Abstract

We discuss a class of singular boundary value problems of fractional -difference equations. Some existence and uniqueness results are obtained by a fixed point theorem in partially ordered sets. Finally, we give an example to illustrate the results.

1. Introduction

In recent years, many papers on fractional differential equations have appeared, because of their demonstrated applications in various fields of science and engineering; see [1–11] and the references therein. Based on the increasingly extensive application of discrete fractional calculus and the development of -difference calculus or quantum calculus (see [12–19] and the references therein), fractional -difference equations have attracted the attention of researchers for the numerous applications in a number of fields such as physics, chemistry, aerodynamics, biology, economics, control theory, mechanics, electricity, signal and image processing, biophysics, blood flow phenomena, aerodynamics, and fitting of experimental data; see [20–23]. Some recent work on the existence theory of fractional -difference equations can be found in [24–29]. However, the study of singular boundary value problems (BVPs) with fractional -difference equations is at its infancy and lots of work on the topic should be done.

Recently, in [25], Ferreira has investigated the existence of positive solution for the following fractional -difference equations BVP by applying a fixed point theorem in cones.

More recently, in [30], Caballero et al. have studied positive solutions for the following BVP: by fixed point theorem in partially ordered sets.

Motivated by the work above, in this paper, we discuss the existence and uniqueness of solutions for the singular BVPs of factional -difference equations given by where and is continuous with (i.e., is singular at ).

2. Preliminary Results

For convenience, we present some definitions and lemmas which will be used in the proofs of our results.

Let and define

The -analogue of the power function with is More generally, if , then Note that if then . The -gamma function is define by and it satisfies .

Following, let us recall some basic concepts of -calculus [12].

Definition 1. For , we define the -derivative of a real-value function as Note that .

Definition 2. The higher order -derivatives are defined inductively as For example, , where is a positive integer and the bracket . In particular, .

Definition 3. The -integral of a function in the interval is given by If and is defined in the interval , its integral from to is define by Similarly as done for derivatives, an operator can be define, namely, Observe that and if is continuous at , then .
We now point out three formulas ( denotes the derivative with respect to variable )

Remark 4. We note that if and , then [24].

Definition 5 (see [21]). Let and be a function defined on . The fractional -integral of the Riemann-Liuville type is and

Definition 6 (see [23]). The fractional -derivative of the Riemann-Liuville type of is defined by and where is the smallest integer greater than or equal to .

Lemma 7 (see [21, 23]). Let and let be a function define on . Then, the next formulas hold:(1),(2).

Lemma 8 (see [24]). Let and let be a positive integer. Then, the following equality holds:

Lemma 9. Let and ; then the BVP has a unique solutionwhere

Proof. By Lemmas 7 and 8, we see thatwhere , and are some constants to be determined. Since , we must have . Now, differentiating both sides of (22) and using (15), we obtain Using and , we must set , and Finally, we obtain The proof is complete.

Lemma 10. Function defined above satisfies the following conditions:(i) is a continuous function on ;(ii) for .

Proof. (i) Obviously, is continuous on .
(ii) Let It is clear that , for . Now, in view of Remark 4, for Therefore, . This proof is complete.

By we denote the class of those functions satisfying the following condition; implies .

Theorem 11 (see [31]). Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that there exists an element with . Suppose that there exists such thatAssume that either is continuous or is such that Besides if then has a unique fixed point.

Let be the Banach space with the classic metric given by .

Notice that this space can be equipped with a partial order given by In [32], it is proved that satisfies condition (29) of Theorem 11. Moreover, for , as the function satisfies condition (30).

3. Main Result

In this section, we will consider the question of positive solutions for BVP (3). At first, we prove some lemmas required for the main result.

Lemma 12. Let , and is a continuous function with . Suppose that is a continuous function on . Then the function defined by is continuous on , where is Green function be given in Lemma 9.

Proof. We will divide the proof into three parts.
Case 1 (). First, . Since is continuous on , we can find a positive constant such that for any . Thus, For , let ; then we obtainHence,where denotes the -beta function.
When , we see that ; that is is continuous at .
Case 2 (). We should prove when . Without loss of generality, we consider (it is the same argument for ). In fact, where When , we verify .
As , then , when . Moreover, We have converges pointwise to the zero function and is bounded by a function belonging to , by Lebesgue’s dominated convergence theorem when .
Now, we prove when .
In fact, as and taking into account that , we get when .
Above all, we obtain , when .
Case 3 (). It is easy to check that and is continuous at . The proof is the same as the proof of Case 1.