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Perturbations of Cauchy differences

Eszter Gselmann, Tomasz Małolepszy, Janusz Matkowski

Abstract

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and also more general cases where the inhomogeneity depends on unknown functions \begin{align*} f(x+y)-f(x)-f(y)&= αx y \\[2.5mm] f(x+y)-f(x)-f(y)&= α(x y)\\[2.5mm] f(x+y)-f(x)-f(y)&= α(x)α(y). \end{align*} Our results extend previous work on the bilinearity of the Cauchy exponential difference by Alzer and Matkowski. We characterize solutions under various structural and regularity assumptions, including additive and exponential Cauchy differences, and show that solutions often reduce to additive functions, exponential polynomials, or combinations thereof. For Levi-Civita type equations, we provide explicit representations of solutions in terms of additive and exponential components. Furthermore, we determine conditions under which real-valued solutions exist and describe their forms. The paper concludes with open problems concerning generalized equations that cannot be solved by the methods presented here, suggesting directions for future research.

Perturbations of Cauchy differences

Abstract

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form where is a biadditive mapping, and also more general cases where the inhomogeneity depends on unknown functions \begin{align*} f(x+y)-f(x)-f(y)&= αx y \\[2.5mm] f(x+y)-f(x)-f(y)&= α(x y)\\[2.5mm] f(x+y)-f(x)-f(y)&= α(x)α(y). \end{align*} Our results extend previous work on the bilinearity of the Cauchy exponential difference by Alzer and Matkowski. We characterize solutions under various structural and regularity assumptions, including additive and exponential Cauchy differences, and show that solutions often reduce to additive functions, exponential polynomials, or combinations thereof. For Levi-Civita type equations, we provide explicit representations of solutions in terms of additive and exponential components. Furthermore, we determine conditions under which real-valued solutions exist and describe their forms. The paper concludes with open problems concerning generalized equations that cannot be solved by the methods presented here, suggesting directions for future research.
Paper Structure (4 sections, 21 theorems, 142 equations)

This paper contains 4 sections, 21 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Additive Cauchy differences
  4. Exponential Cauchy differences

Key Result

Lemma 1

Let $G$ be a commutative group, $n$ be a positive integer and let where $a_{1}, \ldots, a_{n}$ are linearly independent complex-valued additive functions defined on $G$. Then the monomials $\left\{a^{\alpha}\right\}$ for different multi-indices are linearly independent.

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Remark
  • Lemma 1: Lemma 2.7 of Sze91
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • ...and 32 more