On Stability of the Cauchy Equation on Solvable Groups

Abstract

The notion of (ψ, γ)-stability was introduced in [Faȋziev 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 [Faȋziev 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 [Faȋziev 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 [Faȋziev 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.

Nonlinear fuzzy stability of a functional equation related to a characterization of inner product spaces via fixed point technique

“…Since then there have been several new results on stability of various classes of functional equations in the Hyers-Ulam sense or Hyers-Ulam-Rassias sense in normed spaces (see [10,15,16,34] and references cited therein). Stability problems of functional equations on arbitrary groups and on non-abelian groups were treated in [6][7][8][9]. In [20][21][22][23], various stability results concerning Cauchy, Jensen, quadratic and cubic functional equations were investigated in fuzzy normed spaces.…”