Bounded solutions and asymptotic stability of nonlinear difference equations in the complex plane. II. (Q1600391)

The authors investigate asymptotic properties of solutions of a certain \(m\)-th order nonlinear difference equation (which is rather complicated to be given here explicitly). The paper is a continuation of the earlier paper of the same authors [Arch. Math., Brno 36, No. 2, 139--158 (2000; Zbl 1053.39016)]. Comparing with that paper, the nonlinearity in the studied equation of the reviewed paper is allowed to be more general. The investigated equation is transformed into an operator equation in a suitable Banach space and the main results of the paper, the existence and uniqueness statements for solutions in \(l_1\), are proved using the fixed point theorem for holomorphic maps due to \textit{C. J. Earle} and \textit{R. S. Hamilton} [Global Analysis, Proc. Sympos. Pure Math. 16, 61--65 (1970; Zbl 0205.14702)]. The results of the paper are illustrated by a number of examples which seem to be quite nontrivial.