On the approximate solution of D'Alembert type equation originating from number theory

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.peerReviewe