A variational convergence problem with antiperiodic boundary conditions

Existence and uniqueness of anti-periodic solutions for evolution inclusions generated by the subdifferential of a convex lower semicontinuos function appeared in a series of papers (see the references in the present paper). We present here two epigraphical versions of the mentioned results involving new variational convergence techniques and the stable convergence of Young measure [C.Castaing-P.Raynaud de Fitte-M.Valadier, Young measures on Topological Spaces. With applications in Control Theory and Probability Theory, Kluwer (2004)]. After some preliminaries (section 1), in section 2 we summarize the basic results on the convergence of bounded sequences in L^1_H([0,T]), where H is a Hilbert space. In Section 3 we prove some existence and uniqueness results of anti-periodic solutions for a first order evolution inclusions generated by a subdifferential of a convex lower semicontinuous even functions defined on H and present applications to a new existence result of anti-periodic solutions. Section 4 is devoted to the existence of anti-periodic solutions for a second order evolution inclusions via a variational approach involving the biting convergence, Young measures, the characterization of the second dual of L^1_H([0,T]) and other tools.