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Hyperstability of some functional equations in modular spaces

Abderrahman Baza, Mohamed Rossafi, Mohammed Mouniane

Abstract

In this paper, we investigate some hyperstability results, inspired by the concept of Ulam stability, for the following functional equations: \begin{equation} \varphi(x+y)+\varphi(x-y)=2\varphi(x)+2\varphi(y) \end{equation} \begin{equation} \varphi(ax+by) = A\varphi(x)+B\varphi(y)+C \end{equation} \begin{equation}\label{eqnd} f\left(\sum_{i=1}^{m}x_{i}\right)+\sum_{1\leq i<j\leq m}f\big(x_{i}-x_{j}\big)=m\sum_{i=1}^{m}f(x_{i}) \end{equation} in modular spaces.

Hyperstability of some functional equations in modular spaces

Abstract

In this paper, we investigate some hyperstability results, inspired by the concept of Ulam stability, for the following functional equations: \begin{equation} \varphi(x+y)+\varphi(x-y)=2\varphi(x)+2\varphi(y) \end{equation} \begin{equation} \varphi(ax+by) = A\varphi(x)+B\varphi(y)+C \end{equation} \begin{equation}\label{eqnd} f\left(\sum_{i=1}^{m}x_{i}\right)+\sum_{1\leq i<j\leq m}f\big(x_{i}-x_{j}\big)=m\sum_{i=1}^{m}f(x_{i}) \end{equation} in modular spaces.

Paper Structure

This paper contains 5 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Hyperstability of Equation \ref{['eqquad']}
  3. Hyperstability of a general linear functional equation
  4. Hyperstability of the $n$-dimensional quadratic functional equation
  5. Hyperstability of the $n$-dimensional quadratic equation in Banach space

Key Result

Proposition 1

In a modular space,

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • ...and 10 more