgue-measu able unc ion wi h ini e in eg al on
each compac subse o R+, such ha Zε
0
ϕ( )d > 0 o ε > 0. Assume ha
Tis con inuous o Xsa is ies he condi ion
i (xn)is a nondec easing sequence in Xwi h xn→x
hen xn≤x o all n∈N.
I he e exis s x0∈Xwi h x0≤T(x0)and he o bi o x0is bounded, hen
Thas a ixed poin .
No e ha ou ex a assump ion is ha he o bi o x0is bounded.
Jou nal o Con ex Analysis
Volume 17 (2010), No. 2, 597–609
Fixed Poin Theo ems o Mappings
Sa is ying a Condi ion o In eg al Type
in Pa ially O de ed Se s∗
J. Ha jani
Depa amen o de Ma em´a icas, Uni e sidad de Las Palmas de G an Cana ia,
Campus de Ta i a Baja, 35017 Las Palmas de G an Cana ia, Spain
K. Sada angani
Depa amen o de Ma em´a icas, Uni e sidad de Las Palmas de G an Cana ia,
Campus de Ta i a Baja, 35017 Las Palmas de G an Cana ia, Spain
ksada [email p o ec ed]gc.es
Dedica ed o P o esso Jos´e Rod ´ıguez Exp´osi o on he occasion o his 60 h bi hday.
Recei ed: July 14, 2008
Re ised manusc ip ecei ed: June 22, 2009
The pu pose o his pape is o p esen some ixed poin heo ems o mono one ope a o s in a me ic
space endowed wi h a pa ial o de using a gene al con ac i e condi ion o in eg al ype.
Keywo ds: Fixed poin , pa ially o de ed me ic spaces
1991 Ma hema ics Subjec Classi ica ion: 47H10
1. P elimina ies
Recen ly he Banach con ac ion p inciple [8] was discussed in a me ic space endowed
wi h a pa ial o de whe e some applica ions o ma ix equa ions [16] and o o dina y
di e en ial equa ions [11, 13] a e p esen ed. The usual con ac ion condi ion is weak-
ened bu a he expense ha he ope a o is mono one. The main idea in [11, 16]
in ol es combining he ideas in he con ac ion p inciple wi h hose in he mono one
i e a i e echnique [2, 3].
This a icle p esen s new esul s o con ac ions sa is ying a condi ion o in eg al ype
in o de ed me ic spaces and hese esul s a e sligh ex ensions o hose in [11, 16].
Exis ence o ixed poin in pa ially o de ed se s s a s wi h Ta ki’s heo em [18].
Recen ly, a lo o pape s ha e ea ed his equa ion (see, o example [5, 6, 7, 9, 10, 11,
12, 13, 14, 15, 16, 19]).
∗Pa ially suppo ed by Minis e io de Ciencia y Tecnolog´ıa, p ojec MTM 2007-65706.
ISSN 0944-6532 / $ 2.50 c
Helde mann Ve lag
3. Fixed poin heo ems in pa ially o de ed me ic spaces 127
598 J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ...
2. Fixed poin heo ems
Suppose (X, ≤) is a pa ially o de ed se and :X−→ X. We say is non-dec easing
i x, y ∈X,x≤yimplies (x)≤ (y).
In a ecen pape [1], R. Aga wal, M. El-Gebeily and D. O’Regan es ablished he
ollowing heo em.
Theo em 2.1. Le (X, ≤)be a pa ially o de ed se and suppose ha he e is a me ic
don Xsuch ha (X, d)is a comple e me ic space. Assume he e is a non-dec easing
unc ion ψ: [0,∞)−→ [0,∞)wi h limn→∞ ψn( ) = 0 o each > 0and also suppose
F:X−→ Xis a nondec easing mapping wi h
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))]},
o all x≥y. Also suppose ei he Fis con inuous o i (xn)⊂Xis a nondec easing
sequence wi h xn→xin X hen xn≤x o all n∈N.
I he e exis s x0∈Xwi h x0≤F(x0), hen Fhas a ixed poin .
Now, we p esen ou main esul in his pape .
P e iously, we de ine o 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))].
Theo em 2.2. Le (X, ≤)be a pa ially o de ed se and suppose ha he e exis s a
me ic din Xsuch ha (X, d)is a comple e me ic space. Le F:X→Xbe a
con inuous and nondec easing mapping such ha he e exis s k∈[0,1) wi h
Zd(F(x),F (y))
0
ϕ( )d ≤kZm(x,y)
0
ϕ( )d o x≥y, (1)
whe e ϕ:R+−→ R+is a Lebesgue-in eg able mapping such ha Rε
0ϕ( )>0 o ε > 0.
I he e exis s x0∈Xwi h x0≤F(x0) hen Fhas a ixed poin .
P oo . I F(x0) = x0 hen he p oo is inished. Suppose ha x0< F(x0). Since
x0< F(x0) and Fis nondec easing, we ob ain by induc ion ha
x0≤F(x0)≤F2(x0)≤ · · · ≤ Fn(x0)≤Fn+1(x0)≤....
Pu xn+1 =Fn(x0). Then o each in ege n≥1, om (1) and, as he elemen s xnand
xn+1 a e compa able, we ge
Zd(xn,xn+1)
0
ϕ( )d =Zd(F(xn−1),F (xn))
0
ϕ( )d ≤kZm(xn−1,xn)
0
ϕ( )d . (2)
128 Fixed poin heo ems o mappings sa is ying a condi ion o ...
J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ... 599
Taking in o accoun ha
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 ob ain
m(xn−1, xn) = max{d(xn−1, xn), d(xn, xn+1)}.
Subs i u ing in o (2) we ob ain
Zd(xn,xn+1)
0
ϕ( )d ≤kZmax{d(xn−1,xn),d(xn,xn+1)}
0
ϕ( )d
=kmax (Zd(xn−1,xn)
0
ϕ( )d , Zd(xn,xn+1)
0
ϕ( )d ).(3)
I max nRd(xn−1,xn)
0ϕ( )d , Rd(xn,xn+1)
0ϕ( )d o=Rd(xn,xn+1)
0ϕ( ), hen, by (3),
Zd(xn,xn+1)
0
ϕ( )d ≤kZd(xn,xn+1)
0
ϕ( )d .
and, as k∈[0,1), we ha e ha Rd(xn,xn+1)
0ϕ( )d = 0. By ou hypo hesis abou ϕ, we
ge d(xn, xn+1) = 0, o , equi alen ly, xn=xn+1 =F(xn) and xnis a ixed poin o F.
I max nRd(xn−1,xn)
0ϕ( )d , Rd(xn,xn+1)
0ϕ( )d o=Rd(xn−1,xn)
0ϕ( ) hen, om (3), we ge
Zd(xn,xn+1)
0
ϕ( )d ≤kZd(xn−1,xn)
0
ϕ( )d . (4)
Using induc ion we ha e
Zd(xn,xn+1)
0
ϕ( )d ≤kZd(xn−1,xn)
0
ϕ( )d ≤ · · · ≤ knZd(x0,x1)
0
ϕ( )d .
Taking limi as n→ ∞
lim
n→∞ Zd(xn,xn+1)
0
ϕ( )d = 0.(5)
3. Fixed poin heo ems in pa ially o de ed me ic spaces 129

600 J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ...
On he o he hand, by (4), as k∈[0,1),
Zd(xn,xn+1)
0
ϕ( )d ≤kZd(xn−1,xn)
0
ϕ( )d < Zd(xn−1,xn)
0
ϕ( )d
and, as ϕis a non-nega i e unc ion, we ob ain ha {d(xn, xn+1)}is a non-nega i e
and non-inc easing sequence. We pu limn→∞ d(xn, xn+1) = a.
In wha ollows, we will p o e ha a= 0.
Suppose ha a > 0. As 0 < a ≤d(xn, xn+1) o all n, and, aking in o accoun ou
assump ion abou ϕ,
0<Za
0
ϕ( )d ≤Zd(xn,xn+1)
0
ϕ( )d .
Taking limi as n→ ∞ and, om (5),
0<Za
0
ϕ( )d ≤lim
n→∞ Zd(xn,xn+1)
0
ϕ( )d = 0,
which is a con adic ion. The e o e,
lim
n→∞ d(xn, xn+1) = 0.(6)
Now, we show ha {xn}is a Cauchy sequence.
Suppose ha {xn}is no a Cauchy sequence he e exis s an ε > 0 and subsequences
{m(p)}and {n(p)}such ha m(p)< n(p)< m(p+ 1) wi h
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 ha e
lim
p→∞ Zd(xm(p)−1,xm(p))
0
ϕ( )d = lim
p→∞ Zd(xn(p)−1,xn(p))
0
ϕ( )d = 0.(8)
By he iangula inequali y 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), his implies
lim
p→∞ Zd(xm(p)−1,xn(p)−1)
0
ϕ( )d ≤Zε
0
ϕ( )d . (9)
130 Fixed poin heo ems o mappings sa is ying a condi ion o ...
J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ... 601
Again, using he iangula inequali y and (7), we ge
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 in o accoun (6), we ob ain
lim
p→∞ Z1
2[d(xm(p)−1,xn(p))+d(xm(p),xn(p)−1)]
0
ϕ( )d ≤Zε
0
ϕ( )d . (10)
F om (1) and (7), we can ge
Zε
0
ϕ( )d ≤Zd(xm(p),xn(p))
0
ϕ( )d
=Zd(F(xm(p)−1),F (xn(p)−1))
0
ϕ( )d ≤kZm(xm(p)−1,xn(p)−1)
0
ϕ( )d
=kmax Zd(xm(p)−1,xn(p)−1)
0
ϕ( )d , Zd(xm(p)−1,xm(p))
0
ϕ( )d ,
Zd(xn(p)−1,xn(p))
0
ϕ( )d , Z1
2[d(xm(p)−1,xn(p))+d(xm(p),xn(p)−1)]
0
ϕ( )d !,
and, aking limi as p→ ∞, and aking in o accoun (8), (9) and (10), we ob ain
Zε
0
ϕ( )d ≤kZε
0
ϕ( )d .
As k∈[0,1), his implies Rε
0ϕ( )d = 0 which is a con adic ion.
The e o e, {xn}is a Cauchy sequence. Since Xis a comple e me ic space he e exis s
z∈Xsuch ha limn→∞ xn=z.
Finally, we p o e ha z∈Xis a ixed poin o F.
As Fis a con inuous mapping and limn→∞ xn=z, hen
z= lim
n→∞ xn+1 = lim
n→∞ F(xn) = F(z)
and he p oo is comple e.
In wha ollows, we p o e ha Theo em 2.2 is s ill alid o Fno necessa ily con inu-
ous, assuming he ollowing hypo hesis in X(which appea s in Theo em 1 o [1]):
i (xn)⊂Xis a nondec easing sequence wi h xn→x hen xn≤x o all n∈N.(11)
3. Fixed poin heo ems in pa ially o de ed me ic spaces 131
602 J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ...
Theo em 2.3. Le (X, ≤)be a pa ially o de ed se and suppose ha he e exis s a
me ic din Xsuch ha (X, d)is a comple e me ic space. Le F:X−→ Xbe a
nondec easing mapping such ha he e exis s k∈[0,1) wi h
Zd(F(x),F (y))
0
ϕ( )d ≤kZm(x,y)
0
ϕ( )d , o x≥y,
whe e ϕ:R+−→ R+is a Lebesgue-in eg able mapping such ha Rε
0ϕ( )d > 0 o
ε > 0. Assume ha Xsa is ies (11) and he e exis s x0∈Xwi h x0≤F(x0), hen F
has a ixed poin .
P oo . Following he p oo o Theo em 2.2, we only ha e o check ha F(z) = z.
F om (2) and (11), we ha e
Zd(F(z),xn+1)
0
ϕ( )d ≤kZm(z,xn)
0
ϕ( )d
=kmax (Zd(z,xn)
0
ϕ( )d , Zd(z,F (z))
0
ϕ( )d ,
Zd(xn+1,xn)
0
ϕ( )d , Z1
2[d(z,xn+1)+d(xn,F (z))]
0
ϕ( )d ),
and, aking limi as n→ ∞, and, by (5), we ge
Zd(F(z),z)
0
ϕ( )d ≤kZd(F(z),z)
0
ϕ( )d ,
which implies ha Rd(F(z),z)
0ϕ( )d = 0. By ou assump ion abou ϕ, his gi es us
d(F(z), z) = 0
and his p o es ha zis a ixed poin o F.
Rema k 2.4. I we assume ha ϕis a noninc easing unc ion in Theo em 2.2 i s p oo
is less complica ed.
In ac , pe haps he mo e di icul pa in Theo em 2.2 is o p o e ha {xn}is a Cauchy
sequence. Unde assump ion ha ϕis a noninc easing unc ion, o m > n we can ge
Zd(xm,xn)
0
ϕ( )d ≤Zd(xm,xm−1)+d(xm−1,xm−2)+···+d(xn+1,xn)
0
ϕ( )d
=Zd(xn+1,xn)
0
ϕ( )d +Zd(xn+2,xn+1)+d(xn+1,xn)
d(xn+1,xn)
ϕ( )d
+···+Zd(xn+1,xn)+···+d(xm−1,xm−2)+d(xm,xm−1)
d(xn+1,xn)+···+d(xm−1,xm−2)
ϕ( )d .
Applying a simple change o a iables, ou in eg als can be ans o med in
Zd(xm,xn)
0
ϕ( )d ≤
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 poin heo ems o mappings sa is ying a condi ion o ...
J. Ha jani, K. Sada angani / Fixed Poin Theo ems o Mappings Sa is ying ... 603
and, as ϕis a noninc easing unc ion, we can ge
Zd(xm,xn)
0
ϕ( )d
≤
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 in o accoun (4) in he p oo o Theo em 2.2, we ob ain
Zd(xm,xn)
0
ϕ( )d ≤
m
X
i=n+1 Zd(xi,xi−1)
0
ϕ( )d
≤
m
X
i=n+1
ki−1Zd(x0,x1)
0
ϕ( )d = Zd(x0,x1)
0
ϕ( )d !(kn+···+km−1)
≤ Zd(x0,x1)
0
ϕ( )d !kn
1−k.
Taking limi as n→ ∞ we ha e
lim
m,n→∞ Zd(xm.xn)
0
ϕ( )d = 0.(12)
Now, suppose ha {xn}is no a Cauchy sequence. This means ha he e exis s an
ε > 0 such ha o any p∈Nwe can ind m(p), n(p)∈Nwi h m(p), n(p)> p sa is ying
d(xm(p),xn(p))≥ε. Consequen ly,
Zd(xm(p),xn(p))
0
ϕ( )d ≥Zε
0
ϕ( )d > 0,
and, aking limi as p→ ∞, we ge
lim
p→∞ Zd(xm(p),xn(p))
0
ϕ( )d ≥Zε
0
ϕ( )d > 0
and his con adic s o (12).
Rema k 2.5. I we pu ϕ( ) = 1 in (1) o Theo em 2.2, we ha e
d(F(x), F(y)) ≤k m(x, y) o x≥y
and ou Theo em 2.2 is a pa icula case o Theo em 2.2 o [1] o he unc ion ψ( ) = k
wi h k∈[0,1).
Rema k 2.6. I we pu ϕ( ) = 1 in (1) o Theo em 2.2 hen he condi ion d(F(x), F(y))
≤kd(x, y) o x≥yimplies d(F(x), F (y)) ≤k m(x, y) and Theo em 2.1 in [11] and
Theo em 2.1 in [16] a e pa icula cases o ou Theo em 2.2.
3. Fixed poin heo ems in pa ially o de ed me ic spaces 133

