* (35) L. Kuiper, W. Hermsen, P. R. den Hartog, and J. O. Urama, “Temporal and spectral evolution in x- and gamma-rays of magnetar 1e 1547.0–5408 since its 2008 october outburst: The discovery of a transient hard pulsed component after its 2009 january outburst,” The Astrophysical Journal 748 no. 2, (Mar, 2012) 133. http://dx.doi.org/10.1088/0004-637X/748/2/133.
  * (36) N. Rea et al., “The first outburst of the new magnetar candidate SGR 0501+4516,” Mon. Not. Roy. Astron. Soc. 396 (2009) 2419, arXiv:0904.2413 [astro-ph.HE].
  * (37) H. An et al., “NuSTAR Observations of Magnetar 1E 1841–045,” Astrophys. J. 779 (2013) 163, arXiv:1310.6221 [astro-ph.HE].
  * (38) J. K. Vogel et al., “NuSTAR Observations of the Magnetar 1E 2259+586,” Astrophys. J. 789 no. 1, (2014) 75, arXiv:1408.0768 [astro-ph.HE].
  * (39) G. Younes, M. G. Baring, C. Kouveliotou, A. Harding, S. Donovan, E. Göğüş, V. Kaspi, and J. Granot, “The Sleeping Monster: NuSTAR Observations of SGR 1806–20, 11 Years After the Giant Flare,” Astrophys. J. 851 no. 1, (2017) 17, arXiv:1711.00034 [astro-ph.HE].
  * (40) S. Olausen and V. Kaspi, “The McGill Magnetar Catalog,” Astrophys. J. Suppl. 212 (2014) 6, arXiv:1309.4167 [astro-ph.HE].
  * (41) A. M. Beloborodov, “On the mechanism of hard X-ray emission from magnetars,” Astrophys. J. 762 (2013) 13, arXiv:1201.0664 [astro-ph.HE].
  * (42) M. Baring, Z. Wadiasingh, P. Gonthier, and A. Harding, “Hard X-ray Quiescent Emission in Magnetars via Resonant Compton Upscattering,” J. Phys. Conf. Ser. 932 no. 1, (2017) 012021, arXiv:1710.09017 [astro-ph.HE].
  * (43) Z. Wadiasingh, M. G. Baring, P. L. Gonthier, and A. K. Harding, “Resonant Inverse Compton Scattering Spectra from Highly-magnetized Neutron Stars,” Astrophys. J. 854 no. 2, (2018) 98, arXiv:1712.09643 [astro-ph.HE].
  * (44) S. P. Harris, J.-F. Fortin, K. Sinha, and M. G. Alford, “Axions in neutron star mergers,” JCAP 07 (2020) 023, arXiv:2003.09768 [hep-ph].
  * (45) A. Burrows, M. Ressell, and M. S. Turner, “Axions and SN1987A: Axion trapping,” Phys. Rev. D 42 (1990) 3297–3309.
  * (46) F. Fattoyev, C. Horowitz, J. Piekarewicz, and G. Shen, “Relativistic effective interaction for nuclei, giant resonances, and neutron stars,” Phys. Rev. C 82 (2010) 055803, arXiv:1008.3030 [nucl-th].
  * (47) E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, “Evidence for quark-matter cores in massive neutron stars,” Nature Phys. (2020) , arXiv:1903.09121 [astro-ph.HE].
  * (48) V. Dexheimer, L. Soethe, J. Roark, R. Gomes, S. Kepler, and S. Schramm, “Phase Transitions in Neutron Stars,” Int. J. Mod. Phys. E 27 no. 11, (2018) 1830008, arXiv:1901.03252 [astro-ph.HE].
  * (49) S. Han, M. Mamun, S. Lalit, C. Constantinou, and M. Prakash, “Treating quarks within neutron stars,” Phys. Rev. D 100 no. 10, (2019) 103022, arXiv:1906.04095 [astro-ph.HE].
  * (50) S. B. Ruester, M. Hempel, and J. Schaffner-Bielich, “The outer crust of non-accreting cold neutron stars,” Phys. Rev. C 73 (2006) 035804, arXiv:astro-ph/0509325.
  * (51) F. Douchin and P. Haensel, “A unified equation of state of dense matter and neutron star structure,” Astron. Astrophys. 380 (2001) 151, arXiv:astro-ph/0111092.
  * (52) N. Glendenning, Compact Stars: Nuclear Physics, Particle Physics, and General Relativity.  Astronomy and Astrophysics Library. Springer New York, 2000.  https://books.google.com/books?id=BBSOgxe2S1AC.
  * (53) L. F. Roberts and S. Reddy, “Charged current neutrino interactions in hot and dense matter,” Phys. Rev. C 95 no. 4, (2017) 045807, arXiv:1612.02764 [astro-ph.HE].
  * (54) D. Yakovlev, A. Kaminker, O. Y. Gnedin, and P. Haensel, “Neutrino emission from neutron stars,” Phys. Rept. 354 (2001) 1, arXiv:astro-ph/0012122.
  * (55) N. Iwamoto, “Nucleon-nucleon bremsstrahlung of axions and pseudoscalar particles from neutron star matter,” Phys. Rev. D 64 (2001) 043002.
  * (56) B. L. Friman and O. V. Maxwell, “Neutrino emissivities of neutron stars.,” ApJ 232 (Sept., 1979) 541–557.
  * (57) R. Machleidt, “Historical perspective and future prospects for nuclear interactions,” Int. J. Mod. Phys. E 26 no. 11, (2017) 1730005, arXiv:1710.07215 [nucl-th].
  * (58) R. P. Brinkmann and M. S. Turner, “Numerical rates for nucleon-nucleon, axion bremsstrahlung,” Phys. Rev. D 38 (Oct, 1988) 2338–2348. https://link.aps.org/doi/10.1103/PhysRevD.38.2338.
  * (59) P. Carenza, T. Fischer, M. Giannotti, G. Guo, G. Martínez-Pinedo, and A. Mirizzi, “Improved axion emissivity from a supernova via nucleon-nucleon bremsstrahlung,” JCAP 10 no. 10, (2019) 016, arXiv:1906.11844 [hep-ph]. [Erratum: JCAP 05, E01 (2020)].
  * (60) N. Iwamoto, “Axion Emission from Neutron Stars,” Phys. Rev. Lett. 53 (1984) 1198–1201.
  * (61) S. Stoica, B. Pastrav, J. Horvath, and M. Allen, “Pion mass effects on axion emission from neutron stars through NN bremsstrahlung processes,” Nucl. Phys. A 828 (2009) 439–449, arXiv:0906.3134 [nucl-th]. [Erratum: Nucl.Phys.A 832, 148 (2010)].
  * (62) G. Raffelt, Stars as laboratories for fundamental physics: The astrophysics of neutrinos, axions, and other weakly interacting particles.  5, 1996.
  * (63) M. V. Beznogov, E. Rrapaj, D. Page, and S. Reddy, “Constraints on Axion-like Particles and Nucleon Pairing in Dense Matter from the Hot Neutron Star in HESS J1731-347,” Phys. Rev. C 98 no. 3, (2018) 035802, arXiv:1806.07991 [astro-ph.HE].
  * (64) C. Hanhart, D. R. Phillips, and S. Reddy, “Neutrino and axion emissivities of neutron stars from nucleon-nucleon scattering data,” Phys. Lett. B 499 (2001) 9–15, arXiv:astro-ph/0003445.
  * (65) J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108 (Dec, 1957) 1175–1204. https://link.aps.org/doi/10.1103/PhysRev.108.1175.
  * (66) D. Yakovlev, K. Levenfish, and Y. Shibanov, “Cooling neutron stars and superfluidity in their interiors,” Phys. Usp. 42 (1999) 737–778, arXiv:astro-ph/9906456.
  * (67) A. Sedrakian and J. W. Clark, “Superfluidity in nuclear systems and neutron stars,” Eur. Phys. J. A 55 no. 9, (2019) 167, arXiv:1802.00017 [nucl-th].
  * (68) D. Page, J. M. Lattimer, M. Prakash, and A. W. Steiner, “Stellar Superfluids,” arXiv:1302.6626 [astro-ph.HE].
  * (69) S. Beloin, S. Han, A. W. Steiner, and D. Page, “Constraining Superfluidity in Dense Matter from the Cooling of Isolated Neutron Stars,” Phys. Rev. C 97 no. 1, (2018) 015804, arXiv:1612.04289 [nucl-th].
  * (70) H. An, A. Cumming, and V. M. Kaspi, “Flux Relaxation after Two Outbursts of the Magnetar SGR 1627–41 and Possible Hard X-Ray Emission,” Astrophys. J. 859 no. 1, (2018) 16, arXiv:1804.00266 [astro-ph.HE].
  * (71) J. Chen, J. Clark, R. Davé, and V. Khodel, “Pairing gaps in nucleonic superfluids,” Nucl. Phys. A 555 (1993) 59–89.
  * (72) W. C. Ho, K. G. Elshamouty, C. O. Heinke, and A. Y. Potekhin, “Tests of the nuclear equation of state and superfluid and superconducting gaps using the Cassiopeia A neutron star,” Phys. Rev. C 91 no. 1, (2015) 015806, arXiv:1412.7759 [astro-ph.HE].
  * (73) K. P. Levenfish and D. G. Yakovlev, “Specific heat of neutron star cores with superfluid nucleons,” Astronomy Reports 38 no. 2, (Mar., 1994) 247–251.
  * (74) M. G. Alford and K. Pangeni, “Gap-bridging enhancement of modified Urca processes in nuclear matter,” Phys. Rev. C 95 no. 1, (2017) 015802, arXiv:1610.08617 [nucl-th].
  * (75) J. Keller and A. Sedrakian, “Axions from cooling compact stars,” Nucl. Phys. A 897 (2013) 62–69, arXiv:1205.6940 [astro-ph.CO].
  * (76) E. Flowers, M. Ruderman, and P. Sutherland, “Neutrino pair emission from finite-temperature neutron superfluid and the cooling of young neutron stars.,” ApJ 205 (Apr., 1976) 541–544.
  * (77) D. G. Yakovlev and K. P. Levenfish, “Modified URCA process in neutron star cores.,” A&A 297 (May, 1995) 717.
  * (78) P. Coleman, Introduction to Many-Body Physics.  Cambridge University Press, 2015.  https://books.google.com/books?id=PDg1HQAACAAJ.
  * (79) K. A. Maslov, E. E. Kolomeitsev, and D. N. Voskresensky, “Relativistic Mean-Field Models with Scaled Hadron Masses and Couplings: Hyperons and Maximum Neutron Star Mass,” Nucl. Phys. A950 (2016) 64–109, arXiv:1509.02538 [astro-ph.HE].
  * (80) B.-A. Li, B.-J. Cai, L.-W. Chen, and J. Xu, “Nucleon Effective Masses in Neutron-Rich Matter,” Prog. Part. Nucl. Phys. 99 (2018) 29–119, arXiv:1801.01213 [nucl-th].
  * (81) P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner, and K. A. van Bibber, “Experimental Searches for the Axion and Axion-Like Particles,” Ann. Rev. Nucl. Part. Sci. 65 (2015) 485–514, arXiv:1602.00039 [hep-ex].
  * (82) J. A. Pons and D. Viganò, “Magnetic, thermal and rotational evolution of isolated neutron stars,” arXiv:1911.03095 [astro-ph.HE].
  * (83) A. V. Bilous et al., “A $NICER$ View of PSR J0030+0451: Evidence for a Global-scale Multipolar Magnetic Field,” Astrophys. J. Lett. 887 no. 1, (2019) L23, arXiv:1912.05704 [astro-ph.HE].
  * (84) S. Mereghetti, A. Tiengo, P. Esposito, D. Gotz, L. Stella, G. Israel, N. Rea, M. Feroci, R. Turolla, and S. Zane, “A XMM-Newton view of the soft gamma-ray repeater SGR 1806-20: Long term variability in the pre-super giant flare epoch,” Astrophys. J. 628 (2005) 938–945, arXiv:astro-ph/0502417.
  * (85) P. Esposito et al., “SGR 1806-20 about two years after the giant flare: Suzaku, XMM-Newton and INTEGRAL observations,” Astron. Astrophys. 476 (2007) 321, arXiv:0710.2789 [astro-ph].
  * (86) F. C. Zelati, A. Borghese, N. Rea, D. Viganò, T. Enoto, P. Esposito, J. A. Pons, S. Campana, and G. L. Israel, “The long-term enhanced brightness of the magnetar 1E 1547.0–5408,” Astron. Astrophys. 633 (2020) A31, arXiv:1912.09158 [astro-ph.HE].
  * (87) L. Kuiper, P. G. Jonker, M. A. P. Torres, A. Rest, and S. Keek, “The discovery of a new hard X-ray source in the Norma region by INTEGRAL: IGR J16293-4603,” The Astronomer’s Telegram 1774 (Oct., 2008) 1.
  * (88) E. Göǧüs, P. M. Woods, C. Kouveliotou, Y. Kaneko, B. M. Gaensler, and S. Chatterjee, “Spatial, Temporal, and Spectral Properties of X-ray Emission from the Magnetar SGR 0501+4516,” ApJ 722 no. 1, (Oct., 2010) 899–908, arXiv:1008.4089 [astro-ph.HE].
  * (89) L. Lin, C. Kouveliotou, M. G. Baring, A. J. van der Horst, S. Guiriec, P. M. Woods, E. Göğüş, Y. Kaneko, J. Scargle, J. Granot, and et al., “Fermi/gamma-ray burst monitor observations of sgr j0501+4516 bursts,” The Astrophysical Journal 739 no. 2, (Sep, 2011) 87. http://dx.doi.org/10.1088/0004-637X/739/2/87.
  * (90) Z. Qu, Z. Li, Y. Chen, S. Dai, L. Ji, R. Xu, and S. Zhang, “The short bursts in sgr 1806-20, 1e 1048-5937, and sgr 0501+4516,” Publications of the Astronomical Society of the Pacific 127 no. 949, (Mar, 2015) 211–222. http://dx.doi.org/10.1086/680484.
  * (91) A. Camero, A. Papitto, N. Rea, D. Viganò, J. A. Pons, A. Tiengo, S. Mereghetti, R. Turolla, P. Esposito, S. Zane, and et al., “Quiescent state and outburst evolution of sgr 0501+4516,” Monthly Notices of the Royal Astronomical Society 438 no. 4, (Jan, 2014) 3291–3298. http://dx.doi.org/10.1093/mnras/stt2432.
  * (92) O. Benli, S. Caliskan, and U. Ertan, “Long-term evolution, X-ray outburst and optical/infrared emission of SGR 0501+4516,” Mon. Not. Roy. Astron. Soc. 447 (2015) 2282–2286, arXiv:1412.1677 [astro-ph.HE].
  * (93) Y. L. Mong and C. Y. Ng, “X-Ray Observations of Magnetar SGR 0501+4516 from Outburst to Quiescence,” Astrophys. J. 852 no. 2, (2018) 86, arXiv:1801.00937 [astro-ph.HE].
  * (94) Y.-J. Guo, S. Dai, Z.-S. Li, Y. Liu, H. Tong, and R.-X. Xu, “Understanding the X-ray spectrum of anomalous X-ray pulsars and soft gamma-ray repeaters,” Res. Astron. Astrophys. 15 no. 4, (2015) 525–536, arXiv:1403.6739 [astro-ph.HE].
  * (95) Y. E. Nakagawa, T. Enoto, K. Makishima, A. Yoshida, K. Yamaoka, T. Sakamoto, N. Rea, and K. Hurley, “Suzaku Detection of Hard X-ray Emission in SGR 0501+4516 Short Burst Spectrum,” Astrophysics and Space Science Proceedings 21 (Jan., 2011) 323.
  * (96) W. Wang, H. Tong, and Y.-J. Guo, “Hard X-ray emission cutoff in the anomalous X-ray pulsar 4U 0142+61 detected by $INTEGRAL$,” Res. Astron. Astrophys. 14 (2014) 673–682, arXiv:1311.0107 [astro-ph.HE].
  * (97) T. Tamba, A. Bamba, H. Odaka, and T. Enoto, “Temporal and spectral X-ray properties of magnetar SGR 1900+14 derived from observations with NuSTAR and XMM-Newton,” Publ. Astron. Soc. Jap. 71 no. 5, (2019) , arXiv:1906.04406 [astro-ph.HE].
  * (98) L. Ducci, S. Mereghetti, D. Götz, and A. Santangelo, “Ten years of INTEGRAL observations of the hard X-ray emission from SGR 1900+14,” Astron. Astrophys. 583 (2015) A113, arXiv:1510.00940 [astro-ph.HE].
  * (99) S.-S. Weng and E. Göğüş, “Broadband X-ray Spectral Investigations of Magnetars, 4u 0142+61, 1e 1841–045, 1e 2259+586, and 1e 1048.1–5937,” Astrophys. J. 815 no. 1, (2015) 15, arXiv:1511.06614 [astro-ph.HE].
  * (100) M. G. Baring, Z. Wadiasingh, P. L. Gonthier, A. K. Harding, and K. Hu, “The Mysterious Magnetospheres of Magnetars,”  12, 2020.  arXiv:2012.10815 [astro-ph.HE].
  * (101) J. S. Heyl and L. Hernquist, “Birefringence and dichroism of the QED vacuum,” J. Phys. A 30 (1997) 6485–6492, arXiv:hep-ph/9705367.
  * (102) J. Heyl and I. Caiazzo, “Strongly Magnetized Sources: QED and X-ray Polarization,” Galaxies 6 no. 3, (2018) 76, arXiv:1802.00358 [astro-ph.HE].
  * (103) H. S. Krawczynski et al., “Astro2020 Science White Paper: Using X-Ray Polarimetry to Probe the Physics of Black Holes and Neutron Stars,” arXiv:1904.09313 [astro-ph.HE].
  * (104) A. Y. Potekhin, D. G. Yakovlev, G. Chabrier, and O. Y. Gnedin, “Thermal structure and cooling of superfluid magnetic neutron stars,” Astrophys. J. 594 (2003) 404–418, arXiv:astro-ph/0305256 [astro-ph].
  * (105) A. M. Beloborodov and X. Li, “Magnetar heating,” Astrophys. J. 833 no. 2, (2016) 261, arXiv:1605.09077 [astro-ph.HE].
  * (106) A. Y. Potekhin, G. Chabrier, and D. Yakovlev, “Heat blanketing envelopes and thermal radiation of strongly magnetized neutron stars,” Astrophys. Space Sci. 308 (2007) 353–361, arXiv:astro-ph/0611014.
  * (107) A. Potekhin, G. Chabrier, and D. Yakovlev, “Internal temperatures and cooling of neutron stars with accreted envelopes,” Astron. Astrophys. 323 (1997) 415, arXiv:astro-ph/9706148.
  * (108) E. H. Gudmundsson, C. J. Pethick, and R. I. Epstein, “Structure of neutron star envelopes,” ApJ 272 (Sept., 1983) 286–300.
  * (109) A. Potekhin, J. Pons, and D. Page, “Neutron stars - cooling and transport,” Space Sci. Rev. 191 no. 1-4, (2015) 239–291, arXiv:1507.06186 [astro-ph.HE].
  * (110) D. G. Yakovlev and C. Pethick, “Neutron star cooling,” Ann. Rev. Astron. Astrophys. 42 (2004) 169–210, arXiv:astro-ph/0402143.
  * (111) D. Viganò, N. Rea, J. A. Pons, R. Perna, D. N. Aguilera, and J. A. Miralles, “Unifying the observational diversity of isolated neutron stars via magneto-thermal evolution models,” Mon. Not. Roy. Astron. Soc. 434 (2013) 123, arXiv:1306.2156 [astro-ph.SR].
  * (112) A. D. Kaminker, A. Y. Potekhin, D. G. Yakovlev, and G. Chabrier, “Heating and cooling of magnetars with accreted envelopes,” MNRAS 395 no. 4, (June, 2009) 2257–2267, arXiv:0902.4213 [astro-ph.SR].
  * (113) W. C. G. Ho, K. Glampedakis, and N. Andersson, “Magnetars: super(ficially) hot and super(fluid) cool,” MNRAS 422 no. 3, (May, 2012) 2632–2641, arXiv:1112.1415 [astro-ph.HE].
  * (114) S. Shapiro and S. Teukolsky, Black holes, white dwarfs, and neutron stars: The physics of compact objects.  1983\.
  * (115) M. G. Alford and S. P. Harris, “Beta equilibrium in neutron star mergers,” Phys. Rev. C 98 no. 6, (2018) 065806, arXiv:1803.00662 [nucl-th].
  * (116) E. Rrapaj and S. Reddy, “Nucleon-nucleon bremsstrahlung of dark gauge bosons and revised supernova constraints,” Phys. Rev. C 94 no. 4, (2016) 045805, arXiv:1511.09136 [nucl-th].
  * (117) L. Landau and I. Pomeranchuk, “Electron cascade process at very high-energies,” Dokl. Akad. Nauk Ser. Fiz. 92 (1953) 735–738.
  * (118) A. B. Migdal, “Bremsstrahlung and pair production in condensed media at high energies,” Phys. Rev. 103 (Sep, 1956) 1811–1820. https://link.aps.org/doi/10.1103/PhysRev.103.1811.
  * (119) E. van Dalen, A. Dieperink, and J. Tjon, “Neutrino emission in neutron stars,” Phys. Rev. C 67 (2003) 065807, arXiv:nucl-th/0303037.
  * (120) S. K. Dhiman, R. Kumar, and B. K. Agrawal, “Non-rotating and rotating neutron stars in the extended field theoretical model,” Phys. Rev. C 76 (2007) 045801, arXiv:0709.4081 [nucl-th].
  * (121) B. Liu, V. Greco, V. Baran, M. Colonna, and M. Di Toro, “Asymmetric nuclear matter: The Role of the isovector scalar channel,” Phys. Rev. C 65 (2002) 045201, arXiv:nucl-th/0112034.
  * (122) J. K. Bunta and S. Gmuca, “Asymmetric nuclear matter in the relativistic mean field approach with vector cross interaction,” Phys. Rev. C 68 (2003) 054318, arXiv:nucl-th/0309046.
  * (123) R. Utama, W.-C. Chen, and J. Piekarewicz, “Nuclear charge radii: Density functional theory meets Bayesian neural networks,” J. Phys. G 43 no. 11, (2016) 114002, arXiv:1608.03020 [nucl-th].
  * (124) R. Nandi, P. Char, and S. Pal, “Constraining the relativistic mean-field model equations of state with gravitational wave observations,” Phys. Rev. C 99 no. 5, (2019) 052802, arXiv:1809.07108 [astro-ph.HE].


