# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021045
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## On periodic solutions to a class of delay differential variational inequalities

 Department of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: The author is supported by Hanoi National University of Education under grant number SPHN20-06

In this paper, we introduce and study a class of delay differential variational inequalities comprising delay differential equations and variational inequalities. We establish a sufficient condition for the existence of periodic solutions to delay differential variational inequalities. Based on some fixed point arguments, in both single-valued and multivalued cases, the solvability of initial value and periodic problems are proved. Furthermore, we study the conditional stability of periodic solutions to this systems.

Citation: Nguyen Thi Van Anh. On periodic solutions to a class of delay differential variational inequalities. Evolution Equations & Control Theory, doi: 10.3934/eect.2021045
##### References:
 [1] N. T. V. Anh, Periodic Solutions to differential variational inequalities of parabolic-elliptic type, Taiwanese J. Math., 24 (2020), 1497-1527.  doi: 10.11650/tjm/200301.  Google Scholar [2] N. T. V. Anh and T. D. Ke, Asymptotic behavior of solutions to a class of differential variational inequalities, Ann. Polon. Math., 114 (2015), 147-164.  doi: 10.4064/ap114-2-5.  Google Scholar [3] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar [4] E. P. Avgerinos and N. S. Papageorgiou, Differential variational inequalities in ${\bf{R}}^ N$, Indian J. Pure Appl. Math., 28 (1997), 1267-1287.   Google Scholar [5] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.  Google Scholar [6] X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.  Google Scholar [7] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^l(\mu, X)$, Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.1090/S0002-9939-1993-1132408-X.  Google Scholar [8] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Springer-Verlag, New York, 2003.  Google Scholar [9] J. Gwinner, On differential variational inequalities and projected dynamical systems - equivalence and a stability result, Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, Suppl., (2007), 467–476.  Google Scholar [10] J. Gwinner, A note on linear differential variational inequalities in Hilbert Spaces, System Modeling and Optimization, IFIP Adv. Inf. Commun. Technol., Springer, Heidelberg, 391 (2013), 85-91.   Google Scholar [11] A. Hanalay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York, London 1996.  Google Scholar [12] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.  Google Scholar [13] Z. Liu, N. V. Loi and V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350125. doi: 10.1142/S0218127413501253.  Google Scholar [14] N. V. Loi, On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal., 122 (2015), 83-99.  doi: 10.1016/j.na.2015.03.019.  Google Scholar [15] J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475.  doi: 10.1215/S0012-7094-50-01741-8.  Google Scholar [16] N. V. Minh, F. Rabiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory, 32 (1998), 332-353.  doi: 10.1007/BF01203774.  Google Scholar [17] S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral. Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.  Google Scholar [18] J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.  Google Scholar [19] J. S. Pang and D. E. Stewart, Solution dependence on initial conditions in differential variational inequalities, Math. Program., 116 (2009), 429-460.  doi: 10.1007/s10107-007-0117-5.  Google Scholar [20] W. J. Rugh, Linear System Theory, Prentice Hall Information and System Sciences Series. Prentice Hall, Inc., Englewood Cliffs, NJ, 1993.  Google Scholar [21] D. E. Stewart, Dynamics with Inequalities. Impacts and Hard Constraints, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611970715.  Google Scholar [22] X. Wang, Y. Qi, C. Tao and Y. Xiao, A class of delay differential variational inequalities, J. Optim. Theory Appl., 172 (2017), 56-69.  doi: 10.1007/s10957-016-1002-2.  Google Scholar

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##### References:
 [1] N. T. V. Anh, Periodic Solutions to differential variational inequalities of parabolic-elliptic type, Taiwanese J. Math., 24 (2020), 1497-1527.  doi: 10.11650/tjm/200301.  Google Scholar [2] N. T. V. Anh and T. D. Ke, Asymptotic behavior of solutions to a class of differential variational inequalities, Ann. Polon. Math., 114 (2015), 147-164.  doi: 10.4064/ap114-2-5.  Google Scholar [3] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar [4] E. P. Avgerinos and N. S. Papageorgiou, Differential variational inequalities in ${\bf{R}}^ N$, Indian J. Pure Appl. Math., 28 (1997), 1267-1287.   Google Scholar [5] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.  Google Scholar [6] X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.  Google Scholar [7] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^l(\mu, X)$, Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.1090/S0002-9939-1993-1132408-X.  Google Scholar [8] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Springer-Verlag, New York, 2003.  Google Scholar [9] J. Gwinner, On differential variational inequalities and projected dynamical systems - equivalence and a stability result, Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, Suppl., (2007), 467–476.  Google Scholar [10] J. Gwinner, A note on linear differential variational inequalities in Hilbert Spaces, System Modeling and Optimization, IFIP Adv. Inf. Commun. Technol., Springer, Heidelberg, 391 (2013), 85-91.   Google Scholar [11] A. Hanalay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York, London 1996.  Google Scholar [12] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.  Google Scholar [13] Z. Liu, N. V. Loi and V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350125. doi: 10.1142/S0218127413501253.  Google Scholar [14] N. V. Loi, On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal., 122 (2015), 83-99.  doi: 10.1016/j.na.2015.03.019.  Google Scholar [15] J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475.  doi: 10.1215/S0012-7094-50-01741-8.  Google Scholar [16] N. V. Minh, F. Rabiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory, 32 (1998), 332-353.  doi: 10.1007/BF01203774.  Google Scholar [17] S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral. Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.  Google Scholar [18] J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.  Google Scholar [19] J. S. Pang and D. E. Stewart, Solution dependence on initial conditions in differential variational inequalities, Math. Program., 116 (2009), 429-460.  doi: 10.1007/s10107-007-0117-5.  Google Scholar [20] W. J. Rugh, Linear System Theory, Prentice Hall Information and System Sciences Series. Prentice Hall, Inc., Englewood Cliffs, NJ, 1993.  Google Scholar [21] D. E. Stewart, Dynamics with Inequalities. Impacts and Hard Constraints, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611970715.  Google Scholar [22] X. Wang, Y. Qi, C. Tao and Y. Xiao, A class of delay differential variational inequalities, J. Optim. Theory Appl., 172 (2017), 56-69.  doi: 10.1007/s10957-016-1002-2.  Google Scholar
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