• Previous Article
    On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application
  • EECT Home
  • This Issue
  • Next Article
    Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems
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  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 & Control Theory, doi: 10.3934/eect.2020055
References:
[1]

K. T. AndrewsK. L. KuttlerJ. 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.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-valued analysis, in Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Brézis, On some degenerate nonlinear parabolic equations, Proc. Symposia in Pure Math., 18 (1970), 28-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

K. L. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 2000 (2000), 1-20.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. T. AndrewsK. L. KuttlerJ. 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.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-valued analysis, in Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Brézis, On some degenerate nonlinear parabolic equations, Proc. Symposia in Pure Math., 18 (1970), 28-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

K. L. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 2000 (2000), 1-20.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548

[2]

Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287

[3]

Kenneth Kuttler. Measurable solutions for elliptic and evolution inclusions. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020041

[4]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618

[5]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258

[6]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

[7]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629

[8]

Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312

[9]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[10]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[11]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[12]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

[13]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[14]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

[15]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

[16]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117

[17]

Nuno F. M. Martins. Detecting the localization of elastic inclusions and Lamé coefficients. Inverse Problems & Imaging, 2014, 8 (3) : 779-794. doi: 10.3934/ipi.2014.8.779

[18]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[19]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations & Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029

[20]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (16)
  • HTML views (22)
  • Cited by (0)

[Back to Top]