# A new class of differential quasivariational inequalities with an
application to a quasistatic viscoelastic frictional contact problem††thanks:
This work was supported by the National Natural Science Foundation of China
(11671282, 11771067, 12171339, 12171070).

Xu Chua, Tao Chena, Nan-jing Huanga111Corresponding author. E-mail address:
<EMAIL_ADDRESS><EMAIL_ADDRESS>and Yi-bin Xiaob
a. Department of Mathematics, Sichuan University, Chengdu, Sichuan, P.R. China
b. School of Mathematical Sciences, University of Electronic Science and
Technology of China, Chengdu, P.R. China

Abstract: The overarching goal of this paper is to introduce and investigate a
new nonlinear system driven by a nonlinear differential equation, a history-
dependent quasivariational inequality, and a parabolic variational inequality
in Banach spaces. Such a system can be used to model quasistatic frictional
contact problems for viscoelastic materials with long memory, damage and wear.
By using the Banach fixed point theorem, we prove an existence and uniqueness
theorem of solution for such a system under some mild conditions. As a novel
application, we obtain a unique solvability of a quasistatic viscoelastic
frictional contact problem with long memory, damage and wear.

Keywords: Differential quasivariational inequality; frictional contact
problem; viscoelastic materials; long memory; damage and wear.

2020 AMS Subject Classification: 49J40; 34G20; 74M10; 74M15

##  1 Introduction

Assume that $V,X,Y$ and $W$ are separable and reflexive Banach spaces, $V^{*}$
and $Y^{*}$ are the dual spaces of $V$ and $Y$, respectively. Let $Y\subset
Y_{1}\subset Y^{*}$, where $Y_{1}$ is a separable Hilbert space. Suppose that
$K_{V}$ and $K_{Y}$ are closed, convex and nonempty subsets of $V$ and $Y$,
respectively. Let $I:=[0,T]$, where $T>0$ is a constant. In this paper, we are
interested in studying a new class of differential quasivariational
inequalities (DQVIs) with the following form: find $u:I\rightarrow K_{V}$,
$\zeta:I\rightarrow K_{Y}$ and $w:I\rightarrow W$ such that, for all $t\in I$,

$\left\\{\begin{array}[]{l}\dot{w}(t)=F(t,w(t),\dot{u}(t)),\\\ \left\langle
A(t,u(t))+\int_{0}^{t}B(t-s,u(s),\zeta(s))ds+C(t,\dot{u}(t)),v-\dot{u}(t)\right\rangle_{V^{*}\times
V}\\\
\quad\mbox{}+j(w(t),\dot{u}(t),v)-j(w(t),\dot{u}(t),\dot{u}(t))\geq\langle
f(t),v-\dot{u}(t)\rangle_{V^{*}\times V},\quad\forall v\in K_{V},\\\
\langle\dot{\zeta}(t),\eta-\zeta(t)\rangle_{Y_{1}}+a(\zeta(t),\eta-\zeta(t))\geq\langle\phi(t,u(t),\zeta(t)),\eta-\zeta(t)\rangle_{Y_{1}},\quad\forall\eta\in
K_{Y},\\\ u(0)=u_{0},\;w(0)=w_{0},\;\zeta(0)=\zeta_{0}.\\\ \end{array}\right.$
(1.1)

Clearly, if $C,w$ and $\zeta$ are omitted, $K_{V}=V$, $A(t,u(t))=A(u(t))$ and
$j(w(t),\dot{u}(t),v)=j({u}(t),v)$, then (1.1) is reduced to the following
problem: find $u:I\rightarrow V$ such that, for all $t\in I$,

$\left\\{\begin{array}[]{l}\left\langle
A(u(t))+\int_{0}^{t}B(t-s,u(s))ds,v-\dot{u}(t)\right\rangle_{V^{*}\times V}\\\
\quad+j(u(t),v)-j(u,\dot{u}(t))\geq\langle
f(t),v-\dot{u}(t)\rangle_{V^{*}\times V},\quad\forall v\in V,\\\
u(0)=u_{0},\\\ \end{array}\right.$

which is the time-dependent quasivariational inequality with the history-
dependent operator considered by Kasri and Touzaline [15].

The research on differential variational inequalities (DVIs) has a long
history (see, for example, the excellent survey due to Brogliato and Tanwani
[3] and the references therein). It is well known that Pang and Stewart [28]
are the first systematically to consider DVIs in finite-dimensional Euclidean
spaces. Since then, various theoretical results, approximating algorithms and
real applications have been investigated extensively for classical DVIs and
DQVIs under different conditions in the literature; for instance we refer the
reader to [21, 24, 22, 35, 36, 37, 8, 12, 20, 18, 19, 34, 38, 39] and the
references therein.

It is worth mentioning that contact mechanics has important applications in
daily life and industry, such as coupling devices, bearings, ultrasonic
welding and many others. After nearly 40 years of research, contact mechanics
has formed a relatively complete set of mathematical theory. It is well known
that the variational inequalities and quasivariational inequalities as well as
hemivariational inequalities are the most important mathematical tools to
obtain the existence and uniqueness of the solutions for various problems
arising in contact mechanics [4, 11, 14, 16, 26, 31, 17, 2, 10, 9]. In order
to describe the long memory property of materials such as rock and rubber,
Sofonea and Matei [30] introduced and studied a new class of history-dependent
quasivariational inequalities. Moreover, by relaxing the contact condition in
[30], Sofonea and Xiao [33] considered a new class of quasivariational
inequalities involving two history-dependent operators. On the other hand,
applying Archard’s wear laws and Coulomb’s friction laws, Andrews et al. [1]
obtained the existence and uniqueness of the solution of a dynamic
thermoviscoelastic contact problem. Furthermore, Sofonea et al. [32] proposed
a new mathematical model to capture frictional contact problem with wear.
Recently, Chen et al. [6] introduced a hyperbolic quasi-variational inequality
to characterize a dynamic viscoelastic contact problem with friction and wear.
Very recently, in order to model an elastic frictional contact problem with
long memory, damage and wear, Chen et al. [5] considered a new class of
differential nonlinear system driven by a differential equation, a history-
dependent hemivariational inequality and a parabolic variational inequality in
Banach spaces. Nevertheless, in the study of quasistatic viscoelastic
frictional contact problems, it is necessary to consider the properties of
viscoelastic materials with long memory, damage and wear (see Section 4 for
more details). To the best of authors’ knowledge, there are few works
considering quasistatic frictional contact problems for viscoelastic materials
with long memory, damage and wear. Thus, it would be important and interesting
to investigate DQVIs (1.1) which can be used to model quasistatic frictional
contact problems for viscoelastic materials with long memory, damage and wear.

The overarching goal of this paper is to introduce and investigate a new
nonlinear system driven by a nonlinear differential equation, a history-
dependent quasivariational inequality, and a parabolic variational inequality
in Banach spaces, which can be used to describe the quasistatic frictional
contact problems for viscoelastic materials with long memory, damage and wear.
The main contributions of this paper are twofold. One is to deliver some
sufficient conditions for ensuring the existence and uniqueness of solution to
DQVIs (1.1). The other is to show the unique solvability result for a new
quasistatic viscoelastic frictional contact problem with long memory, damage
and wear by applying the obtained results for DQVIs (1.1).

The rest of the paper is structured as follows. The next section recalls some
known definitions and lemmas. After that in Section 3, we show the existence
and uniqueness of solution for DQVIs (1.1) under some mild conditions by
employing the Banach fixed point theorem. In Section 4, we provide a novel
application of our abstract results to a quasistatic viscoelastic frictional
contact problem with long memory, damage and wear.

##  2 Preliminaries

Let $(X,\|\cdot\|_{X})$ be a real Banach space with its dual $X^{*}$ and
$\langle\cdot,\cdot\rangle_{X^{*}\times X}$ denote the duality pairing between
$X^{*}$ and $X$. In this section, we recall some known definitions and lemmas
which will be used to obtain our main results (see [25, 27] for more details).

A functional $j(u):X\rightarrow\mathbb{R}$ is called lower semicontinuous if
and only if for any convergence sequence $\\{u_{n}\\}_{n=1}^{\infty}\subset X$
satisfying $u_{n}\rightarrow u\in X$, one has
$\lim\inf_{n\rightarrow\infty}{j(u_{n})}\geq j(u)$. Let
$j:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be a lower semicontinuous convex
functional with domain $D(j)=\\{u\in X\>|\>j(u)<+\infty\\}$ and $u\in D(j)$.
Then there exists $u^{*}\in X^{*}$ such that $j(v)-j(u)\geq\langle
u^{*},v-u\rangle_{X^{*}\times X}$ holds for all $v\in X$. The set of all such
$u^{*}\in X^{*}$ is called the convex subdifferential of $j$ at $u$ and we
denote it by $\partial j(u)$.

For a set-valued operator $A:X\rightarrow 2^{X^{*}}$, the graph of $A$ is
denoted by $G(A)$, i.e.,

$G(A):=\\{(u,u^{*})\in X\times X^{*}\>|\>u^{*}\in A(u)\\}.$

A set-valued operator $A:X\rightarrow 2^{X^{*}}$ is called monotone if

$\langle u^{*}-v^{*},u-v\rangle_{X^{*}\times X}\geq
0,\;\forall(u,u^{*}),\;(v,v^{*})\in G(A).$

Moreover, a monotone operator $A$ is called maximal monotone if for any
$(u,u^{*})\in X\times X^{*}$ satisfying

$\langle u^{*}-v^{*},u-v\rangle_{X^{*}\times X}\geq 0,\quad\forall(v,v^{*})\in
G(A),$

one has $(u,u^{*})\in G(A)$.

A functional $j:X\rightarrow\mathbb{R}\cup\\{\infty\\}$ is called proper if
$j(v)>-\infty$ for all $v\in X$ and there exists a point $u\in X$ such that
$j(u)<+\infty$. For a proper, convex and lower semicontinuous functional
$j:X\rightarrow\mathbb{R}\cup\\{\infty\\}$, it is well known that $\partial
j:X\to 2^{X^{*}}$ is maximal monotone.

At the end of this section, we recall two known results which will be used to
obtain our main results.

######  Lemma 2.1.

[31, Proposition 3.1] Let $\Lambda:C([0,T];X)\rightarrow C([0,T];X)$ be an
operator satisfying the following property: there exists a constant $h>0$ such
that

$\begin{array}[]{l}\left\|\Lambda u_{1}(t)-\Lambda u_{2}(t)\right\|_{X}\leq
h\int_{0}^{t}\par\left\|u_{1}(s)-u_{2}(s)\right\|_{X}ds,\quad\forall
u_{1},u_{2}\in C([0,T];X),\;t\in[0,T].\end{array}$

Then there exists a unique element $u^{*}\in C([0,T];X)$ such that $\Lambda
u^{*}=u^{*}$.

######  Lemma 2.2.

[23, Lemma 3.3] Let $V$ be a reflexive Banach space and $K$ be a closed convex
nonempty subset of $V$. Suppose that $A:K\to V^{*}$ and $\varphi:K\times
K\rightarrow\mathbb{R}$ satisfy the following hypotheses:

  * H(A):

$A:K\rightarrow V^{*}$ is strongly monotone Lipschitz continuous, i.e.,

    * (a)

$\langle Au_{1}-Au_{2},u_{1}-u_{2}\rangle_{V^{*}\times V}\geq
m\|u_{1}-u_{2}\|^{2}_{V}$ for all $u_{1},u_{2}\in K$ with $m>0$;

    * (b)

$\|Au_{1}-Au_{2}\|_{V^{*}}\leq L\|u_{1}-u_{2}\|_{V}$ for all $u_{1},u_{2}\in
K$ with $L>0$.

