THE GENERAL LINEAR EQUATION IS STABLE

Abstract

The stability of the Cauchy functional equation f(x+y) = f(x)+f(y) was asked by S. Ulam and proved in 1941 by D.H. Hyers :  f(x + y) -f(x)-f(y) is small, then f is close to an additive function. If f is continuous, then f is close to a linear map. The problem of the stability of the general linear equation f(ax + by + c) = Af(x) + Bf(y) + C was raised by Th. M. Rassias and J. Tabor. We prove some results in the case of mappings between Banach spaces. In particular, we obtain the stability (in a certain sense) of the general linear equation.