Nonlinear difference equations with asymptotically stable solutions

Abstract

We establish conditions of asymptotic stability for all solutions of the equation X _n+1=F(X _n ), n≥0, in the Banach space E in the case where r(F′(x))<1 ∀ x ∈ E, r′(x) is the spectral radius of F′(x). An example of an equation with an unstable solution is given.