We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line $y=p$ where $p$ is a fixed point of $\varphi$, with a possible exception for $p=1$. The range of any non-constant continuous solution is an interval whose end-points are fixed by $\varphi$ and which contains in its interior no fixed point except for $1$. We also gave a characterization of the class of continuous...

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 $$\begin{array}{cc}\hfill \sum _{i=1}^k\sum _{j=1}^{\ell }f\left(p_i{q}_j\right)=& \sum _{i=1}^{k}g\left(p_i\right)\sum _{j=1}^{\ell }h\left(q_j\right)\hfill \\ \hfill \text{and}\sum _{i=1}^k\sum _{j=1}^{\ell }F\left(p_i{q}_j\right)=& \sum _{i=1}^{k}G\left(p_i\right)+\sum _{j=1}^{\ell }H\left(q_j\right)+\lambda \sum _{i=1}^{k}G\left(p_i\right)\sum _{j=1}^{\ell }H\left(q_j\right)\hfill \end{array}$$ valid for all complete probability distributions $(p_1,...,p_k)$, $(q_1,...,q_{\ell })$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in ℝ$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.

We solve Matkowski's problem for strictly comparable quasi-arithmetic means.