Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

Abstract

Let S and G be a commutative semigroup and a commutative group, respectively, C and R+ the sets of complex numbers and nonnegative real numbers, respectively, and σ:S→S or σ:G→G an involution. In this paper, we first investigate general solutions of the functional equation f(x+σy)=f(x)g(y)-g(x)f(y) for all x,y∈S, where f,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality |f(x+σy)-f(x)g(y)+g(x)f(y)|≤ψ(y) for all x,y∈G, where f,g:G→C and ψ:G→R+.