A new fixed point approach to hyperstability of radical-type functional equations in quasi-$(2,\beta)$-Banach spaces

TL;DR

This paper introduces quasi-$(2,eta)$-Banach spaces, extends fixed point theorems to these spaces, and applies them to establish hyperstability results for radical-type functional equations.

Contribution

It defines quasi-$(2,eta)$-Banach spaces, extends fixed point theorems to this setting, and applies these results to analyze hyperstability of functional equations.

Findings

Fixed point theorem valid in quasi-$(2,eta)$-Banach spaces

General solution for radical-type functional equations

Hyperstability results established for these equations

Abstract

The main focus of this paper is to define the notion of quasi-$(2,\beta)$-Banach space and show some properties in this new space, by help of it and under some natural assumptions, we prove that the fixed point theorem [16, Theorem 2.1] is still valid in the setting of quasi-$(2,\beta)$-Banach spaces, this is also an extension of the fixed point result of Brzd\k{e}k et al. [12, Theorem 1] in $2$-Banach spaces to quasi-$(2,\beta)$-Banach spaces. In the next part, we give a general solution of the radical-type functional equation (1.2). In addition, we study the hyperstability results for these functional equation by applying the aforementioned fixed point theorem, and at the end of this paper we will derive some consequences.