Ulam stability of an additive-quadratic functional equation in F-space and quasi-Banach spaces

TL;DR

This paper establishes the stability of a specific additive-quadratic functional equation in F-spaces and quasi-Banach spaces using direct and fixed point methods, addressing challenges posed by non-standard norms.

Contribution

It proves Hyers-Ulam stability of the equation in $eta$-homogeneous F-spaces and quasi-Banach spaces, extending stability results to these complex spaces.

Findings

Proved stability in $eta$-homogeneous F-spaces.

Extended stability results to quasi-Banach spaces.

Overcame challenges posed by non-standard norms.

Abstract

By adopting the direct method and fixed point method, we prove that the Hyers-Ulam stability of the following additive-quadratic functional equation \begin{equation} f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in $\beta$-homogeneous $F$-spaces and quasi-Banach spaces. There are some differences that we consider the target space with the $\beta$-homogeneous norm and quasi-norm. Overcoming the $\beta$-homogeneous norm and quasi-norm bottlenecks, we get some new results.