Stability of a generalized trigonometric functional equation

The stability of the functional equation F(x+y)−G(x−y)=2H(x)K(y)F(x+y)−G(x−y)=2H(x)K(y) over the domain of an abelian group GG and the range of the complex field is investigated. Several related results extending a number of previously known ones, such as the ones dealing with the sine functional equation, the d’Alembert functional equation and Wilson functional equation, are derived as direct consequences. Applying the main result to the setting of Banach algebra, it is shown that if their operators satisfy a functional inequality and are subject to certain natural requirements, then these operators must be solutions of some well-known functional equations.