The space of (𝜓,𝛾)–additive mappings on semigroups

Faiziev, V. A.; Th.M., Rassias; Sahoo, Prasanna K.

doi:10.1090/s0002-9947-02-03036-2

Cited by 47 publications

(15 citation statements)

References 25 publications

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“…It was shown in [5] that the equation 3is (ψ, γ)-stable for the pair (G; E) if and only if P AM ψ,γ (G; E) = HOM (G; E). The following result is one of the main results in [5]. Let E 1 and E 2 be Banach spaces over reals.…”

Section: Definition 5 the Cauchy Functional Equationmentioning

confidence: 99%

“…It is clear that if γ is a constant function then P X ψ,γ (G) = P X(G). If f ∈ P X ψ,γ (G), then it known (see [5]) that the (ψ, γ)-pseudocharacter satisfies 3. Some preliminary results.…”

Section: Definition 5 the Cauchy Functional Equationmentioning

confidence: 99%

“…The first paper to extend Hyers' result to a class nonabelian groups and semigroups was [2]. The notion of (ψ, γ)-stability of the Cauchy functional equation was introduced in [5]. In [5] among other results, it was proven that the Cauchy functional equation f (xy) = f (x) + f (y) is (ψ, γ)-stable on any abelian group as well as any meta-Abelian (step-two nilpotent) group.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of (ψ, γ)-stability of the Cauchy functional equation was introduced in [5]. In [5] among other results, it was proven that the Cauchy functional equation f (xy) = f (x) + f (y) is (ψ, γ)-stable on any abelian group as well as any meta-Abelian (step-two nilpotent) group. It was also shown that any arbitrary group A can be embedded into a group G, where the Cauchy functional equation is (ψ, γ)-stable.…”

Section: Introductionmentioning

confidence: 99%

“…We say that a pseudocharacter ϕ of the group G is nontrivial if ϕ / ∈ X(G). Next we recall some notions from [5] that we need for this paper. Let R + 0 = [0, ∞) be the set of nonnegative numbers and R + = (0, ∞) be the set of positive numbers.…”

Section: Introductionmentioning

confidence: 99%

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On Stability of the Cauchy Equation on Solvable Groups

Faiziev

Sahoo

2015

Ukr Math J

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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.

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Section: Definition 5 the Cauchy Functional Equationmentioning

confidence: 99%

Section: Definition 5 the Cauchy Functional Equationmentioning

confidence: 99%