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The uniqueness of solution for a fractional order nonlinear eigenvalue problem with p-Laplacian operator
Advances in Difference Equations volume 2014, Article number: 186 (2014)
In this article, we investigate the uniqueness of solutions for the fractional order differential equation with p-Laplacian operator , , , , , where , , are the standard Riemann-Liouville derivatives with , , , , , with , and the p-Laplacian operator is defined as , . Based on a basic property of the p-Laplacian operator and the Banach contraction mapping principle, the uniqueness of solutions for the fractional order differential equation is established for the cases and .
Differential equations of fractional order have been shown to be valuable tools in the modeling of many phenomena arising from science and engineering, such as the charge transport in amorphous semiconductors , flows through porous media, electrochemistry and material science [2–5]. In the recent years, there has been a significant development in fractional order differential equations involving various boundary conditions. For example, by using the contraction mapping principle, Rehman and Khan  established the existence and uniqueness of positive solutions for the following fractional order multi-point boundary value problem:
where , , , , with . In , Zhang et al. discussed the existence and uniqueness of positive solutions for the following fractional differential equation with derivatives:
where , , , , with , is the standard Riemann-Liouville derivative. is continuous, and may be singular at . By means of monotone iterative technique, the existence and uniqueness of the positive solution for a fractional differential equation with derivatives are established, and the iterative sequence of the solution, an error estimation and the convergence rate of the positive solution are also given.
Recently, some excellent work on nonlocal integral boundary condition for fractional differential equation and system was done by Zhang et al.  and Ahmad and Nieto . In , by establishing some comparison results and combining with a monotone iterative method, the existence of an extremal solution for a nonlinear system involving the right-handed Riemann-Liouville fractional derivative with nonlocal coupled integral boundary conditions was obtained. Ahmad and Nieto  employed standard fixed point theorems to study the uniqueness and existence of solution for a class of Riemann-Liouville fractional differential equations with fractional boundary conditions. Some new existence and uniqueness results are obtained. Here we also refer the reader to some recent work on fractional differential equation (see [10–17]).
Since the turbulent flow in a porous medium is a fundamental mechanics problem, Leibenson  introduced the following p-Laplacian equation to describe the flow of porous medium:
where , . Inspired by Leibenson’s work, Wang et al.  investigated the nonlinear nonlocal boundary value problem
where , , , . By using Krasnosel’skii’s fixed point theorem and the Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. Then, by using upper and lower solutions method, Wu and Zhou  studied the existence of positive solutions for the fractional order eigenvalue problem with the p-Laplacian operator
where , , are the standard Riemann-Liouville derivatives with , , , , , with , the p-Laplacian operator is defined as , . By constructing upper and lower solutions, the existence of positive solutions for the problem is established.
However, because of the stronger nonlinearity of the p-Laplacian operator, the uniqueness of solution for the above problem is still unknown. It is well known that the Banach contraction mapping principle is difficult to apply to the p-Laplacian operator to obtain the uniqueness of solution since it is nonlinear. In this paper, by studying the property of the p-Laplacian operator, we overcome this difficulty and establish the uniqueness of solution for the eigenvalue problem of the fractional differential equation (1.1).
The rest of this article is organized as follows. In Section 2, we present some definitions and preliminary results that are to be used to prove our main results. In Section 3, we present our main results followed by the proofs. Finally, we give an example to demonstrate the application of our main results.
2 Preliminaries and lemmas
In this paper, we restrict our attention to the use of the Riemann-Liouville fractional derivatives. For details of some basic definitions of the fractional calculus, we refer the reader to [19–21] or other texts on basic fractional calculus.
Based on a basic fact of the p-Laplacian operator, we can obtain the following lemma.
Lemma 2.1 (1) If , , then
If , , and , then(2.2)
Lemma 2.2 (see )
Suppose that . Then the following boundary value problem:
is equivalent to the following integral equation:
is the Green function of the boundary value problem (2.3) and
Lemma 2.3 The Green function in Lemma 2.2 has the following properties:
is continuous on and for any ;
Proof (i) is obvious. We prove that (ii) is valid.
In fact, by (2.6), we have
Equation (2.7) is a straightforward consequence of (2.8). The proof is thus completed. □
Let satisfy the relation . We consider the associated linear boundary value problem
for and .
Lemma 2.4 (see )
The associated linear boundary value problem (2.9) has unique positive solution
Let the Banach space be endowed with the norm . By Lemma 2.4, is a solution of the boundary value problem (1.1) if and only if is a solution of the integral equation
We define an operator by
It is easy to see that x is the solution of the boundary value problem (1.1) if and only if x is the fixed point of T. As , we know that is a continuous and compact operator.
3 Main results
In this section, we use the Banach contraction mapping principle to prove the existence and uniqueness of the solution of problem (1.1). Firstly, we give the result on the case . As , if , we have , and we have the following theorem.
Theorem 3.1 Suppose , and the following conditions hold:
(A1) There exists a nonnegative continuous function , which does not vanish for some , and a constant such that for any
(A2) For any ,
(A3) There exists some constant such that and
Then there exists a constant such that for any , the BVP (1.1) has a unique solution.
For any , we shall prove that F is a contraction mapping. By (3.1) and (3.2), we have
By (2.2), (3.3), and (3.5), for any and for , we have
So it follows from (2.7) and (3.5) that
which implies that
and then is a contraction mapping since . By means of the Banach contraction mapping principle, we get the result that F has a unique fixed point in , that is, the BVP (1.1) has a unique solution. □
In the case , as , we get , and we have the following theorem.
Theorem 3.2 Suppose , and the following condition holds:
(A4) There exist some constant and functions , such that , and
Then there exists a constant such that for any , the BVP (1.1) has a unique solution.
Proof In this case, we choose
and prove that F is a contraction mapping for any . It follows from (3.7) and the Hölder inequality that
From (2.1) and (3.9), for any ,
So by (2.7) and (3.10), we have
which implies that
and thus is a contraction mapping since . By means of the Banach contraction mapping principle, we get the result that F has a unique fixed point in , that is, the BVP (1.1) has a unique solution. □
Example Consider the following fractional order differential equation with the p-Laplacian operator:
The BVP (3.11) has a unique solution for any .
then we have
and . Thus (A1)-(A3) all are satisfied, by Theorem 3.1, the BVP (3.11) has a unique solution for any . □
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The authors declare that there is no conflict of interests regarding the publication of this paper.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.