In the present paper we study the generalized Hyers–Ulam stability of the generalized trigonometric functional equations $$f(xy)+\mu (y)f(x\sigma (y))=2f(x)g(y)+2h(y),x,y\in S;$$ $$f(xy)+\mu (y)f(x\sigma (y))=2f(y)g(x)+2h(x),x,y\in S,$$ where S is a semigroup, $\sigma$ : $S⟶S$ is a involutive morphism, and $\mu$ : $S⟶\mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S.$ As an application, we establish the generalized Hyers–Ulam stability theorem on amenable monoids and when $\sigma$ is an involutive automorphism of S.

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g ( x σ ( y ) ) = g ( x ) g ( y ) + f ( x ) f ( y ) , x , y ∈ S , f ( x σ ( y ) ) = f ( x ) g ( y ) + f ( y ) g ( x ) , x , y ∈ S , \matrix{ {g\left( {x\sigma \left( y \right)} \right) = g\left( x \right)g\left( y \right) + f\left( x \right)f\left( y \right),} & {x,y \in S,} \cr {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) + f\left( y \right)g\left( x \right),} & {x,y \in S,} \cr } and f ( x σ ( y ) ) = f ( x ) g ( y ) - f ( y ) g ( x ) , x , y ∈ S , \matrix{ {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) - f\left( y \right)g\left( x \right),} & {x,y \in S,} \cr } where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.

This paper is mainly concerned with the following functional equation where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ-spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.