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December  2020, 9(4): 935-960. doi: 10.3934/eect.2020055

## Measurable solutions to general evolution inclusions

 1 Department of Mathematics and Statistics, Oakland University, Rochester MI 48309 USA 2 Retired 3 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: andrews@oakland.edu

Received  October 2019 Revised  February 2020 Published  December 2020 Early access  May 2020

This work establishes the existence of measurable solutions to evolution inclusions involving set-valued pseudomonotone operators that depend on a random variable $\omega\in \Omega$ that is an element of a measurable space $(\Omega, \mathcal{F})$. This result considerably extends the current existence results for such evolution inclusions since there are no assumptions made on the uniqueness of the solution, even in the cases where the parameter $\omega$ is held constant, which leads to the usual evolution inclusion. Moreover, when one assumes the uniqueness of the solution, then the existence of progressively measurable solutions under reasonable and mild assumptions on the set-valued operators, initial data and forcing functions is established. The theory developed here allows for the inclusion of memory or history dependent terms and degenerate equations of mixed type. The proof is based on a new result for measurable solutions to a parameter dependent family of elliptic equations. Finally, when the choice $\omega = t$ is made, where $t$ is the time and $\Omega = [0, T]$, the results apply to a wide range of quasistatic inclusions, many of which arise naturally in contact mechanics, among many other applications.

Citation: Kevin T. Andrews, Kenneth L. Kuttler, Ji Li, Meir Shillor. Measurable solutions to general evolution inclusions. Evolution Equations and Control Theory, 2020, 9 (4) : 935-960. doi: 10.3934/eect.2020055
##### References:
 [1] K. T. Andrews, K. L. Kuttler, J. Li and M. Shillor, Measurable solutions for elliptic inclusions and quasistatic problems, Comput. Math. Appl., 77 (2019), 2869-2882.  doi: 10.1016/j.camwa.2018.09.025. [2] J.-P. Aubin and H. Frankowska, Set-valued analysis, in Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [3] A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1. [4] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280. [5] H. Brézis, On some degenerate nonlinear parabolic equations, Proc. Symposia in Pure Math., 18 (1970), 28-28. [6] Z. Denkowski, S. Migórski and N. S. Papageorgiu, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-9158-4. [7] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. [8] S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis. Vol. I. Theory, in Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4. [9] K. L. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 2000 (2000), 1-20. [10] K. L. Kuttler and J. Li, Measurable solutions for stochastic evolution equations without uniqueness, Appl. Anal., 94 (2015), 2456-2477.  doi: 10.1080/00036811.2014.989498. [11] K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 2016 (2016), 12 pp. [12] K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math., 1 (1999), 87-123.  doi: 10.1142/S0219199799000067. [13] J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [14] S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [15] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [16] M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799. [17] M. Sofonea, W. Han and M. Shillor, Analysis and approximations of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

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##### References:
 [1] K. T. Andrews, K. L. Kuttler, J. Li and M. Shillor, Measurable solutions for elliptic inclusions and quasistatic problems, Comput. Math. Appl., 77 (2019), 2869-2882.  doi: 10.1016/j.camwa.2018.09.025. [2] J.-P. Aubin and H. Frankowska, Set-valued analysis, in Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [3] A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1. [4] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280. [5] H. Brézis, On some degenerate nonlinear parabolic equations, Proc. Symposia in Pure Math., 18 (1970), 28-28. [6] Z. Denkowski, S. Migórski and N. S. Papageorgiu, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-9158-4. [7] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. [8] S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis. Vol. I. Theory, in Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4. [9] K. L. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 2000 (2000), 1-20. [10] K. L. Kuttler and J. Li, Measurable solutions for stochastic evolution equations without uniqueness, Appl. Anal., 94 (2015), 2456-2477.  doi: 10.1080/00036811.2014.989498. [11] K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 2016 (2016), 12 pp. [12] K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math., 1 (1999), 87-123.  doi: 10.1142/S0219199799000067. [13] J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [14] S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [15] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [16] M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799. [17] M. Sofonea, W. Han and M. Shillor, Analysis and approximations of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.
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