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Title: On the approximate solution of D'Alembert type equation originating from number theory
Authors: Bouikhalene, B.
Elqorachi, E.
Charifi, A.
Keywords: D'Alembert functional equation;Monoid ℝ²;Multiplicative function;Stability;Superstability;Ecuación funcional de D'Alembert;Monoide ℝ²;Función multiplicativa;Estabilidad;Superstabilidad
Issue Date: 2013
Publisher: Universidad de Extremadura
Abstract: We solve the functional equation E(α) : f(x₁x₂+ αy₁y₂, x₁y₂ + x₂y₁) + f(x₁x₂ ̶ αy₁y₂, x₂y₁ ̶ x₁y₂) = 2f(x₁, y₁)f(x₂, y₂), where (x₁, y₁), (x₂, y₂) ∈ ℝ², f : ℝ² → ℂ and α is a real parameter, on the monoid ℝ². Also we investigate the stability of this equation in the following setting: ⃒f(x₁x₂ + αy₁y₂, xy₂ + x₂y₁) + f(x₁x₂ ̶ αy₁y₂, x₂y₁ ̶ x₁y₂) ̶ 2f(x₁, y₁) f (x₂, y₂)⃒ ≤ min{φ(x₁), ψ(y₁), ϕ(x₂), ζ(y₂)}. From this result, we obtain the superstability of this equation.
Appears in Collections:Extracta Mathematicae Vol. 28, nº 2 (2013)

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