Some hyperstability results of a p-radical functional equation related to quartic mappings in non-archimedean Banach spaces

Samir Kabbaj,Ahmed Nuino,Mustapha Esseghyr Hryrou

doi:10.22199/issn.0717-6279-3704

Abstract

The aim of this paper is to introduce and solve the following p-radical functional equation related to quartic mappings where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzd¸ek’s fixed point theorem [13], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to quartic mapping.

Highlights

A classical question in the theory of functional equation is the following:”When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of this equation.”If the answer is affirmative, we say that equation is stable
In the case p < 0, each f : E1 → E2 satisfying (1.1) must be additive. This result is called the hyperstability of Cauchy functional equation
The main purpose of this paper is to achieve the general solution of the functional equation (1.4) and establish some hyperstability results for the considered equation in non-Archimedean Banach space

Summary

Introduction

A classical question in the theory of functional equation is the following:. ”When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of this equation.”. H. Hyers [26] in 1941 under the assumption that G1 and G2 are Banach spaces for the the additive functional equation as follows: Theorem 1.1. In the case p < 0, each f : E1 → E2 satisfying (1.1) must be additive This result is called the hyperstability of Cauchy functional equation. In 2013, Brzdek [15] improved, extended and complemented several earlier classical stability results concerning the additive Cauchy equation (in particular Theorem 1.3). A function || · ||∗ : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: 1. The main purpose of this paper is to achieve the general solution of the functional equation (1.4) and establish some hyperstability results for the considered equation in non-Archimedean Banach space. Kφ(x) − ψ(x)k∗ ≤ sup Λnε(x), n∈N0 x ∈ X

Main results

Consequences