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
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 [1-5]. Some recent results on fractional boundary value problems can be found in [6-14] and references therein. In , the authors studied a boundary value problem of fractional differential equations with fractional separated boundary conditions.
In this article, motivated by , we consider a fractional boundary value problem with fractional integro-differential boundary conditions given by
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.
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).
is given by
Proof It is well known  that the solution of the fractional differential equation in (2.1) can be written as
Using the integral boundary conditions of the problem (2.1) together with (2.3), (2.4), and (2.5) yields
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:
3 Main results
In the sequel, we use the following notation:
Our first result is based on the Leray-Schauder nonlinear alternative .
Lemma 3.1 (Nonlinear alternative for single valued maps)
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.
Consequently, we have
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. □
Next, we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.
Then the boundary value problem (1.1) has a unique solution provided
whereωis given by (3.2).
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 .
Theorem 3.5 (Krasnoselskii’s fixed point theorem)
LetMbe a closed, bounded, convex, and nonempty subset of a Banach spaceX. LetA, Bbe the operators such that (i) whenever; (ii) Ais compact and continuous; (iii) Bis a contraction mapping. Then there existssuch that.
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 . □
Example 4.1 Consider the following boundary value problem:
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:
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 .
The authors declare that they have no competing interests.
Each of the authors, BA and AA contributed to each part of this work equally and read and approved the final version of the manuscript.
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.
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