Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem

Wenzhe Xie,Zhiguo Luo,Jing Xiao

doi:10.1155/2014/540351

Abstract

By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.

Highlights

Fractional calculus, which is a powerful tool for the description of memory and hereditary properties of materials and processes, has attracted the attention of many researchers and has been successfully applied in various fields, such as economics, engineering, and physical sciences
The main results in this paper are the following
We will apply the method of upper and lower solutions combined with fixed point theorem to consider the existence of solutions of FBVP (3)

Summary

Introduction

Fractional calculus, which is a powerful tool for the description of memory and hereditary properties of materials and processes, has attracted the attention of many researchers and has been successfully applied in various fields, such as economics, engineering, and physical sciences. Nagumo conditions play an important role in the boundary value problems with nonlinear term involved in the derivative, since as it is known, for instance, for second order differential equations, the existence of upper and lower solutions, by itself, is not sufficient to ensure the existence of solutions. The studies dealing with the Nagumo conditions are well established by applying the method of upper and lower solutions combined with fixed point theorem or topological degree theory for the case of integer order (see [15,16,17]). Inspirited by the papers mentioned above, in this paper, under Nagumo conditions we aim to apply the method of upper and lower solutions combined with fixed point theorems to discuss the existence of solutions for the following Riemann-Liouville fractional boundary value problem (FBVP for short): D0q+u (t) + f (t, u (t) , D0q+−1u (t)) = 0, t ∈ [0, 1] ,.

Preliminaries

Main Results

Examples