On Hyperstability of the Cauchy Functional Equation in n-Banach Spaces

Janusz Brzdęk,El-Sayed El-Hady

doi:10.3390/math8111886

Janusz Brzdęk, El-Sayed El-Hady

Open Access

https://doi.org/10.3390/math8111886

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Abstract

We present some hyperstability results for the well-known additive Cauchy functional equation f(x+y)=f(x)+f(y) in n-normed spaces, which correspond to several analogous outcomes proved for some other spaces. The main tool is a recent fixed-point theorem.

Highlights

The issue of stability of functional equations has been motivated by a problem raised by S.M
We say that a given functional equation is stable in some class of functions if any function from that class, satisfying the equation approximately, is near an exact solution of the equation
The concept of an approximate solution and the idea of nearness of two functions can be understood in many nonstandard ways, depending on the needs and tools available in a particular situation. One of such non-classical measures of distance can be created using the notion of n-norm, introduced by A

Summary

Introduction

The issue of stability of functional equations has been motivated by a problem raised by S.M. Problems of that type are very natural for difference, differential, functional and integral equations and many examples of recent results concerning their stability as well as further references can be found in [6,7]. The concept of an approximate solution and the idea of nearness of two functions can be understood in many nonstandard ways, depending on the needs and tools available in a particular situation. One of such non-classical measures of distance can be created using the notion of n-norm, introduced by A. We refer to [15] for several examples of investigations of stability of functional equations in the n-normed spaces. We present two possible extensions of the results in [13] to the case of n-normed spaces

Preliminaries

Hyperstability Results

Proofs of Theorems 3 and 4

Conclusions