  * H($\varphi$):

$\varphi:K\times K\rightarrow\mathbb{R}$ is such that

    * (a)

for any $u\in K,\varphi(u,\cdot)$ is convex and lower semicontinuous on $K$;

    * (b)

for any $u_{1},u_{2},v_{1},v_{2}\in K$, there exists $\beta>0$ such that

$\varphi(u_{1},v_{2})-\varphi(u_{1},v_{1})+\varphi(u_{2},v_{1})-\varphi(u_{2},v_{2})\leq\beta\|u_{1}-u_{2}\|_{V}\|v_{1}-v_{2}\|_{V}.$

If $m>\beta$, then for each $f\in V^{*}$, there exists a unique element $u\in
K$ such that

$\langle Au,v-u\rangle_{V^{*}\times V}+\varphi(u,v)-\varphi(u,u)\geq\langle
f,v-u\rangle_{V^{*}\times V},\quad\forall v\in K.$

##  3 Unique solvability for DQVIs (1.1)

In this section, we provide some sufficient conditions for ensuring the
existence and uniqueness of solution for DQVIs (1.1). To this end, we consider
the Gelfand triplet of Banach spaces $\left(V,H,V^{*}\right)$ which have
compact and dense embeddings. We also need the following assumptions.

  * H(A):

The operator $A:I\times V\rightarrow V^{*}$ satisfies

    * (a)

$A(\cdot,v)$ is continuous on $I$ for any given $v\in V$.

    * (b)

$A(t,\cdot)$ is Lipschitz continuous with $L_{A}>0$ on $V$ for any given $t\in
I$, i.e.,

$\left\|A(t,u_{1})-A(t,u_{2})\right\|_{V^{*}}\leq
L_{A}\left\|u_{1}-u_{2}\right\|_{V},\quad\forall(t,u_{1},u_{2})\in I\times
V\times V.$

  * H(B):

The operator $B:I\times V\times Y\rightarrow V^{*}$ satisfies

    * (a)

$B(\cdot,v,\zeta)$ is continuous on $I$ for any given $v\in V$ and $\zeta\in
Y$.

    * (b)

$B(t,\cdot,\cdot)$ is Lipschitz continuous with $L_{B}>0$ on $V\times Y$ for
any given $t\in I$, i.e.,

$\left\|B(t,u_{1},\zeta_{1})-B(t,u_{2},\zeta_{2})\right\|_{V^{*}}\leq
L_{B}(\|u_{1}-u_{2}\|_{V}+\|\zeta_{1}-\zeta_{2}\|_{Y}),\quad\forall t\in
I,\;\forall u_{1},u_{2}\in V,\;\forall\zeta_{1},\zeta_{2}\in Y.$

    * (c)

There exists $\rho\in L^{2}(I;\mathbb{R}^{+})$ such that

$\|B(t,u,\zeta)\|_{V^{*}}\leq\rho(t)(\|\zeta\|_{Y}+\|u\|_{V}),\quad\forall(t,u,\zeta)\in
I\times V\times Y.$

  * H(C):

The operator $C:I\times V\rightarrow V^{*}$ satisfies

    * (a)

$C(t,\cdot)$ is Lipschitz continuous with $L_{C1}>0$ on $V$ for any given
$t\in I$, i.e.,

$\left\|C(t,u_{1})-C(t,u_{2})\right\|_{V^{*}}\leq
L_{C1}\left\|u_{1}-u_{2}\right\|_{V},\quad\forall(t,u_{1},u_{2})\in I\times
V\times V.$

    * (b)

$C(\cdot,u)$ is Lipschitz continuous with $L_{C2}>0$ on $I$ for any given
$u\in V$, i.e.,

$\left\|C(t_{1},u)-C(t_{2},u)\right\|_{V^{*}}\leq
L_{C2}\left\|t_{1}-t_{2}\right\|_{V},\quad\forall(t_{1},t_{2},u)\in I\times
I\times V.$

    * (c)

$C(t,\cdot)$ is strong monotone with $L_{C}>0$ on $V$ for any given $t\in I$,
i.e.,

$\left\langle C(t,u_{1})-C(t,u_{2}),u_{1}-u_{2}\right\rangle_{V^{*}\times
V}\geq m_{C}\|u_{1}-u_{2}\|_{V}^{2},\quad\forall(t,u_{1},u_{2})\in I\times
V\times V.$

  * H(j):

The functional $j:W\times V\times V\rightarrow\mathbb{R}$ satisfies

    * (a)

$j(w,u,\cdot)$ is convex proper and lower semicontinuous on $V$ for any given
$(w,u)\in W\times V$.

    * (b)

There exist $\alpha_{0}>0$ and $\alpha_{1}>0$ such that

$\displaystyle\;j\left(w_{1},u_{1},v_{2}\right)-j\left(w_{1},u_{1},v_{1}\right)+j\left(w_{2},u_{2},v_{1}\right)-j\left(w_{2},u_{2},v_{2}\right)$
$\displaystyle\leq$
$\displaystyle\;\alpha_{0}\left\|w_{1}-w_{2}\right\|_{W}\left\|v_{1}-v_{2}\right\|_{V}+\alpha_{1}\left\|u_{1}-u_{2}\right\|_{V}\left\|v_{1}-v_{2}\right\|_{V},\quad\forall
w_{1},w_{2}\in W,\;\forall u_{1},u_{2},v_{1},v_{2}\in V.$

  * H(F):

The operator $F:I\times W\times V\rightarrow W$ satisfies

    * (a)

$F(\cdot,w,v)$ is continuous on $I$ for any given $(w,v)\in W\times V$.

    * (b)

$F(t,\cdot,\cdot)$ is Lipschitz continuous with $L_{F}>0$ on $V\times Y$ for
any given $t\in I$, i.e.,

$\left\|F(t,w_{1},u_{1})-F(t,w_{2},u_{2})\right\|_{W}\leq
L_{F}(\|u_{1}-u_{2}\|_{V}+\|w_{1}-w_{2}\|_{W}),\quad\forall t\in I,\;\forall
w_{1},w_{2}\in W,\;\forall u_{1},u_{2}\in V.$

  * H($\phi$):

The operator $\phi:I\times V\times Y\rightarrow Y_{1}$ satisfies

    * (a)

$\phi(t,\cdot,\cdot)$ is Lipschitz continuous with $L_{\phi}>0$ on $V\times Y$
for any given $t\in I$, i.e.,

$\left\|\phi(t,u,\zeta)-\phi(t,v,\eta)\right\|_{Y_{1}}\leq
L_{\phi}(\|u-v\|_{V}+\|\zeta-\eta\|_{Y_{1}}),\quad\forall t\in I,\;\forall
u,v\in V,\;\forall\zeta,\eta\in Y.$

    * (b)

$\phi(\cdot,0_{V},0_{Y})\in L^{2}(I;Y_{1}).$

  * H(a):

The functional $a:Y\times Y\rightarrow\mathbb{R}$ satisfies

    * (a)

$a(\cdot,\cdot)$ is a continuous bilinear symmetric coercive functional and
there exist $a_{1}\in\mathbb{R}$ and $a_{2}>0$ such that

$a(\eta,\eta)+a_{1}\|\eta\|_{Y_{1}}^{2}\geq
a_{2}\|\eta\|_{Y}^{2},\quad\forall\eta\in Y.$

The main result of this section can be stated as follows.

######  Theorem 3.1.

Suppose that assumptions H(A), H(B), H(C), H(j), H(F), H($\phi$), H(a) hold
and $m_{C}>\alpha_{1}$. Then DQVIs (1.1) has a unique solution
$(\zeta,u,w)\in(H^{1}(I;Y_{1})\cap L^{2}(I;Y))\times C^{1}(I;K_{V})\times
C^{1}(I;W)$.

We divide the proof of Theorem 3.1 into a series of lemmas. To this end, we
first consider the following auxiliary problem.

######  Problem 3.1.

For any given $w\in C(I;W)$, find $\dot{u}_{w}:I\rightarrow K_{V}$ such that,
for all $t\in I$ and

$\displaystyle\quad\langle
C(t,\dot{u}_{w}(t)),v-\dot{u}_{w}(t)\rangle_{V^{*}\times
V}+j(w(t),\dot{u}_{w}(t),v)-j(w(t),\dot{u}_{w}(t),u\dot{}_{w}(t))$
$\displaystyle\geq\langle f(t),v-\dot{u}_{w}(t)\rangle_{V^{*}\times
V},\quad\forall v\in K_{V}.$ (3.1)

######  Lemma 3.1.

Assume that H(C) and H(j) hold and $m_{C}>\alpha_{1}$. Then for any given
$w\in C(I;W)$ and $f\in C(I;V^{*})$, Problem 3.1 has a unique solution
$\dot{u}_{w}\in C(I,K_{V})$.

###### Proof.

For any fixed $t$, define an operator $A:V\rightarrow V^{*}$ and a function
$\varphi:V\times V\rightarrow\mathbb{R}$ by setting

$Au=C(t,\dot{u}(t)),\quad\varphi(u,v)=j(w(t),\dot{u}(t),v),\quad\forall u,v\in
K,t\in I.$

Clearly, $A$ and $\varphi$ satisfy all the assumptions of Lemma 2.2. Thus, it
follows from Lemma 2.2 that Problem 3.1 has a unique solution
$\dot{u}_{w}(t)\in K_{V}$. Next we prove that $\dot{u}_{w}\in C(I,K_{V})$. For
$t_{1},t_{2}\in I$, denote
$\dot{u}_{w}(t_{i})=\dot{u}_{i},\;f(t_{i})=f_{i},\;w(t_{i})=w_{i}$ with
$i=1,2$. Thus, inequality (3.1) yields

$\displaystyle\langle
C(t_{1},\dot{u}_{1}),\dot{u}_{2}-\dot{u}_{1}\rangle_{V^{*}\times
V}+j(w_{1},\dot{u}_{1}(t),\dot{u}_{2})-j(w_{1},\dot{u}_{1},\dot{u}_{1})\geq\langle
f_{1},\dot{u}_{2}-\dot{u}_{1}\rangle$ (3.2)

and

$\displaystyle\langle
C(t_{2},\dot{u}_{2}),\dot{u}_{1}-\dot{u}_{2}\rangle_{V^{*}\times
V}+j(w_{2},\dot{u}_{2}(t),\dot{u}_{1})-j(w_{2},\dot{u}_{2},\dot{u}_{2})\geq\langle
f_{2},\dot{u}_{1}-\dot{u}_{2}\rangle.$ (3.3)

Adding (3.2) to (3.2), one has

$\displaystyle\langle
C(t_{1},\dot{u}_{1})-C(t_{2},\dot{u}_{2}),\dot{u}_{1}-\dot{u}_{2}\rangle_{V^{*}\times
V}$ $\displaystyle\leq$ $\displaystyle
j(w_{1},\dot{u}_{1}(t),\dot{u}_{2})-j(w_{1},\dot{u}_{1},\dot{u}_{1})+j(w_{2},\dot{u}_{2}(t),\dot{u}_{1})$
$\displaystyle\mbox{}-j(w_{2},\dot{u}_{2},\dot{u}_{2})+\langle
f_{1}-f_{2},u_{1}-u_{2}\rangle$

and so

$\displaystyle\langle
C(t_{1},\dot{u}_{1})-C(t_{1},\dot{u}_{2}),\dot{u}_{1}-\dot{u}_{2}\rangle_{V^{*}\times
V}$ (3.4) $\displaystyle\leq$ $\displaystyle
j(w_{1},\dot{u}_{1}(t),\dot{u}_{2})-j(w_{1},\dot{u}_{1},\dot{u}_{1})+j(w_{2},\dot{u}_{2}(t),\dot{u}_{1})$
$\displaystyle\mbox{}-j(w_{2},\dot{u}_{2},\dot{u}_{2})+\langle
f_{1}-f_{2},\dot{u}_{1}-\dot{u}_{2}\rangle+\langle
C(t_{2},\dot{u}_{2})-C(t_{1},\dot{u}_{2}),\dot{u}_{1}-\dot{u}_{2}\rangle_{V^{*}\times
V}.$

By conditions H(j)(b) and H(C)(b)(c), inequality (3.4) becomes

$\displaystyle(m_{C}-\alpha_{1})\|\dot{u}_{1}-\dot{u}_{2}\|\leq\alpha_{0}\|w_{1}-w_{2}\|_{W}+\|f_{1}-f_{2}\|_{V^{*}}+L_{C2}|t_{1}-t_{2}|.$
(3.5)

Since $w\in C(I;W)$ and $f\in C(I;V^{*})$, it follows from (3.5) that

$\lim_{t_{1}\rightarrow
t_{2}}\|\dot{u}_{w}(t_{1})-\dot{u}_{w}(t_{2})\|_{V}=0,$

i.e., $\dot{u}_{w}\in C(I,K_{V})$. This finishes the proof. ∎

######  Lemma 3.2.

Suppose that H(A), H(B), H(C), H(j) hold and $m_{C}>\alpha_{1}$. Then for any
given $\zeta\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)$, $w\in C(I;W)$, and $f\in
C(I;V^{*})$, the following problem

$\left\\{\begin{array}[]{l}\text{ find }u_{w\zeta}:(0,T)\rightarrow
K_{V}\text{ such that }\\\ \left\langle
A(t,u_{w\zeta}(t))+\int_{0}^{t}B(t-s,u_{w\zeta}(s),\zeta(s))ds+C(t,\dot{u}_{w\zeta}(t)),v-\dot{u}_{w\zeta}(t)\right\rangle_{V^{*}\times
V}\\\
\quad\mbox{}+j(w,\dot{u}_{w\zeta}(t),v)-j(w,\dot{u}_{w\zeta}(t),\dot{u}_{w\zeta}(t))\geq\langle
f(t),v-\dot{u}_{w\zeta}(t)\rangle_{V^{*}\times V},\quad\forall t\in
I,\;\forall v\in K_{V},\\\ \text{ where
}u_{w\zeta}(0)=u_{0}\end{array}\right.$ (3.6)

has a unique solution $u_{w\zeta}\in C^{1}(I;K_{V})$.

###### Proof.

Let us fix $\eta\in C(I;K_{V})$. We consider the following auxiliary for
(3.6).

$\left\\{\begin{array}[]{l}\text{ find }u_{w\zeta\eta}:(0,T)\rightarrow
K_{V}\text{ such that }\\\ \langle
C(t,\dot{u}_{w\zeta\eta}(t)),v-\dot{u}_{w\zeta\eta}(t)\rangle_{V^{*}\times
V}+j(w(t),\dot{u}_{w\zeta\eta}(t),v)-j(w(t),\dot{u}_{w\zeta\eta}(t),\dot{u}_{w\zeta\eta}(t))\\\
\quad\mbox{}\geq\langle
f_{\eta}(t),v-\dot{u}_{w\zeta\eta}(t)\rangle_{V^{*}\times V},\quad\forall t\in
I,\;\forall v\in K_{V},\\\ \text{ where
}u_{w\zeta\eta}(0)=u_{0},\end{array}\right.$ (3.7)

where $f_{\eta}$ is defined by

$f_{\eta}(t):=f(t)-\int_{0}^{t}B\left(t-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)-A\left(t,\int_{0}^{t}\eta(s)ds\right).$

We first claim that $f_{\eta}\in C(I;V^{*})$. For any given $t_{1},\;t_{2}\in
I$ with $t_{1}<t_{2}$, it follows from the definition of $f_{\eta}$ that

$\displaystyle\|f_{\eta}(t_{2})-f_{\eta}(t_{1})\|_{V}^{*}$ (3.8)
$\displaystyle\leq$
$\displaystyle\|f(t_{1})-f(t_{2})\|+\left\|\int_{0}^{t_{2}}B\left(t_{2}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)ds-\int_{0}^{t_{1}}B\left(t_{1}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)ds\right\|$
$\displaystyle\mbox{}+\left\|A\left(t_{2},\int_{0}^{t_{2}}\eta(s)ds\right)-A\left(t_{2},\int_{0}^{t_{1}}\eta(s)ds\right)+A\left(t_{2},\int_{0}^{t_{1}}\eta(s)ds\right)-A\left(t_{1},\int_{0}^{t_{1}}\eta(s)ds\right)\right\|$
$\displaystyle\leq$
$\displaystyle\|f(t_{1})-f(t_{2})\|+\int_{t_{1}}^{t_{2}}\left\|B\left(t_{2}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)\right\|ds$
$\displaystyle\mbox{}+\int_{0}^{t_{1}}\left\|B\left(t_{2}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)ds-B\left(t_{1}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)\right\|ds$
$\displaystyle\mbox{}+\left\|A\left(t_{2},\int_{0}^{t_{1}}\eta(s)ds\right)-A\left(t_{1},\int_{0}^{t_{1}}\eta(s)ds\right)\right\|+L_{A}\left\|\int_{t_{1}}^{t_{2}}\eta(s)ds\right\|.$

By condition H(B)(c) and Hölder’s inequality, we have

$\displaystyle\int_{t_{1}}^{t_{2}}\left\|B\left(t_{2}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)\right\|ds$
(3.9) $\displaystyle\leq$
$\displaystyle\int_{t_{1}}^{t_{2}}|\rho(t_{2}-s)|\left\|\int_{0}^{s}\eta(s)ds\right\|ds+\int_{t_{1}}^{t_{2}}|\rho(t_{2}-s)|\left\|\zeta(s)\right\|ds$
$\displaystyle\leq$ $\displaystyle
T\|\eta\|_{C^{(}I;V)}\left(\sqrt{t_{2}-t_{1}}\right)\left(\int_{t_{1}}^{t_{2}}|\rho(t_{2}-s)|^{2}ds\right)^{\frac{1}{2}}+\left(\int_{t_{1}}^{t_{2}}|\rho(t_{2}-s)|^{2}ds\right)^{\frac{1}{2}}\left(\int_{t_{1}}^{t_{2}}\|\zeta(s)\|^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
T^{\frac{3}{2}}\|\eta\|_{C^{(}I;V)}\|\rho\|_{L^{2}(t_{1},t_{2};\mathbb{R}^{+})}+\|\rho\|_{L^{2}(t_{1},t_{2};\mathbb{R}^{+})}\|\zeta(s)\|_{L^{2}(t_{1},t_{2};Y)}$
$\displaystyle\mbox{}\rightarrow 0\;\mbox{as $|t_{1}-t_{2}|\rightarrow 0$}.$

According to condition H(B)(a), one has

$\displaystyle\int_{0}^{t_{1}}\left\|B\left(t_{2}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)ds-B\left(t_{1}-s,\int_{0}^{s}\eta(s)ds,\zeta(s)\right)\right\|ds\rightarrow
0\;\mbox{as $|t_{1}-t_{2}|\rightarrow 0$}.$ (3.10)

By the continuity of $f$ and $A(t,\cdot)$, we know that
$\lim_{|t_{1}-t_{2}|\rightarrow
0}\|f_{\eta}(t_{2})-f_{\eta}(t_{1})\|_{V}^{*}=0$ and so $f_{\eta}\in
C(I;V^{*})$.

