Results of Positive Solutions for the Fractional Differential System on an Infinite Interval

Ying Wang,Yumei Zi,Jun Yang

doi:10.1155/2020/5174529

Abstract

The chief topic of this paper is to investigate the fractional differential system on an infinite interval. By introducing an appropriate compactness criterion in a special function space and applying the Schauder fixed-point theorem and the Banach contraction mapping principle, we established the results for the existence and uniqueness of positive solutions. An example is then given to show the utilization of the main results.

Highlights

We investigate the following fractional differential system on an infinite interval:
Liang and Zhang [17] applied the fixed-point theorem to obtain the existence of positive solutions for the following fractional differential equation:
Let E be defined as (24) and M be any bounded subset of E. en, M is relatively compact in E if 􏼈(x(t)/1 + tα− 1) : x ∈ M􏼉 is equicontinuous on any finite subinterval of J, and for any given ε > 0, there exists N > 0 such that |(x(t1)/1 + tα1− 1) − (x(t2)/1 + tα2− 1)| < ε uniformly with respect to all x ∈ M, and t1, t2 > N

Summary

Introduction

We investigate the following fractional differential system on an infinite interval:. Under the proper initial or boundary conditions, to study the positive solution of the above models is very necessary; especially, for the boundary value problems on the infinite interval, many authors put their interest in it [7,8,9,10,11,12,13,14,15,16]. Liang and Zhang [17] applied the fixed-point theorem to obtain the existence of positive solutions for the following fractional differential equation:. There are few studies on fractional differential systems of infinite intervals, it is necessary to do so. We use two different techniques: the Schauder fixed-point theorem and the Banach contraction mapping principle, for system (1), to obtain the existence of positive solutions and the uniqueness of positive solutions

Preliminaries and Lemmas

Main Results

An Example