Stability of the Fréchet Equation in Quasi-Banach Spaces

Sang Og Kim

doi:10.3390/math8040490

Abstract

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-Banach spaces. In the proof, a fixed point theorem due to Dung and Hang, which is an extension of a fixed point theorem in Banach spaces, plays a main role.

Highlights

Ulam [1] raised a problem of finding conditions under which there exists an exact additive map near an approximate additive map
We show that the Fréchet equation is not stable for p ∈ {1, 2}
Using a recently developed fixed point theorem, we have proved the Hyers–Ulam stability of the Fréchet equation in quasi-Banach spaces

Summary

Introduction

Ulam [1] raised a problem of finding conditions under which there exists an exact additive map near an approximate additive map. Assume that ( X, +) is a uniquely 2-divisible abelian group, (Y, k · k, κ ) is a quasi-Banach space and L < 1 is a real number. Let the constants 1 < p < 2 and c > 0 be such that the mapping f : X → Y satisfies k f ( x + y + z) + f ( x ) + f (y) + f (z) − f ( x + y) − f (y + z) − f ( x + z)k ≤ c(k x k p + kyk p + kzk p ), for all x, y, z ∈ X.

Nonstability of the Fréchet Equation

Conclusions