Existence and Uniqueness of Solutions for a Fractional Order Antiperiodic Boundary Value Problem with a -Laplacian Operator

Ruihui Huang

doi:10.1155/2013/743538

Abstract

We study the existence and uniqueness of solutions for a class of antiperiodic boundary value problems of the fractional differential equation with a <svg style="vertical-align:-2.29482pt;width:10.2px;" id="M2" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,8.863)"><use xlink:href="#x1D45D"/></g> </svg>-Laplacian operator. Based on the Leray-Schauder nonlinear alternative, several sufficient conditions of the existence and uniqueness of solution of the above problem are established. Our results improve and complement the recent work of Chen and Liu, 2012.

Highlights

Economics is a rich source for mathematical ideas
We study the existence and uniqueness of solutions for a class of antiperiodic boundary value problems of the fractional differential equation with a p-Laplacian operator
Mathematical model is an important tool designed to describe the operation of the economy of a country or a region

Summary

Introduction

Economics is a rich source for mathematical ideas. mathematical model is an important tool designed to describe the operation of the economy of a country or a region. Several theoretical interesting results are available in the literature [6,7,8,9,10,11,12,13,14] for the existence and uniqueness of solution of fractional order differential equations. Assume that (S∗∗) there exists some constant ε ∈ (0, β) such that b(t) ∈ L1/ε([0, 1], [0, +∞)) and a C − N function ψ with 󵄨󵄨󵄨󵄨f (t, x)󵄨󵄨󵄨󵄨 ≤ b (t) ψ (|x|) , a.e. We complement a uniqueness result on the ABVP (2), which is based on the Banach contraction mapping principle and a basic property of the p-Laplacian operator: if q > 2, |x|, |y| ≤ M, . 󵄨󵄨󵄨󵄨f (t, x) − f (t, y)󵄨󵄨󵄨󵄨 ≤ b (t) ψ (󵄨󵄨󵄨󵄨x − y󵄨󵄨󵄨󵄨) , a.e. (t, x) ∈ [0, 1] × R. (12)

Preliminaries and Lemmas

The Proof of the Main Results

Examples