Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

Abstract

This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.

Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay

“…In this manuscript, we deliberate phase spaces B h , B h that are the same as described in [30]. Therefore, we bypass the details.…”

“…Nowadays, existence results of mild solutions for such problems became very attractive and several researchers are working on it. Recently, several papers have been written on the fractional order problems with SDD [23,[25][26][27][28][29][30][31][32][33][34][35][36] and the sources therein.…”

Existence and Controllability of Fractional Neutral Integro-Differential Systems With State-Dependent Delay in Banach Spaces

“…Moreover, B r (x, X) symbolizes the closed ball in X with the middle at x and the distance r. It needs to be outlined that, once the delay is infinite, then we should talk about the theoretical phase space B h in a beneficial way. In this manuscript, we deliberate phase spaces B h which are same as described in [30]. So, we bypass the details.…”

Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps

“…These processes are suitably modeled by impulsive fractional stochastic differential equations. Moreover, the qualitative behavior such as the existence and controllability of fractional dynamical systems are current important issues explored by many researchers, for example see [7], [18], [20]. Tai et al [26] addressed the controllability results of fractional-order impulsive neutral functional infinite delay integro-differential systems in Banach spaces by using Krasnoselskii's fixed-point theorem.…”

“…Here, we move from deterministic impulsive fractional differential equations to stochastic impulsive fractional differential equations with Poisson jumps for the study of existence of solutions and controllability properties. Motivated by few studies [7], [18], [23], [22], the existence of solutions and approximate controllability of the following impulsive fractional stochastic differential system with infinite delay and Poisson jumps remains an untreated topic in the literature:…”