Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2009, Article ID 494720, 9pages doi:10.1155/2009/494720 Research Article Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations Bashir Ahmad1and Juan J. Nieto2 1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Correspondence should be addressed to Bashir Ahmad, bashir [email protected] Received 19 February 2009; Accepted 27 April 2009 Recommended by Paul Eloe We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order qgiven by cDqxtft, xt, 0<t<1, q∈m−1,m,m∈N,m≥2, x00, x00,x 00,...,xm−200, x1αxη. Our results are based on the contraction mapping principle and Krasnoselskii’s fixed point theorem. Copyright q2009 B. Ahmad and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Fractional differential equations involve derivatives of fractional order. They arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electro-dynamics of complex medium, and polymer rheology. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see 1–17 and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored. The subject of multipoint nonlocal boundary value problems, initiated by Ilin and Moiseev 18,19, has been addressed by many authors, for instance, 20–26. The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity, and wave propagation, see 27and the references therein. The multipoint boundary conditions may be understood in the sense that the controllers at the end points dissipate or add energy according to censors located at intermediate positions. 2 Abstract and Applied Analysis For m∈N,m ≥2,and q∈m−1,m,we consider the following nonlinear fractional differential equation of order qwith nonlocal boundary conditions: cDqxtft, t,0<t<1, x00,x 00,x 00,...,x m−200,x 1αxη, 0<η<1,αη m−1/ 1,α∈R, 1.1 where cDis the Caputo fractional derivative and f:0,1×X→Xis continuous. Here, X, ·is a Banach space and CC0,1,Xdenotes the Banach space of all continuous functions from 0,1→Xendowed with a topology of uniform convergence with the norm denoted by ·. By a solution of 1.1, we mean a function x∈Cof class Cm0,1which satisfies the nonlocal fractional boundary value problem 1.1. 2. Preliminaries Let us recall some basic definitions 12,15,17on fractional calculus. Definition 2.1. For a function g:0,∞→R,the Caputo derivative of fractional order qis defined as cDqgt1 Γn−qt 0 t−sn−q−1gnsds, n −1<q<n, nq1,2.1 where qdenotes the integer part of the real number q. Definition 2.2. The Riemann-Liouville fractional integral of order qis defined as Iqgt1 Γqt 0 gs t−s1−qds, q > 0,2.2 provided that the integral exists. Definition 2.3. The Riemann-Liouville fractional derivative of order qfor a function gtis defined by Dqgt1 Γn−qd dtnt 0 gs t−sq−n1ds, n q1,2.3 provided the right hand side is pointwise defined on 0,∞. We remark that the Caputo derivative becomes the conventional nth derivative of the function as q→nand the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer-order derivatives. On the other hand, the Riemann-Liouville fractional derivative could hardly produce the physical interpretation of the initial conditions required for the initial value problems Abstract and Applied Analysis 3 involving fractional differential equations the same applies to the boundary value problems of fractional differential equations. Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see 17. Lemma 2.4 see 28.For q>0,the general solution of the fractional differential equation cDqxt0is given by xtc0c1tc2t2···cn−1tn−1,2.4 where ci∈R,i0,1,2,...,n−1(nq1). In view of Lemma 2.4, it follows that IqcDqxtxtc0c1tc2t2···cn−1tn−1,2.5 for some ci∈R,i0,1,2,...,n−1nq1. Now, we state a known result due to Krasnoselskii 29which is needed to prove the existence of at least one solution of 1.1. Theorem 2.5. Let Mbe a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that iAx By ∈Mwhenever x, y ∈M, iiAis compact and continuous, iiiBis a contraction mapping. Then there exists z∈Msuch that zAz Bz. To study the nonlinear problem 1.1, we first consider the associated linear problem and obtain its solution. Lemma 2.6. For a given σ∈C0,1,the unique solution of the boundary value problem, cDqxtσt,0<t<1,q∈m−1,m ,m∈N,m≥2, x00,x 00,x 00,...,x m−200,x 1αxη, 0<η<1,αη m−1/ 1,α∈R, 2.6 is given by xtt 0 t−sq−1 Γqσsds −tm−1 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds. 2.7 4 Abstract and Applied Analysis Proof. Using 2.5, we have xtt 0 t−sq−1 Γqσsds −c0−c1t−c2t2−···−cm−1tm−1,2.8 where c0,c 1,c 2,...,c m−1∈Rare arbitrary constants. In view of the relations cDqIqxtxt and IqIpxtIqpxtfor q, p > 0,x∈L0,1,we obtain xtt 0 t−sq−2 Γq−1σsds −c1−2c2t−···−m−1cm−1tm−2, xtt 0 t−sq−3 Γq−2σsds −2c2−···−m−1m−2cm−1tm−3,.... 