Research Article Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity

Difference equations with continuous time are popular enough with researches [1–8]. Volterra equations are undoubtedly also very important for both theory and applications [3, 8–12]. 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). Here the conditions from [8, 10] are generalized for nonlinear stochastic difference second-kind Volterra equations with continuous time. All results are obtained by general method of Lyapunov functionals construction proposed by Kolmanovskiı̆ and Shaikhet [8, 13–21]. Let {Ω,F,P} be a probability space and let {Ft, t ≥ t0} be a nondecreasing family of sub-σ-algebras of F, that is, Ft1 ⊂ Ft2 for t1 < t2, let H be a space of Ft-adapted functions x with values x(t) in Rn for t ≥ t0 and the norm ‖x‖2 = supt≥t0 E|x(t)|2. Consider the stochastic difference second-kind Volterra equation with continuous time: