Existence and Hyers–Ulam stability of stochastic integrodifferential equations with a random impulse

Abstract

Abstract The theoretical approach of random impulsive stochastic integrodifferential equations (RISIDEs) with finite delay, noncompact semigroups, and resolvent operators in Hilbert space is investigated in this article. Initially, a random impulsive stochastic integrodifferential system is proposed and the existence of a mild solution for the system is established using the Mönch fixed-point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results including a 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.

Keywords

• Stochastic differential equations
• Random impulse
• Noncompact semigroup
• Hyers–Ulam stability
• Mean-square exponential stability