Now it follows from Lemma 3.1 that problem (3.7) has a unique solution
$\dot{u}_{w\zeta\eta}\in C(I;K_{V})$. Thus, we can define an operator
$\Lambda:\eta\mapsto\dot{u}_{w\zeta\eta}$, where $\eta\in C(I;V)$ and
$\dot{u}_{w\zeta\eta}(t)$ is the unique solution of problem (3.7). Next we
show that $\Lambda$ has a unique fixed point in $C(I;V)$. In fact, let
$\dot{u}_{0}(t)$ and $\dot{u}_{1}(t)$ be two solutions of (3.7). Then

$\displaystyle\langle C(t,\dot{u}_{i}(t)),v-\dot{u}_{i}(t)\rangle_{V^{*}\times
V}+j(w,\dot{u}_{i}(t),v)-j(w,u\dot{}_{i}(t),\dot{u}_{i}(t))\geq\langle
f_{\eta_{i}}(t),v-\dot{u}_{i}(t)\rangle_{V^{*}\times V},\;i=0,1$ (3.11)

for all $v\in K_{V}$ and all $t\in I$. Testing (3.11) with $\dot{u}_{1-i}(t)$,
we obtain

$\displaystyle\langle
C(t,\dot{u}_{0}(t))-C(t,\dot{u}_{1}(t)),\dot{u}_{0}-\dot{u}_{1}(t)\rangle_{V^{*}\times
V}$ (3.12) $\displaystyle\leq$ $\displaystyle
j(w,\dot{u}_{0}(t),\dot{u}_{1}(t))+j(w,\dot{u}_{1}(t),\dot{u}_{0}(t))-j(w,\dot{u}_{1}(t),\dot{u}_{1}(t))$
$\displaystyle\mbox{}-j(w,\dot{u}_{0}(t),\dot{u}_{0}(t))+\langle
f_{\eta_{1}}-f_{\eta_{0}},\dot{u}_{0}(t)-\dot{u}_{1}(t)\rangle_{V^{*}\times
V}$

for all $t\in I$. According to conditions H(A)(b), H(B)(b) and H(j)(b),
inequality (3.12) yields

$\displaystyle m_{C}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|^{2}_{V}$ (3.13)
$\displaystyle\leq$
$\displaystyle\alpha_{1}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|^{2}_{V}+\|f_{\eta_{1}}(t)-f_{\eta_{0}}(t)\|_{V^{*}}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$
$\displaystyle\leq$
$\displaystyle\alpha_{1}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|^{2}_{V}++\left\|A\left(t,\int_{0}^{t}\eta_{0}(s)ds\right)-A\left(t,\int_{0}^{t}\eta_{1}(s)ds\right)\right\|_{V}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$
$\displaystyle\mbox{}+\left\|\int_{0}^{t_{2}}B\left(t-s,\int_{0}^{s}\eta_{0}(s)ds,\zeta(s)\right)ds-\int_{0}^{t_{1}}B\left(t-s,\int_{0}^{s}\eta_{1}(s)ds,\zeta(s)\right)ds\right\|_{V}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$
$\displaystyle\leq$
$\displaystyle\alpha_{1}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|^{2}_{V}+L_{A}\int_{0}^{t}\|\eta_{0}(s)-\eta_{1}(s)\|_{V}ds\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$
$\displaystyle\mbox{}+L_{B}\int_{0}^{t}\int_{0}^{t}\|\eta_{0}(s)-\eta_{1}(s)\|_{V}dlds\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$
$\displaystyle\leq$
$\displaystyle\alpha_{1}\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|^{2}_{V}+(L_{A}+TL_{B})\int_{0}^{t}\|\eta_{0}(s)-\eta_{1}(s)\|_{V}ds\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}$

for all $t\in I$. Hence

$\displaystyle\|\dot{u}_{0}(t)-\dot{u}_{1}(t)\|_{V}\leq\frac{L_{A}+TL_{B}}{(m_{C}-\alpha_{1})}\int_{0}^{t}\|\eta_{1}(s)-\eta_{2}(s)\|_{V}ds.$

From Lemma 2.1 we can see that $\Lambda$ has a unique fixed point
$\dot{u}_{w\zeta}\in C(I;K_{V})$. Since $u_{w\zeta}(0)=u_{0}$, we know that
(3.6) has a unique solution $u_{w\zeta}\in C^{1}(I;K_{V})$. This completes the
proof. ∎

######  Remark 3.1.

We need some useful inequalities for the solution $u_{w\zeta}$ of problem
(3.6). Let $w_{1},\;w_{2}\in C(I;W)$ and $\zeta_{1},\;\zeta_{2}\in
H^{1}(I;Y_{1})\cap L^{2}(I;Y)$. Replacing $w=w_{i}$ and $\zeta=\zeta_{i}$ with
$i=1,\;2$ in problem (3.6) and applying Hölder’s inequality, we have

$\displaystyle(m_{C}-\alpha_{1})\|\dot{u}_{w_{1}\zeta_{1}}(t)-\dot{u}_{w_{2}\zeta_{2}}(t)\|$
$\displaystyle\leq$ $\displaystyle
L_{B}\int_{0}^{t}\int_{0}^{t}\|\dot{u}_{w_{1}\zeta_{1}}(s)-\dot{u}_{w_{2}\zeta_{2}}(s)\|ds+L_{B}\int_{0}^{t}\|\zeta_{1}(s)-\zeta_{2}(s)\|ds$
(3.14)
$\displaystyle\mbox{}+\alpha_{0}\|w_{1}(t)-w_{2}(t)\|+L_{A}\int_{0}^{t}\|\dot{u}_{w_{1}\zeta_{1}}(s)-\dot{u}_{w_{2}\zeta_{2}}(s)\|ds$
$\displaystyle\leq$
$\displaystyle(L_{A}+L_{B}t)\int_{0}^{t}\|\dot{u}_{w_{1}\zeta_{1}}(s)-\dot{u}_{w_{2}\zeta_{2}}(s)\|ds+\mbox{}+\alpha_{0}\|w_{1}(t)-w_{2}(t)\|$
$\displaystyle\mbox{}+L_{B}\sqrt{t}\left(\int_{0}^{t}\|\zeta_{1}(s)-\zeta_{2}(s)\|^{2}ds\right)^{\frac{1}{2}}ds\
$

and so

$\displaystyle\|\dot{u}_{w_{1}\zeta_{1}}(t)-\dot{u}_{w_{2}\zeta_{2}}(t)\|$
$\displaystyle\leq$
$\displaystyle\frac{L_{A}+L_{B}t}{m_{C}-\alpha_{1}}\int_{0}^{t}\|\dot{u}_{w_{1}\zeta_{1}}(s)-\dot{u}_{w_{2}\zeta_{2}}(s)\|ds+\frac{L_{B}\sqrt{t}}{m_{C}-\alpha_{1}}\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I;Y_{1})}$
(3.15)
$\displaystyle\mbox{}+\frac{\alpha_{0}}{m_{C}-\alpha_{1}}\|w_{1}(t)-w_{2}(t)\|.$

Applying Gronwall’s inequality yields

$\displaystyle\|\dot{u}_{w_{1}\zeta_{1}}(t)-\dot{u}_{w_{2}\zeta_{2}}(t)\|$
(3.16) $\displaystyle\leq$
$\displaystyle\frac{(L_{A}+L_{B}t)\alpha_{0}}{(m_{C}-\alpha_{1})^{2}}e^{\frac{2L_{A}t+L_{B}t^{2}}{2(m_{C}-\alpha_{1})}}\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|ds+\frac{\alpha_{0}}{m_{C}-\alpha_{1}}\|w_{1}(t)-w_{2}(t)\|$
$\displaystyle\mbox{}+\left(\frac{L_{B}\sqrt{t}}{m_{C}-\alpha_{1}}+\frac{2L_{B}t^{\frac{3}{2}}(L_{A}+L_{B}t)}{3(m_{C}-\alpha_{1})^{2}}e^{\frac{2L_{A}t+L_{B}t^{2}}{2(m_{C}-\alpha_{1})}}\right)\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I;Y_{1})}.$

In order to prove Theorem 3.1, we also need to introduce the following
auxiliary problem.

######  Problem 3.2.

Find $u_{\zeta}:I\rightarrow K_{V}$ and $w_{\zeta}:I\rightarrow W$ such that,
for all $t\in I$,

$\displaystyle\dot{w}_{\zeta}(t)=F(t,w_{\zeta}(t),\dot{u}_{\zeta}(t)),$
$\displaystyle\left\langle
A(t,u_{\zeta}(t))+\int_{0}^{t}B(t-s,u_{\zeta}(s)),\zeta(s))ds+C(t,\dot{u}_{\zeta}(t)),v-\dot{u}_{\zeta}(t)\right\rangle_{V^{*}\times
V}$
$\displaystyle\quad\mbox{}+j(w_{\zeta}(t),\dot{u}_{\zeta}(t),v)-j(w_{\zeta}(t),\dot{u}_{\zeta}(t),\dot{u}_{\zeta}(t))\geq\langle
f(t),v-\dot{u}_{\zeta}(t)\rangle_{V^{*}\times V},\quad\forall v\in K_{V},$

where $u_{\zeta}(0)=u_{0}$, $w_{\zeta}(0)=w_{0}$ and $\zeta\in C(I;K_{Y})$.

######  Lemma 3.3.

Suppose that H(A), H(B), H(C), H(j) and H(F) hold and $m_{C}>\alpha_{1}$. Then
for any given $\zeta\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)$ and $f\in C(I;V^{*})$,
Problem 3.2 has a unique solution $(u_{\zeta},w_{\zeta})\in
C^{1}(I;K_{V})\times C^{1}(I;W)$.

###### Proof.

Define an operator $G:C^{1}(I;W)\rightarrow C(I;K_{V})$ by
$G(w)(t)=\dot{u}_{w\zeta}(t)$, where $u_{w\zeta}$ is the solution of problem
(3.6). By the proof of Lemma 3.2, we know that $G$ is well-defined. Then we
need to illustrate that there exists a unique $w_{\zeta}\in C^{1}(I;W)$ such
that

$\displaystyle\dot{w}_{\zeta}(t)=F(t,w_{\zeta}(t),G(w_{\zeta})(t)).$ (3.17)

Moreover, $(G(w_{\zeta}),w_{\zeta})\in C(I;K_{V})\times C^{1}(I;W)$ is the
unique solution of Problem 3.2.

For this purpose, we consider an operator $\Lambda:C(I;W)\rightarrow
C^{1}(I;K_{V})$ defined as follows:

$\Lambda
w_{\zeta}(t)=\int_{0}^{t}F(s,w_{\zeta}(s),G(w_{\zeta})(s))ds+w_{0},\;\forall
t\in[0,T].$

According to the definition of operator $G$ and conditions H(F)(a)(b) that
$\Lambda w_{\zeta}\in C^{1}(I;W)$ when $w_{\zeta}\in C(I;W)$. Because the
fixed point of $\Lambda$ is the solution of (3.17), we only need to prove that
$\Lambda$ has a unique fixed point in $C(I;W)$.

Let $w_{\zeta 1},\;w_{\zeta 2}\in C(I;W)$. We can draw a conclusion from $G$,
$\Lambda$ and condition H(F)(b) that

$\displaystyle\quad\|\Lambda w_{\zeta 1}(t)-\Lambda w_{\zeta 2}(t)\|$
$\displaystyle=\left\|\int_{0}^{t}F(s,w_{\zeta 1}(s),G(w_{\zeta
1})(s))ds-\int_{0}^{t}F(s,w_{\zeta 2}(s),G(w_{\zeta 2})(s))ds\right\|$
$\displaystyle\leq\int_{0}^{t}\left\|F(s,w_{\zeta 1}(s),G(w_{\zeta
1})(s))ds-F(s,w_{\zeta 2}(s),G(w_{\zeta 2})(s))\right\|ds.$
$\displaystyle\leq\int_{0}^{t}L_{F}\left(\|w_{\zeta 1}(s)-w_{\zeta
2}(s)\|+\|G(w_{\zeta 1})(s)-G(w_{\zeta 2})(s)\|\right)ds.$ (3.18)

Denote

$c_{q}=\frac{(L_{A}+L_{B}T)\alpha_{0}}{(m_{C}-\alpha_{1})^{2}}e^{\frac{2L_{A}T+L_{B}T^{2}}{2(m_{C}-\alpha_{1})}},\quad
c_{p}=\frac{\alpha_{0}}{m_{A}-\alpha_{1}}.$

Then it follows from Remark 3.1 that

$\|G(w_{\zeta 1})(s)-G(w_{\zeta 2})(s)\|\leq c_{q}\int_{0}^{s}\|w_{\zeta
1}(l)-w_{\zeta 2}(l)\|dl+c_{p}\|w_{\zeta 1}(s)-w_{\zeta 2}(s)\|.$

These inequalities give

$\displaystyle\quad\|\Lambda w_{\zeta 1}(t)-\Lambda w_{\zeta 2}(t)\|$
$\displaystyle\leq(L_{F}+L_{F}c_{p})\int_{0}^{t}\|w_{\zeta 1}(s)-w_{\zeta
2}(s)\|ds+L_{F}c_{q}\int_{0}^{t}\int_{0}^{s}\|w_{\zeta 1}(l)-w_{\zeta
2}(l)\|dlds$ (3.19)

and so

$\displaystyle\|\Lambda w_{\zeta 1}(t)-\Lambda w_{\zeta
2}(t)\|_{\beta}\leq\left(\frac{L_{F}+L_{F}c_{p}}{\beta}\right)\left\|w_{\zeta
1}-w_{\zeta 2}\right\|_{\beta}+\frac{L_{F}c_{q}}{\beta^{2}}\left\|w_{\zeta
1}-w_{\zeta 2}\right\|_{\beta}$ (3.20)

with $\beta>0$ and $\|w\|_{\beta}=\max_{t\in I}e^{-\beta t}\|w(t)\|_{W}$ for
all $w\in C(I;W).$ Thus, $\|\cdot\|_{\beta}$ is an equivalent norm in
$C(I;W)$. From inequality (3.20), we know that $\Lambda$ is a contraction
operator on $C(I;W)$ and $C(I;W)$ endowed with the norm $\|\cdot\|_{\beta}$.
Thus, $\Lambda$ has a unique fixed point $w_{\zeta}\in C(I;W)$ and so
$(w_{\zeta},G(w_{\zeta}))$ solves the Problem 3.2, which concludes the proof.
∎

######  Remark 3.2.

We need some useful inequalities for the solution $u_{\zeta}$ of Problem 3.2.
For any given $\zeta_{1},\;\zeta_{2}\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)$, let
$(\dot{u}_{\zeta_{i}},w_{\zeta_{i}})\in C(I;V)\times C^{1}(I;W)$ be the unique
solution of Problem 3.2 with $\zeta=\zeta_{i}$ for $i=1,\;2.$ From the proof
of Lemma 3.3, we have

$\displaystyle\|w_{\zeta_{1}}(t)-w_{\zeta_{2}}(t)\|$ (3.21)
$\displaystyle\leq$
$\displaystyle(L_{F}+L_{F}c_{p})\int_{0}^{t}\|w_{\zeta_{1}}(s)-w_{\zeta_{2}}(s)\|ds+L_{F}c_{q}\int_{0}^{t}\int_{0}^{s}\|w_{\zeta_{1}}(l)-w_{\zeta_{2}}(l)\|dlds+L_{F}c_{r}T\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})}$
$\displaystyle\leq$
$\displaystyle(L_{F}+L_{F}c_{p}+L_{F}c_{q}T)\int_{0}^{t}\|w_{\zeta_{1}}(s)-w_{\zeta_{2}}(s)\|ds+L_{F}c_{r}T\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})}$

and

$\displaystyle\|\dot{u}_{\zeta_{1}}(t)-\dot{u}_{\zeta_{2}}(t)\|$
$\displaystyle\leq$ $\displaystyle
c_{q}\int_{0}^{t}\|w_{\zeta_{1}}(s)-w_{\zeta_{2}}(s)\|ds+c_{p}\|w_{\zeta_{1}}(t)-w_{\zeta_{2}}(t)\|+c_{r}\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})}.$
(3.22) $\displaystyle\leq$
$\displaystyle(c_{q}T+c_{p})\|w_{\zeta_{1}}(s)-w_{\zeta_{2}}(s)\|+c_{r}\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})},$

where

$c_{r}=\frac{L_{B}\sqrt{t}}{m_{C}-\alpha_{1}}+\frac{2L_{B}t^{\frac{3}{2}}(L_{A}+L_{B}t)}{3(m_{C}-\alpha_{1})^{2}}e^{\frac{2L_{A}t+L_{B}t^{2}}{2(m_{C}-\alpha_{1})}}.$

By Gronwall’s inequality, it follows from (3.21) that

$\displaystyle\|w_{\zeta_{1}}(t)-w_{\zeta_{2}}(t)\|\leq\left(L_{F}c_{r}T\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})}\right)e^{(L_{F}+L_{F}c_{p}+L_{F}c_{q}T)T}.$
(3.23)

Combining (3.22) and (3.23) yields

$\displaystyle\|\dot{u}_{\zeta_{1}}(t)-\dot{u}_{\zeta_{2}}(t)\|_{V}\leq\left((L_{F}c_{q}c_{r}T^{2}+L_{F}c_{p}c_{r}T)e^{(L_{F}+L_{F}c_{p}+L_{F}c_{q}T)T}+c_{r}\right)\|\zeta_{1}-\zeta_{2}\|_{L^{2}(I,Y_{1})}.$

Moreover, for the unique solution $(u_{\zeta},w_{\zeta})\in
C^{1}(I;K_{V})\times C^{1}(I;W)$ of Problem 3.2, we need to consider the
following auxiliary problem.

