On the Hyers-Ulam stabilities of functional equations on n-Banach spaces

Abstract

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.

On approximate homomorphisms of ternary semigroups

“…Some results on the stability of functional equations in n-Banach spaces can be found for example in [4,19] (see also the references given there). Theorem 2.2.…”

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

“…An n-normed space in which every Cauchy sequence is convergent is called n-Banach space. Moreover, we have the following property stated in [23] (see also [15]). Lemma 1.…”

“…Let us now recall some basic definitions and facts concerning n-normed spaces (for more details we refer the reader to [15]; see also [14,[21][22][23]).…”

“…One of such non-classical measures of distance can be created using the notion of n-norm, introduced by A. Misiak [14]. We refer to [15] for several examples of investigations of stability of functional equations in the n-normed spaces.…”

Hyers-Ulam stability of quadratic forms in 2-normed spaces

“…Lemma 1.4. [3] Let (X, ‖•, •‖) be a linear 2-normed space and x ∈ X. Then the following properties hold:…”