Theo y o exis ence and uniqueness o solu ion o b p 237
4.2. F ac ional bounda y alue p oblem
4.2.1. Exis ence and uniqueness o posi i e and non-
dec easing solu ions o a class o singula ac-
ional bounda y alue p oblems
In his pape , we discuss he exis ence and uniqueness o a posi i e and
nondec easing solu ion o he nex ac ional bounda y alue p oblem
(Dα
0+u( ) +  , u( )= 0 ,0< < 1,
u(0) = u0(1) = u00(0) = 0 ,(4.8)
whe e 2 < α ≤3, Dα
0+is he Capu o’s di e en ia ion and : (0,1]×[0,∞)→
[0,∞) wi h lim
→0+ ( , −) = ∞, i.e., is singula a = 0. Ou main ool in
his pape is he ollowing ixed poin heo em in pa ially o de ed me ic
spaces, which is he main esul o he i s pape o Sec ion 1.
Theo em 7. Le (X, ≤)be a pa ially o de ed se and suppose ha he e
exis s a me ic din Xsuch ha (X, d)is a comple e me ic space. Assume
ha Xsa is ies he ollowing condi ion:
i (xn)is a nondec easing sequence in Xsuch ha xn→x
hen xn≤x, o all n∈N.
Le T:X→Xbe a nondec easing mapping such ha
d(Tx, Ty)≤d(x, y)−ψd(x, y), o any x, y ∈Xwi h x≥y ,
whe e ψ)is an al e ing dis ance unc ion.
I he e exis s x0∈Xwi h x0≤Tx0 hen Thas a ixed poin .
Besides, i o any x, y ∈X he e exis s z∈Xwhich is compa able o xand
y, hen he ixed poin is unique.
The i s esul o he pape is:
238 F ac ional bounda y alue p oblem
Theo em 49. Conside P oblem (4.8), whe e 2< α ≤3, unde hese as-
sump ions:
(i) : (0,1]×[0,∞)→[0,∞)is con inuous and sa is ies lim
→0+ ( , −) = ∞.
(ii) The unc ion σ ( , y)is a con inuous unc ion on [0,1]×[0,∞), whe e
0< σ < 1.
(iii) The e exis s 0< λ ≤Γ(α−σ)
Γ(1−σ)such ha , o any x, y ∈[0,∞)wi h y≥x
and ∈[0,1],
0≤ σ ( , y)− ( , x)≤λ·ln(y−x+ 1) .
Then P oblem (4.8) has a unique nonnega i e solu ion.
We no ice in he pape ha Theo em 49 emains alid i we eplace as-
sump ion (iii) by
(iii)’ The e exis s 0 < λ ≤Γ(α−σ)
Γ(1−σ)such ha , o any x, y ∈[0,∞) wi h y≥x
and ∈[0,1],
0≤ σ ( , y)− ( , x)≤λ·ψ(y−x+ 1) ,
whe e ψ: [0,∞)→[0,∞) is a con inuous unc ion such ha ϕ(x) = x−ψ(x)
sa is ies:
(a) ϕ: [0,∞)→[0,∞) is nondec easing.
(b) ϕ(0) = 0.
(c) ϕis posi i e in (0,∞).
Nex , we p obe ha he G een’s unc ion G( , s) associa ed o P oblem (4.8)
which is gi en by
G( , s) = 






(α−1) (1 −s)α−2−( −s)α−1
Γ(α),0≤s≤ ≤1,
(1 −s)α−2
Γ(α−1) ,0≤ ≤s≤1,
Theo y o exis ence and uniqueness o solu ion o b p 239
is s ic ly inc easing in he i s a iable in (0,1).
Ou main esul is he ollowing.
Theo em 50. Unde assump ion o Theo em 49, P oblem (4.8) has a unique
s ic ly inc easing and posi i e solu ion.
Ou esul s can be compa ed wi h he ones ob ained by T. Qiu, Z. Bai,
Exis ence o posi i e solu ions o singula ac ional di e en ial equa ions,
Elec . J. Di . Eq. ol. (2008), 146, 2008, 1–9, whe e he au ho s s udy he
same p oblem, bu hei esul s do no gi e uniqueness o he solu ion and
nei he mono onic cha ac e o his solu ion.
Ou main con ibu ion in his pape is he uniqueness o he solu ion and ,
mo eo e , he s ic ly inc easing cha ac e o his solu ion.
Hindawi Publishing Co po a ion
Bounda y Value P oblems
Volume 2009, A icle ID 421310, 10 pages
doi:10.1155/2009/421310
Resea ch A icle
Exis ence and Uniqueness o Posi i e and
Nondec easing Solu ions o a Class o Singula
F ac ional Bounda y Value P oblems
J. Caballe o Mena, J. Ha jani, and K. Sada angani
Depa amen o de Ma em´
a icas, Uni e sidad de Las Palmas de G an Cana ia, Campus de Ta i a Baja,
35017 Las Palmas de G an Cana ia, Spain
Co espondence should be add essed o K. Sada angani, [email p o ec ed]
Recei ed 24 Ap il 2009; Accep ed 14 June 2009
Recommended by Juan Jos´
e Nie o
We es ablish he exis ence and uniqueness o a posi i e and nondec easing solu ion o a singula
bounda y alue p oblem o a class o nonlinea ac ional diffe en ial equa ion. Ou analysis elies
on a ixed poin heo em in pa ially o de ed se s.
Copy igh q2009 J. Caballe o Mena e al. This is an open access a icle dis ibu ed unde he
C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and
ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
1. In oduc ion
Many pape s and books on ac ional diffe en ial equa ions ha e appea ed ecen ly. Mos
o hem a e de o ed o he sol abili y o he linea ac ional equa ion in e ms o a special
unc ion see, e.g., 1,2 and o p oblems o analy ici y in he complex domain 3. Mo eo e ,
Delbosco and Rodino 4conside ed he exis ence o a solu ion o he nonlinea ac ional
diffe en ial equa ion Dα
0u  , u, whe e 0 <α<1and :0,a×R→R,0<a≤∞
is a gi en con inuous unc ion in 0,a×R. They ob ained esul s o solu ions by using
he Schaude ixed poin heo em and he Banach con ac ion p inciple. Recen ly, Zhang 5
conside ed he exis ence o posi i e solu ion o equa ion Dα
0u  , u, whe e 0 <α<1
and :0,1×0,∞→0,∞is a gi en con inuous unc ion by using he sub- and supe -
solu ion me hods.
In his pape , we discuss he exis ence and uniqueness o a posi i e and nondec easing
solu ion o bounda y- alue p oblem o he nonlinea ac ional diffe en ial equa ion
Dα
0u   , u  0,0< <1,
u0u1u00,
1.1
Theo y o exis ence and uniqueness o solu ion o b p 241

2 Bounda y Value P oblems
whe e 2 <α≤3, Dα
0is he Capu o’s diffe en ia ion and :0,1×0,∞→0,∞wi h
lim →0  , −∞i.e., is singula a 0.
No e ha his p oblem was conside ed in 6whe e he au ho s p o ed he exis ence
o one posi i e solu ion o 1.1by using K asnoselskii’s ixed poin heo em and nonlinea
al e na i e o Le ay-Schaude ype in a cone and assuming ce ain hypo heses on he unc ion
.In6 he uniqueness o he solu ion is no ea ed.
In his pape we will p o e he exis ence and uniqueness o a posi i e and
nondec easing solu ion o he p oblem 1.1by using a ixed poin heo em in pa ially
o de ed se s.
Exis ence o ixed poin in pa ially o de ed se s has been conside ed ecen ly in 7–12.
This wo k is inspi ed in he pape s 6,8.
Fo exis ence heo ems o ac ional diffe en ial equa ion and applica ions, we e e o
he su ey 13. Conce ning he de ini ions and basic p ope ies we e e he eade o 14.
Recen ly, some exis ence esul s o ac ional bounda y alue p oblem ha e appea ed
in he li e a u e see, e.g., 15–17.
2. P elimina ies and P e ious Resul s
Fo he con enience o he eade , we p esen he e some no a ions and lemmas ha will be
used in he p oo s o ou main esul s.
De ini ion 2.1. The Riemman-Liou ille ac ional in eg al o o de α>0o a unc ion :
0,∞→Ris gi en by
Iα
0  1
Γα
0
 −sα−1 sds 2.1
p o ided ha he igh -hand side is poin wise de ined on 0,∞.
De ini ion 2.2. The Capu o ac ional de i a i e o o de α>0 o a con inuous unc ion :
0,∞→Ris gi en by
Dα
0  1
Γn−α
0
ns
 −sα−n1ds, 2.2
whe e n−1<α≤n, p o ided ha he igh -hand side is poin wise de ined on 0,∞.
The ollowing lemmas appea in 14.
Lemma 2.3. Le n−1<α≤n,u∈Cn0,1.Then
Iα
0Dα
0u u −c1−c2 −···−cn n−1,2.3
whe e ci∈R,i1,2,...,n.
242 F ac ional bounda y alue p oblem
Bounda y Value P oblems 3
Lemma 2.4. The ela ion
Iα
0Iβ
0ϕIαβ
0ϕ2.4
is alid when Re β>0,Reαβ>0,ϕx∈L10,b.
The ollowing lemmas appea in 6.
Lemma 2.5. Gi en ∈C0,1and 2<α≤3, he unique solu ion o
Dα
0u   0,0< <1,
u0u1u00,
2.5
is gi en by
u 1
0
G , s sds, 2.6
whe e
G , s⎧
⎪
⎪
⎨
⎪
⎪
⎩
α−1 1−sα−2− −sα−1
Γα,0≤s≤ ≤1,
1−sα−2
Γα−1,0≤ ≤s≤1.
2.7
Rema k 2.6. No e ha G , s>0 o /
0andG0,s0see 6.
Lemma 2.7. Le 0<σ<1,2<α≤3and F:0,1→Ris a con inuous unc ion wi h
lim →0F ∞. Suppose ha σF is a con inuous unc ion on 0,1. Then he unc ion de ined
by
H 1
0
G , sFsds 2.8
is con inuous on [0,1], whe e G , sis he G een unc ion de ined in Lemma 2.5.
Now, we p esen some esul s abou he ixed poin heo ems which we will use la e .
These esul s appea in 8.
Theo em 2.8. Le X, ≤be a pa ially o de ed se and suppose ha he e exis s a me ic din Xsuch
ha X, dis a comple e me ic space. Assume ha Xsa is ies he ollowing condi ion: i {xn}is a
non dec easing sequence in Xsuch ha xn→x hen xn≤x o all n∈N.Le T:X→Xbe a
nondec easing mapping such ha
dTx,Ty≤dx, y−ψdx, y, o x≥y, 2.9
Theo y o exis ence and uniqueness o solu ion o b p 243
4 Bounda y Value P oblems
whe e ψ:0,∞→0,∞is con inuous and nondec easing unc ion such ha ψis posi i e in
0,∞,ψ00and lim →∞ψ ∞. I he e exis s x0∈Xwi h x0≤Tx0 hen Thas a ixed
poin .
I we conside ha X, ≤sa is ies he ollowing condi ion:
o x,y ∈X he e exis s z∈Xwhich is compa able o xand y, 2.10
hen we ha e he ollowing heo em 8.
Theo em 2.9. Adding condi ion 2.10 o he hypo heses o Theo em 2.8 one ob ains uniqueness o
he ixed poin o .
In ou conside a ions, we will wo k in he Banach space C0,1{x:0,1→
R,con inuous}wi h he s anda d no m xmax0≤ ≤1|x |.
No e ha his space can be equipped wi h a pa ial o de gi en by
x,y ∈C0,1,x≤y⇐⇒ x ≤y , o ∈0,1.2.11
In 10i is p o ed ha C0,1,≤wi h he classic me ic gi en by
dx,ymax
0≤ ≤1x −y 2.12
sa is ies condi ion 2o Theo em 2.8. Mo eo e , o x, y ∈C0,1, as he unc ion max{x,y}
is con inuous in 0,1,C0,1,≤sa is ies condi ion 2.10.
3. Main Resul
Theo em 3.1. Le 0<σ<1,2<α≤3, :0,1×0,∞→0,∞is con inuous and
lim →0  , −∞, σ  , yis a con inuous unc ion on 0,1×0,∞. Assume ha he e exis s
0<λ≤Γα−σ/Γ1−σsuch ha o x, y ∈0,∞wi h y≥xand ∈0,1
0≤ σ  , y−  , x≤λ·lny−x13.1
Then one’s p oblem 1.1has an unique nonnega i e solu ion.
P oo . Conside he cone
P{u∈C0,1:u ≥0}.3.2
No e ha , as Pis a closed se o C0,1,Pis a comple e me ic space.
244 F ac ional bounda y alue p oblem
Bounda y Value P oblems 5
Now, o u∈Pwe de ine he ope a o Tby
Tu 1
0
G , s s, usds. 3.3
By Lemma 2.7,Tu ∈C0,1. Mo eo e , aking in o accoun Rema k 2.6 and as σ  , y≥0
o  , y∈0,1×0,∞by hypo hesis, we ge
Tu 1
0
G , ss−σsσ s, usds ≥0.3.4
Hence, TP⊂P.
In wha ollows we check ha hypo heses in Theo ems 2.8 and 2.9 a e sa is ied.
Fi s ly, he ope a o Tis nondec easing since, by hypo hesis, o u≥
Tu 1
0
G , s s, usds
1
0
G , ss−σsσ s, usds
≥1
0
G , ss−σsσ s, sds T  .
3.5
Besides, o u≥
dTu,T max
∈0,1|Tu −T  |
max
∈0,1Tu −T   max
∈0,11
0
G , s s, us − s, sds
max
∈0,11
0
G , ss−σsσ s, us − s, sds
≤max
∈0,11
0
G , ss−σλ·lnus− s1ds
3.6
As he unc ion ϕxlnx1is nondec easing hen, o u≥ ,
lnus− s1≤lnu− 13.7
Theo y o exis ence and uniqueness o solu ion o b p 245
252 F ac ional bounda y alue p oblem
y, hen Thas a unique ixed poin .
Be o e o p esen he main esul o he pape , we p o e some lemmas.
The G een’s unc ion associa ed o P oblem (4.9) is gi en by
G( , s) = 






