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 ( x y ) + μ ( y ) f ( x σ ( y ) ) = 2 f ( x ) g ( y ) + 2 h ( y ) , x , y ∈ S ; f ( x y ) + μ ( y ) f ( x σ ( y ) ) = 2 f ( y ) g ( x ) + 2 h ( 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 ( x σ ( 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