-
Previous Article
Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body
- EECT Home
- This Issue
-
Next Article
A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction
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 |
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.
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. |
show all references
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. |
| [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 and Control Theory, 2020, 9 (4) : 1041-1055. 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 and Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309 |
| [7] |
Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete and 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 and 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 and Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 |
| [10] |
Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087 |
| [11] |
Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 |
| [12] |
Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022010 |
| [13] |
Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455 |
| [14] |
Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 |
| [15] |
Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117 |
| [16] |
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 and Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567 |
| [17] |
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 and Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745 |
| [18] |
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 and Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673 |
| [19] |
Nuno F. M. Martins. Detecting the localization of elastic inclusions and Lamé coefficients. Inverse Problems and Imaging, 2014, 8 (3) : 779-794. doi: 10.3934/ipi.2014.8.779 |
| [20] |
Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 |
2020 Impact Factor: 1.081
Tools
Metrics
Other articles
by authors
[Back to Top]