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An existence and stability theorem for a class of functional equations.

Consider the class of functional equations g[F(x,y)] = H[g(x),g(y)], where g: E --> X, f: E x E --> E, H: X x X --> X, E is a set and (X,d) is a complete metric space. In this paper we prove that, under suitable hypotheses on F, H and ?(x,y), the existence of a solution of the functional inequality d(f[F(x,y)],H[f(x),f(y)]) = ?(x,y), implies the existence of a solution of the above equation.