International Journal of Differential Equations

Volume 2017 (2017), Article ID 4683581, 7 pages

https://doi.org/10.1155/2017/4683581

## Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Correspondence should be addressed to Jian-Ping Sun

Received 11 October 2016; Revised 14 December 2016; Accepted 15 December 2016; Published 29 January 2017

Academic Editor: Julio D. Rossi

Copyright © 2017 Cheng-Min Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: , , , where , , , and . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

#### 1. Introduction

Fractional calculus has wide applications in many fields of science and engineering, for example, fluid flow, biosciences, rheology, electrical networks, chemical physics, control theory of dynamical systems, and optics and signal processing [1].

Recently, nonlinear fractional differential equations have been discussed under the following boundary conditions (BCs for short): (1)Integer derivative BCs: , , , , , , , , , , ; see papers [2–7], respectively. (2)Integer derivative and integral BCs: , , , ; see papers [8, 9], respectively. (3)Integer and fractional derivative BCs: , , , , , , , , , , , , , ; see papers [10–16], respectively.(4)Integer derivative and fractional integral BCs: , , ; see papers [17, 18], respectively.

Besides, there are some other BCs involved in fractional differential equations, such as nonlinear BCs; refer to [19, 20].

Motivated greatly by the above-mentioned works, in this paper, we study the following boundary value problem (BVP for short) of nonlinear fractional differential equation with fractional integral BCs as well as integer and fractional derivativewhere and denote the standard Caputo fractional derivatives and denotes the standard Riemann-Liouville fractional integral. Throughout this paper, we always assume that , , , , and is continuous.

In order to prove our main results, the following well-known fixed point theorems are needed.

Theorem 1 (nonlinear alternative of Leray-Schauder type [21]). *Let be a Banach space with closed and convex. Assume is a relatively open subset of with and is a continuous and compact map. Then either*(a)* has a fixed point in or*(b)*there exists and such that .*

*Theorem 2 (Banach contraction principle [22]). Let be a complete metric space and be contractive. Then has a unique fixed point in .*

*2. Preliminaries*

*In this section, we always assume that , , and denotes the integer part of . Now, for the convenience of the reader, we give the definitions of the Riemann-Liouville fractional integrals and fractional derivatives and the Caputo fractional derivatives on a finite interval of the real line, which may be found in [1].*

*Definition 3. *The Riemann-Liouville fractional integrals and of order on are defined by respectively.

*Definition 4. *The Riemann-Liouville fractional derivatives and of order on are defined by respectively, where

*Definition 5. *Let and be the Riemann-Liouville fractional derivatives of order . Then the Caputo fractional derivatives and of order on are defined by respectively, where

*Lemma 6 (see [23]). If , then the equation , , is satisfied for .*

*Lemma 7 (see [23]). Let . Then the equation , , is satisfied for .*

*Lemma 8 (see [1]). Let be given by (5). Then the following relations hold:(1)For , .(2)If , then .*

*Lemma 9 (see [1]). Let be given by (5) and . Then where , .*

*For any , we define *

*Lemma 10. Let be nonnegative. Then , .*

*Proof. *For any , we have

*3. Main Results*

*3. Main Results*

*In the remainder of this paper, for any nonnegative function , we denote and for any , we use the norm*

*Lemma 11. Let be a given function. Then the BVPhas a unique solution where*

*Proof. *It follows from the equation in (11) and Lemma 9 thatSo, In view of (14), (16), and the BCs , we get and so, Then, by using Lemmas 6, 7, and 8, we may obtainwhich together with the BC implies that Therefore, the BVP (11) has a unique solution

*Lemma 12. Let be nonnegative. Then *

*Proof. *In view of Lemma 10, we have

*Now, we define an operator by Obviously, is a solution of the BVP (1) if and only if is a fixed point of .*

*Theorem 13. Assume that , , and there exist nonnegative functions , nonnegative increasing continuous function defined on , and such thatThen the BVP (1) has one nontrivial solution.*

*Proof. *Let . Since and are continuous on and , respectively, we may denoteFirst, we prove that is continuous. Suppose that (), , and (). Then for any and , we have . This together with (27) and (28) implies that, for any and , By applying Lebesgue dominated convergence theorem, we get which indicates that is continuous.

Next, we show that is compact. Assume that is a subset of . Then for any , we haveIn what follows, we will prove that is relatively compact. On the one hand, for any , there exists such that , and so, it follows from (27), (28), and (31) that which shows that is uniformly bounded. On the other hand, for any , since is uniformly continuous on , there exists such that, for any with ,For any , there exists such that , and so, for any with , it follows from (28), (31), and (33) that which indicates that is equicontinuous. By Arzela-Ascoli theorem, we know that is relatively compact. Therefore, is compact.

