Positive Solutions for Singular 𝑝-Laplacian Fractional Differential System with Integral Boundary Conditions

Abstract

This paper investigates the existence of positive solutions for a class of singular -Laplacian fractional differential equations with integral boundary conditions. By using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions to the boundary value system is guaranteed.

1. Introduction

This paper investigates the singular -Laplacian fractional boundary value problem: where , and , , are real numbers. , , , are nondecreasing functions of bounded variation . and are the standard Riemann-Liouville fractional derivatives of order , respectively, and the integrals in (1) are Riemann-Stieltjes integrals. Here : is Carathéodory function; that is, satisfies the local Carathéodory condition on , and may be singular at the value 0 of all its space variables , , in which , .

We say that satisfies the local Carathéodory condition on ,  , if(i): is measurable for all ;(ii): is continuous for a.e. ;(iii)for each compact set there is a function such that for a.e. and all .

A vector is called positive solution of system (1) if and only if satisfies (1) and or for any .

Fractional differential equation can describe many phenomena in various fields of science and engineering, such as control, porous media, electrochemistry, and electromagnetic. For details, see [1–8] and the references therein. There are also a large number of papers dealing with the solvability of nonlinear fractional differential equations. Papers [9–13] discuss fractional boundary value problems with nonlinearities having singularities in space variables.

Paper [14] is concerned with the existence of positive solutions to the following BVP for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals: where and is positive integer, the integrals are Riemann-Stieltjes integrals, and is a Carathéodory function on . Some existence and multiplicity results of positive solutions are obtained by using the Krasnosel'skii fixed point theorem, the Leray-Schauder nonlinear alternative, and the Leggett-Williams fixed point theorem.

In the past few decades, in order to meet the demands of research, the -Laplacian equation is introduced in some BVP, such as [15, 16].

Paper [15] investigates the existence of solutions for the BVP of fractional -Laplacian equation with the following form: where , and , and , is a -Laplacian operator. is a Caputo fractional derivative. A new result on the existence of solutions for the above fractional boundary value problem is obtained, which generalize and enrich some known results to some extent from the literature, by using the coincidence degree theory.

Paper [16] studies the existence of positive solutions of the following singular fourth-order coupled system with integral boundary conditions: where and are positive parameters, , , , , and are nondecreasing functions of bounded variation, , and the integrals in (4) are Riemann-Stieltjes integrals, : and : are two continuous functions, and may be singular at , while may be singular at ;  : are continuous and may be singular at and/or , in which and . By using the fixed point theory in cones, explicit range for and is derived such that for any and which lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed.

Inspired by above works, our work presented in this paper has the following new features. Firstly, our study is on singular nonlinear differential systems; that is, may be singular at the value 0 of all its space variables , , which bring about many difficulties. Secondly, the techniques used in this paper are approximation methods, and a special cone has been developed to overcome the difficulties due to the singularity and to apply the fixed-point theorem. Finally, we discuss the BVP with integral boundary conditions, that is, system (1) including multipoint and nonlocal boundary value problems as special cases. To our knowledge, very few authors studied the existence of positive solutions for -Laplacian fractional differential equation with boundary conditions involving Riemann-Stieltjes integrals. Hence we improve and generalize the results of previous papers to some degree, and so it is interesting and important to study the existence of positive solutions for system (1).

Throughout the paper, is the norm in the Banach and is the norm in the Banach . Let ; then is a Banach space endowed with the norm . Thus, is a Banach with the norm defined by for .

This paper is organized as follows. In Section 2, we present some results of fractional calculus theory and auxiliary technical lemmas, which are used in the next two sections. Section 3 deals with the approximate problem of (1). We induce the solvability of this problem to the existence of a fixed point of an operator . By the Leggett-Williams fixed point theorem, the existence of at least three fixed points of is obtained. In Section 4, we prove the existence and multiplicity of positive solutions of problem (1) by applying the results of Sections 2 and 3.

