Journal of Function Spaces
Volume 2015 (2015), Article ID 418569, 8 pages
http://dx.doi.org/10.1155/2015/418569

Research Article

Positive Solutions for a Class of Singular Boundary Value Problems with Fractional -Difference Equations

Jufang Wang, Changlong Yu, and Yanping Guo

College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China

Received 6 July 2014; Accepted 21 September 2014

Academic Editor: Kishin Sadarangani

Copyright © 2015 Jufang Wang 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.

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 solution

where

Proof. By Lemmas 7 and 8, we see that

where , 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 that

Assume 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 obtain

Hence,

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.

Lemma 13. Suppose that . Then,

where .

Proof.

Let , .
Since , let ; we can get has a maximum at the point .
Hence,

For the convenience, we denote by .

Next, we denote the class of functions by satisfying

(i) is nondecreasing;(ii) for any ;(iii), where is the class of functions appearing in Theorem 11.

We give our main result as follows.

Theorem 14. Let , is continuous and , and is a continuous function on . Assume that there exists such that for with and ,

where . Then the BVP (3) has a unique positive solution (i.e., for ).

Proof. We define the cone by

It is clear that is a complete metric space as is a closed set of . It is also easy to check that satisfies conditions (29) and (30) of Theorem 11.
We define the operator by

In view of Lemma 12, . Moreover, it follows from the nonnegativeness of and that for . Thus, .
Next, we will prove that assumptions in Theorem 11 are satisfied.
First, for , we have

Hence, the operator is nondecreasing. Besides, for and ,

Since is nondecreasing and ,

Moreover, when , we get

Obviously, the last inequality is satisfied for .
Taking into account that the zero function satisfies , in view of Theorem 11, the operator has a unique fixed point in .
At last, we will prove is a positive solution. We assume that there exists such that . Since of problem (3) is a fixed point of the operator , we have

For the nonnegative character of and , the last relation gives

is continuous and ; then for , we can find , and, for , we have . It is clear that and , where is the Lebesgue measure on . That is to say, a.e. . This is a contradiction because is a rational function in .
Therefore, for .
The proof is complete.

4. Example

Consider the following singular BVP:

Here, , , , and for . Notice that is continuous in and .

At first, we define by

It is clear that is a nondecreasing function; for , we can get

Moreover, for , also satisfies

In fact, when ,

equivalently

Above all, for .

Second, for and , we have

that is, satisfies assumptions of Theorem 14.

Third, we should prove belongs to . By elemental calculus, it is easy to check that is nondecreasing and , for .

In order to prove , we notice that if , then the sequence is a bounded sequence because in contrary case, that is, , we get

Now, we assume that , and then we find such that for each there exists with .

Since is a bounded sequence, we can find a subsequence of with , for certain . When , it implies that

and, as the unique solution of is , we deduce that . Therefore, and this contradicts the fact that for every .

Thus, and this proves that .

Finally, in view of Theorem 14,

that is, when , boundary value problem (52) has a unique positive solution.

Conflict of Interests

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

Acknowledgments

This paper is supported by the Natural Science Foundation of China (11201112), the Natural Science Foundation of Hebei Province (A2013208147), (A2014208152), and (A2015208114), and the Foundation of Hebei Education Department (Z2014062).

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