Abstract
We consider a class of abstract nonlinear evolutionary inclusions of first order with a multivalued Clarke subgradient term. We use a surjectivity result for pseudomonotone multivalued operators in order to prove existence and uniqueness of solutions. Next, we use the Banach fixed point theorem and establish the unique solvability to evolutionary inclusion with history-dependent operators. We apply this result to second order evolutionary inclusions governed by two history-dependent operators, which depend on the solution and its time derivative, respectively. Finally, we specify existence and uniqueness results for nonlinear first and second order hemivariational inequalities with or without history-dependent operators.
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Acknowledgements
This research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under grant no. N N201 604640, the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant no. W111/7.PR/2012, the National Science Center of Poland under Maestro Advanced Project no. DEC-2012/06/A/ST1/00262, and the project Polonium “Mathematical and Numerical Analysis for Contact Problems with Friction” 2014/15 between the Jagiellonian University in Krakow and Université de Perpignan Via Domitia.
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Migórski, S., Ochal, A., Sofonea, M. (2015). Evolutionary Inclusions and Hemivariational Inequalities. In: Han, W., Migórski, S., Sofonea, M. (eds) Advances in Variational and Hemivariational Inequalities. Advances in Mechanics and Mathematics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-14490-0_2
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DOI: https://doi.org/10.1007/978-3-319-14490-0_2
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