α−1(1 −s)α−1−( −s)α−1
Γ(α),0≤s≤ ≤1,
α−1(1 −s)α−1
Γ(α),0≤ ≤s≤1,
being Γ he gamma unc ion.
Lemma 3. Suppose ha 0< σ < 1,1< α ≤2and F: (0,1] →Ris a
con inuous unc ion such ha lim
→0+F( ) = ∞.
I σF( )is a con inuous unc ion on [0,1] hen he unc ion
H( ) = Z1
0
G( , s)F(s)ds ,
is con inuous on [0,1].
Lemma 4. Assume ha 0< σ < 1. Then,
max
0≤ ≤1Z1
0
G( , s)s−σds=Aα−1−Aα−σ
Γ(α)β(1 −σ, α),
whe e A=α−1
α−σ1
1−σand βis he Eule be a unc ion.
By commodi y, we deno e by K=Aα−1−Aα−σ
Γ(α)β(1 −σ, α).
Fo ou main esul , we need he class o unc ions Agi en by φ∈ A i
φ: [0,∞)→[0,∞) and i sa is ies
(i) φis nondec easing.
(ii) φ(x)< x, o any x > 0.
(iii) β(x) = φ(x)
xis such ha β( n)→1⇒ n→0.
The main esul o he pape is he nex heo em.

Theo y o exis ence and uniqueness o solu ion o b p 253
Theo em 52. Suppose ha 0< σ < 1and 1< α ≤2. Unde he ollowing
assump ions:
(i) : (0,1] ×[0,∞)→[0,∞)is a con inuous unc ion such ha
lim
→0+ ( , −) = ∞,
(ii) σ ( , y)is a con inuous unc ion on [0,1] ×[0,∞).
(iii) The e exis 0< λ ≤1
Ksuch ha , o x, y ∈[0,∞)wi h y≥xand
∈[0,1],
0≤ σ( ( , y)− ( , x)) ≤λφ(y−x),
whe e φ∈ A,
P oblem (4.9) has a unique posi i e solu ion.
Finally, we p esen an example illus a ing ou esul s.
Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332
Con en s lis s a ailable a ScienceDi ec
Compu e s and Ma hema ics wi h Applica ions
jou nal homepage: www.else ie .com/loca e/camwa
Posi i e solu ions o a class o singula ac ional bounda y
alue p oblems✩
J. Caballe o∗, J. Ha jani, K. Sada angani
Depa amen o de Ma emá icas, Uni e sidad de Las Palmas de G an Cana ia, Campus de Ta i a Baja, 35017 Las Palmas de G an Cana ia, Spain
a icle in o
Keywo ds:
F ac ional bounda y alue p oblem
Fixed poin heo em
Posi i e solu ion
abs ac
In his pape , we in es iga e he exis ence and uniqueness o posi i e solu ions o he
ollowing singula ac ional bounda y alue p oblem
Dα
0+u( )+ ( ,u( )) =0,0< <1,
u(0)=u(1)=0,
whe e 1 < α ≤2, Dα
0+is he s anda d Riemann–Liou ille di e en ia ion and :(0,1] ×
[0,∞)−→ [0,∞)wi h lim →0+ ( ,−)= ∞ (i.e., is singula a =0). Ou analysis
elies on a ixed poin heo em in pa ially o de ed se s.
©2011 Else ie L d. All igh s ese ed.
1. In oduc ion
Many pape s and books on ac ional di e en ial equa ions ha e appea ed ecen ly (see, o example, [1–12]). Mos o
hem a e de o ed o he sol abili y o linea ac ional equa ions in e ms o a special unc ion (see, e.g., [3,8]) and o p oblems
o analy ici y in he complex domain [7]. Mo eo e , Delbosco and Rodino [4] conside ed he exis ence o a solu ion o he
nonlinea ac ional di e en ial equa ion Dα
0+u= ( ,u), whe e 0 < α < 1 and :[0,a] × R−→ R,0<a≤ +∞
is a gi en con inuous unc ion in (0,a)×R. They ob ained hei esul s by using he Schaude ixed poin heo em and
he Banach con ac ion p inciple. Zhang [11] conside ed he exis ence o posi i e solu ion o he equa ion Dα
0+u= ( ,u),
whe e 0 < α < 1 and :[0,1]×[0,∞)−→ [0,∞), is a gi en con inuous unc ion by using he sub- and supe -solu ion
me hods.
Recen ly, Bai and Lü [1] ha e in es iga ed he exis ence and mul iplici y o posi i e solu ions o he bounda y alue
p oblem
Dα
0+u( )+ ( ,u( )) =0,0< <1
u(0)=u(1)=0,(1)
whe e 1 < α ≤2 and :[0,1]×[0,∞)−→ [0,∞)is con inuous, by using some ixed poin heo ems on cones.
Mo i a ed by [1], in his pape we discuss he exis ence and uniqueness o posi i e solu ions o P oblem (1) assuming
ha :(0,1]×[0,∞)−→ [0,∞)is such ha lim →0+ ( ,−)= ∞(i.e., is singula a =0).
Ou s udy is based on a ixed poin heo em in pa ially o de ed se s. The exis ence o ixed poin s in pa ially o de ed
se s has been conside ed ecen ly in [13–17]. This wo k is inspi ed by pape s [13,2,9].
Fo exis ence heo ems o ac ional di e en ial equa ions and applica ions, we e e o su eys [5,8]. Conce ning he
de ini ions and basic p ope ies, we e e he eade o [10].
✩This esea ch was pa ially suppo ed by ‘‘Minis e io de Educación y Ciencia’’ P ojec MTM 2007/65706.
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (J. Caballe o), [email p o ec ed] (J. Ha jani), [email p o ec ed] (K. Sada angani).
0898-1221/$ – see on ma e ©2011 Else ie L d. All igh s ese ed.
doi:10.1016/j.camwa.2011.04.013
Theo y o exis ence and uniqueness o solu ion o b p 255
1326 J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332
2. P elimina ies and basic ac s
Fo he con enience o he eade , we p esen he e some no a ion and lemmas which will be used in he p oo s o ou
esul s.
De ini ion 1. The Riemann–Liou ille ac ional in eg al o o de α > 0 o a unc ion :(0,∞)→Ris de ined by
Iα
0+ ( )=1
Γ(α) ∫
0
( −s)α−1 (s)ds,
p o ided ha he igh -hand side is poin wise de ined on (0,∞), and whe e Γ(α) deno es he classical gamma unc ion.
De ini ion 2. The Riemann–Liou ille ac ional de i a i e o o de α > 0 o a unc ion :(0,∞)→Ris gi en by
Dα
0+ ( )=1
Γ(n−α) d
d n∫
0
(s)
( −s)α−n+1ds,
whe e n= [α]+1 and [α]deno es he in ege pa o α.
The ollowing wo lemmas can be ound in [10].
Lemma 1. Le α > 0and u ∈C(0,1)∩L1(0,1). Then he ac ional di e en ial equa ion
Dα
0+u( )=0
has
u( )=c1 α−1+c2 α−2+···+cn α−n,
whe e ci∈R(i=1,2,...,n)and n = [α]+1as unique solu ion.
Lemma 2. Assume ha u ∈C(0,1)∩L1(0,1)wi h ac ional de i a i e o o de α > 0 ha belongs o C(0,1)∩L1(0,1). Then
Iα
0+Dα
0+u( )=u( )+c1 α−1+c2 α−2+···+cn α−n,
o some ci∈R(i=1,...,n)and n = [α]+1.
Using Lemma 2, in [1] he ollowing esul is p o ed.
Lemma 3. Gi en ∈C[0,1]and 1< α ≤2, he unique solu ion o
Dα
0+u( )+ ( )=0,0< <1,
u(0)=u(1)=0,
is
u( )=∫1
0
G( ,s) (s)ds,
whe e
G( ,s)=






α−1(1−s)α−1−( −s)α−1
Γ(α) ,0≤s≤ ≤1
α−1(1−s)α−1
Γ(α) ,0≤ ≤s≤1.
Rema k 1. I is easily checked ha G( ,s)is a con inuous unc ion on [0,1]×[0,1]and i sa is ies G( ,s) > 0, o
,s∈(0,1).
In wha ollows, we p esen he ixed poin heo em which we will use la e . This esul appea s in [13].
By Jwe deno e he class o hose unc ions β:[0,∞)−→ [0,1)sa is ying he ollowing condi ion
β( n)→1 implies n→0.
Theo em 1 (Theo em 2.1 o [13]).Le (X,≤)be a pa ially o de ed se and suppose ha he e exis s a me ic d in X such ha
(X,d)is a comple e me ic space. Le T:X−→ X be a nondec easing mapping such ha he e exis s an elemen x0∈X wi h
x0≤Tx0. Suppose ha he e exis s β∈Jsuch ha
d(Tx,Ty)≤β(d(x,y)) ·d(x,y), o x,y∈X wi h x ≥y.
256 F ac ional bounda y alue p oblem
J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332 1327
Assume ha ei he T is con inuous o X is such ha
i {xn}is a nondec easing sequence in X such ha xn→x hen xn≤x o all n ∈N.(2)
Besides, i
o each x,y∈X he e exis s z ∈X which is compa able o x and y,(3)
hen T has a unique ixed poin .
In ou conside a ions, we will wo k in he Banach space C[0,1]={x: [0,1] → R,con inuous}wi h he classical me ic
gi en by d(x,y)=sup0≤ ≤1{|x( )−y( )|}.
No ice ha his space can be equipped wi h a pa ial o de gi en by
x,y∈C[0,1],x≤y⇔x( )≤y( ), o ∈ [0,1].
In [15] i is p o ed ha (C[0,1],≤)sa is ies condi ion (2) o Theo em 1. Mo eo e , o x,y∈C[0,1], as he unc ion
max(x,y)∈C[0,1], (C[0,1],≤)sa is ies condi ion (3).
3. Main esul
Ou s a ing poin o his sec ion is he ollowing lemma.
Lemma 4. Le 0< σ < 1,1< α ≤2and F :(0,1] −→ Ris a con inuous unc ion wi h lim →0+F( )= ∞. Suppose ha
σF( )is a con inuous unc ion on [0,1]. Then he unc ion de ined by
H( )=∫1
0
G( ,s)F(s)ds
is con inuous on [0,1], whe e G( ,s)is he G een unc ion appea ing in Lemma 3.
P oo . We di ide he p oo in o h ee cases.
Case 1: 0=0.
I is easily checked ha H(0)=0. Since σF( )is con inuous on [0,1], we can ind a cons an M>0 such ha
| σF( )| ≤ M o any ∈ [0,1]. Hence
|H( )−H(0)| = |H( )| = ∫1
0
G( ,s)F(s)ds
=∫1
0
G( ,s)s−σsσF(s)ds
=∫
0
α−1(1−s)α−1−( −s)α−1
Γ(α) s−σsσF(s)ds+∫1
α−1(1−s)α−1
Γ(α) s−σsσF(s)ds
=∫1
0
α−1(1−s)α−1
Γ(α) s−σsσF(s)ds−∫
0
( −s)α−1
Γ(α) s−σsσF(s)ds
≤∫1
0
α−1(1−s)α−1
Γ(α) s−σsσF(s)ds+∫
0
( −s)α−1
Γ(α) s−σsσF(s)ds
≤M∫1
0
α−1(1−s)α−1
Γ(α) s−σds+M∫
0
( −s)α−1
Γ(α) s−σds
=M α−1
Γ(α) ∫1
0
(1−s)α−1s−σds+M
Γ(α) ∫
0
( −s)α−1s−σds
=M α−1
Γ(α) ∫1
0
(1−s)α−1s−σds+M α−1
Γ(α) ∫
01−s
α−1
s−σds.(4)
I in he in eg al 
01−s
α−1s−σdswe make he change o a iables u=s
hen we ob ain
∫
01−s
α−1
s−σds= 1−σ∫1
0
(1−u)α−1u−σdu.
Theo y o exis ence and uniqueness o solu ion o b p 257

