Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

Dongyuan Liu,Zigen Ouyang,Huilan Wang

doi:10.1155/2015/263748

Abstract

We consider the following state dependent boundary-value problemD0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0,0<t<1;y(0)=0,ηy(σ(1))=y(1),whereDαis the standard Riemann-Liouville fractional derivative of order1<α<2,0<η<1,p≤0,0<β<1,β+1-α≥0the functiongis defined asg(t,u):[0,1]×[0,∞)→[0,∞), andg(0,0)=0the functionfis defined asf(t,u):[0,1]×[0,∞)→[0,∞)σ(t),τ(t)are continuous ontand0≤σ(t),τ(t)≤t. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.

Highlights

Fractional order differential equations has useful applications in many fields, such as physics, mechanics, chemistry, engineering, biology, and so on
Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references
There has been a significant development in fractional differential equations (e.g., [1,2,3,4,5,6,7,8,9])

Summary

Introduction

Fractional order differential equations has useful applications in many fields, such as physics, mechanics, chemistry, engineering, biology, and so on. Wu [15] used the wavelet operational method for solving fractional partial differential equations numerically. Since it is one of the important fields to be concerned with the boundary value problems for fractional order differential equations, some authors considered the existence of positive solutions for fractional differential equations or systems with boundary value conditions [16,17,18,19,20,21,22,23,24,25] and the stability [26]. Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for boundary value problem (4).

Preliminary

Main Results

Examples