On a linear functional equation of second order
Our aim is to study continuous solutions φ of the classical linear iterative equation φ(f(x,y)) = g(x,y)φ(x,y) + h(x,y), where the given function f is defined as a pair of means. We are interested in the case when f has no fixed points. In turns out that in such a case continuous solutions of (1) depend on an arbitrary function.
In this paper, we obtain all possible general solutions of the sum form functional equations valid for all complete probability distributions , , , fixed integers; , and , , , , , are real valued mappings each having the domain , the unit closed interval.
We solve Matkowski's problem for strictly comparable quasi-arithmetic means.