• Previous Article
    Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $
  • EECT Home
  • This Issue
  • Next Article
    Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
doi: 10.3934/eect.2021045
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. DiestelW. 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. MinhF. 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. WangY. QiC. 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

show all references

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. DiestelW. 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. MinhF. 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. WangY. QiC. 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

[1]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

[2]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[3]

Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021004

[4]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467

[5]

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

[6]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[7]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

[8]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145

[9]

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

[10]

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

[11]

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

[12]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

[13]

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

[14]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121

[15]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044

[16]

Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021027

[17]

Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451

[18]

Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453

[19]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070

[20]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375

2020 Impact Factor: 1.081

Article outline

[Back to Top]