Qualitative Behaviour of Stochastic Integro-differential Equations with Random Impulses

Abstract

In this paper, we study the existence and some stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups in Hilbert spaces via resolvent operators. Initially, we prove the existence of mild solution for the system is established by using Mönch fixed point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results includes continuous dependence of solutions on initial conditions, exponential stability and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results.

Existence Results of Random Impulsive Integrodifferential Inclusions with Time-Varying Delays

“…To encompass a variety of abrupt incidents, incorporating impulsive occurrences, partial DEs driven by stochastic processes or equations with random impulses offer a natural and efficient approach. Many researchers have thought about the study of integro-DEs with random impulses, including those in [6][7][8][9][10]. Dassios explored the singular system within a category of DEs with multiple delays in his work documented in [11].…”

“…Moreover, Li et al, featured in [18], undertook an investigation into the existence of solutions for the Sturm-Liouville DEs featuring random impulses and boundary value problems, employing Dhage's fixed point theorem as a pivotal tool in their analysis. The existence, singularity, and stability of the delay integro-DEs with random impulse were established by [6,7].…”