On the Superstability of a Generalization of the Cosine Equation

The aim of this paper is to investigate the stability problem for the functional equation:
ƒ(xy)+ƒ(xσ(y))=2g(x)ƒ(y), x,y∈G (Eg,ƒ)
and the superstability of the d'Alembert's equation:
ƒ(xy)+ƒ(xσ(y))=2ƒ(x)ƒ(y), x,y∈G (A)
under the conditions from which the differences of each equation are bounded by φ(x), ψ(x) and min(φ(x),ψ(y)) where G is an arbitrary group, not necessarily abelian, ƒ, g are complex valued functions, φ, ψ are real valued functions and σ is an involution of G.