#### Abstract

This paper is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional -difference equations with nonlocal integral boundary conditions. The existence results are obtained by applying some well-known fixed point theorems and illustrated with examples.

#### 1. Introduction

Several kinds of boundary value problems of fractional-order have recently been investigated by many researchers. Fractional derivatives appear naturally in the mathematical modelling of dynamical systems involving fractals and chaos. In fact, the concept of fractional calculus has played a key role in improving the work based on integer-order (classical) calculus in several diverse disciplines of science and engineering. This might have been due to the fact that fractional-differential operators help to understand the hereditary phenomena in many materials and processes in a better way than the corresponding integer-order differential operators. Examples include physics, chemistry, biology, biophysics, blood flow phenomena, control theory, signal and image processing, and economics [1–4]. For some recent results on the topic, see a series of papers [5–12] and the references therein.

Fractional -difference equations, regarded as fractional analogue of -difference equations, have been studied by several researchers. For some earlier work on the subject, we refer to [13, 14], whereas the recent development on the existence theory of fractional -difference equations can be found in [15–25]. In a recent paper [26], the authors investigated a nonlocal boundary value problem of nonlinear fractional -difference equations: where , is the fractional -derivative of the Caputo type, is -derivative, and .

The purpose of the present paper is to study the following nonlocal boundary value problem of nonlinear fractional -difference equations: where , is the fractional -derivative of the Caputo type, and is a real number.

The paper is organized as follows. Section 2 contains some necessary background material on the topic, while the main results are presented in Section 3. We make use of Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative to establish the existence results for the problem at hand.

#### 2. Preliminaries on Fractional -Calculus

Here we recall some definitions and fundamental results on fractional -calculus.

*Definition 1 (see [13]). *Let , let , and let be a function defined on . The fractional -integral of the Riemann-Liouville type is and
where

Recall that , with

More generally, if , then

Lemma 2 (see [27]). *For , , the following is valid:
*

In particular, for , using -integration by parts, we have

For , we define the -derivative of a real valued function as For more details, see [28].

*Definition 3 (see [29]). * The fractional -derivative of the Riemann-Liouville type of order is defined by and
where is the smallest integer greater than or equal to .

*Definition 4 (see [29]). *The fractional -derivative of the Caputo type of order is defined by
where is the smallest integer greater than or equal to .

Now we state some known results involving -derivatives and -integrals.

Lemma 5 (see [29]). *Let , and let be a function defined on . Then *

*(i) *,

*(ii) *,

*(iii) *, .

Lemma 6 (see [19]). *Let and . Then, the following equality holds:
*

In the forthcoming analysis, the following lemma plays a pivotal role.

Lemma 7. *Let be a given function. Then the unique solution of the boundary value problem,
**
is given by
**
where
*

*Proof. *Using Lemma 5, we can write the solution of fractional -difference equation in (13) as

Using the boundary conditions of (13) in (16), we get and
where is given by (15). Substituting the values of in (16), we obtain (14).

Let denote the Banach space of all continuous functions from to endowed with the norm defined by .

By virtue of Lemma 7, we define an operator as and note that the given problem (2) has solutions only if the operator equation has fixed points.

#### 3. Main Results

In the sequel, we assume that is a continuous function and that there exists a -integrable function such that .

For computational convenience, we introduce the notations: where

Our first existence result is based on Leray-Schauder nonlinear alternative.

Lemma 8 (nonlinear alternative for single valued maps [30]). *Let be a Banach space, a closed, convex subset of , and an open subset of with . Suppose that is a continuous, compact (i.e., is a relatively compact subset of ) map. Then either *

*(i)** has a fixed point in *,
* or*

*(ii) there is ** (the boundary of ** in **) and ** with *.

Theorem 9. * Suppose that is a continuous function. In addition, it is assumed that ** there exist functions and a nondecreasing function such that , for .** Then the problem (2) has at least one solution on if there exists a positive number such that
**
where
*

*Proof. *As a first step, it will be shown that the operator (defined by (18)) maps bounded sets into bounded sets in . Notice that is continuous. For a positive number , let be a bounded set in . Then, for any , we find that

This establishes our assertion.

Next, we show that the operator maps bounded sets into equicontinuous sets of . Taking with and together with the inequality: for (see [19] p. 4), we obtain
It is obvious that the right hand side of the above inequality tends to zero independently of as . Therefore the operator is completely continuous by the Arzelá-Ascoli theorem.

Thus the operator satisfies the hypothesis of Lemma 8 and hence by its conclusion, either condition (i) or condition (ii) holds. We claim that the conclusion (ii) is not possible.

Let with given by (21). Then we will show that . Indeed, by means of , we get
Assume that there exist and such that . Then for such a choice of and , we get a contradiction:
Thus it follows by Lemma 8 that has a fixed point which is a solution of the problem (2). This completes the proof.

*Remark 10. *In case we take in to be continuous, then , where is given by (20).

Our next result deals with existence and uniqueness of solutions for the problem (2) and is based on Banach’s fixed point theorem.

Theorem 11. * Suppose that the assumption holds and that , where is given by (19). Then the boundary value problem (2) has a unique solution. *

* Proof. * Fix , ( is given by (20)), and define . We will show that , where is defined by (18). For , it follows by the assumption that
Then, for , we have
which, in view of (19), implies that
This shows that .

Now, for , we have
which, by (19), takes the form
As (the given assumption), therefore is a contraction. Hence, by Banach’s contraction mapping principle, the problem (2) has a unique solution.

If we take ( is a positive constant), the condition becomes and Theorem 11 can be phrased as follows.

Corollary 12. *Let there exists a constant with given by (20), and let be a continuous function satisfying the assumption: , , . Then the boundary value problem (2) has a unique solution. *

Our last result relies on Krasnoselskii’s fixed point theorem [31].

Lemma 13 (Krasnoselskii). * Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let be two operators such that (i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping. Then there exists such that .*

Theorem 14. * Let be a continuous function satisfying . Furthermore, it is assumed that ** there exist a function and a nondecreasing function such that
**If
**
with , then the boundary value problem (2) has at least one solution on .*

* Proof. * Let us consider the set , where is given by
and introduce the operators and on as
In order to show the hypothesis of Krasnoselskii’s fixed point theorem, we proceed as follows. For , we find that
This implies that . From the continuity of , it follows that the operator is continuous. Also, is uniformly bounded on as
Next, for any , and with , we have
which is independent of and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . (iii) From and (33) it follows that is a contraction mapping.

Thus all the conditions of Lemma 13 are satisfied. Hence, by the conclusion of Lemma 13, the problem (2) has at least one solution on .

As a special case, for , there always exists a positive such that (34) holds true. In consequence, we have the following corollary.

Corollary 15. *Let be a continuous function satisfying . Furthermore, , , and . Then the boundary value problem (2) has at least one solution on provided (33) holds. *

#### 4. Examples

*Example 1. * Consider the problem
where . It is easy to find that , and
Obviously . In consequence, , and condition (21) yields . Thus, the hypothesis of Theorem 9 is satisfied. Hence it follows by the conclusion of Theorem 9 that there exists at least one solution for the problem (39).

*Example 2. * Consider the nonlocal boundary value problem given by
Here and is a constant to be fixed later on. With the given data, it is found that , and
Letting
all the conditions of Corollary 12 are satisfied. Therefore, the conclusion of Corollary 12 applies to the problem (41).

#### Conflict of Interests

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

#### Acknowledgments

The authors thank the referees for their useful suggestions that led to the improvement of the original paper. This work has been partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.