Fixed points in functional inequalities (Q1008457)

By using fixed point methods, the author proves the generalized Hyers--Ulam stability of the following functional inequalities \[ \|f(x)+f(y)+f(z)\|\leq \|f(x+y+z)\|,\eqno{(1)} \] \[ \|f(x)+f(y)+2f(z)\|\leq\bigg|\bigg|2f\bigg(\frac{x+y}{2}+z\bigg)\bigg|\bigg|,\eqno{(2)} \] in a Banach space.