Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems Michal Feckan ABSTRACT We present a survey of our recent achievements based on an asymp- totic approximation method carried out by projection and averaging for initial value problems of infinite-dimensional difference equations with small parameters. Applications are given to delay integro-differential equations and to semilinear Schrodinger equations as well. 1 Introduction The purpose of this note is to present our recent results (4) on combination of the Galerkin approximation method [1] with the asymptotic-averaging procedure (2], (13]. We formulate our abstract results for semilinear dif- ference equations with small parameters satisfying certain properties. By using the Galerkin method, we reduce infinite-dimensional semilinear dif- ference equations with initial value conditions to finite-dimensional ones. We also derive asymptotic approximation error bounds. Then we study finite-dimensional semilinear difference equations with initial value condi- tions on the discrete time scale 1/e, where we embed our problem in an ordinary differential equation with the small parameter C. In this way, we are able to apply the classical averaging method [2], [13]. We also study, on the discrete time scales 1/c and 1/e2, infinite-dimensional semilinear difference equations with stable and center linear parts. Abstract results are at first demonstrated on an example of an integro- differential equation with a memory and delay coupled by the small pa- rameter e. Its dynamics is studied on the time scale 1/c2. Then we study a semilinear Schrodinger equation. This part is related to former studies on weakly nonlinear wave equations with fixed ends [5], [8], [9], [11], [12], [14]. The influence of almost periodic perturbations on the dynamics is in- 1991 Mathematics Subject Classification: 34C29, 35A40, 65L60, 65M15. Key words and phrases: Galerkin-averaging method, differential-diference equations, Schrodinger equations. This work was supported by Grant GA-SAV 2/5133/98.