Solution of Generalized Jensen and Quadratic Functional Equations

A. Charifi (), D. Zeglami () and S. Kabbaj ()
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A. Charifi: Ibn Tofail University, Department of Mathematics, Faculty of Sciences
D. Zeglami: Ibn Tofail University, Department of Mathematics, Faculty of Sciences
S. Kabbaj: ENSAM, Moulay Ismail University, Department of Mathematics

A chapter in Frontiers in Functional Equations and Analytic Inequalities, 2019, pp 273-292 from Springer

Abstract: Abstract We obtain in terms of additive and multi-additive functions the general solution f : S → H of each of the functional equations ∑ λ ∈ Φ f ( x + λ y + a λ ) = N f ( x ) , x , y ∈ S , $$\displaystyle \sum _{\lambda \in \varPhi } f(x+\lambda y+a_{\lambda })=Nf(x),\ x,y\in S, $$ ∑ λ ∈ Φ f ( x + λ y + a λ ) = N f ( x ) + N f ( y ) , x , y ∈ S , $$\displaystyle \sum _{\lambda \in \varPhi }f(x+\lambda y+a_{\lambda })=Nf(x)+Nf(y),\ x,y\in S, $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = Φ $$N=\left \vert \varPhi \right \vert $$ designates the number of its elements, a λ , λ ∈ Φ $$ \left \{ a_{\lambda },\lambda \in \varPhi \right \} $$ are arbitrary elements of S, and (H, +) is an abelian group. In addition, some applications are given. These equations provide a common generalization of many functional equations (Cauchy’s, Jensen’s, quadratic, Φ-quadratic equations, …).

Date: 2019
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DOI: 10.1007/978-3-030-28950-8_16

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