Evolution Inclusions with Applications to Hemivariational Inequalities

Stanislaw Migorski^*

We consider a class of abstract nonlinear evolution inclusions of first order with a multivalued Clarke subdifferential term. We prove existence and uniqueness of solutions using a surjectivity result for pseudomonotone multivalued operators. Next, using the Banach fixed point theorem we establish the unique solvability to evolution inclusion with history-dependent operator. We apply this result to second order evolution inclusions involving two history-dependent operators which depend on the unknown and its time derivative. Finally, we specify existence and uniqueness results to second order inclusions with the Volterra-type integral operator. We illustrate the abstract results to prove the existence of a unique solution to first and second order hemivariational inequalities.

Mathematics Subject Classification: 34G25 35K86 35L86

Keywords: Evolution inclusion, hemivariational inequality, history-dependent operator, Clarke subdifferential, pseudomonotone operator.

Minisymposion: Analysis and Control of Evolution Equations and Inclusions