A class of nonlocal problems of fractional differential equations with composition of derivative and parameters

Mei Jia,Xiping Liu,Lin Li,Zhanbing Bai,Junqiu Song

doi:10.1186/s13662-019-2181-6

Abstract

In this paper, we study existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters. By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The results we obtain show the influence of parameter λ and parameter a on the existence of positive solutions. Finally, some examples are given to illustrate our main results.

Highlights

1 Introduction In this paper, we investigate existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters
The theorems we obtain show the influence of parameter λ and parameter a on the existence of positive solutions
We investigate existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters (1.1)

Summary

Introduction

It is meaningful to study the boundary value problem of fractional differential equations with disturbance parameters, see [31–35] and the references therein. We investigate existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters (1.1). 3, we investigate the existence of at least one positive solution for boundary value problem (1.1). Definition 2.1 A function u = u(t) is called a solution of fractional boundary value problem (1.1) if u ∈ E and satisfies (1.1). Lemma 2.3 For any y ∈ Lq[0, 1], the fractional differential initial value problem.

Because t

Proof Denote

Proof Let aλ