On Stability of the Cauchy Equation on Solvable Groups

Abstract

The notion of $(ψ, γ)$-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation $f (xy) = f (x) + f (y)$ is $(ψ, γ)$-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, Publ. Math. Debrecen, 75, 6 (2009)], it was proved that the Cauchy equation is $(ψ, γ)$-stable on step-two solvable groups and step-three nilpotent groups. In the present paper, we prove a more general result and show that the Cauchy equation is $(ψ, γ)$-stable on solvable groups.