On extension of solutions of a simultaneous system of iterative functional equations

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form $$\varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),$$ $$\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),$$ to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541])

Iterations of mean-type mappings and invariant means

It is shown that, under some general conditions, the sequence of iterates of every mean-type mapping on a finite dimensional cube converges to a unique invariant mean-type mapping. Some properties of the invariant means and their applications are presented

On a system of simultaneous iterative functional equations

A system of two simultaneous functional equations in a single variable, related to a generalized Gołąb-Schinzel functional equation, is considered

A remark on periodic entire functions

Periodicity of an entire function is characterized by the behavior of coefficients of its Maclaurin expansion

On the commutation of generalized means on probability spaces

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$. We say that $(f,g)$ is a $(\lambda, \mu)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $$f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;d\mu\right)\right)d \lambda\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;d\lambda\right)\right)d \mu\right)\!,$$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that $(f,g)$ is a $(\lambda, \mu)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$.Comment: 9 pages, no figures. Fixed minor details. Final version to appear in Indagationes Mathematica

Explicit solutions of the invariance equation for means

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in a recent paper of P. Kahlig and J. Matkowski, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed