КОРОТКIПОВIДОМЛЕННЯ UDC 512.5 V. A. Fa ˘ iziev (Tver State Agricultural Academy, Russia), P. K. Sahoo (Univ. Louisville, USA) ON STABILITY OFCAUCHY EQUATION ON SOLVABLE GROUPS * ПРО СТIЙКIСТЬ РIВНЯННЯ КОШI НА РОЗВ’ЯЗНИХ ГРУПАХ The notion of (ψ,γ)-stability was introduced in [Fa ˘ iziev V. A., Rassias Th. M., Sahoo P. K. The space of (ψ,γ)-additive mappings on semigroups // Trans. Amer. Math. Soc. – 2002. – 354. – P. 4455 – 4472]. It was shown that the Cauchy equation f (xy)= f (x)+ f (y) is (ψ,γ)-stable on any Аbelian group as well as any meta-Abelian group. In [Fa ˘ iziev V. A., Sahoo P. K. On (ψ,γ)-stability of Cauchy equation on some noncommutative groups // Publ. Math. Debrecen. – 2009. – 75. – P. 67 – 83], it was proved that the Cauchy equation is (ψ,γ)-stable on step-two solvable groups and step-three nilpotent groups. In our paper, we prove a more general result and show that the Cauchy equation is (ψ,γ)-stable on solvable groups. Поняття (ψ,γ)-стiйкостi введено в роботi [Fa ˘ iziev V. A., Rassias Th. M., Sahoo P. K. The space of (ψ,γ)-additive mappings on semigroups // Trans. Amer. Math. Soc. – 2002. – 354. – P. 4455 – 4472]. Було показано, що рiвняння Кошi f (xy)= f (x)+ f (y) є (ψ,γ)-стiйким як на довiльнiй абелевiй групi, так i на довiльнiй метабелевiй групi. В роботi [Fa ˘ iziev V. A., Sahoo P. K. On (ψ,γ)-stability of Cauchy equation on some noncommutative groups // Publ. Math. Debrecen. – 2009. – 75. – P. 67–83] доведено, що рiвняння Кошi є (ψ,γ)-стiйким як на двоступеневих розв’язних групах, так i на триступеневих нiльпотентних групах. В нашiй роботi доведено бiльш загальний результат i показано, що рiвняння Кошi є (ψ,γ)-стiйким на розв’язних групах. 1. Introduction. In 1940, S.M. Ulam [11] posed the following fundamental problem. Given a group G 1 , a metric group (G 2 ,d) and a positive number ε, does there exist a δ> 0 such that if f : G 1 → G 2 satisfies d(f (xy),f (x)f (y)) <δ for all x,y ∈ G 1 , then a homomorphism T : G 1 → G 2 exists with d(f (x),T (x)) <ε for all x,y ∈ G 1 ? Interested reader should see S. M. Ulam [11] for a discussion of such problems, as well as D. H. Hyers [8], Th. M. Rassias [9], J. Acz´ el and J. Dhombres [1], G. L. Forti [7], and P. K. Sahoo and Pl. Kannappan [10]. The first affirmative answer to this problem was given by D. H. Hyers [8] in 1941. On a group G, the Cauchy functional equation f (x + y)= f (x)+ f (y) takes the form f (xy)= = f (x)+ f (y) for all x,y ∈ G. In connection with Hyers’ result the following question arises. Let G be an arbitrary group and let a mapping f : G → R (the set of reals) be such that the set, D, defined by {f (xy) − f (x) − f (y) | x,y ∈ G} is bounded. Is it true that there is a mapping T : G → R that satisfies T (xy) − T (x) − T (y) = 0 for all x,y ∈ G, and the set {T (x) − f (x) | x ∈ G} is bounded. A negative answer was given in 1987 by G.L. Forti [6]. He constructed a real-valued function f on the free group F 2 of rank two (and also on a free semigroup S 2 of rank two) such that the set D is bounded but for any additive function T, the function f (x) − T (x) is not bounded. It is worth pointing out that in 1987, Fa ˘ iziev in [2] gave a description of all such functions f on the free product of semigroups A ∗ B. In [3], it was established that the Cauchy functional equation is not stable on the free product A ∗ B of groups A and B unless A = B = Z 2 . * This paper was partially supported by an IRIG grant from the Office of the VP for Research, and an A&S grant from the College Arts and Sciences, University of Louisville. c  V. A. FA ˘ IZIEV, P. K. SAHOO, 2015 1000 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7