######  Problem 3.3.

Find $\zeta:I\rightarrow K_{Y}$ such that

$\displaystyle\langle\dot{\zeta}(t),\eta-\zeta(t)\rangle_{Y_{1}}+a(\zeta(t),\eta-\zeta(t))\geq\langle\phi(t,u_{\zeta}(t),\zeta(t)),\eta-\zeta(t)\rangle_{Y_{1}},\quad\forall\eta\in
K_{Y},$ (3.24)

with $\zeta(0)=\zeta_{0}\in K_{Y}.$

######  Remark 3.3.

We observe that if $\zeta\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)$ is a unique
solution of Problem 3.3, then $(\zeta,u_{\zeta},w_{\zeta})\in
H^{1}(I;Y_{1})\cap L^{2}(I;Y)\times C^{1}(I;K_{Y})\times C^{1}(I;W)$ is a
unique solution of DQVIs (1.1).

In order to prove Lemma 3.5, we need the following Lemma given in [13].

######  Lemma 3.4.

[13] Suppose that condition H(a) holds. Then for any given $\lambda\in
L^{2}(I;Y_{1})$, there exists a unique $\zeta\in H^{1}(I;Y_{1})\cap
L^{2}(I;Y)$ such that

$\displaystyle\langle\dot{\zeta}(t),\eta-\zeta(t)\rangle_{Y_{1}}+a(\zeta(t),\eta-\zeta(t))\geq\langle\lambda(t),\eta-\zeta(t)\rangle_{Y_{1}},\quad\forall\eta\in
K_{Y},$ (3.25)

with $\zeta(0)=\zeta_{0}\in K_{Y}$. Moreover, if $\zeta_{i}$ is the unique
solution to problem (3.25) for $\lambda_{i}\in L^{2}(I;Y_{1})$ with $i=1,2$,
then

$\displaystyle\left\|\zeta_{1}(t)-\zeta_{2}(t)\right\|_{Y_{1}}^{2}\leq
d_{1}\int_{0}^{t}\left\|\lambda_{1}(s)-\lambda_{2}(s)\right\|_{Y_{1}}^{2}ds\quad\text{
for a.e. }t\in(0,T)$ (3.26)

with $d_{1}>0.$

######  Lemma 3.5.

Suppose that conditions H(A), H(B), H(C), H(j), H(F), H($\phi$), H(a) hold and
$m_{C}>\alpha_{1}$. Then Problem 3.3 has a unique solution $\zeta\in
H^{1}(I;Y_{1})\cap L^{2}(I;Y).$

###### Proof.

For any given $\theta\in L^{2}(I;Y_{1})$, it follows from Lemma 3.3 that
Problem 3.2 has a unique solution $(u_{\theta},w_{\theta})\in C(I;K_{V})\times
C^{1}(I;W)$. Let $\phi_{\theta}(t):=\phi(t,u_{\theta}(t),\theta(t))$. We claim
that $\phi_{\theta}\in L^{2}(I;Y_{1})$. Indeed, the condition H($\phi$)(a)
derives the following inequality

$\displaystyle\|\phi_{\theta}(t)\|^{2}_{Y_{1}}$ $\displaystyle\leq$
$\displaystyle
2\|\phi(t,u_{\theta}(t),\theta(t))-\phi(t,0_{V},0_{Y_{1}})\|^{2}_{Y_{1}}+2\|\phi(t,0_{V},0_{Y_{1}})\|^{2}_{Y_{1}}$
(3.27) $\displaystyle\leq$ $\displaystyle
2(L_{\phi}(\|u_{\theta}(t)\|_{V}+\|\theta\|_{Y_{1}}))^{2}+2\|\phi(t,0_{V},0_{Y_{1}})\|^{2}_{Y_{1}}$
$\displaystyle\leq$ $\displaystyle
4L_{\phi}^{2}(\|u_{\theta}(t)\|^{2}_{V}+\|\theta\|^{2}_{Y_{1}})+2\|\phi(t,0_{V},0_{Y_{1}})\|^{2}_{Y_{1}}.$

Thus inequality (3.27) and condition H($\phi$)(b) show that

$\displaystyle\|\phi_{\theta}(t)\|^{2}_{L^{2}(I;Y_{1})}\leq
4L^{2}_{\phi}T\|u_{\theta}(t)\|^{2}_{C(I:V)}+4L^{2}_{\phi}\|\theta\|^{2}_{L^{2}(I;Y_{1})}+2\|\phi(t,0_{V},0_{Y_{1}})\|^{2}_{L^{2}(I;Y_{1})}$
(3.28)

and so $\phi_{\theta}(t)\in L^{2}(I;Y_{1})$. Taking $\lambda=\phi_{\theta}$ in
Lemma 3.4, we know that there exists a unique $\zeta_{\theta}\in
H^{1}(I;Y_{1})\cap L^{2}(I;Y)$ such that inequality (3.25) holds.

Next we define an operator $\Lambda:L^{2}(I;Y_{1})\rightarrow
H^{1}(I;Y_{1})\cap L^{2}(I;Y)$ by $\Lambda(\theta)(t)=\zeta_{\theta}(t).$
Because $\left(Y,Y_{1},Y^{*}\right)$ is a Gelfand triplet with dense and
compact embeddings, one has $H^{1}(I;Y_{1})\cap L^{2}(I;Y)\subset
L^{2}(I;Y_{1})$. The rest is to show that $\Lambda$ has a unique fixed point
in $L^{2}(I;Y_{1}).$ To this end, let $\theta_{i}\in L^{2}(I;Y_{1})$ with
$i=1,\;2.$ Then Lemma 3.4 yields

$\displaystyle\|\Lambda(\theta_{1})(t)-\Lambda(\theta_{2})(t)\|^{2}_{Y_{1}}$
$\displaystyle=$
$\displaystyle\|\zeta_{\theta_{1}}(t_{1})-\zeta_{\theta_{2}}(t_{2})\|^{2}_{Y_{1}}$
$\displaystyle\leq$ $\displaystyle
d_{1}\int_{0}^{t}\left\|\phi(s,u_{\theta_{1}}(s),\theta_{1}(s))-\phi(s,u_{\theta_{2}}(s),\theta_{2}(s))\right\|_{Y_{1}}^{2}ds.$

Using condition H($\phi$)(a) to the above inequality obtains

$\displaystyle\|\Lambda(\theta_{1})(t)-\Lambda(\theta_{2})(t)\|^{2}_{Y_{1}}$
$\displaystyle\leq$ $\displaystyle
d_{1}\int_{0}^{t}\left\|\phi(s,u_{\theta_{1}}(s),\theta_{1}(s))-\phi(s,u_{\theta_{2}}(s),\theta_{2}(s))\right\|_{Y_{1}}^{2}ds$
(3.29) $\displaystyle\leq$ $\displaystyle
2d_{1}L_{\phi}\int_{0}^{t}\left(\|u_{\theta_{1}}(s)-u_{\theta_{2}}(s)\|_{V}^{2}+\|\theta_{1}(s)-\theta_{2}(s)\|_{Y_{1}}^{2}\right)ds.$

By Remark 3.2, we can adapt the inequality (3.29) as

$\displaystyle\|\Lambda(\theta_{1})(t)-\Lambda(\theta_{2})(t)\|^{2}_{Y_{1}}$
$\displaystyle\leq$ $\displaystyle
2d_{1}L_{\phi}\int_{0}^{t}\left(\int_{0}^{t}\|\dot{u}_{\theta_{1}}(s)-\dot{u}_{\theta_{2}}(s)\|_{V}^{2}ds+\|\theta_{1}(s)-\theta_{2}(s)\|_{Y_{1}}^{2}\right)ds$
(3.30) $\displaystyle\leq$ $\displaystyle
c\|\theta_{1}-\theta_{2}\|^{2}_{L^{2}(I;Y_{1})},$

where

$c=2d_{1}L_{\phi}\left(\left((L_{F}c_{q}c_{r}T^{2}+L_{F}c_{p}c_{r}T)e^{(L_{F}+L_{F}c_{p}+L_{F}c_{q}T)T}+c_{r}\right)^{2}T^{2}+1\right).$

Thus inequality (3.30) shows that

$\displaystyle\|\Lambda(\theta_{1})(t)-\Lambda(\theta_{2})(t)\|_{L^{(}I;Y_{1})}\leq\sqrt{c}\int_{0}^{t}\|\theta_{1}-\theta_{2}\|_{L^{2}(I;Y_{1})}.$
(3.31)

Now it follows from Lemma 2.1 that there exists a unique element $\theta^{*}$
in $L^{2}(I;Y_{1})$ such that $\theta^{*}$ is the unique fixed point of
$\Lambda$, i.e., $\theta^{*}=\Lambda(\theta^{*})=\zeta_{\theta^{*}}\in
H^{1}(I;Y_{1})\cap L^{2}(I;Y)$. Thus, $\theta^{*}$ is a unique solution of
Problem 3.3. This completes the proof. ∎

Finally, we can give the proof of Theorem 3.1 as follows.

###### Proof.

By Remark 3.3 and Lemma 3.5, it is easy to see that
$(\zeta,u_{\zeta},w_{\zeta})\in(H^{1}(I;Y_{1})\cap L^{2}(I;Y))\times
C^{1}(I;K_{Y})\times C^{1}(I;W)$ is a unique solution of DQVIs (1.1). This
completes the proof of Theorem 3.1. ∎

##  4 An application to a contact problem for viscoelastic materials with long
memory, damage and wear

In this section, we will use the results presented in Section 3 to solve a
frictional contact problem of viscoelastic materials with long memory, damage
and wear. Assume that $\Omega$ is a bounded and open domain in
$\mathbb{R}^{d}(d=2,3)$ and its boundary $\Gamma:=\partial\Omega$ is Lipschitz
continuous. For any $\bm{x}\in\Omega$, we denote it as
$\bm{x}=(x_{1},x_{2},\cdots,x_{d})$. Let $\bm{\nu}$ denote the unit outward
normal vector defined $a.e.$ on $\Gamma$. The space of second order symmetric
tensors on $\mathbb{R}^{d}$ is denoted by $\mathbb{S}^{d}$. Moreover, assume
that $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are equipped with the following
inner products and norms

$\displaystyle\bm{u\cdot
v}=u_{i}v_{i},\qquad\;\;\|\bm{v}\|_{\mathbb{R}^{d}}=(\bm{v\cdot
v})^{\frac{1}{2}},\qquad\forall\bm{u},\bm{v}\in\mathbb{R}^{d},$
$\displaystyle\bm{\sigma\cdot\tau}=\sigma_{ij}\tau_{ij},\qquad\|\bm{\tau}\|_{\mathbb{S}^{d}}=(\bm{\tau\cdot\tau})^{\frac{1}{2}},\qquad\forall\bm{\sigma},\bm{\tau}\in\mathbb{S}^{d},$

where repeated indices represent summation convention.

We use notations $\bm{u}=(u_{i})$, $\bm{\sigma}=(\sigma_{ij})$ and
$\bm{\varepsilon(u)}=(\varepsilon_{ij}(\bm{u}))=(\frac{1}{2}(u_{i,j}+u_{j,i})),i,j=1,2,\cdots,d$
to denote the displacement vector, the stress tensor and the linearized strain
tensor, respectively, where $u_{i,j}\equiv\frac{\partial u_{i}}{\partial
x_{j}}$. Here and below, the spatial derivative is defined in the sense of
distribution. The normal and tangential components of stress field
$\bm{\sigma}$ on $\Gamma$ are denoted by
$\sigma_{\nu}=(\bm{\sigma\nu})\bm{\cdot}\bm{\nu}$ and
$\bm{\sigma_{\tau}}=\bm{\sigma}\bm{\nu}-\sigma_{\nu}\bm{\nu}$, respectively.
The normal and tangential components of the displacement field on $\Gamma$ are
denoted by $u_{\nu}=\bm{u}\bm{\cdot}\bm{\nu}$ and
$\bm{u_{\tau}}=\bm{u}-u_{\nu}\bm{\nu}$, respectively. We are interested in the
evolution of the body on the time interval $I:=[0,T]$, where $T>0$ is a
constant. The time partial derivative for a function $f(\bm{x},t)$ is denoted
by $\dot{f}(\bm{x},t)$. For the sake of simplicity, we usually omit the
variable $\bm{x}$ in functions.

Now we are in the position to describe the physical background of viscoelastic
materials’ quasistatic frictional contact problems with damage, wear and long
memory. We consider a viscoelastic body which occupies $\Omega$. The boundary
$\partial\Omega$ is composed of three disjoint measurable parts $\Gamma_{1}$,
$\Gamma_{2}$ and $\Gamma_{3}$ with $\mbox{meas}(\Gamma_{1})>0$. Volume forces
with density $\bm{f}_{0}$ act on $\Omega$ and surface tractions with density
$\bm{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{1}$, we suppose that the body is
clamped, i.e., the displacement field disappears there. The contact model is
established by a normal compliance condition on $\Gamma_{3}$ with sliding
Coulomb’s law of dry friction and wear. The foundation is made of a perfectly
rigid material. The velocity of the foundation is denoted by
$\bm{v}^{*}(t)\neq\bm{0}$. Let $\bm{n}^{*}$ be the direction of the
foundation’s velocity and $\alpha$ be wear rate which are given respectively
by

$\bm{n}^{*}(t)=-\bm{v}^{*}(t)/\left\|\bm{v}^{*}(t)\right\|,\quad\alpha(t)=k\left\|\bm{v}^{*}(t)\right\|,$

where $k$ stands for the wear coefficient. We use the following abbreviations
to simplify the notations $\mathcal{Q}=\Omega\times I$, $\Sigma=\Gamma\times
I$, $\Sigma_{i}=\Gamma_{i}\times I$, $i=1,2,3.$

Thus, we can present the formulation of the contact problem as follows.

