New uniqueness results for boundary value problem of fractional differential equation

Yujun Cui, Wenjie Ma, Qiao Sun, Xinwei Su Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China cyj720201@163.com State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China School of Science, China University of Mining and Technology, Beijing 10083, China


Introduction
In this paper, we consider the uniqueness of solutions of the following boundary value problems for nonlinear fractional differential equation: Boundary value problems for nonlinear fractional differential equation have been investigated extensively.The motivation for those works arises from both the development of the theory of fractional calculus itself and the study of models of viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [5,7,8,16]).In applications, one is interested in showing the existence and multiplicity of solution (or positive solution).
Consequently, there has been a significant development in the study of boundary value problems of fractional differential equation, see [1,3,6,7,9,11,12,15].In [3], the authors considered the existence and multiplicity of positive solutions of BVP (1) by means of the Krasnosel'skii fixed-point theorem and the Leggett-Williams fixed-point theorem.However, there are few works on the uniqueness for boundary value problems of fractional differential equations [2,4,10,13,14].In [13], the authors studied the following multipoint boundary value problems of fractional order: where and D α t denotes the standard Riemann-Liouville fractional derivative.They proved the uniqueness of solutions to BVP (2) by means of the Banach's contraction mapping principle.Recently, the following nonlinear fractional differential equations with two point boundary conditions was also studied by Cui [4]: = 0, where 2 < p 3 is a real number.By use of u 0 -positive operator, a uniqueness result is proved, provided that f is a Lipschitz continuous function.The novelty of [4] is that the Lipschitz constant is related to the first eigenvalue corresponding to the relevant operator.Motivated by the results mentioned above, in this paper, we study the uniqueness of solutions for BVP (1) based on the Banach's contraction mapping principle and the theory of linear operator.It should be remarked that the method used in [4] is not suitable for BVP (1).

Preliminaries and lemmas
For convenience, we present here the necessary definitions from fractional calculus theory.These definitions can be found in the literature.Definition 1. (See [12].)The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, ∞) → R is given by provided that the right-hand side is pointwise defined on (0, ∞).
Let C[0, 1] denote the Banach space of real-valued continuous function with norm where By Lemma 1, BVP (1) can be converted into a fixed-point problem x = Ax, where Clearly, BVP (1) has a solution if and only if the associated fixed-point problem x = Ax has a fixed point.
Proof.By the Lagrange mean value theorem, we have This implies that E = ∅ and M α.
On the other hand, substituting the value t = 0 to the inequality a(1−t) 1+t α+1 − 2t α , we get a 1 and M 1.This completes the proof.
Y. Cui et al.

Main results
We first prove a uniqueness result based on the Banach's contraction mapping principle.
Next, we prove a uniqueness result by means of the theory of linear operator.
Theorem 2. Suppose that f : [0, 1] × R → R is a continuous function and there exists a constant L > 0 such that Proof.We define an operator In the following, we separate the proof into the following four steps.
From the proof of Theorems 1 and 2 we have Using this and Lemma 2, we get that is,