1. Pompeiu’s functional equations between unital algebras.
- Author
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Aissi, Y., Zeglami, D., and Mouzoun, A.
- Abstract
Let A and B be unital algebras, that need not be abelian, over fields K and K ′ respectively, let α , a , b , c ∈ K , β ∈ K ′ and λ ∈ C . The present work aims to determine the general solution Φ : A → B of the functional equation Φ (x + y + α x y) = Φ (x) + Φ (y) + β Φ (x) Φ (y) , x , y ∈ A ,
and to describe the solutions Φ : A → M 2 (C) of the functional equation Φ (a x + b y + c x y) = Φ (x) + Φ (y) + λ Φ (x) Φ (y) , x , y ∈ A.
We also show that, when (A , ·) (as a semigroup) is commutative and regular (for instance when dim A = 1 ), the explicit forms of the solutions of the last equation can be given. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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