2. Preliminaries

For the convenience of the reader, we present here the necessary definitions and lemmas from fractional calculus theory. These definitions and lemmas can be found in the recent literatures [14, 17–19].

For convenience, we list the following signs and assumptions for system (1).

We note that(1),  ,  ,  , ;(2),  .

We assume that(H₁),  , and  .

Obviously, ,  , .

Definition 1 (see [17]). The fractional integral of order of a function : is defined by provided that the right-hand side exists.

Definition 2 (see [17]). The Riemann-Liouville fractional order derivative of order of a function : is defined by provided that the right-hand side exists.

Lemma 3 (see [17]). Let and . Then where ,  .

Lemma 4 (see [18]). Suppose that , and , . Then ,, and

Lemma 5. If (H₁) holds, then for any and the boundary value problem has a unique solution where

Proof. By Lemma 3, we can see that Considering that solutions satisfy and  , we can get and . As a result,

Lemma 6 (see [14]). If (H₁) holds, then for any and the boundary value problem has a unique solution where

Lemma 7 (see [18]). Let and be as defined in (12) and (18). Then(1) and on ,(2) for ,(3) and on ,(4) for ,(5) and on ,(6) for ,(7) and on ,(8) for .

Proof. For notational convenience, we denote by From paper [19], (7) and (8) hold. Likewise, we have , since ; for , we have , so we have , and from [18], (1)–(6) hold.

Lemma 8. Let , be as defined in (11) and (17), . Then(1) and on ,(2) for ,(3) and on ,(4) for ,(5) and on ,(6) for ,(7) and on ,(8) for .

Proof . By (H₁), (11), and Lemma 7, we have , so (2) holds; likewise, (1) and (3)–(8) hold.

3. Auxiliary Regular Problem

To overcome singularity, we consider the following approximate problem of (1): where is a positive integer and Clearly, , .

Define a cone in as By Lemma 4, we can obtain that for and .

For each , let us define operators , and by and . By Lemmas 5 and 6, we know that fixed points of are positive solution of the system (20).

Lemma 9. is a completely continuous operator.

Proof. We divide the proof into three steps.
Step 1. We prove that is well defined.
For all , let . Then by (21) and (23), we have , . It follows from Lemma 8 that , , , are nonnegative and continuous on . Therefore, we get , and , , and on . As a result, ; then we can get .
Step 2. We prove that : is continuous.
Let be a convergent sequence. Suppose that . That is, and . Then and uniformly on , where . Since, by Lemma 4, for and , we have and uniformly on . In addition, it follows from (8), for and , that
Let
Since , , and and uniformly on , we have , for . Since and are bounded in and , inequalities (26) imply that and are bounded in . As a result, there exists such that , for and all ; from (8) in Lemma 8, we have ; for and all . From the Lebesgue dominated convergence theorem, we can obtain Since is continuous on , we have is uniformly continuous on ; that is, for all , , such that for all , and , we have For (28) we can say that for above and , for each , we have Since Lemma 8 and , for , all , we have From (29), (30), and (31), we can get so and likewise, Then That is, , so .
We prove that : is continuous.
Step 3. We prove that is compact; that is, for all , let be a bounded set, and is relatively compact in . For notational convenience, we denote by .
(I) We prove is uniformly bounded in .
For all , is a bounded set; then , such that for all , we have ; that is, , , and from (26), for and , we have and . Since , for some , such that , for and all , then and then ; that is is uniformly bounded in , so is also uniformly bounded in .
(II) We prove is equicontinuous on , where .
From Lemma 8, we can get , , and is uniformly continuous on ; that is, for all , such that for all , , and , we have ,, and  .
For all , we have likewise That is, is equicontinuous on , so is also equicontinuous on , .
Applying the Arzelà-Ascoli theorem, is relatively compact in , so we prove that is compact.
From Steps 1–3, we can get is a completely continuous operator.
The proof of Lemma 9 is completed.

