Existence of a mild solution for an impulsive nonlocal non-autonomous neutral functional differential equation

Abstract

This paper considers an impulsive neutral differential equation with nonlocal initial conditions in an arbitrary Banach space E. The existence of the mild solution is obtained by using Krasnoselskii's fixed point theorem and approximation techniques without imposing the strong restriction on nonlocal function and impulsive functions. An example is also provided at the end of the paper to illustrate the abstract theory.

Approximate Controllability of impulsive neutral integro-differential equations with deviated arguments and infinite delay in Banach spaces

“…One of the fundamental issues considered in the hypothesis of differential equations with deviated arguments is to set up advantageous conditions. The outcome, we will demonstrate in this paper sums up a few ones acquired before in [2,13,14].…”

“…This theory in recent years has attracted the attention of vast number of researchers, interested in both in the theory and its applications. For more details, we refer [2,3,8,9,16,17] By the motivation of above mentioned literature we have proved the existence of mild solutions for an impulsive neutral integro-differential equation with infinite delay and with deviated argument in a Banach space (E, · ) through the utilization of the Schauder fixed point theorem. In section 2, we gave some definitions, preliminaries, some lemmas and theorems.…”

On mild solutions for integrodifferential systems with integral impulses in Banach spaces

“…The theory of impulsive equattions has been widely developed, especially concerning differential equations (see for example the papers of Wang and Ezzinbi [20], Mophou [18], Suganya and Arjunan [19], Hernandez et al [15], Chadha and Pandey [8] and the book of Benchora et al [3]).…”

Existence results for non-autonomous neutral integro-differential systems with impulsive and nonlocal conditions

“…In particular, in [15], author has demonstrated the controllability of a system of impulsive semilinear non-autonomous differential equations via Rothe's type fixed-point theorem. By applying approximation techniques and fractional operator, the existence of the mild solution for different class of impulsive functional integro-differential equations have been established by many authors [16][17][18][19][20][21]. Recently, in [19], the authors investigate the existence of a mild solution for an impulsive nonlocal non-autonomous neutral functional differential equation in Banach space by utilizing the approximation techniques, fractional powers of operators and Krasnoselskii's fixed-point theorem.…”

“…Motivated by above mentioned works [13,19], the main purpose of this paper is to prove the existence of mild solutions for the following impulsive non-autonomous neutral partial integro-differential equations in a Banach space E:…”