Positive Solutions of a Fractional Boundary Value Problem with Changing Sign Nonlinearity

Yongqing Wang,Yonghong Wu,Lishan Liu

doi:10.1155/2012/149849

Yongqing Wang, Yonghong Wu + Show 1 more

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https://doi.org/10.1155/2012/149849

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Abstract

We discuss the existence of positive solutions of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.

Highlights

Much attention has been paid to the existence of solutions for fractional differential equations due to its wide range of applications in engineering, economics, and many other fields, and for more details see, for instance, 1–17 and the references therein
If u is a positive solution of P, u is a fixed point of A in Q, ut u, u t ≤ Kt, 2.15 where K
We consider the following boundary value problem: cD0α utλft, u t − λω t r t 0, 0 < t < 1, 2.23 u 0 u 1 u 0 0, where λ > 0, ω t is defined in Lemma 2.6, u t − λω t Let max{u t − λω t, 0}

Summary

Introduction

Much attention has been paid to the existence of solutions for fractional differential equations due to its wide range of applications in engineering, economics, and many other fields, and for more details see, for instance, 1–17 and the references therein. By using the Krasnosel’skii fixed-point theorem and the Leray-Schauder nonlinear alternative, Bai and Qiu 14 consider the positive solution for the following boundary value problem: cD0α u t f t, u t 0, 0 < t < 1, P u 0 u 1 u 0 0, where 2 < α ≤ 3 is a real number, cD0α is the Caputo fractional derivative, f : 0, 1 × 0, ∞ → 0, ∞ is continuous and singular at t 0. The purpose of this paper is to establish the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem: cD0α u t λf t, u t 0, 0 < t < 1, 1.1 u 0 u 1 u 0 0, where 2 < α ≤ 3 is a real number, cD0α is the Caputo fractional derivative, λ is a positive parameter, and f may change sign and may be singular at t 0, 1.

Basic Definitions and Preliminaries

Existence of Positive Solutions

Examples