######  Problem 4.1.

Find a displacement field $\bm{u}:\mathcal{Q}\rightarrow\mathbb{R}^{d}$, a
stress field $\bm{\sigma}:\mathcal{Q}\rightarrow\mathbb{S}^{d}$, a damage
field $\zeta:\Sigma_{3}\rightarrow[0,1]$ and a wear function
$w:\Sigma_{3}\rightarrow\mathbb{R}$ such that

$\displaystyle\bm{\sigma}(t)=\mathcal{A}\left(t,\bm{\varepsilon}(\bm{u}(t))\right)+\int_{0}^{t}\mathcal{B}\left(t-s,\bm{\varepsilon}(\bm{u}(s)),\zeta(s)\right)ds+\mathcal{C}(t,\bm{\varepsilon}(\dot{\bm{u}}(t)))\quad$
$\displaystyle\quad\mbox{in}\quad\;\mathcal{Q},$ (4.1)
$\displaystyle\dot{\zeta}-\kappa\Delta\zeta+\partial
I_{[0,1]}(\zeta)\ni\phi(\bm{\varepsilon}(\bm{u}(t)),\zeta)\quad$
$\displaystyle\quad\mbox{in}\quad\;\mathcal{Q},$ (4.2)
$\displaystyle\frac{\partial\zeta}{\partial\nu}=0\quad$
$\displaystyle\quad\mbox{on}\quad\;\Sigma,$ (4.3)
$\displaystyle-\mbox{Div}\bm{\sigma}(t)=\bm{f}_{0}(t)\quad$
$\displaystyle\quad\mbox{in}\quad\;\mathcal{Q},$ (4.4)
$\displaystyle\bm{u}(t)=\bm{0}\quad$
$\displaystyle\quad\mbox{on}\quad\Sigma_{1},$ (4.5)
$\displaystyle\bm{\sigma}(t)\bm{\nu}=\bm{f}_{2}(t)\quad$
$\displaystyle\quad\mbox{on}\quad\Sigma_{2},$ (4.6)
$\displaystyle\dot{u}_{\nu}(t)\leq
g,\;\sigma_{\nu}(t)+p(w(t),\dot{u}_{\nu}(t))\leq 0\quad$
$\displaystyle\quad\mbox{on}\quad\Sigma_{3},$ (4.7)
$\displaystyle(\dot{u}_{\nu}(t)-g)(\sigma_{\nu}(t)+p(w(t),\dot{u}_{\nu}(t)))=0\quad$
$\displaystyle\quad\mbox{on}\quad\Sigma_{3},$ (4.8)
$\displaystyle-\bm{\sigma}_{\tau}(t)=\mu
p(w(t),\dot{u}_{\nu})\cdot\bm{n^{*}}(t)\quad$
$\displaystyle\quad\mbox{on}\quad\;\Sigma_{3},$ (4.9)
$\displaystyle\dot{w}(t)=\alpha(t)p(w(t),\dot{u}_{\nu}(t))\quad$
$\displaystyle\quad\mbox{on}\quad\;\Sigma_{3},$ (4.10)
$\displaystyle\bm{u}(0)=\bm{u}_{0},w(0)=0,\zeta(0)=\zeta_{0}\in(0,1)\quad$
$\displaystyle\quad\mbox{on}\quad\;\Gamma_{3}.$ (4.11)

######  Remark 4.1.

The conditions (4.7) and (4.8) can be rewritten as

$-\sigma_{\nu}(t)\in\partial I_{K}(\dot{u}_{\nu})+p(w(t),\dot{u}_{\nu}),$

where $K:=\\{s\in\mathbb{R}\>|\>s\leq g\\}$, $I_{K}$ is the indication
function of $K$ and $\partial I_{K}$ is the convex subdifferential of $I_{K}$.
From the definition of convex subdifferential, one has

$(-\sigma_{\nu}(t)-p(\dot{u}_{\nu}(t)-w(t)))\cdot(s-\dot{u}_{\nu})\leq
I_{K}(s)-I_{K}(\dot{u}_{\nu})=0,\quad\forall s\in K.$

Equation (4.1) stands for the viscoelastic constitutive law with damage effect
and long memory, where $\zeta$ is the damage function, $\mathcal{A}$,
$\mathcal{B}$ and $\mathcal{C}$ are elasticity, relaxation and viscosity
operators, respectively. The evolution law of damage function can be modeled
from a parabolic differential inclusion (4.2), where $\phi$ is the mechanical
source of damage which depends on the damage itself and the strain,
$I_{[0,1]}$ is the indication function of the interval $[0,1]$, $\partial
I_{[0,1]}$ is the convex subdifferential of $I_{[0,1]}$ and $\kappa>0$ is a
constant mircrocrack diffusion coefficient. We select the following damage
source function

$\phi(\varepsilon(\bm{u}),\zeta)\equiv\phi_{\mathrm{Fr}}(\|\varepsilon(\bm{u})\|,\zeta)=\lambda_{D}\left(\frac{1-\zeta}{\zeta}\right)-\frac{1}{2}\lambda_{E}\|\varepsilon(\bm{u})\|^{2}+\lambda_{w},$

where $\lambda_{D},\lambda_{E},$ and $\lambda_{w}$ are three positive numbers
([9, 10]). Equation (4.3) represents a homogeneous Neumann condition of
$\zeta$ on $\Sigma$ and equation (4.4) describes the equilibrium equation.
Equations (4.5) and (4.6) denote the clamped boundary condition and the
tractive boundary conditions, respectively. Equations (4.7) and (4.8) are
called normal damped condition with unilateral constraint, which have been
studied in [7] without wear. In the previous literature, the normal compliance
function $p(\dot{u}_{\nu})$ was used to model the lubricated contact. In this
paper, we assume that the lubrication is incomplete and the normal compliance
depends on wear, i.e., $p$ has the form of $p(w,\dot{u}_{\nu})$ with $w$ being
the wear. Equation (4.9) gives the functional relationship between the wear
coefficient $k$ and the friction coefficient which denoted $\mu$. Here we
assume that $\bm{v^{*}}(t)$ is much larger than the tangential body velocity
$\bm{\dot{u}_{\tau}}(t)$. Equation (4.10) is related to the development of the
wear function which can be obtained from the Archard’s law with
$\alpha(t)=k\|\bm{v^{*}}(t)\|$. For more details concerned with equations
(4.7)-(4.10), we can refer to the literature [32]. Finally, equation (4.11)
gives the initial conditions for the displacement field, the wear function and
the damage field.

In order to get the variational formulation of Problem 4.1, we need the
following spaces. Let $W^{k,p}(\Omega;\mathbb{R}^{d})$ denote the Sobolev
space of all functions. On $\Omega$, the weak derivatives of functions with
order less than or equal to $k$ are $p$-integrable. Specially, let
$H^{1}:=W^{1,2}(\Omega;\mathbb{R}^{d})$ and $H=L^{2}(\Omega;\mathbb{R}^{d}).$
Let $V=\\{\bm{v}\in H^{1}|\bm{v}=0\;\mbox{a.e. on}\;\Gamma_{3}\\}$ endowed
with the norm

$\|\bm{u}\|_{V}:=\|\bm{u}\|_{H^{1}}=\|\bm{u}\|_{L^{2}(\Omega;\mathbb{R}^{d})}+\|\triangledown\bm{u}\|_{L^{2}(\Omega;\mathbb{R}^{d\times
d})},$

where $\nabla\bm{u}=(\frac{\partial u_{i}}{\partial x_{j}})$ for
$i,j=1,\cdots,d$ with $\bm{u}\in H^{1}.$ Let
$\mbox{Div}\bm{\sigma}=(\sigma_{ij,j})=(\frac{\partial\sigma_{ij}}{\partial
x_{j}})$ with $\bm{\sigma}\in W^{1,2}(\Omega;\mathbb{S}^{d})$. Then we have
the following Green formula:

$(Div\bm{\sigma},\bm{v})_{H}+(\bm{\sigma},\bm{\varepsilon(u)})_{L^{2}(\Omega;\mathbb{S}^{d})}=\int_{\Gamma}\bm{\sigma\nu}\cdot\bm{v}d_{\Gamma},$

where
$(\bm{\sigma},\bm{\varepsilon(u)})_{L^{2}(\Omega;\mathbb{S}^{d})}=\int_{\Omega}\bm{\sigma}\bm{\cdot}\bm{\varepsilon(u)}d\Omega$.
From the assumption of $\mbox{meas}(\Gamma_{1})>0$, we know that Korn’s
inequality holds, i.e.,

$\|\bm{\varepsilon}(\bm{v})\|_{L^{2}(\Omega;\mathbb{S}^{d})}\geq
c\|\bm{v}\|_{V},\quad\forall\bm{v}\in V.$

Here and then, $c$ represents a positive constant which may change from line
to line. We can endow with inner product on $V$ as

$\langle\bm{u},\bm{v}\rangle_{V}=\langle\bm{\varepsilon}(\bm{u}),\bm{\varepsilon}(\bm{v})\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}.$

Let $Y=H^{1}(\Omega;\mathbb{R})$ and $Y_{1}=L^{2}(\Omega;\mathbb{R})$ endowed
with the inner products

$(w,z)_{Y_{1}}=\int_{\Omega}wzd_{\Omega},\quad(w,z)_{Y}=(w,z)_{Y_{1}}+\int_{\Omega}\nabla
w\cdot\nabla zd_{\Omega}.$

Denote $K_{V}=\\{\bm{v}\in V\>|\>v_{\nu}\leq g\;\mbox{a.e. on}\;\Gamma_{3}\\}$
and $K_{Y}=\\{u\in Y\>|\>0\leq u\leq 1\;\mbox{a.e. in}\;\Omega\\}$. Then
$K_{V}$ and $K_{Y}$ are both convex. Suppose that
$\\{\bm{u},\bm{\sigma},\zeta,w\\}$ are sufficiently smooth functions solving
(4.1)-(4.11) with $t\in I$. Then, we can employ the following method to derive
the variational formulation of Problem 4.1. Firstly, we use the Green formula
and the equation (4.1) to obtain

$\langle\bm{\sigma}(t),\bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm{\dot{u}}}(t))\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}=\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Omega;\mathbb{R}^{d})}+\int_{\Gamma}\bm{\sigma}(t)\bm{\nu}\cdot(\bm{v}-{\bm{\dot{u}}}(t))d\Gamma.$
(4.12)

Considering the boundary conditions (4.5), (4.6), (4.9) and the formula

$\bm{\sigma}(t)\bm{\nu}\cdot\bm{v}=\sigma_{\nu}(t)v_{\nu}+\bm{\sigma}_{\tau}(t)\cdot\bm{v}_{\tau},$

the equation (4.12) can be transformed as

$\displaystyle\langle\bm{\sigma}(t),\bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\dot{u}}(t))\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}$
(4.13) $\displaystyle=$
$\displaystyle\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Omega;\mathbb{R}^{d})}+\int_{\Gamma_{2}}\bm{f_{2}}(t)\cdot(\bm{v}-{\bm{\dot{u}}}(t))d\Gamma-\int_{\Gamma_{3}}\mu
p(w(t),\dot{u}_{\nu}(t))\bm{n^{*}}(t)\cdot(\bm{v}_{\tau}-{\bm{\dot{u}}_{\tau}}(t))d\Gamma$
$\displaystyle\mbox{}+\int_{\Gamma_{3}}{\sigma}_{\nu}(t)\cdot(v_{\nu}-{\dot{u}_{\nu}}(t))d\Gamma.$

From Remark 4.1 and $\bm{v}\in K_{V}$, one has

$\int_{\Gamma_{3}}{\sigma}_{\nu}(t)\cdot(v_{\nu}-{\dot{u}_{\nu}}(t))d\Gamma\geq\int_{\Gamma_{3}}-p(w(t),\dot{u}_{\nu}(t))(v_{\nu}-{\dot{u}_{\nu}}(t))d\Gamma.$

Now (4.13) becomes

$\displaystyle\langle\bm{\sigma}(t),\bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm{\dot{u}}}(t))\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}+\int_{\Gamma_{3}}\mu
p(w(t),\dot{u}_{\nu}(t))\bm{n^{*}}(t)\cdot(\bm{v}_{\tau}-{\bm{\dot{u}}_{\tau}}(t))d\Gamma$
(4.14)
$\displaystyle\mbox{}+\int_{\Gamma_{3}}p(w(t),\dot{u}_{\nu}(t))(v_{\nu}-{\dot{u}_{\nu}}(t))d\Gamma$
$\displaystyle\geq$
$\displaystyle\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Omega;\mathbb{R}^{d})}+\int_{\Gamma_{2}}\bm{f_{2}}(t)\cdot(\bm{v}-{\bm{\dot{u}}}(t))d\Gamma.$

Let

$j(w,\gamma\bm{\dot{u}},\gamma\bm{v}):={\int_{\Gamma_{3}}\mu
p(w(t),\dot{u}_{\nu}(t))\bm{n^{*}}(t)\cdot\bm{v}_{\tau}d\Gamma+\int_{\Gamma_{3}}p(w(t),\dot{u}_{\nu}(t))v_{\nu}d\Gamma.}$

Then it follows from (4.14) that

$\displaystyle\langle\bm{\sigma}(t),\bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm{\dot{u}}}(t))\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}+j(w,\gamma\bm{\dot{u}},\gamma\bm{v})-j(w,\gamma\bm{\dot{u}},\gamma\bm{\dot{u}})$
(4.15) $\displaystyle\geq$
$\displaystyle\left\langle\bm{f_{2}}(t),\gamma\bm{v}-\gamma{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Gamma_{3};\mathbb{R}^{d})}+\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Omega;\mathbb{R}^{d})},$

where $\gamma:V\rightarrow L^{2}(\Gamma_{3};\mathbb{R}^{d})$ is the trace
operator. Applying Korn’s inequality, the Riesz representation theorem and the
trace theorem, we know that there exists $\bm{f}:I\rightarrow V^{*}$ such that

$\langle\bm{f}(t),\bm{v}-\bm{\dot{u}}\rangle_{V^{*}\times
V}=\left\langle\bm{f_{2}}(t),\gamma\bm{v}-\gamma{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Gamma_{3};\mathbb{R}^{d})}+\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^{2}(\Omega;\mathbb{R}^{d})}.$

Similarly, from the integration by parts and the definition of convex
subdifferential of $I_{[0,1]}$, it follows that

$\displaystyle\langle\phi(t,\bm{\varepsilon}(\bm{u}(t)),\zeta(t))-\dot{\zeta}(t),\eta-\zeta(t)\rangle_{L^{2}(\Omega;\mathbb{R})}\leq
a(\zeta,\eta-\zeta),$ (4.16)

where $a(\zeta,\eta)=\kappa\int_{\Omega}\nabla\zeta\cdot\nabla\eta dx$ for all
$\zeta,\eta\in Y.$

Now, by integrating these relations and inequalities, the variational
formulation of Problem 4.1 can be obtained as follows.

######  Problem 4.2.

Find $\bm{u}:I\rightarrow K_{V}$, $\zeta:I\rightarrow K_{Y}$ and
$w:I\rightarrow L^{2}(\Gamma_{3};\mathbb{R})$ such that, for a.e. $t\in I$,

$\displaystyle\bm{\sigma}(t)=\mathcal{A}(t,\bm{\varepsilon}(\bm{u}(t)))+\int_{0}^{t}\mathcal{B}(t-s,\bm{\varepsilon}(\bm{u}(s)),\zeta(s))ds+\mathcal{C}(t,\bm{\varepsilon}(\bm{\dot{u}}(t)))\quad\text{
in }\Omega,\quad$ (4.17)
$\displaystyle\langle\bm{\sigma}(t),\bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm{\dot{u}(t)}})\rangle_{L^{2}(\Omega;\mathbb{S}^{d})}+j(w,\gamma\bm{\dot{u}},\gamma\bm{v})-j(w,\gamma\bm{\dot{u}},\gamma\bm{\dot{u}})\geq\left\langle\bm{f}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{V^{*}\times
V},\;\forall\bm{v}\in K_{V},\quad$ (4.18)
$\displaystyle\langle(\dot{\zeta}(t),\eta-\zeta(t)\rangle_{Y_{1}}+a(\zeta,\eta-\zeta)\geq\langle\phi(t,\bm{\varepsilon}(\bm{u}(t)),\zeta(t)),\eta-\zeta(t)\rangle_{Y_{1}},\;\forall\eta\in
K_{Y},\quad$ (4.19)
$\displaystyle\dot{w}(t)=\alpha(t)p(\dot{u}_{\nu}(t)-w(t))\quad\mbox{ in
}\Gamma_{3},\quad$ (4.20)
$\displaystyle\bm{u}(0)=\bm{u}_{0},w(0)=0,\;\zeta(0)=\zeta_{0}\in(0,1).\quad$
(4.21)

To solve Problem 4.2, we need the following assumptions.

