Differentiable solutions of functional equations in Banach spaces

We deal with the functional equation where functions F and g are given, defined in open subsets of Banach spaces and taking values in Banach spaces as well. We prove theorems on the existtence and uniqueness of solutions of the equation in classes of differentiable functions. As corollaries we get some results on the conjugacy of diffeomorphisms. Analogous results have been known in the finite dimensional case only