Existence and Uniqueness of Positive Solutions for a New System of Fractional Differential Equations

Hongyu Li,Yang Chen

doi:10.1155/2020/2617272

Abstract

By virtue of a recent existing fixed point theorem of increasing φ−h,e-concave operator by Zhai and Wang, we consider the existence and uniqueness of positive solutions for a new system of Caputo-type fractional differential equations with Riemann–Stieltjes integral boundary conditions.

Highlights

In this paper, we consider the following nonlinear Caputotype fractional system:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨cDθ0+1 x(t) + f1(t, x(t), cDθ0+2 y(t) + f2(t, x(t), x(0) x′′(0) 0, y(t)) y(t)) a1(t), a2(t), ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x(1) y(0) y(1)􏽚 x(t)dA1(t), y′′(0) 0,􏽚 y(t)dA2(t), t ∈ [0, 1], t ∈ [0, 1], (1)In recent decades, fractional-order calculus has been widely used in engineering, biology, physics, and so on.Based on it, many scholars have been interested in the study of the existence of nontrivial solutions for various fractional boundary value problems
Many scholars have been interested in the study of the existence of nontrivial solutions for various fractional boundary value problems
In [16], the authors obtained the existence of two positive solutions for a nonlinear Caputo-type fractional system by virtue of fixed point index theory

Summary

Introduction

We consider the following nonlinear Caputotype fractional system:. cDθ0+1 x(t) + f1(t, x(t), cDθ0+2 y(t) + f2(t, x(t), x(0) x′′(0) 0, y(t)) y(t)). In [10], by virtue of Guo–Krasnosel’skii fixed point theorem, Ma and Cui studied the following fractional boundary value problem:. In [16], the authors obtained the existence of two positive solutions for a nonlinear Caputo-type fractional system by virtue of fixed point index theory. In [21], by using Guo–Krasnosel’skii fixed point theorem, the authors studied the existence of positive solutions for an infinite system of fractional Caputo-type differential equations. Ere are few papers about the application of φ− (h, e)-concave operator in nonlinear Caputo-type fractional boundary value problems. In this paper, we use the recent fixed point theorems of φ − (h, e)-concave operator by Zhai and Wang to study system (1). e result of the existence and uniqueness of positive solutions for system (1) is obtained

Preliminaries

Main Results

Application