Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions

J Caballero,K Sadarangani,I Cabrera

doi:10.1155/2012/303545

Abstract

We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions, , , where , and is the Caputo fractional derivative and is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.

Highlights

Many papers and books on fractional differential equations have appeared recently see, for example, 1–22
We present the fixed point theorem which we will be use later
If we look at the proof of Theorem 2.2 in 32 we notice that the condition about the continuity of ψ is redundant

Summary

Introduction

Many papers and books on fractional differential equations have appeared recently see, for example, 1–22. Integral boundary conditions have various applications in chemical engineering, thermo-elasticity, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to some recent papers see, 23–30 and the references therein. Cabada and Wang in 31 investigated the existence of positive solutions for the fractional boundary value problem. The main tool used in 31 is the well-known Guo-Krasnoselskii fixed point theorem and the question of uniqueness of solutions is not treated. We consider our paper as an alternative answer to the results of 31. The fixed point theorem in partially ordered sets is the main tool used in our results. The existence of fixed points in partially ordered sets has been considered recently see, e.g. 32–34

Preliminaries and Basic Facts

Main Result

Some Remarks and Examples