Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives

Xiaoping Zhang,Yongping Sun

doi:10.1155/2014/254012

Abstract

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.

Highlights

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions
We discuss the existence and computing method of positive solutions of the αth-order fractional boundary value problem consisting of the nonlinear fractional differential equation
Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering, and biological sciences

Summary

Introduction

We discuss the existence and computing method of positive solutions of the αth-order fractional boundary value problem consisting of the nonlinear fractional differential equation. There have been many papers dealing with the solutions or positive solutions of boundary value problems for nonlinear fractional differential equations. Nonlinear fractional differential equations with two-point boundary conditions (2) have been studied by several authors. By using a fixed point theorem due to Krasnosel’skii, the author established the existence results for at least one positive solution. Yuan [27] gave sufficient conditions for the existence of multiple positive solution for the semipositone (n, p)-type boundary value problems of nonlinear fractional differential equations. To the best of our knowledge, few authors utilize the monotone methods to study the existence of positive solutions for nonlinear fractional boundary value problems.

Preliminaries

Main Results

Examples