Evolutionary Inclusions and Hemivariational Inequalities

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In this chapter we study evolutionary inclusions of second order. These are multivalued relations which involve the second-order time derivative of the unknown. We start with a basic existence result for such inclusions. Then we provide results on existence and uniqueness of solutions to evolutionary inclusions of the subdifferential type, i.e., inclusions involving the Clarke subdifferential operator of locally Lipschitz functionals. We also prove an existence and uniqueness result for integro-differential evolutionary inclusions. Next, we consider a class of hyperbolic hemivariational inequalities for which we provide a theorem on existence of solutions and, under stronger hypotheses, their uniqueness. We conclude this chapter with a result on existence and uniqueness of solutions to the evolutionary integro-differential hemivariational inequality with the Volterra integral term. The results provided below represent the dynamic counterparts of theorems presented in Chap. 4 and will be used in the study of the dynamic frictional contact problems in Chap. 8.