1328 J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332
By aking (4) in o accoun ,
|H( )| ≤ M α−1
Γ(α) ∫1
0
(1−s)α−1s−σds+M α−1
Γ(α) 1−σ∫1
0
(1−u)α−1u−σdu
=M α−1
Γ(α) +M α−σ
Γ(α) ·β(1−σ , α),
whe e βdeno es he be a unc ion.
In he las exp ession, when →0 we see ha |H( )| → 0 and his p o es he con inui y o Ha 0=0.
Case 2: 0∈(0,1).
We ake n→ 0and we ha e o p o e ha H( n)→H( 0). Wi hou loss o gene ali y, we conside n> 0( he
same a gumen wo ks o n< 0).
In ac ,
|H( n)−H( 0)| = ∫ n
0
α−1
n(1−s)α−1−( n−s)α−1
Γ(α) s−σsσF(s)ds
+∫1
n
α−1
n(1−s)α−1
Γ(α) s−σsσF(s)ds−∫1
0
α−1
0(1−s)α−1
Γ(α) s−σsσF(s)ds
−∫ 0
0
α−1
0(1−s)α−1−( 0−s)α−1
Γ(α) s−σsσF(s)ds
=∫1
0
α−1
n(1−s)α−1
Γ(α) s−σsσF(s)ds−∫ n
0
( n−s)α−1
Γ(α) s−σsσF(s)ds
−∫1
0
α−1
0(1−s)α−1
Γ(α) s−σsσF(s)ds+∫ 0
0
( 0−s)α−1
Γ(α) s−σsσF(s)ds
=∫1
0
( α−1
n− α−1
0)(1−s)α−1
Γ(α) s−σsσF(s)ds−
−∫ 0
0
( n−s)α−1−( 0−s)α−1
Γ(α) s−σsσF(s)ds−∫ n
0
( n−s)α−1
Γ(α) s−σsσF(s)ds
≤M·( α−1
n− α−1
0)
Γ(α) ∫1
0
(1−s)α−1s−σds+M
Γ(α) ∫ 0
0
(( n−s)α−1−( 0−s)α−1)s−σds
+M
Γ(α) ∫ n
0
( n−s)α−1s−σds
≤M( α−1
n− α−1
0)
Γ(α) β(1−σ , α) +M
Γ(α)I1
n+M
Γ(α)I2
n,(5)
whe e
I1
n=∫ 0
0
(( n−s)α−1−( 0−s)α−1)s−σds
I2
n=∫ n
0
( n−s)α−1s−σds.
We claim ha I1
n→0 when n→ ∞.
In ac , as n→ 0, hen
(( n−s)α−1−( 0−s)α−1)s−σ−→ 0,when n→ ∞.
Mo eo e ,
(( n−s)α−1−( 0−s)α−1)s−σ≤(| n−s|α−1+| 0−s|α−1)s−σ≤2s−σ
and, as
∫1
0
2s−σds=2s−σ+1
−σ+1]1
0=2
1−σ<∞,
258 F ac ional bounda y alue p oblem
J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332 1329
we ha e ha he sequence (( n−s)α−1−( 0−s)α−1)s−σcon e ges poin wise o he ze o unc ion and |( n−s)α−1−
( 0−s)α−1|s−σis bounded by a unc ion belonging o L1[0,1], hen by Lebesgue’s domina ed con e gence heo em
I1
n→0 when n→ ∞.(6)
This p o es he claim.
Now, we p o e ha I2
n→0, when n→ ∞.
In ac , as
I2
n=∫ n
0
( n−s)α−1s−σds
≤∫ n
0
s−σds=1
1−σ( 1−σ
n− 1−σ
0)
and, aking in o accoun ha n→ 0, om he las exp ession we ge
I2
n→0,when n→ ∞.(7)
Finally, om (5)–(7) we ob ain
|H( n)−H( 0)| −→ 0,when n→ ∞.
Case 3: 0=1.
I is easily checked ha H(1)=0. Following he same lines in he p oo o Case 1, we can p o e he con inui y o H
a 0=1. 
Lemma 5. Suppose ha 0< σ < 1. Then,
max
0≤ ≤1∫1
0
G( ,s)s−σds=Aα−1−Aα−σ
Γ(α) β(1−σ , α),
whe e G( ,s)is he G een unc ion appea ing in Lemma 3and A =α−1
α−σ1
1−σ.
P oo . In ac , aking in o accoun Case 1 o Lemma 4, we ge
∫1
0
G( ,s)s−σds=∫
0
α−1(1−s)α−1−( −s)α−1
Γ(α) s−σds+∫1
α−1(1−s)α−1
Γ(α) s−σds
=∫1
0
α−1(1−s)α−1
Γ(α) s−σds−∫
0
( −s)α−1
Γ(α) s−σds
= α−1
Γ(α) ∫1
0
(1−s)α−1s−σds−1
Γ(α) ∫
0
( −s)α−1s−σds
= α−1
Γ(α)β(1−σ , α) − α−σ
Γ(α)β(1−σ , α) = α−1− α−σ
Γ(α) ·β(1−σ , α).
Now, using elemen al calculus we can p o e ha he unc ion ( )= α−1− α−σhas a maximum a he poin 0=A=
α−1
α−σ1
1−σ.
This says us ha
max
0≤ ≤1∫1
0
G( ,s)s−σds=Aα−1−Aα−σ
Γ(α) β(1−σ , α). 
Now, we in oduce he ollowing class o unc ions. By Awe deno e he class o unc ions φ:[0,∞)−→ [0,∞)
sa is ying:
(i) φis nondec easing,
(ii) φ(x) < x, o any x>0,
(iii) β(x)=φ(x)
x∈J, whe e Jis he class o unc ions appea ing in Theo em 1.
Examples o unc ions φ∈Aa e φ(x)=µx, wi h 0 ≤µ < 1, φ(x)=x
1+xand φ(x)=ln(1+x).
Deno e by K he cons an appea ing in Lemma 5, i.e.,
K=max
0≤ ≤1∫1
0
G( ,s)s−σds=Aα−1−Aα−γ
Γ(α) β(1−σ , α).
In wha ollows, we p esen ou main esul .
Theo y o exis ence and uniqueness o solu ion o b p 259
1330 J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332
Theo em 2. Le 0< σ < 1,1< α ≤2, :(0,1] × [0,∞)−→ [0,∞)is con inuous and lim →0+ ( ,·)= ∞, σ ( ,y)
is a con inuous unc ion on [0,1] × [0,∞). Assume ha he e exis s 0< λ ≤1
Ksuch ha , o x,y∈ [0,∞)wi h y ≥x and
∈ [0,1],
0≤ σ( ( ,y)− ( ,x)) ≤λφ(y−x),
whe e φ∈A.
Then, P oblem (1) has a unique posi i e solu ion ( his means ha x( ) > 0, o ∈(0,1)).
P oo . Conside he cone
P= {u∈C[0,1]:u( )≥0}.
No ice ha , as Pis a closed se o C[0,1],Pis a comple e me ic space. I is easily checked ha Psa is ies condi ions (2) and
(3) o Theo em 1.
Now, o u∈Pwe de ine he ope a o Tby
(Tu)( )=∫1
0
G( ,s) (s,u(s))ds=∫1
0
G( ,s)s−σsσ (s,u(s))ds.
By Lemma 4,Tu ∈C[0,1]. Mo eo e , in iew o nonnega i eness o G( ,s)and σ ( ,y), o u∈Pwe ha e Tu ∈P. Hence,
T:P−→ P.
In wha ollows, we check ha assump ions in Theo em 1 a e sa is ied.
Fi s , he ope a o Tis nondec easing. In ac , aking in o accoun ou assump ion, o u≥ we ha e
(Tu)( )=∫1
0
G( ,s) (s,u(s))ds=∫1
0
G( ,s)s−σsσ (s,u(s))ds
≥∫1
0
G( ,s)s−σsσ (s, (s))ds=(T )( ).
Besides, o u≥ and u=
d(Tu,T ) =max
∈[0,1]|(Tu)( )−(T )( )|
=max
∈[0,1]((Tu)( )−(T )( )) =max
∈[0,1][∫1
0
G( ,s)( (s,u(s)) − (s, (s)))ds]
=max
∈[0,1][∫1
0
G( ,s)s−σsσ( (s,u(s)) − (s, (s)))ds]
≤max
∈[0,1][∫1
0
G( ,s)s−σλφ(u(s)− (s))ds].
Taking in o accoun ha φis nondec easing, om las inequali y we ge
d(Tu,T ) ≤max
∈[0,1][∫1
0
G( ,s)s−σλφ(u(s)− (s))ds]
≤max
∈[0,1][∫1
0
G( ,s)s−σλφ(d(u, ))ds]
=λφ(d(u, )) max
∈[0,1]∫1
0
G( ,s)s−σds.
Now, Lemma 5 and he ac ha 0 < λ ≤Kgi e us
d(Tu,T ) ≤λφ(d(u, )) ·max
∈[0,1]∫1
0
G( ,s)s−σds≤φ(d(u, ))
=φ(d(u, ))
d(u, ) ·d(u, ) =β(d(u, )) ·d(u, ).
Ob iously, he las inequali y is sa is ied o u= .
Now, aking in o accoun ha he ze o unc ion sa is ies 0 ≤T0, Theo em 1 says us ha he ope a o Thas a unique
ixed poin in K, o , equi alen ly, P oblem (1) has a unique nonnega i e solu ion x( )∈C[0,1].
In wha ollows, we will p o e ha x( )is posi i e solu ion.
260 F ac ional bounda y alue p oblem
J. Caballe o e al. / Compu e s and Ma hema ics wi h Applica ions 62 (2011) 1325–1332 1331
In con a y case, he e exis s 0 < ∗<1 such ha x( ∗)=0. As he nonnega i e solu ion x( )o P oblem (1) is a ixed
poin o he ope a o T, his says us ha
x( )=∫1
0
G( ,s) (s,x(s))ds, o 0 < <1,
and, pa icula ly,
x( ∗)=∫1
0
G( ∗,s) (s,x(s))ds=0.
The nonnega i e cha ac e o G( ,s)and (s,u)and he las ela ion gi e
G( ∗,s)· (s,x(s)) =0 a.e. (s). (8)
Taking in o accoun lim →0+ ( ,0)= ∞ means ha o M>0, we can ind δsuch ha , o s∈ [0,1] ∩ (0, δ) we ha e
(s,0) > M. Obse e ha [0,1]∩(0, δ) ⊂ {s∈ [0,1] : (s,x(s)) > M}and µ([0,1]∩(0, δ)) > 0, whe e µis he Lebesgue
measu e on [0,1]. This and (8) gi e us ha
G( ∗,s)=0 a.e. (s)
and his is a con adic ion because G( ∗,s)is a a ional unc ion in he a iable s.
The e o e, x( ) > 0, o ∈(0,1).
This inishes he p oo . 
In wha ollows, we p esen an example which illus a es Theo em 2.
Example 1. Conside he ollowing singula ac ional bounda y alue p oblem
D
3
2
0+u( )+λ( 2+1)a c an(u( ))
√ =0,0< <1 and λ > 0,
u(0)=u(1)=0.(9)
In his case, ( ,u)=λ( 2+1)a c an u
√ , o ( ,u)∈(0,1]×[0,∞). No ice ha is con inuous in (0,1]×[0,∞)and
lim →0+ ( ,·)= ∞. Mo eo e , σ=1
2and α=3
2.
Now we p o e ha ( ,u)sa is ies assump ions o Theo em 2. P e iously, we p o e ha he unc ion φ, de ined by
φ:[0,∞)−→ 0,π
2
φ(x)=a c an x,
sa is ies ha , o u≥ ,
φ(u)−φ( ) ≤φ(u− ).
In ac , as φ(x)=a c an xis a nondec easing unc ion because φ′(x)=1
1+x2>0 and, consequen ly, o u≥ ,
0≤φ(u)−φ( ).
Pu φ(u)=a c an u=αand φ( ) =a c an =β(no ice ha he nondec easing cha ac e o φgi es us α≥β o
u≥ ).
Taking in o accoun he igonome ic o mula
an(α −β) = an α− an β
1+ an α an β
and, as an α=uand an β= belong o [0,∞), we can ob ain
an(α −β) ≤ an α− an β.
As φis nondec easing, he las inequali y gi es us
α−β≤a c an( an α− an β)
o , equi alen ly,
φ(u)−φ( ) =a c an u−a c an ≤a c an(u− ) =φ(u− ).
This p o es ou claim.
Now, we check ha ( ,u)sa is ies assump ions appea ing in Theo em 2.
Theo y o exis ence and uniqueness o solu ion o b p 261
2 Abs ac and Applied Analysis
In 17 he au ho s s udied he ollowing wo-poin bounda y alue p oblem o
ac ional o de :
Dα
0u a   , u  0,0< <1,1<α≤2,
u0u10,1.1
and hey p o ed he exis ence o posi i e solu ions by means o he K asnosel’skii ixed poin
heo em and Legge -Williams ixed poin heo em.
In 18 he au ho in es iga ed he exis ence o solu ions o
cDα
0u   , u ,0< <1,1<α≤2,
u0ν/
0,u
1ρ/
0.1.2
Since bounda y alues a e nonze o, he Riemann-Liou ille ac ional de i a i e Dα
0is no
sui able and he au ho used he Capu o ac ional de i a i e cDα
0.
Mo i a ed by hese wo ks, in his pape we discuss he exis ence and uniqueness o
posi i e solu ions o he ollowing nonlinea bounda y alue p oblem o ac ional o de :
Dα
0u   , u  0,0< <1,3<α≤4,
u0u0u0u10.1.3
This p oblem was s udied in 21, whe e he au ho s use lowe and uppe solu ion
me hod and he Schaude ixed poin heo em which canno ensu e he uniqueness o he
solu ion. The p ac ical ele ance o 3 <α≤4 appea s in p oblems ela ed wi h o he a eas as
physics and economics which can be modeled by hese ac ional bounda y alues p oblems.
Pa icula ly, hese p oblems appea in he Hamil onian o mula ion o he lag angians
depending on ac ional de i a i es o coo dina es when he sys ems a e nonconse a i e
see, e.g., 7.
Ou main in e es in his pape is o gi e an al e na i e answe o he main esul s o
he pape 21.
The main ool used in ou s udy is a ixed poin heo em in pa ially o de ed se s which
gi es us uniqueness o he solu ion.
2. P elimina ies and P e ious Resul s
Fo he con enience o he eade , we p esen he e some de ini ions, lemmas and basic esul s
ha will be used in he p oo s o ou heo ems.
De ini ion 2.1. The Riemann-Liou ille ac ional in eg al o o de α>0o a unc ion :
0,∞→Ris gi en by
Iα
0  1
Γα
0
 −sα−1 sds 2.1
p o ided ha he igh -hand side is poin wise de ined on 0,∞and whe e Γαdeno es he
gamma unc ion.
268 F ac ional bounda y alue p oblem

