Bifurcation of fixed points from a manifold of trivial fixed points in the infinite dimensional case

We obtain a necessary as well as a sufficient condition for the existence of bifurcation points of a coincidence equation, and, in particular, of a parametrized fixed point problem. In both cases the trivial solutions are assumed to form a nite dimensional submanifold of a Banach manifold. An application is given to a delay differential equation on a manifold: we detect periodic solutions that rotate close to an equilibrium point.