Abstract
Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained.
Highlights
Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered
Sufficient conditions for mean square summability of solutions of linear stochastic difference second-kind Volterra equations were obtained by authors in [10] and [8]
All results are obtained by general method of Lyapunov functionals construction proposed by Kolmanovskiı and Shaikhet [8, 13,14,15,16,17,18,19,20,21]
Summary
Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained. Sufficient conditions for mean square summability of solutions of linear stochastic difference second-kind Volterra equations were obtained by authors in [10] (for difference equations with discrete time) and [8] (for difference equations with continuous time). The conditions from [8, 10] are generalized for nonlinear stochastic difference second-kind Volterra equations with continuous time. Consider the stochastic difference second-kind Volterra equation with continuous time:. Η ∈ H, h0, h1, . . . are positive constants, φ is an Ft0-adapted function for θ ∈ Θ, such that φ
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