The Uniqueness of Solution for a Class of Fractional Order Nonlinear Systems with p-Laplacian Operator

Abstract

We are concerned with the uniqueness of solutions for a class of p-Laplacian fractional order nonlinear systems with nonlocal boundary conditions. Based on some properties of the p-Laplacian operator, the criterion of uniqueness for solutions is established.

1. Introduction

Fractional order differential systems arise from many branches of applied mathematics and physics, such as gas dynamics, Newtonian fluid mechanics, nuclear physics, and biological process [1–12]. In the recent years, there has a significant development in fractional calculus. For example, by using the contraction mapping principle, ur Rehman and Khan [13] established the existence and uniqueness of positive solutions for the fractional order differential equation with multipoint boundary conditions: where , , , and , with . In [14], by using the fixed point theorem of mixed monotone operator, Zhang et al. studied the existence and uniqueness of positive solution for the following fractional order differential systems with multipoint boundary conditions: where , , , , and , with and ; is the standard Riemann-Liouville derivative. Some interesting results were also obtained by Zhang et al. [1, 2, 5, 7, 9], Goodrich [15–17], and Ahmad and Nieto [18].

On the other hand, the -Laplacian equation where , , can describe the turbulent flow in a porous medium; see [19]. Recently, by using Krasnoselskii’ s fixed point theorem and the Leggett-Williams theorem, Wang et al. [20] investigated the existence of positive solutions for the nonlocal fractional order differential equation with a -Laplacian operator: where , , , and . And then, by looking for a more suitable upper and lower solution, Ren and Chen [21] established the existence of positive solutions for four points fractional order boundary value problem: where and are the standard Riemann-Liouville derivatives, -Laplacian operator is defined as , , and the nonlinearity may be singular at both and .

Inspired by the above work, in this paper, we study the uniqueness of positive solutions for the following fractional order differential system with -Laplacian operator: where , , , and are the standard Riemann-Liouville derivatives with , , and , and are positive parameters, -Laplacian operator is defined as , , , and , , , . In the rest of paper, we assume that are continuous.

Normally, we cannot apply the contraction mapping principle for solving the BVP (1) like ur Rehman and Khan [13] since -Laplacian operator is nonlinear. In this paper, by using a property of the -Laplacian operator, we overcome this difficulty and establish the uniqueness of solution for the eigenvalue problem of the fractional differential system (6).

2. Preliminaries and Lemmas

We firstly list the necessary definitions from fractional calculus theory here, which can be found in [10–12].

Definition 1. Let . The fractional integral operator of a function is given by

Definition 2. Let . The Riemann-Liouville fractional derivative of a function is given by where , denotes the integer part of the number , and denotes the gamma function.

Property 1. Letting and , then(1) (2) where and is the smallest integer greater than or equal to .
The main results of this paper are based on the following property of -Laplacian operator, which is easy to be proved.

Lemma 3. (1) If and , then
(2) If , , and , then
Applying Definitions 1 and 2 and Property 1, we have the following lemma.

Lemma 4. Let , , , and . The fractional order boundary value problem, has the unique solution where and .

Similar to (14), the fractional order boundary value problem, has unique solution where and .

Lemma 5. Let , , and . The functions , are continuous on and satisfy(i) for ;(ii)for ,  where (iii)For ,  where

Proof. The proof is obvious; we omit the proof.

The basic space used in this paper is , where is a real number set. Obviously, the space is a Banach space if it is endowed with the norm as follows: for any . By Lemma 4, is a solution of the fractional order system (1) if and only if is a solution of the integral equation

We define an operator by where

It is easy to see that is the solution of the boundary value problem (6) if and only if is the fixed point of . As , we know that is a continuous and compact operator.

3. Main Results

Now we here introduce a new concept: the -contraction mapping.

Definition 6. A function is called a nonlinear -contraction mapping if it is continuous and nondecreasing and satisfies .

Theorem 7. Suppose that , if there exist nonnegative functions , such that and the following conditions are satisfied:for any , there exist -contraction mappings , as Then the fractional order differential system (6) has a unique solution provided that

Proof. In the case , we have , . Now we prove that is a contraction mapping. By (27)-(28) and Lemma 5, we have By (12), (28), and (31), for any and for , we have Similarly, we also have So it follows from (14), (17), and (31)-(32) that Hence where Noticing that , we obtain that is a contraction mapping. By means of the Banach contraction mapping principle, we get that has a unique fixed point in which implies that the fractional order differential system (6) has a unique solution.

Theorem 8. Suppose that , , if there exist nonnegative functions , , such that and the following conditions are satisfied:for any , there exist -contraction mappings , as Then the fractional order differential system (6) has a unique solution provided that

Proof. In the case , , we get ; here we still prove that is a contraction mapping if the conditions of theorem are satisfied. By (37)-(38) and Lemma 5, for any , we have By (11), (39), and (41), for any and for , we have Similarly, we also have So it follows from (14), (17), and (42)-(43) that Hence where Noticing that , we obtain that is a contraction mapping. By means of the Banach contraction mapping principle, we get that has a unique fixed point in which implies that the fractional order differential system (6) has a unique solution.

It follows from Theorems 7 and 8 that the following corollaries for mixed cases hold.

Corollary 9. Suppose that and if there exist nonnegative functions , such that and the following conditions are satisfied:for any , there exist -contraction mappings , as Then the fractional order differential system (6) has a unique solution provided that

Corollary 10. Suppose that and if there exist nonnegative functions , , such that and the following conditions are satisfied:for any , there exist -contraction mappings , as Then the fractional order differential system (6) has a unique solution provided that

Conflict of Interests

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

Acknowledgment

The authors were supported financially by Ministry of Education, State Administration of Foreign Experts “111 Project of Innovation and Intelligence Introducing Planning” (B08039).