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

Let A : D(A) → E, D(A) ⊂ E, be an infinitesimal generator either of an analytic compact semigroup or of a contractive C0-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 ẋ = Ax + f(t, x) + εg(t, x, ε) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to ε = 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.