Stability of generalized additive functional inequalities in Banach spaces

Abstract

In this paper, we study the following generalized additive functional inequality \begin{eqnarray} \|af(x)+bf(y)+cf(z)\| \le \|f(\alpha x+ \beta y+\gamma z)\| , \end{eqnarray} associated with linear mappings in Banach spaces. Moreover, we prove the Hyers--Ulam--Rassias stability of the generalized additive functional inequality (0.1), associated with linear mappings in Banach spaces.