Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

Chen Yang,Jieming Zhang

doi:10.1155/2012/672543

Abstract

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

Highlights

In this paper, we are interested in the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem: D0α u t f t, u, u, . . . , u n−2 g t 0, 0 < t < 1, n − 1 < α ≤ n, n ≥ 2, 1.1 u 0 u 0 · · · u n−2 0 u n−2 1 0, where D0α is the standard Riemann-Liouville fractional derivative and g : 0, 1 → 0, ∞ is continuous.Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, engineering, and biological sciences; see 1–15, for example
For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proof of our theorem
We present some basic concepts in ordered Banach spaces for completeness and a fixed-point theorem which we will be used later

Summary

Introduction

We are interested in the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem: D0α u t f t, u, u , . . . , u n−2 g t 0, 0 < t < 1, n − 1 < α ≤ n, n ≥ 2, 1.1 u 0 u 0 · · · u n−2 0 u n−2 1 0, where D0α is the standard Riemann-Liouville fractional derivative and g : 0, 1 → 0, ∞ is continuous. We are interested in the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem: D0α u t f t, u, u , . When g t ≡ 0, Yang and Chen 22 investigated the existence and uniqueness of positive solutions for the problem 1.1 by means of a fixed-point theorem for u0 concave operators In a recent paper 28 , Zhai et al considered the following operator equation: They established the existence and uniqueness of positive solutions for the above equation, and they present the following interesting result. We apply Theorem 2.5 to study the problem 1.1 , and we obtain a new result on the existence and uniqueness of positive solutions.

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