Approximation of an additive mapping in various normed spaces

Abstract

In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability <br />of the following Cauchy-Jensen additive functional equation: <br />begin{equation}label{main} <br />fleft(frac{x+y+z}{2}right)+fleft(frac{x-y+z}{2}right)=f(x)+f(z)end{equation} <br />in various normed spaces. <br />The concept of Hyers-Ulam-Rassias stability originated from Th. M. <br />Rassias' stability theorem that appeared in his paper: On the <br />stability of the linear mapping in Banach spaces, Proc. Amer. <br />Math. Soc. 72 (1978), 297-300.