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

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