# Stability of the cosine-sine functional equation with involution

Document Type: Original Article

Authors

Abstract

Let $S$ and $G$ be a commutative semigroup and a commutative group respectively, $\Bbb C$ and $\Bbb R^+$ the sets of complex numbers and nonnegative real numbers respectively, $\sigma : S \to S$ or $\sigma : G \to G$ an involution
and $\psi : G\to \Bbb R^+$ be fixed.
In this paper, we first investigate general solutions of the equation
\begin{align}
g(x+\sigma y)=g(x)g(y)+f(x)f(y)\nonumber
\end{align}
for all $x,y \in S$, where $f, g : S \to \Bbb C$  are unknown functions to be determined. Secondly, we consider the Hyers-Ulam stability of the equation,
i.e., we study the functional inequality
\begin{align}
|g(x+\sigma y)-g(x)g(y)-f(x)f(y)|\le \psi(y)\nonumber
\end{align}
for all $x,y \in G$, where $f, g : G\to \Bbb C$.

Keywords

### Histroty

• Receive Date: 27 June 2017
• Revise Date: 11 September 2017
• Accept Date: 11 September 2017