Long-time behavior of delay differential quasi-variational–hemivariational inequalities and application to contact problems
Abstract
In this article, we study a class of differential quasi-variational–hemivariational inequalities involving time delays. We establish new systems and prove the solvability and the existence of decay solutions. Moreover, we are concerned with long-time behavior of solutions by showing the existence of a compact global attractor to m-semiflow associated with delay differential quasi-variational–hemivariational inequalities. An application to the contact problems driven by dynamic systems is discussed to demonstrate our theoretical results.
References
[1]
Akhmerov RR, Kamenskii MI, Potapov AS, Rodkina AE, and Sadovskii BN Measures of Noncompactness and Condensing Operators 1992 Birkhäuser Boston-Basel-Berlin
[2]
Anh NTV Periodic solutions to differential variational inequalities of parabolic–elliptic type Taiwan. J. Math. 2020 24 6 1497-1527
[3]
Anh NTV On periodic solutions to a class of delay differential variational inequalities Evol. Equ. Control Theory 2022 11 4 1309-1329
[4]
Anh NTV Asymptotically periodic solutions for fractional differential variational inequalities Fixed Point Theory. 2023 24 2 459-486
[5]
Anh NTV and Ke TD Asymptotic behavior of solutions to a class of differential variational inequalities Ann. Polon. Math. 2015 114 2 147-164
[6]
Anh NTV and Ke TD On the differential variational inequalities of parabolic–elliptic type Math. Methods Appl. Sci. 2017 40 4683-4695
[7]
Anh NTV and Ke TD On the differential variational inequalities of parabolic–parabolic type Acta Appl. Math. 2021 176 5 25
[8]
Anh NTV and Thuy TV On the delay differential variational inequalities of parabolic-elliptic type Complex Var. Elliptic Equ. 2022 67 12 3048-3073
[9]
Bothe D Multivalued perturbations of m-accretive differential inclusions Isr. J. Math. 1998 108 109-138
[10]
Caraballo T and Kloeden PE Non-autonomous attractor for integro-differential evolution equations Discrete Contin. Dyn. Syst. Ser. S 2009 2 17-36
[11]
Cen J, Khan AA, Motreanu D, and Zeng S Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems Inverse Prob. 2022 38 6 065006, 28
[12]
Chen X and Wang Z Differential variational inequality approach to dynamic games with shared constraints Math. Program. 2014 146 379-408
[13]
Clarke F Optimization and Nonsmooth Analysis 1983 Hoboken Wiley/Interscience (Reprinted: SIAM Publications, Classics in Applied Mathematics 5, 1990)
[14]
Efendiev MA and Ôtani M Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium Ann. Inst. H. Poincaré C Anal. Non Linéaire 2011 28 4 565-582
[15]
Engel KJ and Nagel R One-Parameter Semigroups for Linear Evolution Equations 2000 New York Springer
[16]
Fan K Some properties of convex sets related to fixed point theorems Math. Ann. 1984 266 519-537
[17]
Gasinski L, Migórski S, and Ochal A Existence results for evolution inclusions and variational–hemivariational inequalities Appl. Anal. 2015 94 1670-1694
[18]
Giannessi F and Khan AA Regularization of non-coercive quasi-variational inequalities Control. Cybern. 2000 29 91-110
[19]
Gwinner J On a new class of differential variational inequalities and a stability result Math. Program. 2013 139 205-221
[20]
Halanay A Differential Equations, Stability, Oscillations, Time Lags 1996 New York Academic Press
[21]
Han L and Pang J-S Non-zenoness of a class of differential quasi-variational inequalities Math. Program. 2010 121 171-199
[22]
Han W, Migórski S, and Sofonea M A class of variational–hemivariational inequalities with applications to frictional contact problems SIAM J. Math. Anal. 2014 46 3891-3912
[23]
Hintermüller M, Kovtunenko VA, and Kunisch K Obstacle problems with cohesion: a hemivariational inequality approach and its efficient numerical solution SIAM J. Optim. 2011 21 491-516
[24]
Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces. In: de Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter, Berlin, New York (2001)
[25]
Ke TD, Loi NV, and Obukhovskii V Decay solutions for a class of fractional differential variational inequalities Fract. Calc. Appl. Anal. 2015 18 531-553
[26]
Li X and Liu ZH Sensitivity analysis of optimal control problems described by differential hemivariational inequalities SIAM J. Control Optim. 2018 56 3569-3597
[27]
Li X, Liu ZH, and Papageorgiou NS Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications Nonlinearity 2023 36 1323-1348
[28]
Liu ZH, Loi NV, and Obukhovskii V Existence and global bifurcation of periodic solutions to a class of differential variational inequalities Int. J. Bifur. Chaos 2013 23 1350125
[29]
Liu ZH, Zeng S, and Motreanu D Evolutionary problems driven by variational inequalities J. Differ. Equ. 2016 260 6787-6799
[30]
Liu ZH, Migórski S, and Zeng S Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces J. Differ. Equ. 2017 263 3989-4006
[31]
Liu ZH, Motreanu D, and Zeng S Nonlinear evolutionary systems driven by mixed variational inequalities and its applications Nonlinear Anal. Real World Appl. 2018 42 409-421
[32]
Liu ZH, Motreanu D, and Zeng S On the well-posedness of differential mixed quasivariational-inequalities Topol. Methods Nonlinear Anal. 2018 51 135-150
[33]
Melnik VS and Valero J On attractors of multivalued semi-flows and differential inclusions Set-Valued Anal. 1998 6 83-111
[34]
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear inclusions and hemivariational inequalities. In: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26, Springer, New York (2013)
[35]
Migórski S, Bai Y, and Zeng S A new class of history-dependent quasi variational–hemivariational inequalities with constraints Commun. Nonlinear Sci. Numer. Simul. 2022 114
[36]
Migórski S, Yao JC, and Zeng S A class of elliptic quasi-variational-hemivariational inequalities with applications J. Comput. Appl. Math. 2023 421 114871
[37]
Obukhovskii V and Yao J-C On impulsive functional differential inclusions with Hille–Yosida operators in Banach spaces Nonlinear Anal. 2010 73 1715-1728
[38]
Pang J-S and Stewart DE Differential variational inequalities Math. Program. Ser. A 2008 113 2 345-424
[39]
Sofonea M and Migórski S Variational–hemivariational inequalities with applications 2017 New York Chapman and Hall/CRC
[40]
Vrabie, I. I.: -semigroups and applications. In: North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam (2003)
[41]
Wang X, Qi YW, Tao CQ, et al. A class of delay differential variational inequalities J. Optim. Theory Appl. 2017 172 1 56-69
[42]
Zeng S, Migórski S, and Liu ZH Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities SIAM J. Optim. 2021 31 4 2829-2862
[43]
Zeng S, Migórski S, and Khan AA Nonlinear quasi-hemivariational inequalities: existence and optimal control SIAM J. Control Optim. 2021 59 2 1246-1274
[44]
Zeng S, Motreanu D, and Khan AA Evolutionary quasi-variational–hemivariational inequalities I: existence and optimal control J. Optim. Theory Appl. 2022 193 950-970
[45]
Zeng S, Papageorgiou NS, and Rǎdulescu VD Nonsmooth dynamical systems: from the existence of solutions to optimal and feedback control Bull. Sci. Math. 2022 176 103131, 39
[46]
Anh, N.T.V.: Optimal control problem of fractional evolution inclusions with Clarke subdifferential driven by quasi-hemivariational inequalities. Commun. Nonlinear Sci. Numer. Simul. 128, 16 (2024)
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Birkhauser Verlag
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Publication History
Published: 15 March 2024
Accepted: 24 January 2024
Revision received: 15 December 2023
Received: 28 May 2023
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- Vingroup Innovation Foundation
- Vietnam Ministry of Education and Training
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