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

Tibor Kiss, Péter Tóth

Abstract

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.

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

Abstract

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 . Improving previous results we assume differentiability on each involved function, eliminate a former condition on and , 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.
Paper Structure (9 sections, 21 theorems, 115 equations)

This paper contains 9 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Solutions, where $F$ affine on its entire domain
  4. Solutions, where $F$ is partially affine
  5. The Auxiliary Equation
  6. Complementarity Results
  7. Main Result of the Partially Affine Case
  8. Solutions, where $F$ is nowhere affine
  9. Open questions concerning our equation

Key Result

Proposition 2.1

Assume that $F:I\to\mathbb{R}$ is affine on some nonempty open subinterval $J\subseteq I$ and let $g_1,g_2:I\to\mathbb{R}$ be continuous and strictly monotone. Then $(F,f_k,G,g_k)$ solves equation eq-Invariance-eq over $J$ if and only if there exist an additive function $B:\mathbb{R}\to\mathbb{R}$ a hold on $J$ and on $g_1(J)+g_2(J)$, respectively, where $\beta=\beta_1+\beta_2$.

Theorems & Definitions (43)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 33 more