On The Functional Equation p(x)=h(x, ^[f(x)])

Abstract

We shall consider the system of functional equations (1) ^(x) = h^x,^ [f(x)] .... ), i=1,...,m, where the functions h^ and f of the type Rm+"' —»-R and Rn—*Rn, respectively, are given and are unknown functions. The fundamental theorems regarding the uniqueness and the existence of solutions of the class Cr in the case m=1 are. due to B.Choczewski ( [l] , [2]). This theory has been further extended by J.Matkowski [6]. Our theorem (see §3) generalizes also some result of the author obtained in the case of functions of real variable [4]. On the other hand, the system (1) may be treated as a generalization of Schroeder's equation. Therefore the results of this paper correspond to others contained in [3], [5] and [8] (Fragment tekstu).