Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional -Difference Equation

Abstract

A class of boundary value problems of Caputo fractional -difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.

1. Introduction

Some researchers have paid close attention to the research of -difference equation since the -difference calculus and quantum calculus were discovered by Jackson [1, 2]. After the fractional -difference calculus was developed by Al-Salam et al. [3–6], many papers on the fractional -difference equation kept emerging, such as the papers [7–21] and their references. Among them, Li and Yang [7] established the existence of positive solutions for a class of nonlinear fractional -difference equations with integral boundary conditions by applying monotone iterative method. Koca [8] provided an analytical method that can be used to solve analytically the Caputo fractional -differential equations with initial condition . The advantage of the method is that it can be applied to the integer order -difference equations. By applying the monotone iterative technique combined with the method of lower and upper solutions, Wang et al. [9] obtained the existence of extremal solutions for fractional -difference equation with initial value problem.

There are also many papers about boundary value problems of fractional -difference equations; see [10–19] and the references therein. These experts did researches about the existence of a positive solution and multiple positive solutions to this problem by applying some well-known fixed-point theories such as Krasnosel’skii and Schauder fixed-point theorems. Thereinto, in [7, 15–20], the authors focused on the fractional -difference equation with integral boundary value conditions.

Motivated by the methods of [21–23] and the above works, we study the criteria of three positive solutions for a Caputo fractional -difference equation with integral boundary value conditions by employing properties of Green’s function and the Leggett-Williams fixed-point theorem in this paper. We mainly consider the following problem: where denotes the Caputo fractional -derivative of order and   is continuous function.

In Section 2, basic definitions and some lemmas that will be used in the latter part are presented. In Section 3, some results for the existence of three positive solutions to problem (1)-(2) are established. And some examples to corroborate our results are given in Section 4.

2. Background Materials and Preliminaries

In this piece, we show some basic definitions and some lemmas that will be used to demonstrate our main results in the latter section.

Setting , we define

The -analogue of the power function with is

If , then

We define the -gamma function as follows: which satisfies

For , the -derivative of a function is defined by

The higher order -derivatives are defined by

The -integral of a function defined on the interval is given by provided that the series converges.

If is defined on the interval and , its -integral from to is defined by

The higher order -integrals are defined by

We note that and if is continuous at , we get For more details on the basic material of -calculus, the readers can refer to [1–6].

Now let us give definitions of fractional -integral and -derivative.

Definition 1 (see [4, 6]). Let and let be a function defined on . The fractional -integral of the Riemann-Liouville type is and where denotes the classical Banach space consisting of measurable functions on that are integrable.

Definition 2 (see [6]). The fractional -derivative of the Riemann-Liouville type of order is given by and where is the smallest integer greater than or equal to .

Definition 3 (see [13]). The fractional -derivative of Caputo type of order for a function is defined by

Lemma 4 (see [13]). Let and let be the smallest integer greater than or equal to . Then, for , the following equality holds:

Lemma 5 (see [13]). Let , and the following is valid:

Next, Green’s function for integral boundary value problem (1)-(2) is derived and the properties of Green’s function are concluded. These properties will be used to demonstrate the main results in Section 3.

Lemma 6. Given , the unique solution of the following problem is where and Here is called Green’s function of boundary value problem (17)-(18).

Proof. By Lemma 4, it is clear that (17) is equivalent to for some Applying the boundary condition , there is , and then Using the condition , there is Substituting into (22), we get Let ; integrating equality (24) with respect to from to and then exchanging integral order, we obtain that Solving the above equation, then Substituting into (24), we get The proof is complete.

Remark 7. It is obvious that for all and

Lemma 8. Suppose and Then the function defined by (20) satisfies the following inequalities:

Proof. According to the expression of , we get For the case , it is clear that For the case , it is easy to see that On the other hand, Therefore When , inequalities (28) are obvious. In conclusion, inequalities (28) are fulfilled. The proof is complete.

Corollary 9. If , then , for

Definition 10 (see [24]). If is a cone of the real Banach space , a mapping is continuous and with it is called a nonnegative concave continuous functional on .
Assuming that , , are positive constants, we will employ the following notations: Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.

Lemma 11 (see [24]). Let be a Banach space, be a cone of , and be a constant. Suppose there exists a concave nonnegative continuous functional on with for . Let be a completely continuous operator. Assume there are numbers , , and with such that(i)the set is nonempty and for all ;(ii) for ;(iii) for all with . Then has at least three fixed points ,  , and . Furthermore, we have

3. Existence of Three Positive Solutions

In this section, the above lemmas will be applied to obtain the main results of this paper.