$H(1):$ The elasticity operator $\mathcal{A}:\Omega\times
I\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ satisfies

$\displaystyle\begin{cases}(a)\;\mathcal{A}(\cdot,t,\bm{\varepsilon})\;\mbox{is
measurable on}\;\Omega,\mbox{ for all }(t,\bm{\varepsilon})\in
I\times\mathbb{S}^{d};\\\ (b)\;\mathcal{A}(\bm{x},\cdot,\cdot)\;\mbox{is
continuous on }I\times\mathbb{S}^{d}\mbox{ for a.e. }\;\bm{x}\in\Omega;\\\
(c)\;\mathcal{A}(\bm{x},t,\cdot)\mbox{ is Lipschitz continous
with}\;m_{\mathcal{A}}>0\;\mbox{for all }\;t\in I,\;i.e.,\\\
\quad\left\|\mathcal{A}\left(\bm{x},t,\bm{\varepsilon}_{1}\right)-\mathcal{A}\left(\bm{x},t,\bm{\varepsilon}_{2}\right)\right\|\leq
L_{\mathcal{A}}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|,\;\forall\bm{\varepsilon}_{1},\bm{\varepsilon}_{2}\in\mathbb{S}^{d},\mbox{
a.e. }\bm{x}\in\Omega;\\\ \end{cases}$ (4.22)

$H(2):$ The relaxation operator $\mathcal{B}:\Omega\times
I\times\mathbb{S}^{d}\times\mathbb{R}\rightarrow\mathbb{S}^{d}$ satisfies

$\displaystyle\begin{cases}(a)\mathcal{B}(\cdot,t,\bm{\varepsilon},\zeta)\;\mbox{is
measurable on}\;\Omega,\mbox{ for all
}\bm{\varepsilon}\in\mathbb{S}^{d},\;t\in I\mbox{ and }\zeta\in\mathbb{R};\\\
(b)\mathcal{B}(\bm{x},\cdot,\bm{\varepsilon},\zeta)\mbox{ is continuous on
}I\\\ \quad\mbox{ for a.e. }\bm{x}\in\Omega\mbox{ and all
}(\bm{\varepsilon},\zeta)\in\mathbb{S}^{d}\times\mathbb{R};\\\
(c)\mathcal{B}(\bm{x},t,\cdot,\cdot)\;\mbox{is Lipschitz continuous
with}\;L_{\mathcal{B}}>0\mbox{ for all }t\in I\mbox{ and
a.e.}\;\bm{x}\in\Omega,i.e.,\\\
\quad\|\mathcal{B}(\bm{x},t,\bm{\varepsilon}_{1},\zeta_{1})-\mathcal{B}(\bm{x},t,\bm{\varepsilon}_{2},\zeta_{2})\|\leq
L_{\mathcal{B}}(\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\|+|\zeta_{1}-\zeta_{2}|),\\\
\quad\forall\bm{\varepsilon}_{1},\bm{\varepsilon}_{2}\in\mathbb{S}^{d},\;\zeta_{1},\zeta_{2}\in\mathbb{R},\mbox{
a.e. }\bm{x}\in\Omega;\\\ (d)\mbox{For all }(t,\bm{\varepsilon},\zeta)\in
I\times\mathbb{S}^{d}\times\mathbb{R}\mbox{ and
a.e.}\;\bm{x}\in\Omega,\;\mbox{there exixsts a function }\\\
\quad\rho_{\mathcal{B}}\in L^{2}(I;\mathbb{R}^{+})\mbox{ such that
}\|\mathcal{B}(\bm{x},t,\bm{\varepsilon},\zeta)\|\leq\rho_{\mathcal{B}}(t)(|\zeta|+\|\varepsilon\|).\end{cases}$
(4.23)

$H(3):$ The viscosity operator $\mathcal{C}:\Omega\times
I\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ satisfies

$\displaystyle\begin{cases}(a)\;\mathcal{C}(\cdot,t,\bm{\varepsilon})\;\mbox{is
measurable on}\;\Omega\mbox{ for all }(t,\bm{\varepsilon})\in
I\times\mathbb{S}^{d};\\\ (b)\;\mathcal{C}(\bm{x},\cdot,\cdot)\;\mbox{is
continuous on }I\times\mathbb{S}^{d}\mbox{ for a.e. }\;\bm{x}\in\Omega;\\\
(c)\;\mathcal{C}(\bm{x},t,\cdot)\mbox{ is strongly monotone
with}\;m_{\mathcal{C}}>0\;\mbox{for all }\;t\in I,\;i.e.,\\\
\quad\left(\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{1}\right)-\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{2}\right)\right)\cdot\left(\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right)\geq
m_{\mathcal{C}}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|^{2},\;\forall\bm{\varepsilon}_{1},\bm{\varepsilon}_{2}\in\mathbb{S}^{d},\mbox{
a.e. }\bm{x}\in\Omega;\\\ (d)\;\mathcal{C}(\bm{x},t,\cdot)\mbox{ is Lipschitz
continous with}\;L_{\mathcal{C}1}>0\;\mbox{for all }\;t\in I,\;i.e.,\\\
\quad\left\|\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{1}\right)-\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{2}\right)\right\|\leq
L_{\mathcal{C}1}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|,\;\forall\bm{\varepsilon}_{1},\bm{\varepsilon}_{2}\in\mathbb{S}^{d},\mbox{
a.e. }\bm{x}\in\Omega;\\\
(e)\;\mathcal{C}(\bm{x},\cdot,\bm{\varepsilon})\mbox{ is Lipschitz continous
with}\;L_{\mathcal{C}2}>0\;\mbox{for all
}\;\bm{\varepsilon}\in\mathbb{S}^{d},\;i.e.,\\\
\quad\left\|\mathcal{C}\left(\bm{x},t_{1},\bm{\varepsilon}\right)-\mathcal{C}\left(\bm{x},t_{2},\bm{\varepsilon}\right)\right\|\leq
L_{\mathcal{C}2}\left\|t_{1}-t_{2}\right\|,\;\forall t_{1},t_{2}\in I,\mbox{
a.e. }\bm{x}\in\Omega;\\\ \end{cases}$ (4.24)

$H(4):$ The normal compliance function
$p:\Gamma_{3}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{+}$
satisfies

$\displaystyle\begin{cases}(a)\;p(\cdot,w,u)\;\mbox{is measurable
on}\;\Gamma_{3}\mbox{ for all }w,\,u\in\mathbb{R};\\\
(b)\;p(\bm{x},\cdot,\cdot)\;\mbox{is Lipschitz continuous
with}\;\tilde{L}_{p}>0\mbox{, a.e.}\;\bm{x}\in\Omega,i.e.,\\\
\quad|p(\bm{x},w_{1},u_{1})-p(\bm{x},w_{2},u_{2})|\leq\tilde{L}_{p}(|w_{1}-w_{2}|+|u_{1}-u_{2}|),\;\forall
w_{1},w_{2},u_{1},u_{2}\in\mathbb{R};\\\ (c)\;p(\bm{x},0,0)=0\mbox{ for a.e.
}\bm{x}\in\Omega\mbox{ and all }r\leq 0.\end{cases}$ (4.25)

$H(5):$ The damage source function $\phi:\Omega\times
I\times\mathbb{S}^{d}\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies

$\displaystyle\begin{cases}(a)\phi(\cdot,t,\bm{\varepsilon},\zeta)\;\mbox{is
measurable on}\;\Omega,\;\mbox{for all }t\in
I,\;\bm{\varepsilon}\in\mathbb{S}^{d}\mbox{ and}\;\zeta\in\mathbb{R};\\\
(b)\phi(\bm{x},t,\cdot,\cdot)\;\mbox{is Lipschitz continuous
with}\;L_{\phi}>0\mbox{ for all }t\in I\mbox{ a.e.}\;\bm{x}\in\Omega,i.e.,\\\
\quad\|\phi(\bm{x},\bm{\varepsilon}_{1},\zeta_{1})-\phi(\bm{x},\bm{\varepsilon}_{2},\zeta_{2})\|\leq\tilde{L}_{\phi}(\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\|+|\zeta_{1}-\zeta_{2}|)\\\
\quad\forall\bm{\varepsilon}_{1},\bm{\varepsilon}_{2}\in\mathbb{S}^{d},\;\zeta_{1},\zeta_{2}\in\mathbb{R},\mbox{
a.e. }\bm{x}\in\Omega;\\\
(c)\phi(\bm{x},t,\bm{0}_{\mathbb{S}^{d}},0_{\mathbb{R}})\in
L^{2}(I;L^{2}(\Omega;\mathbb{R})).\end{cases}$ (4.26)

Moreover, suppose the volume forces and surface tractions satisfy

$\bm{f}_{0}\in C(I;H),\quad\bm{f}_{2}\in
C\left(I;L^{2}\left(\Gamma_{2};\mathbb{R}^{d}\right)\right).$ (4.27)

To make the sliding condition (4.9) and wear condition (4.10) reasonable, the
coefficient of friction, wear coefficient and velocity of the foundation are,
respectively, assumed to be

$\mu\in L^{\infty}(\Gamma_{3};\mathbb{R}),\;\mu(\bm{x})\geq
0\;a.e.\;\bm{x}\in\Gamma_{3},$ (4.28) $k\in
L^{\infty}(\Gamma_{3};\mathbb{R}),\;k(\bm{x})\geq
0\;a.e.\;\bm{x}\in\Gamma_{3},$ (4.29)

and

$\left\\{\begin{array}[]{l}\bm{v}^{*}\in C\left(I;\mathbb{R}^{3}\right)\text{
and there exist }v_{1},v_{2}>0\text{ such that }\\\
v_{1}\leq\left\|\bm{v}^{*}(t)\right\|\leq v_{2},\;\forall t\in
I.\end{array}\right.$ (4.30)

Clearly, the conditions (4.29) and (4.30) suggest that

$\bm{n}^{*}\in C\left(I;\mathbb{R}^{3}\right),\;\alpha\in
C(I;L^{\infty}(\Gamma_{3};\mathbb{R})).$ (4.31)

Moreover, $g\geq 0$ yields that $\bm{0}_{V}\in K_{V}$.

For Banach spaces $V=\\{\bm{u}\in
H^{1}(\Omega;\mathbb{R}^{d})\>|\>\bm{u}=0\mbox{ on }\Gamma_{3}\\}$ and
$Y=H^{1}(\Omega;\mathbb{R})$, Hilbert spaces $H=L^{2}(\Omega;\mathbb{R}^{d})$
and $Y_{1}=L^{2}(\Omega;\mathbb{R})$, by the basic theory of Sobolev spaces,
we know that $(V,H,V^{*})$ and $(Y,Y_{1},Y^{*})$ form two Gelfond triplets.
Let $X=L^{2}(\Gamma_{3};\mathbb{R}^{d})$ and $W=L^{2}(\Gamma_{3};\mathbb{R})$.

Now we give a unique solvability result for Problem 4.2 as follows.

######  Theorem 4.1.

Let assumptions (4.22)-(4.31) hold. If

$m_{\mathcal{C}}>\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}\tilde{L}(p)+\tilde{L}(p),$

then Problem 4.2 has a unique solution
$(\zeta,\bm{u}_{\zeta},w_{\zeta})\in(H^{1}(I;Y_{1})\cap L^{2}(I;Y))\times
C(I;K_{V})\times C^{1}(I;W)$.

###### Proof.

For any $t\in I$, define operators $A(t,\cdot):V\rightarrow V^{*}$,
$B(t,\cdot,\cdot):V\times Y\rightarrow V^{*}$, $C(t,\cdot):V\rightarrow
V^{*}$, $F(t,\cdot,\cdot):W\times V\rightarrow W$,
$\phi(t,\cdot,\cdot):V\times Y\rightarrow Y^{*}$, a functional
$j(\cdot,\cdot,\cdot):W\times X\times X\rightarrow\mathbb{R}$ and a symmetric
bilinear form $a(\cdot,\cdot):Y\times Y\rightarrow\mathbb{R}$ by setting

$\displaystyle\begin{cases}\langle A(t,\bm{u}),\bm{v}\rangle_{V^{*}\times
V}=\int_{\Omega}\mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}))\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\\
\langle B(t,\bm{u},\zeta),\bm{v}\rangle_{V^{*}\times
V}=\int_{\Omega}\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}),\zeta)\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\\
\langle C(t,\bm{u}),\bm{v}\rangle_{V^{*}\times
V}=\int_{\Omega}\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}))\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\\
F(t,w,\bm{u})=\alpha(t)p(w,u_{\nu}),\\\
\phi(t,\bm{u},\zeta)=\lambda_{D}\left(\frac{1-\zeta}{\zeta}\right)-\frac{1}{2}\lambda_{E}\|\bm{\varepsilon}(\bm{u})\|^{2}+\lambda_{w},\\\
j(w,\bm{r}_{1},\bm{r}_{2})=\int_{\Gamma_{3}}\mu
p(w,{r_{1}}_{\nu})\bm{n^{*}}{\bm{r}_{2}}_{\tau}d\Gamma+\int_{\Gamma_{3}}p(w,{r_{1}}_{\nu}){r_{2}}_{\nu}d\Gamma\end{cases}$

for all $\bm{u},\;\bm{v}\in V$, $\bm{r}_{1},\;\bm{r}_{2}\in X$, $w\in W$ and
$\zeta\in Y$. Then Problem 4.2 can be transformed as follows

$\displaystyle\dot{w}(t)=F(t,w(t),\bm{u}(t)),$ $\displaystyle\left\langle
A(t,{\bm{u}}(t))+\int_{0}^{t}B(t-s,{\bm{u}}(s),\zeta(s))ds+C(t,\dot{\bm{u}}(t)),\bm{v}-\dot{\bm{u}}(t)\right\rangle_{V^{*}\times
V}$
$\displaystyle\qquad+j(w(t),\dot{\bm{u}},\bm{v})-j(w(t),\dot{\bm{u}}(t),\dot{\bm{u}}(t))\geq\langle
f(t),\bm{v}-\dot{\bm{u}}(t)\rangle_{V^{*}\times V},\quad\forall\bm{v}\in
K_{V},$
$\displaystyle\langle\dot{\zeta}(t),\eta-\zeta(t)\rangle_{Y_{1}}+a(\zeta(t),\eta-\zeta(t))\geq\langle\phi(t,u(t),\zeta(t)),\eta-\zeta(t)\rangle_{Y_{1}},\quad\forall\eta\in
K_{Y},$ $\displaystyle\bm{u}(0)=\bm{u}_{0},w(0)=0,\zeta(0)=\zeta_{0}\in(0,1).$

Next we show that assumption conditions H(A)-H(a) are satisfied.

I). It is easy to see that $A(\cdot,\cdot)$ is continuous on $I\times V$ for
all $t\in I$. From (4.22)(c) and the Holder’s inequality, we have

$\displaystyle\langle
A(t,\bm{u}_{1})-A(t,\bm{u}_{2}),\bm{v}\rangle_{V^{*}\times V}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\|\mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{1}))-\mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{2}))\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$
$\displaystyle\leq$ $\displaystyle
L_{\mathcal{A}}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_{1})-\bm{\varepsilon}(\bm{u}_{2})\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$

for all $\bm{u}_{1},\;\bm{u}_{2},\;\bm{v}\in V$ with $t\in I$. It follows that

$\|A(t,\bm{u}_{1})-A(t,\bm{u}_{2})\|_{V^{*}}\leq
L_{\mathcal{A}}\|\bm{u}_{1}-\bm{u}_{2}\|_{V}.$

This shows that $L_{A}=L_{\mathcal{A}}$ in (4.22)(c) and so assumption
condition H(A) holds.

II). Condition H(B)(a) can be verified from (4.23)(b). In addition, by
Hölder’s inequality and (4.23)(c), one has

$\displaystyle\langle
B(t,\bm{u}_{1},\zeta)-B(t,\bm{u}_{2},\eta),\bm{v}\rangle_{V^{*}\times V}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\|\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{1}),\zeta)-\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{2}),\eta)\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$
$\displaystyle\leq$
$\displaystyle\sqrt{2}L_{\mathcal{B}}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_{1})-\bm{\varepsilon}(\bm{u}_{2})\|^{2}+\|\zeta-\eta\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$

and

$\displaystyle\langle B(t,\bm{u},\zeta),\bm{v}\rangle_{V^{*}\times V}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\|\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}),\zeta)\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$
$\displaystyle\leq$
$\displaystyle\sqrt{2}\rho_{\mathcal{B}}(t)\left(\int_{\Omega}\|\bm{\varepsilon(u)}\|^{2}+\|\zeta\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$

for all $\bm{u}_{1},\;\bm{u}_{2},\;\bm{u},\;\bm{v}\in V$ and $\zeta,\;\eta\in
Y$ with $t\in I$. It follows that

$\begin{cases}\|B(t,\bm{u}_{1},\zeta)-B(t,\bm{u}_{2},\eta)\|_{V^{*}}\leq\left(\sqrt{2}L_{\mathcal{B}}\right)\left(\|\bm{u}_{1}-\bm{u}_{2}\|_{V}+\|\zeta-\eta\|_{Y}\right),\\\
\|B(t,\bm{u},\zeta)\|_{V^{*}}\leq\left(\sqrt{2}\rho_{\mathcal{B}}(t)\right)\left(\|\bm{u}\|_{V}+\|\zeta\|_{Y}\right).\end{cases}$

Thus, $L_{B}=\sqrt{2}L_{\mathcal{B}}$ and
$\rho(t)=\sqrt{2}\rho_{\mathcal{B}}(t)$ in H(B)(b)(c) and so assumption
condition H(B) is true.

