Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

In the present paper we study the generalized Hyers–Ulam stability of the generalized trigonometric functional equations f (xy) + μ(y) f (xσ(y)) = 2 f (x)g(y) + 2h(y), x, y ∈ S; f (xy) + μ(y) f (xσ(y)) = 2 f (y)g(x) + 2h(x), x, y ∈ S, where S is a semigroup, σ: S −→ S is a involutive morphism, and μ: S −→ C is a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ S. As an application, we establish the generalized Hyers–Ulam stability theorem on amenable monoids and when σ is an involutive automorphism of S.

Wilson's functional equation f (xy) + µ(y) f (xσ(y)) = 2 f (x)g(y), x, y ∈ S (h = 0).D'Alembert's functional equation (g = f , h = 0).D'Alembert's functional Equation (5) with µ = 1 is also known as the cosine functional equation and has been studied extensively for a long period of time tracing back to d'Alembert [1].This functional equation plays a crucial role in determining the sum of two vectors in various Euclidean and non-Euclidean geometries.The continuous solutions f : R −→ C of d'Alembert's functional Equation (5) with µ = 1 are known: A part from the trivial solution f = 0, the solutions of ( 5) are where the parameter λ ranges over C (see for example [2]).
Several authors have succeeded to determine the general solution f : S −→ C of d'Alembert's functional Equation (5) in the abelian as well as non abelian case.Probably the very first result obtained for a non abelian group was presented by Kannappan [3].Under the condition that f is abelian: f (zxy) = f (zyx) for all x, y, z ∈ S, the solutions of the Equation ( 5) with µ = 1 are of the form , where γ : S −→ C is multiplicative.
In recent years, the theory of d'Alembert's functional Equation ( 5) with µ = 1 has witnessed important development.For example, for the case of non abelian groups, as shown in works by Y. Dilian about compact groups [4][5][6], Stetkaer [7] for step 2-nilpotent groups, Friis [8] for results on Lie groups and Davison [9,10] for general groups, even monoids.
In [11], Stetkaer obtained the complex valued solutions of d'Alembert's functional Equation (5) for the case when µ is a character of the group S. The non-zero solutions of the Equation ( 5) are the normalized traces of certain representations of the group S on C 2 Furthermore, in [12] Ebanks and Stetkaer presented some new results on groups regarding the solutions of Wilson's functional Equation ( 4) with µ = 1.We shall now also refer to Wilson's first generalization of d'Alembert's functional equation: The formulas constituting the solutions of this equation for the case of abelian groups are known, cf.Aczél [2], Sections 3.2.1 and 3.2.2.
In recent work, Stetkaer ([13,14]) studied the solutions of Wilson's functional Equation (4) and in particular he proved that if f , g are solutions of (4) with f = 0 then g satisfies d'Alembert's functional Equation (5) [15].Determining the solution formulas of f is still an open problem.
A variety of stability results regarding trigonometric functional equations and their generalizations are obtained (cf.[27,57]).
The main purpose of the present paper is to study the stability of the functional Equations ( 1) and ( 6).In the sequel, we obtain some properties of the stability of Equation ( 1) as well as Equation ( 6).As an application we prove the generalized Hyers-Ulam stability of Equations ( 1) and ( 6) on amenable monoids S and when σ is an involutive automorphism of S.