Abs ac and Applied Analysis 3
De ini ion 2.2. The Riemann-Liou ille ac ional de i a i e o o de α>0o a unc ion :
0,∞→Ris gi en by
Dα
0  1
Γn−αd
d n
0
s
 −sα−n1ds, 2.2
whe e nα1andαdeno es he in ege pa o α.
The ollowing wo lemmas can be ound in 17,22.
Lemma 2.3. Le α>0and u∈C0,1∩L10,1. Then ac ional diffe en ial equa ion
Dα
0u 02.3
has
u c1 α−1c2 α−2···cn α−n2.4
o some ci∈R(i1,2,...n) and nα1as unique solu ion.
Lemma 2.4. Assume ha u∈C0,1∩L10,1wi h a ac ional de i a i e o o de α>0 ha
belongs o C0,1∩L10,1.Then
Iα
0Dα
0u u c1 α−1c2 α−2···cn α−n,2.5
o some ci∈Ri1,...,nand nα1.
Using Lemma 2.4,in21 he ollowing esul is p o ed.
Lemma 2.5. Gi en ∈C0,1and  ≥0, he unique nonnega i e solu ion o
Dα
0u   0,0< <1,3<α≤4,
u0u0u0u10
2.6
is
u 1
0
G , s sds, 2.7
whe e
G , s⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
α−11−sα−3− −sα−1
Γα,0≤s≤ ≤1,
α−11−sα−3
Γα,0≤ ≤s≤1.
2.8
Theo y o exis ence and uniqueness o solu ion o b p 269
4 Abs ac and Applied Analysis
In he sequel, we p esen he ixed-poin heo ems which we will use la e . These
esul s appea in 23.
Theo em 2.6. Le X, ≤be a pa ially o de ed se and suppose ha he e exis s a me ic din Xsuch
ha X, dis a comple e me ic space. Assume ha Xsa is ies he ollowing condi ion
i {xn}is a nondec easing sequence in Xsuch ha xn→x, hen xn≤x∀n∈N.2.9
Le T:X→Xbe a nondec easing mapping such ha
dTx,Ty≤dx, y−ψdx, y, o x≥y, 2.10
whe e ψ:0,∞→0,∞is a con inuous and nondec easing unc ion such ha ψis posi i e in
0,∞,ψ00and lim →∞ψ ∞. I he e exis s x0∈Xwi h x0≤Tx0 hen Thas a ixed
poin .
Mo eo e , i X, ≤sa is ies he ollowing condi ion:
o x, y ∈X he e exis s z∈Xwhich is compa able o xand y, 2.11
which appea s in 24, he ollowing esul is p o ed 23.
Theo em 2.7. Adding condi ion 2.11 o he hypo heses o Theo em 2.6 one ob ains he uniqueness
o he ixed poin .
Rema k 2.8. In Theo ems 2.6 and 2.7 he condi ion lim →∞ψ ∞is edundan .
In ou conside a ions we will wo k in he Banach space C0,1{x:0,1→R,
con inuous}wi h he s anda d no m xsup{|x |: ∈0,1}.
No ice ha his space can be equipped wi h a pa ial o de gi en by
x,y ∈C0,1,x≤y⇐⇒ x ≤y , o ∈0,1.2.12
In 24i is p o ed ha C0,1,≤wi h he classical me ic gi en by
dx,ysup
0≤ ≤1x −y 2.13
sa is ies condi ion 2.9o Theo em 2.6. Mo eo e , o x, y ∈C0,1, as he unc ion
maxx, y∈C0,1,C0,1,≤sa is ies condi ion 2.11.
Finally, by Fwe deno e he class o unc ions ψ:0,∞→0,∞con inuous, nonde-
c easing, posi i e in 0,∞and ψ00.
By Jwe deno e he class o unc ions ϕ:0,∞→0,∞con inuous, nondec easing,
sa is ying ha I−ϕ∈F, whe e Ideno es he iden i y mapping on 0,∞.
270 F ac ional bounda y alue p oblem
Abs ac and Applied Analysis 5
3. Main Resul
The main esul o he pape is he ollowing.
Theo em 3.1. P oblem 1.3has a unique posi i e solu ion u i he ollowing condi ions a e
sa is ied.
H1 :0,1×0,∞→0,∞is con inuous and nondec easing wi h espec o he second
a gumen .
H2The e exis s 0∈0,1such ha  0,0>0.
H3The e exis s 0<λ≤α−2Γα1/2such ha , o x,y ∈0,∞wi h y≥xand
∈0,1,
 , y−  , x≤λ·ψy−x,3.1
whe e ψ∈J.
Be o e he p oo o Theo em 3.1, we will need some p ope ies o G een’s unc ion
appea ing in Lemma 2.5.
Lemma 3.2. G , s≥0, and Gis a con inuous unc ion on 0,1×0,1.
P oo . The con inui y o Gis easily checked. In o de o p o e he nonnega i ness o G , s,
o 0 ≤ ≤s≤1, i is ob ious ha
G , s α−11−sα−3
Γα≥0.3.2
In he case o 0 ≤s≤ ≤1wi h /
0, we ha e
G , s1
Γα α−11−sα−3− −sα−1
1
Γα α−11−sα−3−1−s
α−1.
3.3
As s≤s/ , we ha e 1 −s≥1−s/ and, consequen ly,
1−sα−3≥1−s
α−3.3.4
Taking in o accoun ha he unc ion gα αwi h α>0and ∈0,1is dec easing we ha e
1−sα−3≥1−s
α−3≥1−s
α−1.3.5
The las inequali y and 3.3gi e us G , s≥0wi h /
0. Finally, no ice ha G0,s0, and
his inishes he p oo .
Theo y o exis ence and uniqueness o solu ion o b p 271
6 Abs ac and Applied Analysis
Lemma 3.3. One has
sup
∈0,11
0
G , sds 2
α−2Γα1.3.6
P oo . Since
1
0
G , sds 
0
G , sds 1
G , sds
1
Γα
0 α−11−sα−3− −sα−1ds 1
Γα1
α−11−sα−3ds
1
Γα α−1
α−2−1
α α
3.7
and i we pu ϕ 1
0G , sds 1/Γα α−1/α−2−1/α α, hen, as
ϕ 1
Γαα−1
α−2 α−2− α−1>0, o >0,3.8
we deduce ha ϕ 1
0G , sds is s ic ly inc easing and, consequen ly,
sup
∈0,11
0
G , 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 he sequel, we gi e he p oo o Theo em 3.1.
P oo o Theo em 3.1.Conside he cone
P{u∈C0,1:u ≥0}.3.10
Ob iously, Pis a closed se o C0,1,and, hus,Pis a comple e me ic space wi h he dis ance
gi en by du, sup ∈0,1{|u −  |}.Pcan be equipped wi h a pa ial o de de ined by
x, y ∈P, x ≤y⇐⇒ x ≤y , o ∈0,1.3.11
Using a simila a gumen o ha in 24, i can be p o ed ha P, ≤sa is ies condi ion 2.9
o Theo em 2.6. Mo eo e , as o x, y ∈P he unc ion maxx, y∈P,P, ≤sa is ies condi ion
2.11.
272 F ac ional bounda y alue p oblem
Abs ac and Applied Analysis 7
Now, we conside he ope a o Tde ined on Pand gi en by
Tu 1
0
G , s s, usds, o u∈P. 3.12
By H1and Lemma 3.2,Tapplies Pin o i sel .
In he sequel we check ha Tsa is ies he assump ions o Theo em 2.6.
Fi s ly, we p o e ha Tis a nondec easing ope a o . In ac , by H1, o u, ∈Pwi h
u≥ and ∈0,1, we ha e
Tu 1
0
G , s s, usds ≥1
0
G , s s, sds T  .3.13
Now, we p o e ha Tsa is ies he con ac i e condi ion appea ing in Theo em 2.6.
In ac , o u, ∈Pand u≥ , aking in o accoun assump ion H3,wege
dTu,T sup
∈0,1{|Tu −T  |} sup
∈0,1
Tu −T  
sup
∈0,11
0
G , s s, us − s, sds
≤sup
∈0,11
0
G , sλ·ψus− sds.
3.14
As ψ∈Jand, hus, ψis nondec easing and by Lemma 3.3, om he las inequali y we ob ain
dTu,T ≤λψdu,  ·sup
∈0,11
0
G , sds
λ·ψdu,  ·2
α−2Γα1.
3.15
Using he ac ha λ≤2/α−2Γα1assump ion H3, we ha e
dTu,T ≤ψdu,  du, −du, −ψdu, .3.16
Pu ϕxx−ψx,Asψ∈J, his means ha ϕ∈F. The las inequali y gi es us
dTu,T ≤du, −ϕdu, .3.17
This p o es ha Tsa is ies he con ac i e condi ion o Theo em 2.6.
Finally, as G , s≥0Lemma 3.2and ≥0assump ion H1, we ha e
T0 1
0
G , s s, 0ds ≥0,3.18
whe e 0 deno es he ze o unc ion.
Theo y o exis ence and uniqueness o solu ion o b p 273

