Existence of Positive Solutions to Boundary Value Problem of Caputo Fractional Differential Equation

Yaqiong Cui,Zhiping Liu,Shugui Kang

doi:10.1155/2015/708053

Abstract

We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problemD0+αCut+fut,u't=0, t∈0,1, u0=u1=u″0=0, wheref:0,+∞×R→0,+∞is continuous,α∈2,3, andD0+αCis the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.

Highlights

We consider the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem (BVP): CD0α+u (t) + f (u (t), u󸀠 (t)) = 0, t ∈ (0, 1), (1)
If α ∈
Motivated by the above results and [8–10], to cover up this gap, if α ∈ (2, 3], we mainly discuss the existence of positive solutions to fractional differential equation which is under the boundary value conditions u(0) = u(1) = u󸀠󸀠(0) = 0

Summary

Introduction

We consider the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem (BVP): CD0α+u (t) + f (u (t) , u󸀠 (t)) = 0, t ∈ (0, 1) , (1). There has been especially an increased interest in studying the existence of positive solutions for the continuous fractional calculus concerning the Riemann-Liouville and Caputo derivatives; see [1–7] and references therein. Motivated by the above results and [8–10], to cover up this gap, if α ∈ (2, 3], we mainly discuss the existence of positive solutions to fractional differential equation which is under the boundary value conditions u(0) = u(1) = u󸀠󸀠(0) = 0. Based on Schauder’s fixed point theorem, the cone expansion or the cone compression fixed point theorem, and an extension of Krasnoselskii’s fixed point theorem, we obtain the existence of positive solutions and give some examples to illustrate our results. The solutions especially are the interior points of cone; the solutions have better properties

Preliminary

Main Results

Conflict of Interests