- Open Access
The uniqueness of solution for a fractional order nonlinear eigenvalue problem with p-Laplacian operator
© Yang and Zhong; licensee Springer. 2014
- Received: 5 May 2014
- Accepted: 26 June 2014
- Published: 22 July 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 .
- eigenvalue problem
- p-Laplacian operator
- fractional differential equation
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]).
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.
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.
- (2)If , , and , then(2.2)
Lemma 2.2 (see )
is continuous on and for any ;
Proof (i) is obvious. We prove that (ii) is valid.
Equation (2.7) is a straightforward consequence of (2.8). The proof is thus completed. □
for and .
Lemma 2.4 (see )
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.
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:
Then there exists a constant such that for any , the BVP (1.1) has a unique solution.
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:
Then there exists a constant such that for any , the BVP (1.1) has a unique solution.
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. □
The BVP (3.11) has a unique solution for any .
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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