# Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions

Boundary Value Problems20122012:124

DOI: 10.1186/1687-2770-2012-124

Accepted: 4 October 2012

Published: 24 October 2012

## Abstract

We study the existence of solutions for a class of nonlinear Caputo-type fractional boundary value problems with nonlocal fractional integro-differential boundary conditions. We apply some fixed point principles and Leray-Schauder degree theory to obtain the main results. Some examples are discussed for the illustration of the main work.

MSC:34A08, 34A12, 34B15.

### Keywords

fractional differential equations fractional boundary conditions separated boundary conditions fixed point theorems

## 1 Introduction

Nonlocal boundary value problems of fractional differential equations have been extensively studied in the recent years. In fact, the subject of fractional calculus has been quite attractive and exciting due to its applications in the modeling of many physical and engineering problems. For theoretical and practical development of the subject, we refer to the books [15]. Some recent results on fractional boundary value problems can be found in [614] and references therein. In [11], the authors studied a boundary value problem of fractional differential equations with fractional separated boundary conditions.

In this article, motivated by [11], we consider a fractional boundary value problem with fractional integro-differential boundary conditions given by
(1.1)

where denotes the Caputo fractional derivative of order α, f is a given continuous function, and , , () are suitably chosen real constants.

The main aim of the present study is to obtain some existence results for the problem (1.1). As a first step, we transform the given problem to a fixed point problem and show the existence of fixed points for the transformed problem which in turn correspond to the solutions of the actual problem. The methods used to prove the existence results are standard; however, their exposition in the framework of the problem (1.1) is new.

## 2 Preliminaries

Let us recall some basic definitions of fractional calculus [1, 2].

Definition 2.1 For -times absolutely continuous function , the Caputo derivative of fractional order q is defined as

where denotes the integer part of the real number q.

Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as

provided the integral exists.

To define the solution of the boundary value problem (1.1), we need the following lemma, which deals with a linear variant of the problem (1.1).

Lemma 2.3 For a given , the unique solution of the linear fractional boundary value problem
(2.1)
is given by
(2.2)
where
(2.3)
Proof It is well known [2] that the solution of the fractional differential equation in (2.1) can be written as
(2.4)
Using (b is a constant), , , (2.4) gives
(2.5)
Using the integral boundary conditions of the problem (2.1) together with (2.3), (2.4), and (2.5) yields

Substituting the values of , in (2.4), we get (2.2). This completes the proof. □

Remark 2.4 Notice that the solution (2.2) is independent of the parameter , which distinguishes the present work from the one containing the fractional differential equation of (2.1) with the boundary conditions of the form:
(2.6)

In case , the boundary conditions in (2.1) coincide with (2.6) and consequently the corresponding solutions become identical.

## 3 Main results

Let denote the Banach space of all continuous functions from into ℝ endowed with the usual supremum norm.

In view of Lemma 2.3, we define an operator by
(3.1)

Observe that the problem (1.1) has solutions if and only if the operator equation has fixed points.

In the sequel, we use the following notation:
(3.2)

where , with () given by (2.3).

Our first result is based on the Leray-Schauder nonlinear alternative [15].

Lemma 3.1 (Nonlinear alternative for single valued maps)

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
1. (i)

F has a fixed point in , or

2. (ii)

there is a (the boundary of U in C) and with .

Theorem 3.2 Let be a jointly continuous function. Assume that:

() there exist a function and a nondecreasing function such that , ;

() there exists a constant such that

Then the boundary value problem (1.1) has at least one solution on .

Proof Consider the operator defined by (3.1). We show that F maps bounded sets into bounded sets in . For a positive number r, let be a bounded set in . Then
Next, we show that F maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then we obtain

Obviously, the right-hand side of the above inequality tends to zero independently of as . As ℱ satisfies the above assumptions, therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.

Let x be a solution. Then for , using the computations in proving that ℱ is bounded, we have
Consequently, we have
In view of (), there exists M such that . Let us set

Note that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that ℱ has a fixed point which is a solution of the problem (1.1). This completes the proof. □

In the special case when and (κ and N are suitable constants) in the statement of Theorem 3.2, we have the following corollary.

Corollary 3.3 Let be a continuous function. Assume that there exist constants , where ω is given by (3.2) and such that for all , . Then the boundary value problem (1.1) has at least one solution.

Next, we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.

Theorem 3.4 Suppose that is a continuous function and satisfies the following assumption:

() , , , .

Then the boundary value problem (1.1) has a unique solution provided
(3.3)

where ω is given by (3.2).

Proof With , we define , where and ω is given by (3.2). Then we show that . For , we have
Using , the above expression yields
where we used (3.2). Now, for and for each , we obtain

Note that ω depends only on the parameters involved in the problem. As , therefore, ℱ is a contraction. Hence, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on . □

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii’s fixed point theorem [16].

Theorem 3.5 (Krasnoselskii’s fixed point theorem)

Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (i) whenever ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists such that .

Theorem 3.6 Let be a jointly continuous function satisfying the assumption (). In addition we assume that:

() , , and .

Then the problem (1.1) has at least one solution on if
(3.4)
Proof Letting , we choose a real number satisfying the inequality
and consider . We define the operators and on as
For , we find that
Thus, . It follows from the assumption () together with (3.4) that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as

Now, we prove the compactness of the operator .

In view of (), we define , and consequently, for , we have

which is independent of x. Thus, is equicontinuous. Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Theorem 3.5 are satisfied. So, the conclusion of Theorem 3.5 implies that the boundary value problem (1.1) has at least one solution on . □

## 4 Examples

Example 4.1 Consider the following boundary value problem:
(4.1)
Here, , , , , , , , , , , and
Clearly,
Clearly, and

Thus, all the conditions of Corollary 3.3 are satisfied and consequently the problem (4.1) has at least one solution.

Example 4.2 Consider the following fractional boundary value problem:
(4.2)

where α, p, , , , () η, σ are the same as given in (4.1) and . Clearly, and thus, for , all the conditions of Theorem 3.4 are satisfied. Hence, the boundary value problem (4.2) has a unique solution on .

## Declarations

### Acknowledgements

The authors thank the reviewers for their useful comments that led to the improvement of the original manuscript. This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University

## References

1. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
2. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
3. Magin RL: Fractional Calculus in Bioengineering. Begell House, Redding; 2006.
4. Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.
5. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.
6. Agarwal RP, Belmekki M, Benchohra M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009., 2009: Article ID 981728
7. Baleanu D, Mustafa OG, Agarwal RP: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 2010, 23: 1129-1132. 10.1016/j.aml.2010.04.049
8. Hernandez E, O’Regan D, Balachandran K: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 2010, 73(10):3462-3471. 10.1016/j.na.2010.07.035
9. Ahmad B, Ntouyas SK: A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 22
10. Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 36
11. Ahmad B, Ntouyas SK: A note on fractional differential equations with fractional separated boundary conditions. Abstr. Appl. Anal. 2012., 2012: Article ID 818703
12. Aghajani A, Jalilian Y, Trujillo JJ: On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 2012, 15(2):44-69.MathSciNet
13. Ahmad B, Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with strip conditions. Bound. Value Probl. 2012., 2012: Article ID 55
14. Ahmad B, Nieto JJ: Anti-periodic fractional boundary value problem with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 2012, 15: 451-462.MathSciNet
15. Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2005.
16. Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.MathSciNet