Existence and uniqueness of solutions for systems of fractional differential equations with Riemann\u2013Stieltjes integral boundary condition

Xinqiu Zhang,Lishan Liu,Yonghong Wu,Yumei Zou

doi:10.1186/s13662-018-1650-7

Abstract

In this article, we first establish an existence and uniqueness result for a class of systems of nonlinear operator equations under more general conditions by means of the cone theory and monotone iterative technique. Furthermore, the iterative sequence of the solution and the error estimation of the system are given. Then we use this new result to study the existence and uniqueness of the solution for boundary value problems of systems of fractional differential equations with a Riemann–Stieltjes integral boundary condition in real Banach spaces. The results obtained in this paper are more general than many previous results and complement them.

Highlights

1 Introduction In this paper, we first study the following system of nonlinear operator equations in a real Banach space E by means of the cone theory and monotone iterative technique:
In this paper, we first study the following system of nonlinear operator equations in a real Banach space E by means of the cone theory and monotone iterative technique: ⎧ ⎨x = A(x, x), ⎩x = B(x, x), (1.1)where A, B : D × D → E are two nonlinear operators, D is a subset of E
The existence and uniqueness theorems of solutions for boundary value problems of nonlinear fractional differential equations have been studied extensively in the literature, mainly by using the fixed point theorem of the mixed-monotone operator, a priori estimate method and a maximal principle, the Banach contraction mapping principle and the Krasnose’skii fixed point theorem

Summary

Introduction

1 Introduction In this paper, we first study the following system of nonlinear operator equations in a real Banach space E by means of the cone theory and monotone iterative technique: In [37], by using the cone theory and Banach contraction mapping principle, Zhang investigated the existence and uniqueness of solutions for a class of nonlinear operator equations x = Ax in real Banach spaces.

Results

Conclusion