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Báo cáo hóa học: " Research Article Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 65012 13 pages doi 10.1155 2007 65012 Research Article Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity Beatrice Paternoster and Leonid Shaikhet Received 25 December 2006 Accepted 8 May 2007 Recommended by Roderick Melnik 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. Copyright 2007 B. Paternoster and L. Shaikhet. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Definitions and auxiliary results 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 Kolmanovskii and Shaikhet 8 13-21 . Let O F P be a probability space and let Ft t to be a nondecreasing family of sub-ơ-algebras of F that is Ft1 c Ffe for t1 t2 let H be a space of Ft-adapted functions x with values x t in R for t t0 and the norm xh2 supt to E x t 2. Consider the stochastic difference second-kind Volterra equation with continuous time At h0 At h0 F t x t At - hA At