Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II

Abstract

Let X be a commutative normed algebra with a unit element e (or a normed field of characteristic different from 2), where the associated normis sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, f (x) - g(y) = (x - y)h(sx + ty), where f, g, h : X -> X are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove theHyers-Ulamstability of that functional equation under some additional conditions.