On the new concept of solutions and existence results for impulsive fractional evolution equations

Abstract

Abstract. In this paper we discuss the existence of P C-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a P C-mild solution for our problem is introduced. Our main techniques based on fractional calculus and fixed point theorems. Some concrete applications to partial differential equations are considered.

Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

“…Therefore, in recent years, many authors have proposed and proved results of existence and uniqueness for the solutions of such equations, using different methods; Kilbas et al, Podlubny, Zhou, Benchohra and Lazreg, Furati and Kassim, Gu and Trujillo, Mâagli et al, Abbas et al, Yang and Wang, Abbas et al, Zhou et al, and so on are just a few of the most classical references on the subject. Furthermore, these works have paved the way for several new lines of study, such as the fractional theory applied to the so‐called impulsive equations …”

A new estimation method for controllability of semilinear impulsive integro-differential systems

“…Over the past few decades, various types of semilinear and nonlinear systems prescribed by ordinary differential equations have been studied extensively by researchers. Particularly, the existence, controllability and other properties of solutions of various systems have attracted many authors, for instance, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [12], [13], [15], [16], [17], [19], [20], [21], [22], [24], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], and the references cited therein. Using differential techniques, existence and controllability results for a variety of nonlinear differential systems are obtained, and this topic is very important in various fields.…”

Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems

“…Denote C[0, 1] by the set of all continuous functions from [0, 1] into R. impulsive fractional Cauchy problems in Fečkan et al[31], one can make straightforward fractional calculations to show that the system (2) can be written as the following integral system in C[0, 1] × C[0, 1]:…”