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

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

Superstability of approximate cosine type functions on the monoid ℝ²

In this paper, we study the superstability problem for the cosine type functional equation f(x₁x₂, x₁y₂ + x₂y₁) + f(x₁x₂, y₁x₂ ‒ x₁y₂) = 2 f(x₁, y₁) f(x₂, y₂) on the commutative monoid (ℝ²,X). As a result we obtain cosine type functions satisfying the equation approximately.peerReviewe

Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation

AbstractWe use the operatorial approach to obtain, in non-Archimedean spaces, the Hyers–Ulam stability of the Pexider K-quadratic functional equation∑k∈Kf(x+k·y)=κg(x)+κh(y),x,y∈E,where f,g,h:E→F are applications and K is a finite subgroup of the group of automorphisms of E and κ is its order

Superstability of Approximate Cosine Type Functions on the Monoid R²

In this paper, we study the superstability problem for the cosine type functional equation on the commutative monoid (R 2 , ×). As a result we obtain cosine type functions satisfying the equation approximately

On the stability of a class of Cosine type functional equations

The aim of this paper is to investigate the stability problem for the pexiderized trigonometric functional equation f1(xy) + f2(xσ(y)) = 2g1(x)g2(y), x, y ∈ G, (E) where G is an arbitrary group, f1, f2, g1and g2 are complex valued functions on G and σ is an involution of G. Results of this paper also can be extended to the setting of monoids (that is, a semigroup with identity) that need not be abelian. © Soc. Paran. de Mat

On the Approximate Solution of D'Alembert Type Equation Originating from Number Theory

We solve the functional equation and α is a real parameter, on the monoid R 2 . Also we investigate the stability of this equation in the following setting: From this result, we obtain the superstability of this equation

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

Superstability of Approximate Cosine Type Functions on the Monoid R²

In this paper, we study the superstability problem for the cosine type functional equation on the commutative monoid (R 2 , ×). As a result we obtain cosine type functions satisfying the equation approximately