This paper is a generalization of recent results of the papers [l-3] where the following problem was investigated: assume either an ordinary differential (see [2]) or abstract operator equation (see [l, 31) is invariant under a group of symmetries. Then certain conditions on those differential and operator equations are established in these papers to ensure that both the dynamical behaviour of that differential equation and solutions of that operator equation are restricted on the space of all symmetric elements of that group. In this paper, we consider a family of mappings mt, .)LcI (1.1) for a subset Z c R such that sup Z = co. Here II(t, a): X -+ X, Xis a Banach space. We assume rqt, Y) c Y vtez (1.2) for a closed subspace Y c X possessing a continuous projection P: X + Y. We shall find a general condition based on a Lyapunov-like method, which ensures that the asymptotic behaviour of lJ(t, *) as t --t 00 is reduced on Y, i.e. we show limIQII(t,x)I = 0 VXEX, (1.3) t-m where / * 1 is a norm on X and Qx = x Px v x E X. In most of applications, H is a discrete or continuous semidynamical system. At first, we shall apply our abstract results to difference-difference equations, which are discrete versions of difference-differential equations studied in [2], of the form xn(t + 1) = x,(t) + &%+lu) x,(t)) + 4x,(t)) nez, tc N, where a is a constant, d: R” + IR” is globally Lipschitz continuous with the constant Md. As a by-product we find lower bounds of periods of periodic solutions of such equations. Further, there is the following application to discrete-deterministic nonlinear systems [4] x(t + 1) = d-et), u(t)) r(t) = w(t)) tEN (I.41 u(t) = KY(t)),