On Gajda’s type quadratic equation on a locally compact abelian group

Fadli, Brahim; Zeglami, Driss; Kabbaj, Samir

doi:10.1016/j.indag.2015.05.001

Cited by 6 publications

(3 citation statements)

References 16 publications

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“…which means that the functions f, h ∈ C(G) satisfy the equation (2). According to Theorem (1), there exist symmetric k−additive mappings A k ∈ C(G), k ∈ {1, ...., κ} such that…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

On a quadratic type functional equation on locally compact abelian groups

Dimou

Aribou

Chahbi

et al. 2018

Acta Universitatis Sapientiae, Mathematica

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Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported complex-valued Borel measure on G such that $\mu ({\rm{G}}) = {1 \over \kappa }$ . We find the continuous solutions f, g : G → ℂ of the functional equation $$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$ in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).

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“…which means that the functions f, h ∈ C(G) satisfy the equation (2). According to Theorem (1), there exist symmetric k−additive mappings A k ∈ C(G), k ∈ {1, ...., κ} such that…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

On a quadratic type functional equation on locally compact abelian groups

Dimou

Aribou

Chahbi

et al. 2018

Acta Universitatis Sapientiae, Mathematica

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“…where ϕ is a surjective endomorphism of S. Equation (2.2) generalizes several equations which are studied in the literature such as Jensen's, Drygas' or the quadratic equations on abelian monoids. In [4], with ϕ = − id, the functional equation (2.2) was studied in the case where µ = ν − and ν is a regular compactly supported the complex-valued Borel measure on a locally compact abelian Hausdorff group G such that ν(G) = 1 2 and f : G → C is continuous. If (S, +) is a monoid with a neutral element 0 and µ = ν = 1 2 δ 0 , equation (2.2) becomes respectively in the cases where h = f and h = 0 the following generalized Jensen and quadratic functional equations:…”

Section: Introductionmentioning

confidence: 99%

An integral functional equation on abelian semigroups

Akkaoui

2022

Publ Inst Math (Belgr)

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We investigate a generalization of many functional equations. Namely, we consider the following functional equation ?S f(x + y + t) d?(t) + ?S f(x + ?(y) + t) d?(t) = f(x) + h(y), x, y ? S, where (S,+) is an abelian semigroup, ? is a surjective endomorphism of S, E is a linear space over the field K ? {R, C} and ?,? are linear combinations of Dirac measures. Under appropriate conditions on ? and ? and based on Stetkar?s result [9], we find and characterize solutions of the previous functional equation.

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“…We will prove that these contrast the solutions of the functional equation (2.1), where the non-abelian phenomena like solutions of (2.3) may occur. For other similar functional equations we refer to [1,2,5,[7][8][9]11,12,16].…”

Section: Introductionmentioning

confidence: 99%