Solvability of boundary value problems of nonlinear fractional differential equations

Weiqi Chen,Yige Zhao

doi:10.1186/s13662-015-0373-2

Abstract

In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem $D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0 $ , $0< t<1$ , $u(0)=u(1)=u'(0)=0$ , where $2<\alpha\leq3$ is a real number, $D^{\alpha}_{0^{+}}$ is the Riemann-Liouville fractional derivative. By the properties of the Green’s function, the lower and upper solution method and the Leggett-Williams fixed point theorem, some new existence criteria are established. As applications, examples are presented to illustrate the main results.

Highlights

Fractional differential equations have been of great interest
Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self-similar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science; see [ – ]
There have appeared some papers dealing with the existence of solutions of fractional differential equations by the use of techniques of nonlinear analysis; see [ – ]

Summary

Introduction

Fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications. Bai and Lü [ ] studied the following two-point boundary value problem of fractional differential equations: Dα + u(t) + f t, u(t) = , < t < , u( ) = u( ) = , where < α ≤ is a real number and Dα + is the standard Riemann-Liouville fractional derivative. Zhang [ ] considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem

Results

Conclusion