Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

TL;DR

This paper finds the general solutions and stability conditions for a specific functional equation involving cubic, quadratic, and additive components in quasi-Banach spaces, extending the understanding of such equations' behavior.

Contribution

It provides the first comprehensive solution and stability analysis for a generalized mixed-type functional equation in quasi-Banach spaces.

Findings

Derived the general solution for the functional equation

Established Hyers-Ulam-Rassias stability in quasi-Banach spaces

Extended previous results to a broader class of functional equations

Abstract

In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.