Let be the space of all continuous real functions defined on with the maximum norm . We can know it is a Banach space. Define the cone as follows:

From Lemma 6, we know that is a solution of boundary value problem (1)-(2) if and only if it satisfies That is to say, the positive solutions of problem (1)-(2) are equivalent to the fixed points of in defined by Then and using the Ascoli-Arzelà theorem, we are able to confirm that is completely continuous.

We shall use Lemma 11 to discuss the existence of three fixed points to . We then obtain sufficient conditions for the existence of three positive solutions to problem (1)-(2). To establish our main results, we take a positive number , letting the nonnegative concave continuous function on be defined by Denote And suppose that the function satisfies the following condition:(C) is a nonnegative continuous function on and there exists such that

Theorem 12. Assume that condition (C) holds and there exist constants such that(C1) for ;(C2) for , where ;(C3) for , where are positive numbers. Then the boundary value problem (1)-(2) has at least three positive solutions , and

Proof. Set , and then, for , we have from (28) That is, . Therefore is a completely continuous operator. By (C1), we can get Hence condition (ii) of Lemma 11 is satisfied.
We choose for ; then , which implies . Hence, if , then for Thus From the above inequality, we see that for all This affirms that condition (i) of Lemma 11 is satisfied.
Finally, for with , we get This confirms that condition (iii) of Lemma 11 is fulfilled. By virtue of Lemma 11, the boundary value problem (1)-(2) has at least three solutions , and . Taking into account the fact that condition (C) holds, we have The proof is complete.

Theorem 13. Let condition (C) hold. Assume that there exist constants such that (C1), (C2), and (C4) are satisfied, where (C4) for . Then the boundary value problem (1)-(2) has at least three positive solutions , and such that

Proof. From (C4), we get Therefore, The remainder of the proof is similar to the proof of Theorem 12 and is therefore omitted. By Lemma 11, the boundary value problem (1)-(2) has at least three positive solutions , and satisfying The proof is complete.

Theorem 14. Let condition (C) hold. Assume that there exist constants such that (C1) and (C2) are satisfied, and function satisfies(C5) Then the boundary vale problem (1)-(2) has at least three positive solutions.

Proof. From hypothesis (C5), there exist and ; when , we have Set ; consequently we get This shows that condition (C3) of Theorem 12 is satisfied. By Theorem 12, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.

Theorem 15. Assume that there exist two positive constants such that conditions (C), (C2), and (C4) hold. And function satisfies(C6) Then the boundary value problem (1)-(2) has at least three positive solutions.

Proof. In line with (C6), it is easy to see that there exists a positive constant such that, for , we have That is to say, This implies that conditions of Theorem 13 are satisfied. By Theorem 13, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.

In light of the proof of Theorems 14 and 15, we obtain one theorem and four corollaries as follows.

Theorem 16. Assume that the function satisfies conditions (C), (C2), (C5), and (C6). Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 17. Assume that conditions (C), (C2), and (C3) hold. The function satisfies Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 18. Assume that conditions (C), (C1), and (C2) hold. The function satisfies Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 19. Assume that conditions (C), (C2), and (C4) hold. The function satisfies Then the boundary value problem (1)-(2) has at least three positive solutions.

Corollary 20. Assume that conditions (C) and (C2) hold. The function satisfies and Then the boundary value problem (1)-(2) has at least three positive solutions.

4. Examples

In this section, we present three examples to illustrate our results. We take , and by estimation, we then have

Consider the Caputo fractional -difference with the boundary conditions

Example 1. We take There exist constants and such that All the conditions of Theorem 12 hold. Thus, at this moment, by virtue of Theorem 12 we know that the boundary value problem (53)-(54) has three positive solutions.

Example 2. We take There exist constants and such that All the conditions of Theorem 14 hold. Thus, in this case, by Theorem 14 we know that the boundary value problem (53)-(54) has three positive solutions.

Example 3. We seek There exists constant such that All the conditions of Theorem 16 hold. Thus, in this case, by using Theorem 16 we know that the boundary value problem (53)-(54) has three positive solutions.

5. Conclusions

The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractional -difference equation with integral boundary value conditions are discussed. The study in the paper was to provide an analytical method: The Leggett-Williams fixed-point theorem can be used to solve fractional -difference equation. In order to use the Leggett-Williams fixed-point theorem, Green’s function and its properties were derived. By applying these properties and the Leggett-Williams fixed-point theorem, we presented the existence of three positive solutions of this class of fractional -difference equations with integral boundary value conditions. An important advantage of this method is that it can be used to study three positive solutions for integer order -differential equations and fractional differential equation, and so forth.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant no. 11271235), the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111117, and 20111020), and Shanxi Datong University Institute (2016K9 and 2017K4).