Let be an abelian group and let X be a Banach space. If f: is a function such that for every x, and some , then there exists a unique function g: satisfying the equation for every x, such that for every
In the second part there is a short proof of a stability theorem of D. H. Hyers and S. M. Ulam [Proc. Am. Math. Soc. 3, 821-828 (1952; Zbl 0047.295)] for the inequality Finally, the author gives a counterexample for Jensen-convex functions.