8 Abs ac and Applied Analysis
Now, Theo em 2.6 shows ha p oblem 1.3has a leas one nonnega i e solu ion. As
P, ≤sa is ies condi ion 2.11, we ob ain he uniqueness o he solu ion.
In wha ollows, we will p o e ha his solu ion is posi i e  his means ha x >0,
o ∈0,1.
Finally, we will p o e ha he ze o unc ion is no he solu ion o p oblem 1.3.In
ac , in con a y case, he ze o unc ion is a ixed poin o Tand, hus, we ha e
01
0
G0,s
 s, 0ds, o ∈0,1.3.19
The nonnega i e cha ac e o he unc ions Gand and he las exp ession gi e us
G , s· s, 00a.e.s, o ∈0,1.3.20
This and he ac ha G , s/
0a.e.s o any ∈0,1because G , sis gi en by a polyno-
mial implies
s, 00a.e.s.3.21
Taking in o accoun assump ion H2,  0,0>0 o ce ain 0∈0,1. By he con inui y o
we can ind a se A⊂0,1wi h 0∈Aand μA>0, whe e μis he Lebesgue measu e, such
ha  , 0>0 o ∈A. This con adic s 3.21.
This p o es ha he ze o unc ion is no he solu ion o p oblem 1.3. Now, we will
p o e ha he solu ion xis posi i e.
In he con a y case, we ind 0 <
∗<1 such ha x ∗0. As he solu ion xis a ixed
poin o he ope a o T, his means ha
x ∗1
0
G ∗,s
 s, xsds 0.3.22
Since x∈Pand, hus, x≥0 and by he ac ha is nondec easing in he second a iable
and G , s≥0, we can ge
0x ∗1
0
G ∗,s
 s, xsds ≥1
0
G ∗,s
 s, 0ds ≥0,3.23
and his inequali y implies
x ∗1
0
G ∗,s
 s, 0ds 0.3.24
Using a simila easoning o he one abo e used we ob ain a con adic ion.
The e o e, x >0, o ∈0,1.
This inishes he p oo .
274 F ac ional bounda y alue p oblem
Abs ac and Applied Analysis 9
Rema k 3.4. In Theo em 3.1, condi ion H2seems o be a s ong condi ion in o de o ob ain
a posi i e solu ion o p oblem 1.3, bu when he e is uniqueness o solu ion one will see
ha his condi ion is a e y adjus ed one. Mo e p ecisely, unde he assump ion ha p oblem
1.3has a unique nonnega i e solu ion x one has
 0,0>0 o ce ain 0∈0,1i and only i x is a posi i e solu ion.3.25
In ac , i  0,0>0 o ce ain 0∈0,1 he a gumen used in he p oo o Theo em 3.1 gi e
us ha x is a posi i e solu ion.
Fo he o he implica ion, suppose ha  , 00 o any ∈0,1. Unde his assump-
ion, ou p oblem 1.3admi s as solu ions he unc ion x and he ze o unc ion and his
con adic s he hypo hesis abou uniqueness o solu ion o p oblem 1.3. The e o e,  0,0>
0 o ce ain 0∈0,1.
Rema k 3.5. No ice ha he assump ions in Theo em 3.1 a e in a ian by addi i e pe u -
ba ions. Mo e p ecisely, i  , 00 o any ∈0,1and sa is ies H1and H3o
Theo em 3.1, hen g , ua   , u,wi ha:0,1→0,∞a nondec easing con inuous
unc ion wi h a 0/
0 o ce ain 0∈0,1,sa is iesH1,H2,andH3o Theo em 3.1 and
he ollowing nonlinea bounda y alue p oblem o ac ional o de :
Dα
0u g , u  0,0< <1,3<α≤4,
u0u0u0u10
3.26
has a unique posi i e solu ion by Theo em 3.1.
In he sequel we p esen an example whe e he esul s can be applied.
Example 3.6. Conside he ac ional bounda y alue p oblem
D7/2
0u  21ln2u  0,0< <1,
u0u0u0u10.
3.27
In his case,  , u 21ln2u o  , u∈0,1×0,∞. Ob iously, is a con inuous
unc ion and  , 0 21ln 2 /
0 o ∈0,1.As∂ /∂u  211/2u >0 o
u∈0,∞, is nondec easing wi h espec o he second a iable.
Besides, o u≥ and ∈0,1, we ha e
 21ln2u−ln2  21·ln2u
2 
 21ln2 u−
2  21ln1u−
2 
≤ 21ln1u−  ≤2ln
1u− .
3.28
A s aigh o wa d calcula ion gi es us ha ψxln1xsa is ies ha ψ∈J.
Theo y o exis ence and uniqueness o solu ion o b p 275
10 Abs ac and Applied Analysis
Mo eo e , in his case λ2, α7/2 and we ha e
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, Theo em 3.1 p o es he exis ence and uniqueness o a posi i e solu ion o p oblem
3.27.
4. A Final Rema k
In connec ion wi h p oblem 1.3, he main esul in 21is he ollowing.
Theo em 4.1 see 21, Theo em 3.1.P oblem 1.3has a posi i e solu ion u i he ollowing
condi ions a e sa is ied:
H   , u∈C0,1×0,∞,Ris nondec easing ela i e o u,  , p  /
0 o ∈0,1,
whe e p 1
0G , sds 1/Γα α−1/α−2−1/α α, and he e exis s a posi i e
cons an μ<1such ha
kμ  , u≤  , ku,∀0≤k≤1.4.1
In he sequel, we p esen an example which can be ea ed by Theo em 3.1 and i
canno be co e ed by Theo em 4.1.
Example 4.2. Conside he ac ional bounda y alue p oblem
D7/2
Ou  21ρu c0,0≤ ≤1,
u0u0u0u10,
4.2
wi h c>0and0<ρ<1.
In his case,  , u 21ρuc, o  , u∈0,1×0,∞. Ob iously, is con inuous
and nondec easing wi h espec o he second a iable since ∂ /∂u ρ 21>0.
Besides, i u≥ and ∈0,1, we ha e
 , u−  ,  21ρu c−ρ c
 21ρu− ≤2ρu− .
4.3
In his case, ψxρx and i is easily seen ha ϕxx−ψx1−ρxbelongs o F.
276 F ac ional bounda y alue p oblem
Abs ac and Applied Analysis 11
Mo eo e , in his case, λ2and,asα7/2, we ha e
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  , 0c 21>0 o any ∈0,1,Theo em 3.1 gi es us he exis ence and uniqueness
o posi i e solu ion o p oblem 4.2.
On he o he hand, we will show ha  , u 21ρu cwi h 0 <ρ<1andc>0
does no sa is y H o Theo em 4.1. In ac , suppose ha he e exis s 0 <μ<1 such ha
kμ  , u≤  , ku, o any 0 ≤k≤1.4.5
This implies ha
kμ≤  , ku
 , u 21ρku c
 21ρu cρku c
ρu c.4.6
No ice ha limu→∞ρku c/ρu ck, and, consequen ly, aking limi as u→∞in he las
inequali y, we ge
kμ≤k, 4.7
his is alse because 0 <μ<1 and he unc ion hαkαis dec easing when 0 <k<
1. The e o e, p oblem 4.2can be co e ed by Theo em 3.1 and i canno be s udied by
Theo em 4.1.
5. Conclusions
Ou main con ibu ion in his pape is o p o e unde ce ain assump ions he exis ence and
he uniqueness o posi i e solu ion o p oblem 1.3which was ea ed in 21.In21 he
ques ion o uniqueness o solu ion was no conside ed. Mo eo e , we p esen an example
which can be co e ed by he esul s o his pape and canno be ea ed by he ones ob ained
in 21.
Acknowledgmen
This pape was pa ially suppo ed by Minis e io de Educaci´
on y Ciencia P ojec MTM
2007/65706.
Re e ences
1K. Die helm and A. D. F eed, “On he solu ions o nonlinea ac ional o de diffe en ial equa ions
used in he modelling o iscoplas ici y,” in Scien i ice Compu ing in Chemical Enginee ing II:
Theo y o exis ence and uniqueness o solu ion o b p 277
In [22], i is p o ed ha (C[0, 1], ≤) wi h he abo e-men ioned me ic sa is ies condi-
ion (4) o Theo em 1. Mo eo e , o x,yÎC[0, 1], as he unc ion max(x,y)ÎC[0,
1], (C[0, 1], ≤) sa is ies condi ion (5).
By
F
, we deno e he class o unc ions ψ:[0,∞)®[0, ∞) con inuous, nondec eas-
ing, posi i e in (0, ∞) and ψ(0) = 0, and by
J
he class o unc ions : [0, ∞)®[0, ∞)
con inuous, nondec easing, and sa is ying ha I−
ϕ
∈
F
,whe eIdeno es he iden i y
mapping on [0, ∞).
3 Main esul
Ou s a ing poin o his sec ion is he ollowing esul abou G een’s unc ion appea -
ing in Sec ion 2.
Lemma 4. max ∈[0,1]
1

0
G( ,s)ds =1
(α+1) α−1
αα−1−α−1
α

P oo . In ac ,
1

0
G( ,s)ds=

0
G( ,s)ds+
1

G( ,s)ds
=

0
α−1(1 −s)α−1−( −s)α−1
(α)ds+
1

α−1(1 −s)α−1
(α)d
s
=
1

0
α−1(1 −s)α−1
(α)ds−

0
( −s)α−1
(α)ds
=1

(
α
)
α−1
α− α
α

=1

(
α+1
)
( α−1− α)
By an elemen al calcula ion, i can be p o ed ha he maximum o
h( )=1
0G( ,s)ds=1

(
α+1
)
( α−1− α
)
is eached a 0=α−1
α
, hus,
max
0
≤ ≤1
1

0
G( ,s)ds=1
(α+1)α−1
αα−1
−α−1
αα

□
In he sequel, we p esen he main esul o his pape .
Fo con enience, we pu A=1
(α+1)

α−1
αα−1−α−1
αα

.
Theo em 3.Ou P oblem (2) has a unique nonnega i e solu ion u( )i he ollowing
condi ions a e sa is ied:
(H1) : [0, 1] × [0, ∞)®[0, ∞)is con inuous and nondec easing espec o he second
a gumen .
(H2) The e exis s 0<λ≤
1
A
such ha , o x,yÎ[0, ∞)wi h y ≥x and Î[0, 1],
(
,y
)
−
(
,x
)
≤λϕ
(
y−x
),
whe e
ϕ
∈
J
.
P oo . Conside he cone
P={u∈C[0, 1] : u
(
)
≥0}
.
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284 F ac ional bounda y alue p oblem

Ob iously, (P,d) wi h d(x,y) = sup{|x( )-y( )|: Î[0, 1]} is a comple e me ic space
sa is ying condi ions (4) and (5).
Conside he ope a o de ined by
(Tx)( )=
1

0
G( ,s) (s,x(s))ds, o x∈P
,
whe e G( ,s) is he G een’s unc ion appea ing in Sec ion 2. Ob iously, Tapplies P
in o i sel since ( ,x) and G( ,s) a e nonnega i e con inuous unc ions.
In wha ollows we check ha assump ions in Theo em 2 a e sa is ied.
Fi s ly, he ope a o Tis nondec easing.
Indeed, by (H1), o u, ÎP,u≥ , and Î[0, 1], we ha e
(Tu)( )=1
0
G( ,s) (s,u(s))ds≥1
0
G( ,s) (s, (s))ds=(T )( )
.
Now, we p o e ha Tsa is ies he con ac i e condi ion appea ing in Theo em 1.
In ac , o u, ÎPand u≥ and, aking in o accoun assump ion (H2), we ge
d(Tu,T )= sup
∈[0,1]{|Tu( )−T ( )|}
=sup
∈[0,1]{(Tu( )−T ( ))}
=sup
∈[0,1]
1

0
G( ,s)( (s,u(s)) − (s, (s)))d
s
≤sup
∈[0,1]
1

0
G( ,s)λϕ(u(s)− (s))ds.
As
ϕ
∈
J
,is nondec easing, and, aking in o accoun (H2) and Lemma 4, we
ob ain
d(Tu,T )≤λϕ(d(u, )) ·sup
∈[0,1]
1

0
G( ,s)ds
=λϕ
(
d
(
u,
))
·A≤ϕ
(
d
(
u,
))
=d
(
u,
)
−
(
d
(
u,
)
−ϕ
(
d
(
u,
))).
Pu ψ(x)=x-(x). As
ϕ
∈
J
, his means ha
ψ
∈
F
and om he las inequali y
d(
Tu,T
)
≤
d(
u,
)
−ψ
(d(
u,
)).
This p o es ha Tsa is ies he con ac i e condi ion o Theo em 1.
Finally, he nonnega i e cha ac e o he unc ion G( ,s) and ( ,x) [assump ion (H1)]
gi es us
(T0)( )=
1

0
G( ,s) (s,0)ds≥0
,
whe e 0 deno es he ze o unc ion.
The e o e, Theo em 2 says us ha P oblem (2) has a unique nonnega i e solu ion.
□
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Theo y o exis ence and uniqueness o solu ion o b p 285
In he sequel, we p esen a su icien condi ion o he exis ence and uniqueness o
posi i e solu ions o P oblem (2) (posi i e solu ion means x( )>0 o Î(0,1)).The
p oo o his condi ion is simila o he p oo o Theo em 2.3 o [23]. We p esen his
p oo o comple eness.
Theo em 4.Unde assump ions o Theo em 3 and suppose ha (
0
,0)≠0 o ce -
ain
0
Î[0, 1]. Then, P oblem (2) has a unique posi i e solu ion.
P oo . Conside he nonnega i e solu ion x( ) o P oblem (2) whose exis ence is
gua an eed by Theo em 3.
In he sequel, we will p o e ha x( ) is a posi i e solu ion.
Fi s ly, no ice ha x( ) is a ixed poin o he ope a o (Tu)( )=1
0
G( ,s) (s,u(s))d
s
and, consequen ly,
x( )=
1

0
G( ,s) (s,x(s))ds
.
Now, suppose ha he e exis s 0 < * < 1 such ha x( *) = 0. This means ha
x( ∗)=
1

0
G( ∗,s) (s,x(s))ds=0
.
Using ha x( ) is a nonnega i e unc ion, ( ,y) is nondec easing wi h espec o he
second a gumen and he nonnega i e cha ac e o G( ,s), we ge
0=x( ∗)=
1