2.9 Applying the boundary conditions for 2.6,wefindthatc00,c 10,...,c m−20,and cm−11 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds.2.10 Substituting the values of c0,c 1,...,c m−1in 2.8,weobtain xtt 0 t−sq−1 Γqσsds −tm−1 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds. 2.11 This completes the proof. 3. Main Results For the forthcoming analysis, we need the following assumptions: A1ft, x−ft, y≤Lx−y,forall t∈0,1,x,y∈X; A2ft, x≤μt,forall t, x∈0,1×X, μ ∈L10,1,R . In relation to the nonlocal problem 1.1, we define the constants: Λ L Γq1λ, λ L1|α|ηq Γq11−αηm−1 .3.1 Theorem 3.1. Assume that f:0,1×X→Xis a jointly continuous function and satisfies the assumption A1.Then the boundary value problem 1.1has a unique solution provided Λ<1,where Λis given by 3.1. Abstract and Applied Analysis 5 Proof. Define :C→Cby xtt 0 t−sq−1 Γqfs, xsds −tm−1 1−αηm−1 ×1 0 1−sq−1 Γqfs, xsds −αη 0η−sq−1 Γqfs, xsds,t∈0,1. 3.2 Let us set supt∈0,1ft, 0M, and choose r≥M 1−βΓq111|α|ηq 1−αηm−1,3.3 where βis such that Λ≤β<1.Now we show that Br⊂Br,where Br{x∈C:x≤r}. For x∈Br,we have xt≤t 0 t−sq−1 Γq fs, xs ds tm−1 1−αηm−11 0 1−sq−1 Γq fs, xs ds |α|η 0η−sq−1 Γqfs, xsds ≤t 0 t−sq−1 Γq fs, xs −fs, 0  fs, 0 ds tm−1 1−αηm−11 0 1−sq−1 Γqfs, xs −fs, 0fs, 0ds |α|η 0η−sq−1 Γq fs, xs −fs, 0 fs, 0ds ≤Lr Mt 0 t−sq−1 Γqds tm−1 1−αηm−11 0 1−sq−1 Γqds |α|η 0η−sq−1 Γqds ≤L Γq1L1|α|ηq Γq11−αηm−1rM Γq111|α|ηq 1−αηm−1 ΛrM Γq111|α|ηq 1−αηm−1 ≤Λ1−βr≤r. 3.4 6 Abstract and Applied Analysis Now, for x, y ∈Cand for each t∈0,1,we obtain xt−yt ≤t 0 t−sq−1 Γqfs, xs −fs, ysds tm−1 1−αηm−11 0 1−sq−1 Γqfs, xs −fs, ysds |α|η 0η−sq−1 Γqfs, xs −fs, ysds ≤L x−y t 0 t−sq−1 Γqds tm−1 1−αηm−11 0 1−sq−1 Γqds |α|η 0η−sq−1 Γqds L x−y tq Γq1tm−1 1−αηm−11|α|ηq Γq1 ≤L x−y  1 Γq111|α|ηq 1−αηm−1 Λ  x−y . 3.5 Clearly Λdepends on the parameters q, m,α,η,Linvolved in the problem. As Λ<1, therefore, is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. Theorem 3.2. Let f:0,1×X→Xbe a jointly continuous function mapping bounded subsets of 0,1×Xinto relatively compact subsets of X. Further, the assumptions A1−A2hold with λ<1, where λis given by 3.1. Then the boundary value problem 1.1has at least one solution on 0,1. Proof. Let us fix r≥ μ L1 Γq11|α|ηq−1 1−αηm−1,3.6 and consider Br{x∈C:x≤r}.We define the operators Φand Ψon Bras Φxt1 Γqt 0 t−sq−1fs, xsds, Ψxt−tm−1 1−αηm−11 0 1−sq−1 Γqfs, xsds −αη 0η−sq−1 Γqfs, xsds. 3.7 Abstract and Applied Analysis 7 For x, y ∈Br,we find that  ΦxΨy ≤ μ L1 Γq11|α|ηq−1 1−αηm−1≤r. 3.8 Thus, ΦxΨy∈Br.It follows from the assumption A1that Ψis a contraction mapping for λ<1.Continuity of fimplies that the operator Φis continuous. Also, Φis uniformly bounded on Bras Φx≤ μ L1 Γq.3.9 To show that the operator Φis compact, we use the classical Arzela-Ascoli theorem. Let Abe a bounded subset of C.We have to show that ΦAis equicontinuous and for each t, the set ΦAtis relatively compact in X. In view of A1,A2,we define supt,x∈0,1×Brft, x fmax,and consequently we have Φxt1−Φxt2      1 Γqt1 0t2−sq−1−t1−sq−1fs, xsds t2 t1 t2−sq−1fs, xsds     ≤fmax Γq12t2−t1qtq 1−tq 2, 3.10 which is independent of x. Thus, Φis equicontinuous. Using the fact that fmaps bounded subsets into relatively compact subsets, we have that ΦAtis relatively compact in X for every t. Therefore, Φis relatively compact on Br.Hence, By Arzela-Ascoli theorem, Φ is compact on Br.Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the boundary value problem 1.1has at least one solution on 0,1. Example 3.3. Consider the following boundary value problem: cDqxt1 t72 x 1x,2<q≤3,t∈0,1, x00,x 00,x 1x1 2. 3.11 Here, m3,ft, xt  1/t72x/1x,α 1,η 1/2.As ft, x−ft, y≤ 1/49x−y,therefore, A1is satisfied with L1/49.Further, Λ L Γq111|α|ηq |1−ηm−1|1 49Γq114 311 2q<1,2<q≤3.3.12 Thus, by Theorem 3.1, the boundary value problem 3.11has a unique solution on 0,1. 8 Abstract and Applied Analysis Acknowledgments The authors thank the reviewers for their useful comments. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM200761724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR. References 1B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear functional differential equation with three-point boundary value problem,” preprint. 2B. Ahmad and J. J. 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