Stability of the sine addition-subtraction law

Résumé

In this paper, we investigate the stability of the functional equation

\[

f(xy) = f(x)g(y) + \beta\, g(x)f(y) + \gamma\, f(x)f(y), \qquad x, y \in S,

\]

where \(S\) is a semigroup, \(f, g : S \to \mathbb{C}\) are two unknown functions, \(\beta \in \mathbb{C} \setminus \{0\}\) and \(\gamma \in \mathbb{C}\) are fixed constants. We extend our analysis to the functional equation

\[

f(x\sigma(y)) = f(x)g(y) + \beta\, g(x)f(y) + \gamma\, f(x)f(y), \qquad x, y \in S,

\]

where \(\sigma : S \to S\) is an involutive automorphism.