Communications of the
Korean Mathematical Society CKMS

In this paper, we first introduce the notions of $(2,\beta)$-Banach spaces and we will reformulate the fixed point theorem \cite[Theorem 1]{22} in this space. We also show that this theorem is a very efficient and convenient tool for proving the new hyperstability results of the general linear equation in $(2,\beta)$-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. Our results are improvements and generalizations of the main results of Piszczek \cite{p21}, Brzd\k{e}k \cite{18,brz180} and Bahyrycz et al. \cite{ba1} in $(2,\beta)$-Banach spaces.

Keywords: hyperstability, general linear equation, fixed point theorem, $(2,\beta)$-normed spaces

MSC numbers: Primary 39B82, 39B62; Secondary 47H14, 47H10