Lemma 10 (see [20, 21]). Let be a cone in a real Banach space , , let be a nonnegative continuous concave functional on such that for all , and . Suppose that  : is completely continuous and there exist positive constants such that(C₁) and for ,(C₂) for ,(C₃) for with .Then has at least three fixed points , , and satisfying

Remark 11. If , then condition of Lemma 10 implies condition of Lemma 10.

We note , , and a nonnegative continuous concave functional on the cone defined by in which . Since , we get , and is well defined.

For notational convenience, we introduce the following constants:

We work with the following conditions on in (1), .

,  ,   and , such that satisfies the following conditions:(H₂) < , for  ×  ×  ×  × ,(H₃) ≤ , for  ×  ×  ×  × ,(H₄) > , for  ×  ×  ×  × , where , are defined by (43).

Since , we can get satisfies the following conditions:(H₂)′ < , for  ×  ×  ×  × ,(H₃)′ ≤ , for  ×  ×  ×  × ,(H₄)′ > , for  ×  ×  ×  × .

Theorem 12. Assume that (H₁)–(H₄) hold. Then, for , BVP (20) have at least three positive solutions , , and in , satisfying

Proof. We will show that all conditions of Lemma 10 are satisfied.
Firstly, we prove : is completely continuous.
For all , then . So we have , , and , . By (26) we have and . By condition (H₃), we can get (H₃)′ hold, so it follows from condition (H₃) that Thus, for any , by (45), we have which means that , so we have and . Therefore, : . By Lemma 9, we know that : is completely continuous.
Next, similar to (46), it follows from condition (H₂) that if , then . So the condition (C₂) of Lemma 10 holds.
Now, we take and , where ; then which means that and , This proves that .
On the other hand, if , then and , where and . By Lemma 4, for , we have and similarly, we can get , where . By condition (H₄), we can get (H₄)′ hold, so it follows from condition (H₄) that since in which .
Hence, by(49), we have so we can get Hence, by (50), (51), and (52), we have which implies that , for . This shows that condition (C₁) of Lemma 10 is also satisfied.
By Remark 11, condition (C₃) of Lemma 10 holds; that is, we show that all conditions of Lemma 10 are satisfied. So we obtain that BVP (20) have at least three positive solutions , , and in , such that , , , and , hence satisfying
The proof of Theorem 12 is completed.

Lemma 13. Assume that (H₁)–(H₄) hold. Let in be a solution of problem (20). Then the sequence is relatively compact in .

Proof. Firstly, we prove is uniformly bounded. Note that Since is in , we can get is uniformly bounded.
Next, we prove is equicontinuous on . Since is in , we have ,  , and by (26), we have and . From Lemma 8, we can get , , and are uniformly continuous on , that is, for all , such that for all and , , , and we have By (H₂) and Lemma 8, we have and similar to (57), we get That is, is equicontinuous on . Likewise, we can get is equicontinuous on . That is, is equicontinuous on .
Applying the Arzelà-Ascoli theorem, sequence is relatively compact in .
The proof of Lemma 13 is completed.

4. Main Results

Theorem 14. Assume that (H₁)–(H₄) hold. Then, for , BVP (1) have at least three positive solutions , , and in , satisfying

Proof. If (H₁)–(H₄) hold, by Theorem 12 and Lemma 13, we get BVP (20) have at least three positive solutions , , and in , satisfying for and for and the sequence , , and are relatively compact in .
We first consider the situation ; without loss of generality, assume that is convergent in and ; that is, and . Similar to the proof of Lemma 9, we have and uniformly on . Then Since , then there exists , such that for and all . Hence we can get By (61), (63), and the Lebesgue dominated convergence theorem, we can obtain Similarly, we can also get By Lemmas 5 and 6, we can obtain is positive solution of BVP (1).
Similarly, we can also obtain and are positive solutions of BVP (1).
Since for and for , now, we pass to the limit as in (66). Hence, we have
The proof of Theorem 14 is completed.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11071001 and 11201109) and the Natural Science Foundation of Anhui Province (1208085MA13).