Combined system of additive functional equations in Banach algebras

Abstract

In this study, we solve the system of additive functional equations: h ( x + y ) = h ( x ) + h ( y ) , g ( x + y ) = f ( x ) + f ( y ) , 2 f x + y 2 = g ( x ) + g ( y ) , \left\{\begin{array}{l}h\left(x+y)=h\left(x)+h(y),\\ g\left(x+y)=f\left(x)+f(y),\\ 2f\left(\frac{x+y}{2}\right)=g\left(x)+g(y),\end{array}\right. and we investigate the stability of (homomorphism, derivation)-systems in Banach algebras.