The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions

Wenjie Ma,Yujun Cui

doi:10.1155/2018/2176809

Abstract

In this paper, we investigate the eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions Dc0+θp(y)+μf(t,p(y))=0, y∈[0,1], p(0)=p′′(0)=0, p(1)=∫01p(y)dA(y), where Dc0+θ is Caputo fractional derivative, θ∈(2,3], and f:[0,1]×[0,+∞)→[0,+∞) is continuous. By using the Guo-Krasnoselskii’s fixed point theorem on cone and the properties of the Green’s function, some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained.

Highlights

We investigate the eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions cDθ0+p(y) + μf(t, p(y)) = 0, y ∈ [0, 1], p(0) = p󸀠󸀠(0) = 0, p(1) = ∫01 p(y)dA(y), where cDθ0+ is Caputo fractional derivative, θ ∈ (2, 3], and f : [0, 1] × [0, +∞) 󳨀→ [0, +∞) is continuous
By using the Guo-Krasnoselskii’s fixed point theorem on cone and the properties of the Green’s function, some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained
The experience of the last few years has fully borne out the fact that the integer order calculus is not as widely used as fractional order calculus in some fields such as chemistry, control theory, and signal processing

Summary

Introduction

The experience of the last few years has fully borne out the fact that the integer order calculus is not as widely used as fractional order calculus in some fields such as chemistry, control theory, and signal processing. X [1] = μ ∫ x (s) ds, where α ∈ (2, 3], μ ∈ [0, 1), and cDα is the Caputo derivative They solved the above problem by means of classical fixed point theorems. X󸀠 [1] = αx󸀠󸀠 [1] , Journal of Function Spaces where q ∈ [2, 3], σ ∈ [1, 2], f : [0, 1] × R × R 󳨀→ R is a given function, and cDq0+ denotes the Caputo differentiation The author investigated this problem by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green’s function, and Guo-Krasnoselskii fixed point theorem on cone. Our proof is based on the properties of the Green’s function and the Guo-Krasnosel’skii fixed point theorem on cone

Preliminaries

Existence of Positive Solutions

Nonexistence of Positive Solutions

Example