Approximate linear derivations and functional inequalities with applications

In this paper, we prove that any approximate linear derivation on a semisimple Banach algebra is continuous. We deal with the functional inequalities associated with additive mappings and some stability theorems are proved. Based on these facts, we obtain some results for the functional inequalities corresponding to the additive mappings and the equation f(xy)=xf(y)+f(x)yf(xy)=xf(y)+f(x)y.