Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

Yue Liang,Xiaoyan Gao,Yu Ma

doi:10.1155/2013/781404

Abstract

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup , ; in a Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates an equicontinuous semigroup, and is continuous. In the case where satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.

Highlights

The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [1,2,3,4,5,6]
We study the partial differential equation of parabolic type of fractional order
The existence of solutions to fractional evolution equations in Banach space has been studied by several authors; see [7,8,9,10,11,12,13,14,15,16,17]

Summary

Introduction

The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [1,2,3,4,5,6]. In [7, 8], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density They investigated the existence, uniqueness, and regularity of solutions to fractional integrodifferential equations in [9, 10]. To make the things more applicable, in this work, we will prove the existence of mild solutions of the IVP(1) under the measure of noncompactness conditions. We obtain the existence of positive mild solutions of the IVP(1) in this work, which is studied seldom before.

Preliminaries

Existence of Positive Mild Solutions

Positive Mild Solutions of Parabolic Equations