Stability of the cosine-sine functional equation with involution

Document Type: Original Article

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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 $f, g : G\to \Bbb C$.

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