Existence of Uniqueness and Nonexistence Results of Positive Solution for Fractional Differential Equations Integral Boundary Value Problems

Yongqing Wang

doi:10.1155/2018/1547293

Abstract

In this paper, we consider a class of fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are obtained by means of the iterative technique. The interesting point lies in that the assumption on nonlinearity is closely associated with the spectral radius corresponding to the relevant linear operator.

Highlights

We consider the existence of uniqueness and nonexistence of positive solution for the following fractional differential equations: Dα0+u (t) + f (t, u (t)) = 0, 0 < t < 1, u (0) = u󸀠 (0) = ⋅ ⋅ ⋅ = u(n−2) (0) = 0, (1)
While there are a lot of works dealing with the existence and multiplicity of solutions for nonlinear fractional differential equations, the results dealing with the uniqueness of solution are relatively scarce
The main tool used in most of the papers dealing with the uniqueness of solution is the Banach contraction map principle provided that the nonlinearity f is a Lipschitz continuous function

Summary

Introduction

We consider the existence of uniqueness and nonexistence of positive solution for the following fractional differential equations: Dα0+u (t) + f (t, u (t)) = 0, 0 < t < 1, u (0) = u󸀠 (0) = ⋅ ⋅ ⋅ = u(n−2) (0) = 0, (1). The main tool used in most of the papers dealing with the uniqueness of solution is the Banach contraction map principle provided that the nonlinearity f is a Lipschitz continuous function. Motivated by the above work, the aim of this paper is to establish the existence of uniqueness and nonexistence of positive solution to problem (1). The nonexistence results of positive solution are obtained under conditions concerning the spectral radius of the relevant linear operator

Preliminaries

Main Results

Example

Conclusions