About Lyapunov functionals construction for difference equations with continuous time

Stability investigation of hereditary systems is connected often with construction of Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and developed in [1-9] both for differential equations with aftereffect and for difference equations with discrete time. Here, some modification of Lyapunov-type stability theorem is proposed, which allows one to use this method for difference equations with continuous time also. (~) 2004 Elsevier Ltd. All rights reserved. 1. STABILITY THEOREM Consider the difference equation in the form x(t+ho)=F(t,x(t),x(t-hl),x(t-h2) with the initial condition x(O) = ¢(0), 0 c 0 = [to-ho-maxhj,to]. A = Eaj < c~. (1.3) j =o j =0 A solution of problem (1.1),(1.2) is a process x(t) = x(t;t0, ¢), which is equal to the initial function ¢(t) from (1.2) for t < to and is defined by equation (1.1) for t > to. DEFINITION 1.1. The trivial solution of equation (1.1),(1.2) is called stable if for any e > 0 and to >_ 0 there exists a 6 = 6(e, to) > O, such that [x(t;to,¢)l < e, for all t >_ to if []¢]l = sup0~o I¢(0)1 < a.