Now, we will prove that (a) of Theorem 1 is fulfilled. Suppose on the contrary that (b) of Theorem 1 is satisfied; that is, there exists and such that . Then, in view of (25), (26), and Lemma 12, we have which shows that This contradicts the fact .

So, it follows from Theorem 1 that has a fixed point , which is a desired solution of the BVP (1). At the same time, since , we know that the zero function is not a solution of the BVP (1). Therefore, is a nontrivial solution of the BVP (1).

*Theorem 14. Assume that there exists a nonnegative function such thatThen the BVP (1) has a unique solution.*

*Proof. *For any , in view of (37) and Lemma 12, we have This indicates that which together with (38) implies that is contractive. So, it follows from Theorem 2 that has a unique fixed point, and so, the BVP (1) has a unique solution.

*Example 15. *We consider the BVP

*Let , . Then is continuous and , .*

*If we choose , , , and , , then it is easy to verify that (25) is satisfied.*

*Since , , , and , a direct calculation shows that If we choose , then (26) is fulfilled.*

*Therefore, it follows from Theorem 13 that the BVP (41) has one nontrivial solution.*

*Example 16. *We consider the BVP

*Let , . Then is continuous.*

*If we choose , then we may assert that (37) is satisfied. In fact, for any , if , then (37) is obvious. When , we may suppose that . In this case, by Lagrange mean value theorem, there exists such that, for any , that is, (37) is satisfied.*

*On the other hand, in view of , we know that (38) is fulfilled.*

*Therefore, it follows from Theorem 14 that the BVP (43) has a unique solution.*

*Competing Interests*

*Competing Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This paper is supported by the National Natural Science Foundation of China (11661049).*

*References*

*References*

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet - D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 2, pp. 710–719, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,”
*Electronic Journal of Differential Equations*, vol. 2006, no. 36, pp. 1–12, 2006. View at Google Scholar - Z. Bai and T. Qiu, “Existence of positive solution for singular fractional differential equation,”
*Applied Mathematics and Computation*, vol. 215, no. 7, pp. 2761–2767, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J.-R. Yue, J.-P. Sun, and S. Zhang, “Existence of positive solution for BVP of nonlinear fractional differential equation,”
*Discrete Dynamics in Nature and Society*, vol. 2015, Article ID 736108, 6 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,”
*Georgian Mathematical Journal*, vol. 16, no. 3, pp. 401–411, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - K. Zhao and P. Gong, “Existence of positive solutions for a class of higher-order Caputo fractional differential equation,”
*Qualitative Theory of Dynamical Systems*, vol. 14, no. 1, pp. 157–171, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Yang, “Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions,”
*Journal of Applied Mathematics and Computing*, vol. 44, no. 1-2, pp. 39–59, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. Zhang, L. Wang, and Q. Sun, “Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter,”
*Applied Mathematics and Computation*, vol. 226, pp. 708–718, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Guezane-Lakoud and R. Khaldi, “Existence results for a fractional boundary value problem with fractional Lidstone conditions,”
*Journal of Applied Mathematics and Computing*, vol. 49, no. 1-2, pp. 261–268, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Guezane-Lakoud and S. Bensebaa, “Solvability of a fractional boundary value problem with fractional derivative condition,”
*Arabian Journal of Mathematics*, vol. 3, no. 1, pp. 39–48, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Li, “Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition,”
*Advances in Difference Equations*, vol. 2014, article 292, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z.-W. Lv, “Positive solutions of m-point boundary value problems for fractional differential equations,”
*Advances in Difference Equations*, vol. 2011, Article ID 571804, 2011. View at Publisher · View at Google Scholar - Y. Ji, Y. Guo, J. Qiu, and L. Yang, “Existence of positive solutions for a boundary value problem of nonlinear fractional differential equations,”
*Advances in Difference Equations*, vol. 2015, article 13, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1363–1375, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Xu, Z. Wei, and W. Dong, “Uniqueness of positive solutions for a class of fractional boundary value problems,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 590–593, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. K. Ntouyas, “Existence results for first order boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions,”
*Journal of Fractional Calculus and Applications*, vol. 3, no. 9, pp. 1–14, 2012. View at Google Scholar - A. Guezane-Lakoud and R. Khaldi, “Solvability of a fractional boundary value problem with fractional integral condition,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 2692–2700, 2012. View at Publisher · View at Google Scholar · View at Scopus - W. Feng, S. Sun, X. Li, and M. Xu, “Positive solutions to fractional boundary value problems with nonlinear boundary conditions,”
*Boundary Value Problems*, 2014:225, 15 pages, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Xie, J. Xiao, and Z. Luo, “Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions,”
*Applied Mathematics Letters*, vol. 41, pp. 46–51, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Guo and V. Lakshmikantham,
*Nonlinear Problems in Abstract Cones*, Academic Press, San Diego, Calif, USA, 1988. View at MathSciNet - A. Granas and J. Dugundji,
*Fixed Point Theory*, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet

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