Existence, uniqueness and Hyers-Ulam stability of random impulsive stochastic integro-differential equations with nonlocal conditions

Abstract

<abstract><p>In this article, we study the existence and stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups and resolvent operators in Hilbert spaces. Initially, we prove the existence of mild solutions using Hausdorff measures of noncompactness and M$ \ddot{o} $nch fixed point theorem. Then, we explore the stability results which includes continuous dependence of initial conditions, Hyers-Ulam stability and mean-square stability of the system by developing some new analysis techniques and establishing an improved inequality. Finally, we propose an example to validate the obtained results.</p></abstract>

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

“…The existence and stability of impulsive stochastic integro-differential equations which has mild random solutions with some proprieties of semigroup and some operators in suitable space, [6], and integral equation of mild solutions of impulsive fractional differential equations presented in [18]. The impulsive fractional integro-differential equations with mixed boundary defined as Riemann-Liouville fractional integral conditions studied in [24], and some of papers studied the solutions by using fixed point theorems such as [5],also, the nonlocal boundary value of impulsive integro-differential equations as a system studied in [23], interesting existence result, and the concept of Ulam stability have studied in [8].…”

Some new existence results for fractional partial random nonlocal differential equations with delay

“…When we have statistical understanding about the parameters of a dynamic system, that is, when the knowledge is probabilistic, the most popular strategy in mathematical modelling of such systems is to employ random differential equations or stochastic differential equations. As natural extensions of deterministic differential equations, random differential equations appear in numerous applications and have been studied by several mathematicians; see the monographs [17,25,35] and the papers of Baleanu et al [16], Boumaaza et al [11,34], Harikrishnan et al [20], Karapinar et al [23,30], Liu et al [26], and references therein.…”