0
G( ∗,s) (s,x(s))ds≥
1

0
G( ∗,s) (s,0)ds≥0
.
This gi es us x( ∗)=1
0
G( ∗,s) (s,0)ds=
0
.
As G( ,s)≥0 and (s,0)≥0, he las exp ession implies
G
(
∗,s
)
(
s,0
)
=0 a.e
(
s
).
As G( *, s)≠0 a.e (s) (because G( *, s) is gi en by a polynomial), we can ob ain
(
s,0
)
=0 a.e
(
s
).
(6)
On he o he hand, as (
0
,0)≠0 o ce ain
0
Î[0, 1], he nonnega i e cha ac e o
( ,y) gi es us (
0
,0)>0.As ( ,y) is a con inuous unc ion, we can ind a se A⊂[0,
1] wi h
0
ÎA,μ(A)>0,whe eμis he Lebesgue measu e and ( ,0)>0 o any Î
A. This con adic s (6).
The e o e, x( ) > 0 o Î(0, 1). This inishes he p oo . □
Rema k 3. In Theo em 4, he condi ion (
0
,0)≠0 o ce ain
0
Î[0, 1] seems o be
a s ong condi ion in o de o ob ain a posi i e solu ion o P oblem (2), bu when he
solu ion is unique, we will see ha his condi ion is e y adjus ed one. In ac , suppose
ha P oblem (2) has a unique nonnega i e solu ion x( ) hen
(
,0
)
=0 o each ∈[0, 1] i and only i x
(
)
≡0
.
In ac , i ( , 0) = 0 o each Î[0, 1], i is easily seen ha he ze o unc ion sa is ies
P oblem (2) and he uniqueness o he solu ion gi es us x( ) = 0. The e e se implica-
ion is ob ious.
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286 F ac ional bounda y alue p oblem
Rema k 4. No ice ha he hypo heses in Theo em 3 a e in a ian by con inuous pe -
u ba ion. Mo e p ecisely, i ( , 0) = 0 o any Î[0, 1] and sa is ies (H1) and (H2)
o Theo em 3 hen g( ,x)=a( )+ ( ,x)wi ha: [0, 1] ®[0, ∞) con inuous and a≠
0, sa is ies assump ions o Theo em 4, and his means ha he ollowing bounda y
alue p oblem
Dα
0+u( )+g( ,u( )) = 0, 0 < <1
u(0) = u(1) = u(0) = 0

has a unique posi i e solu ion.
Now, we p esen an example ha illus a es ou esul s.
Example 1. Conside he bounda y alue p oblem
D
5
2
0+u( )+c+λ·a c g u( )=0, 0< <1, c,λ>0
u(0) = u(1) = u(0) = 0
⎫
⎪
⎬
⎪
⎭
(7)
In his case,
α
=
5
2
and ( ,u)=c+l·a c g u. I is easily seen ha ( ,u)sa is ies
(H1) o Theo em 3.
In he sequel, we p o e ha ( ,u) sa is ies (H2) o Theo em 3.
P e iously, we conside he unc ion j:[0,∞)®[0, ∞) gi en by j(u)=a c g u and
we will see ha jsa is ies
φ
(
u
)
−φ
(
)
≤φ
(
u−
)
, o u≥
.
In ac , pu j(u)=a c ag u =aand j( )=a c g =b(no ice ha , as u≥ and jis
nondec easing, a≥b).
Then, om
g(α−β)= gα− g
β
1+
g
α·
g
β
and, as α,β∈[0, π
2), hen ga, gbÎ[0, ∞), we can ob ain
g
(
α−β
)
≤ gα− gβ
.
Applying j o he las inequali y and aking in o accoun he nondec easing cha ac-
e o j, we ob ain
α
−β≤a c g
(
gα− gβ
),
o , equi alen ly,
φ
(
u
)
−φ
(
)
=a c g u −a c g =α−β≤a c g
(
u−
)
=φ
(
u−
).
This p oo ou p e ious claim.
Now, o u≥ and Î[0, 1], we ha e,
(
,u
)
−
(
,
)
=λ
(
a c g u −a c g
)
≤λa c g
(
u−
).
Now, we p o e ha j(u)=a c g u belongs o
J
.Ob iously,j:[0,∞)®[0, ∞)isa
con inuous and nondec easing unc ion. Mo eo e , ψ(u)=u-j(u)=u-a c g u is
also con inuous and nondec easing and sa is ies ψ(u)>0 o u> 0 and ψ(0) = 0. Con-
sequen ly,
φ
∈
J
.
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Theo y o exis ence and uniqueness o solu ion o b p 287
Finally, as ( ,0)=c+a c g 0=c> 0, by Theo em 4, P oblem (7) has a unique posi-
i e solu ion o
0<λ≤

1
(52+1)

3
53/2
−3
55/2

−1
≈17.8682
.
4 Some ema ks
In a ecen pape [18], he au ho s s udy he exis ence o posi i e solu ions o a pa i-
cula case o P oblem (2). Mo e p ecisely, hey s udy he ollowing ac ional au ono-
mous bounda y alue p oblem
Dα
0+u( )+λ (u( )) = 0, 0 < <
1
u
(
0
)
=u
(
1
)
=u
(
0
)
=0, (8)
whe e 2 <a≤3, lis a posi i e pa ame e and :(0,∞)®(0, ∞) is con inuous. The
main ool used by he au ho s in his pape is Guo-K anosel’skii ixed-poin heo em
on cones. In [18], he ques ion abou he uniqueness o solu ions is no ea ed.
One o he esul s o [18] is he ollowing heo em.
Theo em 5.[[18],Theo em3.2]I he e exis s l Î(0, 1) such ha q(l)c
2
0
>F
∞
c
1
holds hen, o each lÎ((q(l)c
2
0
)
-1
,(F
∞
c
1
)
-1
), he bounda y alue p oblem (8) has a
leas one posi i e solu ion.
He e, we conside (q(l)c
2
0
)
-1
=0i
0
=∞and (F
∞
c
1
)
-1
=∞i F
∞
=0,whe e
F∞= limu→+∞sup
(u)
u
,F∞= limu→+∞sup
(u)
u
,q( )=
a-1
(1 - ), 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 p esen he ollowing example.
Example 2. Conside he bounda y alue p oblem ha is a a ian o Example 1.
D5/2
0+u( )+λ(c+a c g u( )) = 0, 0 < <1, c,λ>0
,
u
(
0
)
=u
(
1
)
=u
(
0
)
=0, (9)
In his case,
α
=5
2
and (u)=c+a c g u. Then, we ha e F
∞
=0and
0
=∞.Mo e-
o e , 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
0
>F
∞
c
1
holds. Theo em 5 gi es us he exis ence o a posi i e solu ion o P o-
blem (9) o each lÎ(0, ∞). The ques ion o uniqueness canno be ea ed by he
esul s o [18].
On he o he hand, ollowing a simila easoning ha in Example 1, Theo em 4 gi es
us he exis ence o a unique posi i e solu ion o P oblem (9) when
0<λ≤

1
(5/2+1)

3
53/2−3
55/2

−1
≈17.868
2
.
Ou main con ibu ion is he uniqueness o posi i e solu ion o P oblem (9) when 0
<l≤17.8682.
Now, we p esen an example ha canno be s udied by he esul s o [18], and i can
be ea ed by he ones ob ained in his pape .
Example 3. Conside he ollowing bounda y alue p oblem
D5/2
0+u( )+λ( +a c g u( )) = 0, 0 < <1, λ>0
,
u
(
0
)
=u
(
1
)
=u
(
0
)
=0, (10)
Caballe o e al.Bounda y Value P oblems 2011, 2011:25
h p://www.bounda y aluep oblems.com/con en /2011/1/25
Page 8 o 9
288 F ac ional bounda y alue p oblem
In his case, he bounda y alue p oblem is nonau onomous, and hus, his p oblem
canno be s udied by he esul s o [18].
On he o he hand, using a simila a gumen ha in example 1, and using Theo em
4, we ob ain he exis ence o a unique posi i e solu ion o P oblem (10) when 0 <l≤
17.868.
Acknowledgemen s
This esea ch was pa ially suppo ed by “Minis e io de Educación y Ciencia”P ojec MTM 2007/65706.
Au ho s’con ibu ions
We a e pa o he same esea ch g oup and wo k oge he he e o e, we can a i m ha he con en s o his pape
has been p epa ed by all he au ho s: JC, JH, and KS. All au ho s ead and app o ed he inal manusc ip .
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
Recei ed: 28 Feb ua y 2011 Accep ed: 18 Sep embe 2011 Published: 18 Sep embe 2011
Re e ences
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Anal. 71, 3403–3410 (2009). doi:10.1016/j.na.2009.01.240
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equa ion. No heas No mal Uni e si y, Changchun, Jilin P o ince, Republic o China PRC (2009)
22. Nie o, JJ, Rod íguez-López, R: Con ac i e mapping heo ems in pa ially o de ed se s and applica ions o o dina y
di e en ial equa ions. O de . 22, 223–239 (2005). doi:10.1007/s11083-005-9018-5
23. Ha jani, J, López, B, Sada angani, K: On posi i e solu ions o a nonlinea ou h o de bounda y alue p oblem ia a
ixed poin heo em in o de ed se s. Dyn Sys Appl. 19, 625–634 (2010)
doi:10.1186/1687-2770-2011-25
Ci e his a icle as: Caballe o e al.: On exis ence and uniqueness o posi i e solu ions o a class o ac ional
bounda y alue p oblems. Bounda y Value P oblems 2011 2011:25.
Caballe o e al.Bounda y Value P oblems 2011, 2011:25
h p://www.bounda y aluep oblems.com/con en /2011/1/25
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Theo y o exis ence and uniqueness o solu ion o b p 289

