Asymptotic formulae for nonlinear functional difference equations

In this paper, we show asymptotic formulae for solutions of the nonlinear functional difference equation y(n + 1) = L(y(n)) + V(n, y(n))+ f(n, y(n)), where y(n)(theta) = y(n + theta) for theta epsilon {-r, -r + 1, ..., 0}, L, V(n,.) are linear applications, and f(n,.) is not necessarily linear defined from {-r, -r + 1, ..., 0} to C-N. We ask for a trichotomic spectral condition on L, \V(n,.)\ --> 0 as n --> +infinity, \V(n+ 1,.) - V(n,.)\ epsilon l(1), f(n, 0) = 0, and there is gamma epsilon l(1) such that \f(n, x) - f(n, y)\ less than or equal to gamma (n)\x - y \. (C) 2001 Elsevier Science Ltd. All rights reserved.