Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbations in Banach spaces

Let A:D(A)\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-periodic solutions for the equation x'=Ax+f(t,x)+e g(t,x,e) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to e=0. We show that by means of a suitable modification of the classical Mel'nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov-Schmidt reduction to the present nonsmooth case.