Existence of positive solutions for a system of nonlinear Caputo type fractional differential equations with two parameters

Yang Chen,Hongyu Li

doi:10.1186/s13662-021-03650-z

Abstract

The main purpose of this paper is to prove the existence of positive solutions for a system of nonlinear Caputo-type fractional differential equations with two parameters. By using the Guo–Krasnosel’skii fixed point theorem, some existence theorems of positive solutions are obtained in terms of different values of parameters. Two examples are given to illustrate the main results.

Highlights

Fractional-order calculus, which is an important branch of mathematics, was introduced in 1695
Boundary value problems of fractional differential equations have appeared with applications of fractional-order calculus; so far, there have been many literature works about boundary value problems of fractional differential equations
In [10], the authors used the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem to obtain the existence of positive solutions to the nonlinear Caputo fractional q-difference equation with integral boundary conditions

Summary

Introduction

Fractional-order calculus, which is an important branch of mathematics, was introduced in 1695. In [10], the authors used the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem to obtain the existence of positive solutions to the nonlinear Caputo fractional q-difference equation with integral boundary conditions. In [23], the authors investigated a coupled system of Caputo fractional differential equations with coupled non-conjugate Riemann–Stieltjes type integro-multipoint boundary conditions They obtained some new theorems by using the Leray–Schauder nonlinear alternative, the Krasnosel’skii fixed point theorem, and Banach’s contraction mapping principle. In [26, 27], the authors used the Guo–Krasnosel’skii fixed point theorem to investigate the existence of positive solutions for systems of fractional differential equations nonlocal boundary value problems with two parameters, and the existence of positive solutions were obtained.

Caputo fractional differential equation

We define the cone

From and

Then we have

Then we get

We can get