Stability of a quadratic functional equation in quasi--Banach spaces

Bull. Korean Math. Soc. 2008 Vol. 45, No. 3, 587-600 Printed September 1, 2008

Abbas Najati and Fridoun Moradlou University of Mohaghegh Ardabili, University of Tabriz

Abstract : In this paper we establish the general solution and investigate the Hyers--Ulam--Rassias stability of the following functional equation in quasi-Banach spaces. $$ \sum_{ \begin{subarray}{c} 1 \leq i < j \leq 4 \\ 1 \leq k < l \leq 4 \\
k, l \in I_{ij} \end{subarray}} f(x_{i}+x_{j}-x_{k}-x_{l}) =2\sum_{\small{ \begin{subarray}{c} 1 \leq i < j \leq 4
\end{subarray}}}f(x_{i}-x_{j}),$$ where $I_{ij}=\{1,2,3,4\}\setminus\{i,j\}$ for all $1 \leq i < j \leq4$. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. $\bf 72$ (1978), 297--300.

Keywords : Hyers--Ulam--Rassias stability, quadratic function, quasi-Banach space, $p$-Banach space