Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations

Jinhua Wang,Zhigang Liu,Hongjun Xiang

doi:10.1155/2010/186928

Abstract

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

Highlights

Fractional differential equation can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic
There are many papers dealing with the existence and uniqueness of solution for nonlinear fractional differential equation; see, for example, 1–5
From Lemma 3.2, T is completely continuous, by Banach fixed point theorem, the operator T has a unique fixed point in P , which is the unique positive solution of system 1.1

Summary

Introduction

Fractional differential equation can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic. There are many papers dealing with the existence and uniqueness of solution for nonlinear fractional differential equation; see, for example, 1–5. As far as we know, concerning the existence of positive solution for coupled system of nonlinear fractional differential equations with nonzero boundary values. We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations: International Journal of Differential Equations. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, some sufficient conditions for the existence and uniqueness of positive solution to the above coupled boundary values problem are obtained.

Basic Definitions and Preliminaries

Main Result