Hyers–Ulam–Rassias Stability on Amenable Groups

In this chapter, we study the Ulam–Hyers–Rassias stability of the generalized cosine-sine functional equation: $$\displaystyle{\int _{K}\int _{G}f(xtk \cdot y)d\mu (t)dk = f(x)g(\,y) + h(\,y),\;x,y \in G,}$$ where f, g, and h are continuous complex valued functions on a locally compact group G, K is a compact subgroup of morphisms of G, dk is the normalized Haar measure on K, and μ is a complex measure with compact support. Furthermore, we prove a stability theorem in the case where G is amenable, K is a finite subgroup of the automorphisms of G, and μ is a finite K-invariant complex measure, and we obtain also the Hyers–Ulam–Rassias stability of the generalized cosine-sine functional equation: $$\displaystyle{\,f(xy) + f(x\sigma (\,y)) = 2f(x)g(\,y) + 2h(\,y),x,y \in G,}$$ where G is amenable, $$\sigma$$ is an involution of G.