Generalized Hyers-Ulam Stability of Equation (1) on Non-Abelian Semigroups
In the present section, we obtain properties of the stability of Equation ( 1).Theorem 1.Let σ: S −→ S be an involutive morphism of the semigroup S. Let µ: S −→ C be a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. Suppose that the functions f , g, h: S −→ C satisfy the functional inequality for all x, y ∈ S and for some function φ: S −→ R + .Under these assumptions the following statements hold: (1) If σ is an involutive anti-automorphism and f is unbounded, then g is a solution of the long d'Alembert functional equation for all x, y ∈ S.
(2) If σ is an involutive automorphism and f is unbounded, then g is a solution of the short d'Alembert functional Equation (5). Proof.
(1) Let f , g, h satisfy Inequality (7) with σ an involutive anti-automorphism.Then for all x, y, z ∈ S we have Since f is assumed to be unbounded, then g satisfies the functional Equation ( 8). ( 2) If σ is an involutive automorphism, then, by using Inequality (7), µ(xσ(x)) = 1 and the triangle inequality, we obtain The mapping f is assumed to be unbounded, so g is a solution of the short d'Alembert functional Equation (5).This completes the proof.Theorem 2. Let σ: S −→ S be an involutive automorphism of the amenable semigroup S. Let µ: S −→ C be a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. Suppose that the functions f , g, h: S −→ C satisfy the functional inequality for all x, y ∈ S and for some function φ: S −→ R + .Under the additionally assumption that f is unbounded, there is a mapping H : S −→ C such that for all x, y ∈ S.
Proof.For each y fixed in S, the function is bounded.Since S is an amenable semigroup, then, from [58], there is an invariant mean on B(S, C)-the space of the complex-valued bounded functions on S, which we denote by m.We can now define the following mapping H : S −→ C by where f x (y) = f (yx), x, y ∈ S. For all x, y ∈ S, we have From Theorem 2 (2), g is a solution of the short d'Alembert functional Equation ( 5), so we obtain Now, by using the definition of H, Inequality (9) and the definition of m, we obtain for all y ∈ S.This completes the proof.
Theorem 3. Let M be a monoid (a semigroup with identity element e).Let σ: M −→ S be an involutive automorphism of the amenable monoid M. Let µ: S −→ C be a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ M. Suppose that the functions f , g, h: S −→ C satisfy the functional inequality for all x, y ∈ M and for some function φ: S −→ R + .Under the additionally assumption that f is unbounded, there are mappings F, H : 16) for all x, y ∈ M.
Proof.From Theorem 2, there is an H: M −→ C such that for all x, y ∈ M. By replacing x by e in Inequality ( 12), we obtain for all y ∈ M. If we set F = f (e)g + H, we obtain for all x ∈ M.
On the other hand, we have

Generalized Hyers-Ulam Stability of Equation (6) on Non-Abelian Semigroups
In this section, we obtain the stability of Equation ( 6) on an amenable monoid.
Theorem 4. Let σ: S −→ S be an involutive automorphism of the semigroup S. Let µ: S −→ C be a bounded multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. Suppose that the functions f , g, h: S −→ C satisfy the functional inequality for all x, y ∈ S and for some function φ: S −→ R + .Under the additionally assumption that f is unbounded, g is a solution of the short d'Alembert functional Equation (5).
Proof.By using Inequality (18), µ(xσ(x)) = 1, and σ an involutive automorphism, we obtain for all x, y, z ∈ S. The mapping f is assumed to be unbounded and µ is bounded, so g is a solution of the short d'Alembert functional Equation ( 5).This completes the proof.
Theorem 5. Let σ: S −→ S be an involutive automorphism of the amenable semigroup S. Let µ: S −→ C be a bounded multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. Suppose that the functions f , g, h : S −→ C satisfy the functional inequality for all x, y ∈ S and for some function φ: S −→ R + .Under the additionally assumption that f is unbounded, there is a mapping H : S −→ C such that and for all x, y ∈ S.
Proof.For a mapping l: S −→ C, we define the new functions x l and l µ by x l(y) = l(xy) and l µ (x) = µ(x)l(σ(x)) for all x, y ∈ S.
From Inequality (19) for each fixed x in S, the function is bounded.Since, S is amenable semigroup, then there is an invariant mean m on B(S, C).By replacing m by M with M(l) = m(l µ ), we can choose m such that m(l µ ) = m(l) for all l ∈ B(S, C).
The following mapping is well defined on S.
On the other hand, we obtain Therefore, we have ( xy f ) µ = ( y ( x f )) µ for all x, y ∈ S.
Therefore, we have µ(y)( xσ(y) f ) µ = y ( x f ) µ for all x, y ∈ S. By using the definition of H, we obtain From Theorem 4, g is a solution of the short d'Alembert functional Equation ( 5).Since m is additive, by using the above relations, we obtain Since m is invariant and m(l µ ) = m(l) for all bounded functions l on S, then we obtain for all x, y ∈ S. Finally, from Inequality (19) and the definition of H, we have for all y ∈ S.This completes the proof.= 2 f (e)g(x)g(y) + 2H(y)g(x) + 2H(x) = 2F(y)g(x) + 2H(x) for all x, y ∈ M.This completes the proof.