Multiplicity Solutions for Integral Boundary Value Problem of Fractional Differential Systems

Shiying Song,Yujun Cui

doi:10.1155/2020/2651845

Abstract

This paper deals with the existence and multiplicity of solutions for the integral boundary value problem of fractional differential systems:D0+α1u1t=f1t,u1t,u2t,D0+α2u2t=f2t,u1t,u2t,u10=0, D0+β1u10=0, D0+γ1u11=∫01D0+γ1u1ηdA1η,u20=0, D0+β2u20=0, D0+γ2u21=∫01D0+γ2u2ηdA2η,, wherefi:0,1×0,∞×0,∞⟶0,∞is continuous andαi−2<βi≤2,αi−γi≥1,2<αi≤3,γi≥1i=1,2.D0+αis the standard Riemann–Liouville’s fractional derivative of order α. Our result is based on an extension of the Krasnosel’skiĭ’s fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in 2019. The main results are explained by the help of an example in the end of the article.

Highlights

With the deepening of people's understanding of mathematics, the knowledge of mathematics is more and more closely related to the way of production and life of human beings
In [22], the authors obtained the uniqueness results of positive solution by the contraction map principle and obtained some existence results of positive solution through the fixed-point index theory, which is as follows: Dα0+u(t) + f(t, u(t)) 0, 0 < t ≤ 1
We investigate the existence and multiplicity of positive solutions for the integral boundary value problem of higher-order fractional differential systems. is result is based on an extension of the Krasnosel’skiǐ’s fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in [46]

Summary

Introduction

With the deepening of people's understanding of mathematics, the knowledge of mathematics is more and more closely related to the way of production and life of human beings. Fractional calculus is very active in the field of applied mathematics It can be applied in biochemistry, mathematical physics equation, physical science experiment, and other academic fields and in precision production [1–3]. In recent years, many scholars have begun to use the fixed-point index to study the existence and multiplicity of operator equation and operator systems [44–51]. In [46], the authors use the fixed-point index in cones to study the existence, localization, and multiplicity of positive solutions to operator systems of the following form:. Erefore, in this paper, we will apply an extension of the Krasnosel’skiǐ’s fixed-point theorem to investigate the existence and multiplicity of solutions for a class of fractional differential systems.

Preliminaries

Main Results

Conclusions