Positive Solutions of Fractional Differential Equations with -Laplacian

Abstract

The multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with -Laplacian operator is investigated. By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solution are obtained. An example is given to illustrate the main results.

1. Introduction

In recent years, boundary value problems of nonlinear fractional differential equations have been studied extensively (see [1–23] and the references therein). By the use of some fixed-point theorem, the existence results of positive solutions are obtained for singular factional problem [2, 4, 24, 25], impulsive fractional problem [3, 20], nonlocal problem [1, 12, 14, 16, 22, 23], numerical solution problem [17, 18], initial value problem [19, 20], Dirichlet value problem [6, 9, 15], iterative solution problem [5], and so on. Cui [7] considered the following boundary values problems: where is a real number and is the standard Riemann-Liouville differentiation. Under the assumption that is a Lipschitz continuous function, by the use of -positive operator, they studied the uniqueness results for the fractional differential equation.

On the other hand, because of the wide mathematical and physical background, the existence of positive solutions for nonlinear integer-order boundary values problems with -Laplacian operator has received wide attention (see [8, 12, 13, 21, 24–29]). For example, Su et al. [28] considered the following four-point boundary values problems with -Laplacian operator: where . Liu et al. [13], Dong et al. [8], and Zhang et al. [25] studied -Laplacian boundary value problems with fractional derivative. By using the fixed-point index theory, they obtained the existence of positive solutions.

In this paper, we investigate the multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with -Laplacian operator: where and and is the standard Riemann-Liouville differentiation and . By using the Leggett-Williams fixed-point theorem on a cone, the multiplicity results of positive solution are obtained.

2. Preliminaries

Lemma 1 (see [10]). Assume that with a fractional derivative of order that belongs to . Then where is the smallest integer greater than or equal to .

Lemma 2. If and , then the problem has a unique solution where

Proof. Suppose is a solution of the problem (5). By Lemma 1, there is for some . Taking into account the fact that , we have and Thus, By , (10), it holds that So, Applying Lemma 1, we can reduce (13) to an equivalent integral equation for some . With the condition , there is . Consequently, By (15), one has So, By , (17), we have So, the unique solution of problem (5) is The proof is completed.

Lemma 3. Suppose that . The functions and satisfy the following:(1) for ,(2) for ,(3)There exists a positive function such that

Proof.  
The Proof of the Statement (1). From definitions, it is clear that . By , we have If , let It is obvious that for . Hence, we have By using the analogous argument, it holds that for other situations. Hence, for .
If , let It is obvious that for . Hence, we have Similarly, it holds that  for or or . Hence, for .
The Proof of the Statement (2). Let It is easy to check that and are increasing with respect to on . We will show that and are decreasing with respect to on . For , let then we have For , let We have If there exists such that , then Therefore, we have In fact, taking into account the fact that and , one has So, we have , which implies that . Hence is decreasing with respect to on .
For , let ; then we have which implies that is decreasing with respect to on .
We can conclude that is increasing with respect to for and is decreasing with respect to for . Hence, for .
By using the analogous method, we can conclude that is increasing with respect to for and is decreasing with respect to for . Hence, for .
The Proof of the Statement (3). Taking into account the definition of , there is where . It is easy to see that is continuous on and for all . Let where Then The proof is completed.

Lemma 4 (see [11]). Let be a cone in a real space be a nonnegative continuous concave functional on a cone such that , for all , and Suppose is completely continuous and there exist constants such that() and for ;() for ;() for with . Then has at least three fixed points satisfying  and with .

3. Main Result

Let be a Banach space with . Define the cone by Let the nonnegative continuous concave functional on the be defined by

Lemma 5. Let be an operator defined by Then is completely continuous.

Proof. is continuous in view of nonnegativity and continuity of and . Furthermore it is easy to see that by the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem is completely continuous.

For convenience, we introduce the following notations:

Theorem 6. Suppose and there exist constants such that() , for ;() , for ;() , for . Then problem (3) possesses at least three positive solutions , and with

Proof. In order to apply Lemma 4, we divide the proof into four steps.
Step  1. If , then . Assumption implies for . Consequently, Hence .
Step  2. We claim that the condition of Lemma 4 is satisfied. Let . we can easily see that ; consequently, . Hence, if , then for . From assumption , we have So Then, we have This implies that condition of Lemma 4 is satisfied.
Step  3. We now prove the condition of Lemma 4 is satisfied. If , then . Assumption implies for . Thus Hence, the condition of Lemma 4 is also satisfied.
Step  4. Finally, we prove that the condition of Lemma 4 is satisfied. If , by Step  2, we have . Hence, the condition of Lemma 4 is also satisfied.
By Lemma 4, problem (3) has at least three positive solutions , and with The proof is complete.

Example 7. Consider the following boundary value problem: where Choose It is easy to check that and ; by simple computation, we have Let ; one can check that the function satisfies() , for ;() , for ;() , for .That is to say that all the conditions of Theorem 6 hold. Thus Theorem 6 implies the problem (50) has at least three positive solutions , and such that

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The Project is supported by NSFC (11571207), the Scientific Research Foundation of Hunan Provincial Education Department (16A198), the Hunan Provincial Natural Science Foundation of China (2015JJ6101), and the Construct Program of the Key Discipline in Hunan Province.