On continuous solutions of a functional equation

Abstract

This paper discusses continuous solutions of the functional equation φ[f(x)] = g(x, φ(x)) in topological spaces. Let us consider the equation (1) φ[f(x)] = g(x, φ(x)) with φ : X → Y as unknown function. In order to obtain a solution of equation (1), it is enough to extend a function defined on a set which for every x contains exactly one element of the form f(x), where k = 0,±1,±2, . . . and f(x) denotes the kth iterate of the function f (cf. [3] and [4]). In the case when X is an open interval and Y is a Banach space, it is well known under what conditions these extensions are continuous (cf. [5]). Paper [6] by M. Sablik brings theorems which answer the above question for X and Y contained in some Banach spaces ([6, Th. 2.1, Th. 2.2]). In the case when X and Y are locally convex vector spaces the continuity of similar extensions was examined by W. Smajdor in [7] but for the Schroder equation (i.e. φ[f(x)] = sφ(x), 0 < |s| < 1). We are going to adopt the method given in that paper to the more general situation. We shall employ Baron’s Extension Theorem proved in [1] (cf. also [2]). This theorem concerns extending solutions of functional equations from a neighbourhood of a distinguished point (Lemma 7). We shall deal with the following hypotheses: (i) X is a Hausdorff topological space; ξ is a given (and fixed) point of X; Y is a topological space. (ii) The function f maps X into X in such a manner that (2) f is homeomorphism of X onto f(X); (3) ξ ∈ int f(X); 1991 Mathematics Subject Classification: Primary 39B52.