III). It is easy to see that $C(\cdot,\cdot)$ is continuous on $I\times V$ for
all $t\in I$. From (4.24)(c)(d)(e) and Hölder’s inequality, we have

$\displaystyle\langle
C(t,\bm{u})-C(t,\bm{v}),\bm{u}-\bm{v}\rangle_{V^{*}\times V}$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left(\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}))-\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{v}))\right)\bm{\cdot}\left(\bm{\varepsilon}(\bm{u})-\bm{\varepsilon}(\bm{v})\right)d\Omega$
$\displaystyle\geq$ $\displaystyle
m_{\mathcal{C}}\int_{\Omega}\|\bm{\varepsilon}(\bm{u})-\bm{\varepsilon}(\bm{v})\|^{2}d\Omega$
$\displaystyle=$ $\displaystyle m_{\mathcal{C}}\|\bm{u}-\bm{v}\|^{2}_{V},$
$\displaystyle\langle
C(t,\bm{u}_{1})-C(t,\bm{u}_{2}),\bm{v}\rangle_{V^{*}\times V}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\|\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{1}))-\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}_{2}))\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$
$\displaystyle\leq$ $\displaystyle
L_{\mathcal{C}1}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_{1})-\bm{\varepsilon}(\bm{u}_{2})\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V},$
$\displaystyle\langle
C(t_{1},\bm{u})-C(t_{2},\bm{u}),\bm{v}\rangle_{V^{*}\times V}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\|\mathcal{C}(\bm{x},t_{1},\bm{\varepsilon}(\bm{u}))-\mathcal{C}(\bm{x},t_{2},\bm{\varepsilon}(\bm{u}))\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$
$\displaystyle\leq$ $\displaystyle
L_{\mathcal{C}2}\left(\int_{\Omega}\|t_{1}-t_{2}\|^{2}d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_{V}$

for all $\bm{u}_{1},\;\bm{u}_{2},\;\bm{u},\;\bm{v}\in V$ with
$t,t_{1},t_{2}\in I$. It follows that

$\|C(t,\bm{u}_{1})-C(t,\bm{u}_{2})\|_{V^{*}}\leq
L_{\mathcal{C}1}\left(\|\bm{u}_{1}-\bm{u}_{2}\|_{V}\right),\quad\|C(t_{1},\bm{u})-C(t_{2},\bm{u})\|_{V^{*}}\leq
L_{\mathcal{C}2}\left(\|t_{1}-t_{2}\|_{V}\right).$

Thus, $L_{C1}=L_{\mathcal{C}1}$, $L_{C2}=L_{\mathcal{C}2}$ and
$m_{C}=m_{\mathcal{C}}$ in H(C)(a)(b)(c) and so assumption condition H(C) is
satisfied.

IV). Since

$j(w,\bm{r}_{1},\bm{r}_{2})=\int_{\Gamma_{3}}\mu
p(w,{r_{1}}_{\nu})\bm{n^{*}}{\bm{r}_{2}}_{\tau}+p(w,{r_{1}}_{\nu}){r_{2}}_{\nu}d\Gamma,$

we know that $j(w,\bm{r}_{1},\cdot)$ is a convex proper and lower
semicontinuous functional with respect to $\bm{r}_{2}$. Moreover, the convex
subdifferential of $j(w,\bm{r}_{1},\bm{r}_{2})$ with respect to its third
variable can be given by

$\partial j(w,\bm{r}_{1},\bm{r}_{2})=\int_{\Gamma_{3}}\mu
p(w,{r_{1}}_{\nu})\bm{n^{*}}+p(w,{r_{1}}_{\nu})\bm{\nu}d\Gamma.$

This fact combined with (4.25)(b)(c) yields that

$\displaystyle\|\partial j(w,\bm{r}_{1},\bm{r}_{2})\|_{X^{*}}$
$\displaystyle=$
$\displaystyle\left(\int_{\Gamma_{3}}\mu^{2}p^{2}({w,r_{1}}_{\nu})+p^{2}(w,{r_{1}}_{\nu})d\Gamma\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{1+\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}^{2}}\left(\int_{\Gamma_{3}}(p({w,r_{1}}_{\nu})-p(0,0))^{2}d\Gamma\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{1+\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}^{2}}\widetilde{L}_{p}\left(\int_{\Gamma_{3}}(|w|+|{r_{1}}_{\nu}|)^{2}d\Gamma\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{2+2|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}^{2}}\widetilde{L}_{p}(\|\bm{r}_{1}\|_{X}+\|w\|_{W}).$

Taking
$\sqrt{2+2\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}^{2}}\widetilde{L}_{p}=c_{1}$,
there exists a constant $c_{1}>0$ such that

$\displaystyle\|\partial j(w,u,v)\|_{X}\leq
c_{1}(\|w\|_{W}+\|u\|_{X}),\quad\forall(u,v,w)\in X\times X\times W.$ (4.32)

Letting $w_{1},\;w_{2}\in W$ and $\bm{s}_{1},\;\bm{s}_{2}\in X$, one has

$\displaystyle
j(w_{1},\bm{r}_{1},\bm{s}_{2})-j(w_{1},\bm{r}_{1},\bm{s}_{1})+j(w_{2},\bm{r}_{2},\bm{s}_{1})-j(w_{2},\bm{r}_{2},\bm{s}_{2})$
(4.33) $\displaystyle=$ $\displaystyle\int_{\Gamma_{3}}\left[\mu
p(w_{1},{r_{1}}_{\nu})\bm{n^{*}}({\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau})+p(w_{1},{r_{1}}_{\nu})({s_{2}}_{\nu}-{s_{1}}_{\nu})+\mu
p(w_{2},{r_{2}}_{\nu})\bm{n^{*}}({\bm{s}_{1}}_{\tau}-{\bm{s}_{2}}_{\tau})\right.$
$\displaystyle\mbox{}\left.+p(w_{2},{r_{2}}_{\nu})({s_{1}}_{\nu}-{s_{2}}_{\nu})\right]d\Gamma$
$\displaystyle=$
$\displaystyle\int_{\Gamma_{3}}\mu\left(p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})\right)\bm{n^{*}}({\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau})d\Gamma+\int_{\Gamma_{3}}(p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu}))({s_{2}}_{\nu}-{s_{1}}_{\nu})d\Gamma$
$\displaystyle\leq$
$\displaystyle\int_{\Gamma_{3}}|\mu||p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})|\|{\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau}\|d\Gamma+\int_{\Gamma_{3}}|p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})||{s_{2}}_{\nu}-{s_{1}}_{\nu}|d\Gamma$
$\displaystyle\leq$
$\displaystyle\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}\int_{\Gamma_{3}}|p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})|\|{\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau}\|d\Gamma$
$\displaystyle\mbox{}+\int_{\Gamma_{3}}|p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})||{s_{2}}_{\nu}-{s_{1}}_{\nu}|d\Gamma.$

The condition (4.25)(b) combined with Hölder’s inequality shows that

$\displaystyle\int_{\Gamma_{3}}|p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})|\|{\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau}\|d\Gamma$
$\displaystyle\leq$
$\displaystyle\tilde{L}_{p}\int_{\Gamma_{3}}(|{r_{1}}_{\nu}-{r_{2}}_{\nu}|+|w_{1}-w_{2}|)\|{\bm{s}_{2}}_{\tau}-{\bm{s}_{1}}_{\tau}\|d\Gamma$
$\displaystyle\leq$
$\displaystyle\tilde{L}_{p}(\|{r_{1}}_{\nu}-{r_{2}}_{\nu}\|_{X}+|w_{1}-w_{2}|_{W})\|\bm{s}_{2}-\bm{s}_{1}\|_{X}$

and

$\displaystyle\int_{\Gamma_{3}}|p(w_{1},{r_{1}}_{\nu})-p(w_{2},{r_{2}}_{\nu})|\|{s_{2}}_{\nu}-{s_{1}}_{\nu}\|d\Gamma\leq\tilde{L}_{p}(\|{r_{1}}_{\nu}-{r_{2}}_{\nu}\|_{X}+|w_{1}-w_{2}|_{W})\|\bm{s}_{2}-\bm{s}_{1}\|_{X}.$
(4.34)

Then inequality (4.33) becomes

$\displaystyle
j(w_{1},\bm{r}_{1},\bm{s}_{2})-j(w_{1},\bm{r}_{1},\bm{s}_{1})+j(w_{2},\bm{r}_{2},\bm{s}_{1})-j(w_{2},\bm{r}_{2},\bm{s}_{2})$
(4.35) $\displaystyle\leq$
$\displaystyle(\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}+1)\tilde{L}_{p}\left(\|w_{1}-w_{2}\|_{W}\|\bm{s}_{1}-\bm{s}_{2}\|_{X}+\|\bm{r}_{1}-\bm{r}_{2}\|_{X}\|\bm{s}_{1}-\bm{s}_{2}\|_{X}\right).$

This implies that H(j)(b) holds with

$\alpha_{0}=(\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}+1)\tilde{L}_{p},\quad\alpha_{1}=(\|\mu\|_{L^{\infty}(\Gamma_{3};\mathbb{R})}+1)\tilde{L}_{p}.$

and so assumption condition H(j) is satisfied.

V). It follows from (4.34) that

$\displaystyle\|F(t,w_{1},\bm{u}_{1})-F(t,w_{2},\bm{u}_{2})\|^{2}_{W}$
$\displaystyle=$
$\displaystyle\int_{\Gamma_{3}}\alpha(t)^{2}(p(w_{1},{u_{1}}_{\nu})-p(w_{2},{u_{2}}_{\nu}))^{2}d\Gamma$
$\displaystyle\leq$ $\displaystyle
2\|\alpha\|_{C(I;L^{\infty}(\Gamma_{3};\mathbb{R}))}^{2}\tilde{L}_{p}\left(\|\bm{u}_{1}-\bm{u}_{2}\|_{X}^{2}+\|w_{1}-w_{2}\|_{W}^{2}\right)$

and so

$\displaystyle\|F(t,w_{1},\bm{u}_{1})-F(t,w_{2},\bm{u}_{2})\|_{W}\leq\sqrt{2\tilde{L}_{p}}\|\alpha\|_{C(I;L^{\infty}(\Gamma_{3};\mathbb{R}))}(\|\bm{u}_{1}-\bm{u}_{2}\|_{V}+\|w_{1}-w_{2}\|_{W}).$

Thus, H(F)(b) holds with

$L_{p}=\max\left\\{\sqrt{2\tilde{L}_{p}}\|\alpha\|_{C(I;L^{\infty}(\Gamma_{3};\mathbb{R}))},\sqrt{2\tilde{L}_{p}}\|\alpha\|_{C(I;L^{\infty}(\Gamma_{3};\mathbb{R}))}\right\\}.$

Clearly, H(F)(a) follows from condition (4.31) and so assumption condition
H(F) is fulfilled.

VI). Based on (4.26), we can conclude that $H(\phi)$ holds with
$L_{\phi}=\sqrt{2}\tilde{L}_{\phi}$.

VII). Finally, we can check that H(a) holds. Indeed, from

$\|\zeta\|_{Y}^{2}=\|\zeta\|^{2}_{Y_{1}}+\int_{\Omega}\nabla\zeta\cdot\nabla\zeta
d\Omega,$

it follows that
$a(\zeta,\zeta)+\kappa\|\zeta\|^{2}_{Y_{1}}=\kappa\|\zeta\|_{Y}^{2}$ and so
H(a) is satisfied with $a_{1}=a_{2}=\kappa.$

Thus, combining I)-VII), we know that Theorem 4.1 is a direct consequence of
Theorem 3.1. ∎

## Declarations

Conflicts of interest/Competing interests The authors have no known competing
financial interests or personal relationships that could have appeared to
influence the work reported in this manuscript.

## References

  * [1] K.T. Andrews, M. Shillor, S. Wright, A. Klarbring and M. Sofonea, A dynamic thermoviscoelastic contact problem with friction and wear, International Journal of Engineering Science, 35(1997), 1291-1309.
  * [2] T. Angelov, On a rolling problem with damage and wear, Mechanics Research Communications, 26(1999), 281-286.
  * [3] B. Brogliato, A. Tanwani, Dynamical systems coupled with monotone set-valued operators: formalisms applications, well-posedness, and stability, SIAM Review, 62(2020), 3-129.
  * [4] A. Capatina, Variational Inequalities and Frictional Contact Problems, Springer, New York, 2014.
  * [5] T. Chen, N.J. Huang, X.S. Li, Y.Z. Zou, A new class of differential nonlinear system involving parabolic varitional and history-dependent hemi-variational inequalities arising in contact mechanics, Communications in Nonlinear Science and Numerical Simulation, 101(2021), 105886.
  * [6] T. Chen, N.J. Huang, Y.B. Xiao, Variational and numerical analysis of a dynamic viscoelastic contact problem with friction and wear, Optimization, 69(2020), 2003-2031.
  * [7] C. Eck, J. Jarus̆ek, M. Sofonea, A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction, European Journal of Applied Mathematics, 21(2010), 229-251.
  * [8] X.J Chen, Z.Y. Wang, Differential variational inequality approach to dynamic games with shared consteaints, Mathematical Programming, 146(1)(2014), 379-408.
  * [9] M. Frémond, B. Nedjar, Damage in concrete: the unilateral phenomenon, Nuclear Engineering and Design, 156(1995), 323-335.
  * [10] M. Frémond, B. Nedjar, Damage, gradient of damage and principle of virtual work, International Journal of Solids and Structures, 33(1996), 1083-1103.
  * [11] L. Gasiński, A. Ochal, M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Zeitschrift für Analysis und ihre Anwendungen, 34(2015), 251-276.
  * [12] J. Gwinner, On a new class of differential variational inequalities and a stability result, Mathematical Programming, 139(1-2)(2013), 205-221.
  * [13] J. Han, S. Migórski, A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage, Journal of Mathematical Analysis and Applyciations, 443(2016), 57-80.
  * [14] W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Somerville, 2001.
  * [15] A Kasri, A Touzaline, A quasistatic frictional contact problem for viscoelastic materials with long memory, Georgian Mathematical Journal, 27(2)(2020), 249-264.
  * [16] A.S. Kravchuk, P.J. Neittaanmäki, Variational and Quasi-variational Inequalities in Mechanics, Springer, Dordrecht, 2007.
  * [17] A. Kulig, Variational-hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage, Nonlinear Analysis: Real World Applications, 44(2018), 401-416.
  * [18] W. Li, X. Wang, N.J. Huang, Differential inverse variational inequalities in finite dimensional spaces, Acta Mathematica Scientia, 35(2015), 407-422.
  * [19] W. Li, Y.B. Xiao, N.J. Huang, Y.J. Cho, A class of differential inverse quasivariational inequalities in finite dimensional spaces, Journal of Nonlinear Sciences and Applications, 10(2017), 4532-4543.
  * [20] X.S. Li, N.J. Huang, D. O’Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Analysis: Theory Methods and Applications, 72(2010), 3875-3886.
  * [21] Z. Liu, S. Migórski, S. Zeng, A class of history-dependent differential variational inequalities with application to contact problems, Optimization 69 (2020), 743-775.
  * [22] Z.H. Liu, D. Motreanu, S.D. Zeng, Generalized penalty and regularization method for differential variational hemivariational inequalities, SIAM Journal on Optimization, 31(2021), 1158-1183.
  * [23] Z.H. Liu, M. Sofonea, Differential quasivariational inequalities in contact mechanics, Mathematics and Mechanics of Solids, 24(2019), 845-861.
  * [24] Z.H. Liu, S.D. Zeng, Penalty method for a class of differential variational inequalities, Applicable Analysis, 100(2021), 1574-1589.
  * [25] S. Migórski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Springer, New York, 2013.
  * [26] S. Migórski, A. Ochal, M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Analysis: Real World Applications, 22(2015), 604-618.
  * [27] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, CRC Press, New York, 1994.
  * [28] J.S. Pang, D.E. Stewart, Differential variational inequalities, Mathematical Programming, 113(2008), 345-424.
  * [29] M. Shillor, M. Sofonea, J.J. Telega, Models and Analysis of Quasistatic Contact, Springer, Berlin, 2004.
  * [30] M. Sofonea, A. Matei, History-depemdent quasivariational inequalities arising in cantact mechanics, European Journal of Applied Mathematics, 22(2011), 471-491.
  * [31] M. Sofonea, A. Matei, Mathematical Models in Contact Mechanics, Cambridge University Press, Cambridge, 2012.
  * [32] M. Sofonea, F. Pătrulescu, Y. Souleiman, Analysis of a contact problem with wear and unilateral constraint, Applicable Analysis, 95(2016), 2590-2607.
  * [33] M. Sofonea, Y.B. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Applicable Analysis, 95(2016), 2464-2484.
  * [34] X. Wang, Y.W Qi, C.Q. Tao, Y.B. Xiao, A new class of delay differential variational inequalities, Journal of Optimization Theory and Applications, 172(1)(2017), 56-69.
  * [35] Y.H. Weng, T. Chen, N.J. Huang, A new fracrional nolinear system driven by a quasi-hemivariational inequality with an application, Journal of Nonlinear and Convex Analysis, 22(3)(2021), 559-586.
  * [36] Y.H. Weng, T. Chen, X.S. Li, N.J. Huang, Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application, Computers and Mathematics with Applications, 98(2021), 118-138.
  * [37] S.D. Zeng, Z.H. Liu, S. Migórski, A class of fractional differential hemivariational inequalities with application to contact problem, Zwitscgrift für Angewandte Mathematik und Physik, 69(2018), 36.
  * [38] S.D. Zeng, S. Migórski, A.A. Khan, Nonlinear quasi-hemivariational inequalities: existence and optimal control, SIAM Journal on Control and Optimization, 59(2021), 1246-1274.
  * [39] S.D. Zeng, S. Migórski, Z.H. Liu, J.C. Yao, Convergence of a generalized penalty method for variational-hemivariational inequalities, Communications in Nonlinear Science and Numerical Simulation, 92(2021), 105476.