tailieunhanh - LYAPUNOV FUNCTIONALS CONSTRUCTION FOR STOCHASTIC DIFFERENCE SECOND-KIND VOLTERRA EQUATIONS WITH

LYAPUNOV FUNCTIONALS CONSTRUCTION FOR STOCHASTIC DIFFERENCE SECOND-KIND VOLTERRA EQUATIONS WITH CONTINUOUS TIME LEONID SHAIKHET Received 4 August 2003 The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem, this method can be successfully used also for stochastic difference Volterra equations with continuous time usable in mathematical models. The theoretical results are illustrated by numerical calculations. 1. Stability theorem Construction of Lyapunov functionals is usually used for investigation of stability of hereditary. | LYAPUNOV FUNCTIONALS CONSTRUCTION FOR STOCHASTIC DIFFERENCE SECOND-KIND VOLTERRA EQUATIONS WITH CONTINUOUS TIME LEONID SHAIKHET Received 4 August 2003 The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem this method can be successfully used also for stochastic difference Volterra equations with continuous time usable in mathematical models. The theoretical results are illustrated by numerical calculations. 1. Stability theorem Construction of Lyapunov functionals is usually used for investigation of stability of hereditary systems which are described by functional differential equations or Volterra equations and have numerous applications 3 4 8 21 . The general method ofLyapunov functionals construction for stability investigation of hereditary systems was proposed and developed see 2 5 6 7 9 10 11 12 13 17 18 19 for both stochastic differential equations with aftereffect and stochastic difference equations. Here it is shown that after some modification of the basic Lyapunov-type stability theorem this method can also be used for stochastic difference Volterra equations with continuous time which are popular enough in researches 1 14 15 16 20 . Let O F P be a probability space Ft t t0 a nondecreasing family of sub-ơ-algebras of F that is Ft1 c Ft2 for t1 t2 and H a space of Ft-measurable functions x t G R t t0 with norms xh2 sup E x t 2 xH1 sup E x t 2. t t0 te t0 t0 h0 Consider the stochastic difference equation x t h0 n t h0 F t x t x t - h1 x t - h2 . t t0 - h0 Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 1 2004 67-91 2000 Mathematics Subject Classification 39A11 37H10 URL http S1687183904308022 68 Difference Volterra equations with continuous .