Theo y o exis ence and uniqueness o solu ion o b p 291
4.2.5. Posi i e and nondec easing solu ions o a singu-
la bounda y alue p oblem o nonlinea ac-
ional di e en ial equa ions
In his pape , we discuss he exis ence and uniqueness o a posi i e and
nondec easing solu ion o he ollowing ac ional bounda y alue p oblem
(Dα
0+u( ) +  , u( )= 0,0< <1,
u(0) = u0(1) = u00(0) = 0,(4.14)
wi h 2 < α ≤3, and lim
→0+ ( , ·) = ∞, ha is is singula a = 0.
We need he class Ao hose unc ions φ: [0,∞)→[0,∞) sa is ying he
condi ions:
(i) φis nondec easing.
(ii) φ(x)< x, o any x > 0.
(iii) β(x) = φ(x)
xis such ha b( n)→1 implies n→0.
Ou main esul is he nex heo em.
Theo em 57. Suppose ha 0< σ < 1and 2< α ≤3. Unde hese assum-
p ions:
(1) : (0,1] ×[0,∞)→[0,∞)is a con inuous unc ion sa is ying
lim
→0+ ( , −) = ∞,
(2) σ ( , y)is a con inuous unc ion on [0,1] ×[0,∞),
(3) he e exis s 0< λ ≤Γ(α−σ)
Γ(1−σ)and φ∈ A such ha
0≤ σ( ( , y)− ( , x)) ≤λφ(y−x),
o any x, y ∈[0,∞)wi h y≥xand any ∈[0,1],
P oblem (4.14) has a unique nonnega i e solu ion.
Mo eo e , his solu ion is s ic ly inc easing.
292 F ac ional bounda y alue p oblem
The same p oblem was ea ed by T. Qiu, Z. Bai, in Exis ence o posi i e
solu ions o singula ac ional di e en ial equa ions, Elec onic Jou nal o
Di e en ial Equa ions, 146, (2008), 1–9, using he nex heo em.
Theo em 58. Le 0< σ < 1,2< α ≤3, : (0,1] ×[0,+∞)→[0,+∞)is
con inuous and lim
→0+ ( , ·)=+∞, σ ( , y)is con inuous unc ion on [0,1]×
[0,+∞). Assume ha he e exis wo dis inc posi i e cons an ρ, µ(ρ>µ)
such ha
(H1) σ ( , ω)≤ρΓ(α−σ)
Γ(1−σ), o ( , ω)∈[0,1] ×[0, ρ];
(H2) σ ( , ω)≥µΓ(α−σ)
Γ(1−σ), o ( , ω)∈[0,1] ×[0, µ].
Then (1.1) has a leas one posi i e solu ion.
Ou esul imp o es he ones ob ained by hem, since he uniqueness and
he mono onici y o he solu ion canno be deduced by hei pape .
We p esen an example which can be ea ed by Theo em 57 and i canno
be s udied by Theo em .
Communica ions 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
1Depa amen o de Ma em´a icas, Uni e sidad de Las Palmas de G an Cana ia,
Campus de Ta i a Baja, 35017. Las Palmas de G an Cana ia, Spain.
E-mail: jmen[email p o ec ed]
E-mail: [email p o ec ed]
E-mail: ksada [email protected]
This pape is dedica ed o P o esso Je Webb on he occasion o his e i emen
ABSTRACT. In his pape we es ablish he exis ence and uniqueness o a posi i e and nonde-
c easing solu ion o a singula bounda y alue p oblem o a class o nonlinea ac ional di e en ial
equa ions. Ou analysis elies on a ixed poin heo em in pa ially o de ed se s.
AMS (MOS) Subjec Classi ica ion. 45M99,47H09.
1. INTRODUCTION
Many pape s and books on ac ional di e en ial equa ions ha e appea ed e-
cen ly. Mos o hem a e de o ed o he sol abili y o he linea ac ional equa ion
in e ms o a special unc ion (see, o example [3, 12]) and o p oblems o analy ici y
in he complex domain [11]. Mo eo e , Delbosco and Rodino [7] conside ed he exis-
ence o a solu ion o he nonlinea ac ional di e en ial equa ion Dα
0+u= ( , u),
whe e 0 < α < 1 and : [0, a]×R→R, 0 < a ≤+∞is a gi en con inuous unc-
ion in (0, a)×R. They ob ained exis ence esul s by using he Schaude ixed poin
heo em and he Banach con ac ion p inciple. Recen ly, Zhang [19] conside ed he
exis ence o a posi i e solu ion o he equa ion Dα
0+u= ( , u), whe e 0 < α < 1
and : [0,1] ×[0,∞)→[0,∞) is a gi en con inuous unc ion by using he sub and
supe -solu ion me hod.
In his pape , we discuss he exis ence and uniqueness o a posi i e and nonde-
c easing solu ion o he bounda y alue p oblem
Dα
0+u( ) + ( , u( )) = 0,0< < 1,
u(0) = u′(1) = u′′(0) = 0,(1.1)
Recei ed Sep embe 21, 2010 1083-2564 $15.00 c
Dynamic Publishe s, Inc.
Theo y o exis ence and uniqueness o solu ion o b p 293
272 J. CABALLERO, J. HARJANI, AND K. SADARANGANI
[3] L.M.C.M. Campos, On he solu ion o some simple ac ional di e en ial equa ion, In . J.
Ma h. Sci. 13 (1990), 481-496.
[4] M. Belmekki; J.J. Nie o; R. Rod ´ıguez-L´opez, Exis ence o Pe iodic Solu ions o a Nonlinea
F ac ional Di e en ial Equa ion, Bounda y Value P oblems, ol. 2009, A icle ID 324561,
(2009).
[5] Y.K. Chang; J.J. Nie o, Some new exis ence esul s o ac ional di e en ial inclusions wi h
bounda y condi ions, Ma hema ical and Compu e Modelling, 49 (2009), 605-609.
[6] L. Ci i´c; N. Caki´c; M. Rajo i´c; J.S. Ume, Mono one gene alized nonlinea con ac ions in
pa ially o de ed me ic spaces, Fixed Poin Theo y and Applica ions, ol. 2008 A icle 131294.
[7] D. Delbosco; L. Rodino, Exis ence and uniqueness o a nonlinea ac ional di e en ial equa-
ion, J. Ma h. Anal. Appl. 204 (1996), 609-625.
[8] J. Ha jani; K. Sada angani, Fixed poin heo ems o weakly con ac i e mappings in pa ially
o de ed se s, Nonlinea Anal. 71 (2009) 3403-3410.
[9] J. Caballe o, J. Ha jani, K. Sada angani, Exis ence and uniqueness o posi i e and nonde-
c easing solu ions o a class o singula ac ional bounda y alue p oblems, Bounda y alue
p oblems ol. 2009, A icle ID 421310 (2009).
[10] A.A. Kilbas; J.J. T ujillo, Di e en ial equa ions o ac ional o de : me hods, esul s and
p oblems-I, Applicable Analysis 78 (2001), 153-192.
[11] Y. Ling; S. Ding, A class o analy ic unc ions de ined by ac ional de i a i e, J. Ma h. Anal.
Appl. 186 (1994), 504-513.
[12] K. S. Mille ; B. Ross, An In oduc ion o he F ac ional Calculus and F ac ional Di e en ial
Equa ion, Wiley, New Yo k, 1993.
[13] J.J. Nie o; R.L. Pouso; R. Rod ´ıguez-L´opez, Fixed poin heo ems in o de ed abs ac spaces,
P oc. Ame . Ma h. Soc. 135 (2007), 2505-2517.
[14] J.J. Nie o; R. Rod ´ıguez-L´opez, Con ac i e mapping heo ems in pa ially o de ed se s and
applica ions o o dina y di e en ial equa ions, O de 22, (2005), 223-239.
[15] J.J. Nie o; R. Rod ´ıguez-L´opez, Exis ence and uniqueness o ixed poin in pa ially o de ed
se s and applica ions o o dina y di e en ial equa ions, Ac a Ma h. Sinica 23 (2007), 2205-
2212.
[16] D. O’Regan; A. Pe usel, Fixed poin heo ems o gene alized con ac ions in o de ed me ic
spaces, J. Ma h. Anal. Appl. 341 (2008), 1241-1252.
[17] T. Qiu; Z. Bai, Exis ence o posi i e solu ions o singula ac ional di e en ial equa ions,
Elec onic Jou nal o Di e en ial Equa ions, 146 (2008), 1-9.
[18] S. G. Samko; A.A. Kilbas; O.I. Ma iche , F ac ional In eg al and De i a i e. Theo y and
Applica ions, Go don and B each, 1993.
[19] S.Q. Zhang, The exis ence o a posi i e solu ion o a nonlinea ac ional di e en ial equa ion,
J. Ma h. Anal. Appl. 252 (2000), 804-812.
300 F ac ional bounda y alue p oblem

Cap´ı ulo 5
Fu u e lines o esea ch
301
302
5.1. Ope a o s o cyclical ype . . . . . . . . . . . . . 303
5.2. Bes p oximi y poin : app oxima ion and op i-
miza ion........................ 305
5.3. Fixed poin s o dec easing ope a o s and appli-
ca ions......................... 307
5. Fu u e lines o esea ch 303
5.1. Ope a o s o cyclical ype
The ope a o s o cyclical ype we e in oduced by Ki k, S ini asan and
Vee amani in [W. A. Ki k, P.S. S ini asan, P. Vee amani, Fixed poin s o
mappings sa is ying cyclical con ac i e condi ions, Fixed Poin Theo y, 4,
(1), (2003), 79-89] and hey a e de ined o he ollowing o m. Le Xbe a
nonemp y se , ma posi i e in ege and T:X→Xa mapping. X=∪m
i=1Ai
is said o be a cyclic ep esen a ion o Xwi h espec o Ti
(i) Ai, i = 1,2, . . . , m a e nonemp y se s.
(ii) T(A1)⊂A2, . . . , T(Am−1)⊂Am, T(Am)⊂A1.
In [99] he au ho s p o e some ixed poin heo ems o his ype o ope a o s
using se e al con ac i e condi ions.
Ou aim is o s udy some ixed poin heo ems o ope a o s o cyclical ype
in he con ex o pa ially o de ed me ic spaces.
We ha e achie ed some esul s in his a ea as e idenced by he ollowing
pape s:
[HLS1] J. Ha jani, B. L´opez, K. Sada angani, Fixed poin heo ems o cyclic
ϕ-con ac ions in o de ed me ic spaces, Fixed Poin Theo y (accep-
ed).
[HLS2] J. Ha jani, B. L´opez, K. Sada angani, Fixed poin heo ems o cyclic
weak con ac ions in compac me ic spaces, J. Nonlinea Sci. Appl.
(accep ed).
[HSS1] J. Ha jani, F. Sabe ghadam, K. Sada angani, Fixed poin heo em
o cyclic weak con ac ions in pa ially o de ed se s endowed wi h a
comple e me ic, Ca pa hian J. Ma h. (accep ed).
[HSS2] J. Ha jani, F. Sabe ghadam, K. Sada angani, Fixed poin s o map-
pings o cyclical ype in o de ed me ic spaces, Fixed Poin Theo y
(accep ed).
304 Ope a o s o cyclical ype
[KS] E. Ka apina , K. Sada angani, Fixed poin heo y o cyclic (ϕ−ψ)-
con ac ion, Fixed Poin Theo y and Applica ions, ol. 2011, doi:
10.1186/1687-1812-2011-69.
5. Fu u e lines o esea ch 305
5.2. Bes p oximi y poin : app oxima ion
and op imiza ion
Le Aand B wo nonemp y subse s o a me ic space (X, d) and le
T:A→Bbe a mapping. Since Tis no a sel mapping, he equa ion Tx =x
is unlikely o ha e a solu ion. The e o e, is in e es ing ques ion is o ind an
elemen x∈A ha in some sense is closes o Tx. Bes app oxima ion heo-
ems and bes p oximi y poin heo ems a e ele an unde his pe spec i e.
I is clea ha d(x, Tx)≥d(A, B), and an absolu e op imal app oxima e so-
lu ion is an elemen x o which he e o d(x, Tx) assumes he leas possible
alue which is d(A, B).
The poin s x∈Asa is ying d(x, Tx) = d(A, B) a e known as bes p oximi y
poin o T.
In his ield, we ha e w i en he ollowing p ep in :
J. Caballe o, J. Ha jani, K. Sada angani, Bes p oximi y poin heo ems o
non-sel o de con ac ions o Ge agh y- ype in pa ially o de ed me ic spa-
ces.
Some e e ences abou his opic a e:
[AS] M. A. Al-Thaga i, N. Shahzad, Con e gence and exis ence esul s o
bes p oximi y poin s, Nonlinea Anal. 70, (10), (2009), 3665–3671.
[A] A. Amini-Ha andi, Bes p oximi y poin s o p oximal gene alized con-
ac ions in me ic spaces, Op im. Le e . doi: 10.1007./s11590-012-
0470-z.
[EV] A. An hony Eld ed, P. Vee ameni, Exis ence and con e gence o bes
p oximi y poin s, J. Ma h. Anal. Appl. 323, (2006), 1001–1006.
[BSV] C. Di Ba i, T. Suzuki, C. Ve o, Bes p oximi y poin s o cyclic Mei -
Keele con ac ions, Nonlinea Anal. 69, (11), (2008), 3790–3794.
[KA] S. Ka pagam, S. Aga wal, Bes p oximi y poin heo ems o ρ-cyclic
Mei -Keele con ac ions, Fixed Poin Theo y Appl. (2009), A . ID
197308, 9 pages.

306 Bes p oximi y poin : app oxima ion and op imiza ion
[KRV] W. A. Ki k, S. Reich, P. Vee amani, P oximinal e ac s and bes
po ximi ypai heo ems, Nume . Func . Anal. Op im. 24, (2003), 851–
862.
[B1] S. Sadiq Basha, Bes p oximi y poin s: global op imal app oxima e so-
lu ions, J. Glob. Op im. 49, (2011), 15–21.
[B2] S. Sadiq Basha, Bes p oximi y poin heo ems, J. App ox. Theo y 163,
(2011), 1772–1781.
[B3] S. Sadiq Basha, Disc e e op imiza ion in pa ially o de ed se s, J. Glob.
Op im. doi: 10.1007./s10898-011-9774-2.
[BSJ] S. Sadiq Basha, N. Shahzad, R. Jeya aj, Bes p oximi y poin s: ap-
p oxima ion and op i miza ion, Op im. Le . doi: 10.1007/s11590-011-
0404-1.
[BV] S. Sadiq Basha, P. Vee amani, Bes po ximi y pai heo ems o mul i-
unc ions wi h open ib es, J. App ox. Theo y 103, (2000), 119–129.
5. Fu u e lines o esea ch 307
5.3. Fixed poin s o dec easing ope a o s and
applica ions
In he pape [J. J. Nie o, R. Rod ´ıguez-L´opez, Exis ence and uniqueness
o ixed poin in pa ially o de ed se s and applica ions o o dina y di e en ial
equa ions, Ac a Ma h. Sinica 23, 12, (2007), 2205–2212] he au ho s p o e
he ollowing esul .
Theo em 59. Le (X, ≤)be a pa ially o de ed se such ha o each x, y ∈
X he e exis z∈Xwhich is compa able o xand y. Suppose ha he e exis s
a me ic din Xsuch ha (X, d)is a comple e me ic space.
Le T:X→Xbe a noninc easing mapping such ha he e exis s k∈[0,1)
sa is ying
d(Tx, Ty)≤k d(x, y) o x, y ∈Xwi h x≥y.
Suppose also ha ei he Tis con inuous o Xis such ha
i (xn)⊂Xwi h xn→xand consecu i e e ms a e compa able
hen he e exis s a subsequence xnko xnsuch ha
e e y e m is compa able o he limi x.
(5.1)
I he e exis s x0∈Xwi h x0≤Tx0o x0≥Tx0 hen Thas a unique ixed
poin .
No ice ha a di icul ques ion is o ind pa ially o de ed me ic spaces
sa is ying (5.1) and, consequen ly, Theo em 59 is no use ul when Tis no
con inuous.
In he p ac ice, when we wan o in es iga e he exis ence and uniqueness o
solu ions o bounda y alue p oblems whe e he da a unc ion is dec easing
appea s he abo e men ioned ques ion and Theo em 59 does no wo k.
In he li e a u e, he e exis ixed poin heo ems o dec easing ope a o s
which can be applied o bounda y alue p oblems.
Ou aim is o s udy hese ixed poin heo ems and o apply hem o bounda y
308 Fixed poin s o dec easing ope a o s and applica ions
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Fixed Poin Theo ems in Pa ially
O de ed Spaces and Applica ions
Tesis Doc o al
Jackie Ha jani Sauco
Las Palmas de G an Cana ia
Mayo de 2014
Fixed Poin Theo ems in Pa ially O de ed Spaces and Applica ions
Jackie Ha jani Sauco
Depa amen o de Física