Generalized Hyers-Ulam stability of functional equations

Abstract

In this paper, we prove the generalized Hyers-Ulam stability of the following linear functional equations \begin{eqnarray*} f(x+i y)+ f(x-i y) + f(y+ix) + f(y-ix) = 2 f(x) +2 f(y) \end{eqnarray*} and $f((1+i)x)= (1+i) f(x)$, and of the following quadratic functional equations \begin{eqnarray*} f(x+i y) + f(x-i y) + f(y+ix) + f(y-ix) = 0 \end{eqnarray*} and $f((1+i)x) = 2i f(x)$ in complex Banach spaces.