Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

Abstract

We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.

Stability for a class of semilinear fractional stochastic integral equations

“…Several researchers have studied stability of stochastic systems via Lyapunov function techniques. An example of this is the paper of Li et al [22], who proves stability in probability for Itô-Volterra integral equation, also Zhang and Li [47] have stated a stochastic type stability criteria for stochastic integrodifferential equations with infinite retard, and, Zhang and Zhang [48] have dealed with conditional stability of Skorohod Volterra type equations with anticipative kernel. Nguyen [32] present the solution via the fundamental solution for linear stochastic differential equations with time-varying delays to obtain the exponential stability of these systems.…”

Stability for some linear stochastic fractional systems

“…This method is similar to Adomian one. The Lyapunov function techniques have been used to deal with stability in probability for Itô-Volterra integral equation (see Li et al [16]), with some stochastic type stability criteria for stochastic integrodifferential equations with infinite delay (see Zhang and Li [35]) and with conditional stability of Skorohod Volterra type equations with anticipative kernel (see Zhang and Zhang [36]). The mean square stability for Volterra-Itô equations with a function as initial condition has been established by Bao [4] by means of Gronwall lemma.…”