On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses

Bo Zhu,Lishan Liu,Wenguang Yu,Baoyan Han

doi:10.1186/s13661-020-01451-z

Abstract

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.

Highlights

1 Introduction In this paper, we study the following initial boundary value problem of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses:
By the general Banach contraction mapping principle, we get that the operator defined by (3.1) has a unique fixed point u∗ ∈ PC(J, E), which means that problem (1.1) has a unique PC-mild solution
Remark 3.1 In Theorem 3.1, we prove the existence and uniqueness of the PC-mild solutions for problem (1.1) using the general Banach contraction mapping principle

Summary

Introduction

We study the following initial boundary value problem of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses:. In [19, 20], Chen, Zhang, and Li studied the non-autonomous evolution equations with non-instantaneous impulses and obtained the main results of the existence. Guo [28] studied the existence and uniqueness of the following integer nonlinear integro-differential equations of mixed type in a Banach space E:. Zhang, and Li [19] studied the existence of the following fractional non-autonomous integro-differential evolution equations of mixed type:. To the best of our knowledge, we have not found the relevant results that study the initial boundary value problem for the fractional partial integro-differential equations of mixed type with non-instantaneous impulses. Our main results of this paper generalize and improve some corresponding results

Preliminaries

Conclusion