arXiv subjects

Péter Tóth

Publications and source records attributed to Péter Tóth.

4 recordsLinked to original sources

arXiv2026

Generalized inverses of strictly monotone transformations

Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

math.CA↗

arXiv2026

Improved regularity for a composite functional equation stemming from the theory of means

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we assume differentiability on each involved function, eliminate a former condition on $ g'_1 $ and $ g'_2$, moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.

math.CA↗

arXiv2025

Measurable solutions of an alternative functional equation

In this paper we investigate the functional equation \[ φ\left( \frac{x+y}{2} \right) \left( ψ_1(x) - ψ_2(y) \right) = 0 \hspace{20mm} \left( \mbox{ for all } x \in I_1 \mbox{ and } y \in I_2 \right) \] where $ I_1 \,, I_2 $ are open intervals of $ \mathbb{R} $, $ J = \frac{1}{2} \left( I_1 + I_2 \right) $ moreover $ ψ_1 : I_1 \rightarrow \mathbb{R} $, $ ψ_2 : I_2 \rightarrow \mathbb{R} $ and $ φ: J \rightarrow \mathbb{R} $ are unknown functions. We describe the structure of the possible solutions assuming that $ φ$ is measurable. In the case when $ φ$ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.

math.CA↗

arXiv2021

Interval chains and completeness in ultrapowers of ordered sets

The ultrapower $T^{\ast}$ of an arbitrary ordered set $T$ is introduced as an infinitesimal extension of $T$. It is obtained as the set of equivalence classes of the sequences in $T$, where the corresponding relation is generated by an ultrafilter on the set of natural numbers. It is established that $T^{\ast}$ always satisfies Cantor's property, while one can give the necessary and sufficient conditions for $T$ so that $T^{\ast}$ would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of $T^{\ast}$ is determined by the cardinality of $T$.

math.GM↗