An Application of Krasnoselskii Fixed Point Theorem to the Asymptotic Behavior of Solutions of Difference Equations in Banach Spaces1

In this paper we consider the first order difference equation ` ` D xns anif xnqi.q bnig xnqi.qyn is0 is0 and the second order difference equation D qnD xn.qrnf xn.qsng xn.qzns0, where f is a Lipschitz mapping and g is a compact operator, both defined on a Banach space X. We give sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given by A. Drozdowicz and J. Popenda 1987, Proc. Amer. Math. Soc. 99, 135]140., J. Popenda and E. Schmeidel 1994, Publ. Mat. 38, 3]9; 1997, Indian J. Pure Appl. Math. 28, 319]327., and E. Schmeidel 1997, Demonstratio Math. 30, 193]197; 1997; Comm. Appl. Nonlinear Anal. 4, 87]92..