Stability of a mixed additive and quadratic functional equation in quasi-Banach spaces

Abstract

We study the stability of functional equations in quasi-Banach spaces where the quasi-norm is not assumed to be a p-norm. To overcome the modulus of concavity greater than 1 and the discontinuity of quasi-norms we use the squeeze inequality presented in an explicit revision of Aoki–Rolewicz Theorem [13, Theorem 1]. As illustrations, we prove an extension of the stability of a mixed additive and quadratic functional equation in p-Banach spaces to quasi-Banach spaces with better approximation. The technique may be used to prove extensions of other results on the stability of functional equations in p-Banach spaces to quasi-Banach spaces.