Existence and Uniqueness of Mild Solutions to Impulsive Nonlocal Cauchy Problems

Abstract

In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced. We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.

Existence of solution for Volterra–Fredholm type stochastic fractional integro‐differential system of order μ ∈ (1, 2) with sectorial operators

“…Shukla et al worked on the approximate controllability of the linear system, which was then extended to stochastic system with nonlocal conditions [29]. For more details on nonlocal conditions, refer to [38][39][40][41][42][43][44][45]. In [46], the author studies the controllability results of Sobolev type Volterra-Fredholm integro-differential fractional system.…”

Impulsive conformable fractional stochastic differential equations with Poisson jumps

“…According to [15], an R n -value stochastic process ℵ(τ ) is called a unique solution of equation ( 1) if ℵ(τ ) satisfies the following:…”

“…Also, I assume that the mild solution of ( 1) is guaranteed under assumptions (H1)-(H2) (see [15,16]).…”