December  2020, 9(4): 1089-1114. doi: 10.3934/eect.2020047

Fully history-dependent evolution hemivariational inequalities with constraints

1. 

College of Sciences, Beibu Gulf University, Qinzhou, Guangxi 535000, China

2. 

Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland

3. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

4. 

College of Sciences, Beibu Gulf University, Qinzhou, Guangxi 535000, China

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  December 2020 Early access  March 2020

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

Citation: Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations and Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047
References:
[1]

J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity, IMA J. Numer. Anal., 29 (2009), 43-71.  doi: 10.1093/imanum/drm029.

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145. 

[3]

S. Carl, V.K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics. Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[4]

S. CarlV.K. Le and D. Motreanu, Evolutionary variational-hemivariational inequalities: Existence and comparison results, J. Math. Anal. Appl., 345 (2008), 545-558.  doi: 10.1016/j.jmaa.2008.04.005.

[5]

O. ChadliQ.H. Ansari and S. Al-Homidan, Existence of solutions for nonlinear implicit differential equations: An equilibrium problem approach, Numer. Func. Anal. Optim., 37 (2016), 1385-1419.  doi: 10.1080/01630563.2016.1210164.

[6]

O. ChadliQ.H. Ansari and J.-C. Yao, Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.  doi: 10.1007/s10957-015-0707-y.

[7]

F.H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[8]

M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. angew. Math. Phys., 53 (2002), 1099-1109.  doi: 10.1007/PL00012615.

[9]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.

[11] C. EckJ. Jarušek and M. Krbeč, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, 270. Chapman/CRC Press, New York, Boca Raton, FL, 2005.  doi: 10.1201/9781420027365.
[12]

C. EckJ. Jarušek and M. Sofonea, A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction, European J. Appl. Math., 21 (2010), 229-251.  doi: 10.1017/S0956792510000045.

[13]

E.-H. Essoufi and M. Kabbaj, Existence of solutions of a dynamic Signorini's problem with nonlocal friction for viscoelastic piezoelectric materials, Bull. Math. Soc. Sc. Math. Roumanie (N.S.), 48 (2005), 181-195. 

[14]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I. Unilateral Analysis and Unilateral Mechanics, Nonconvex Optimization and its Applications, 69. Kluwer Academic Publishers, Boston, MA, 2003.

[15]

D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications. Vol. Ⅱ: Unilateral Problems, Nonconvex Optimization and its Applications, 70. Kluwer Academic Publishers, Boston, MA, 2003.

[16]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.  doi: 10.1080/02331930801951116.

[17]

J.F. HanS. Migórski and H.D. Zeng, Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response, Nonlinear Anal. Real World Appl., 28 (2016), 229-250.  doi: 10.1016/j.nonrwa.2015.10.004.

[18]

W.M. HanS. Migórski and M. Sofonea, Analysis of a general dynamic history-dependent variational-hemivariational inequality, Nonlinear Anal. Real World Appl., 36 (2017), 69-88.  doi: 10.1016/j.nonrwa.2016.12.007.

[19] W.M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002. 
[20]

J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5.

[21]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.  doi: 10.1007/BF00932654.

[22]

A. Kulig and S. Migórski, Solvability and continuous dependence results for second order nonlinear inclusion with Volterra-type operator, Nonlinear Analysis, 75 (2012), 4729-4746.  doi: 10.1016/j.na.2012.03.023.

[23]

K. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electron. J. Differ. Equ., 2004 (2004), 21 pp.

[24]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[25]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0.

[26]

S. MigórskiA. Ochal and M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 12 (2011), 3384-3396.  doi: 10.1016/j.nonrwa.2011.06.002.

[27]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[28]

S. MigórskiA. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 22 (2015), 604-618.  doi: 10.1016/j.nonrwa.2014.09.021.

[29]

S. MigórskiA. Ochal and M. Sofonea, Evolutionary inclusions and hemivariational inequalities, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 39-64.  doi: 10.1007/978-3-319-14490-0_2.

[30]

S. Migórski and J. Ogorzaly, A class of evolution variational inequalities with memory and its application to viscoelastic frictional contact problems, J. Math. Anal. Appl., 442 (2016), 685-702.  doi: 10.1016/j.jmaa.2016.04.076.

[31]

S. Migórski and J. Ogorzaly, Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics, Z. angew. Math. Phys., 68 (2017), Art. 15, 22 pp. doi: 10.1007/s00033-016-0758-4.

[32]

S. MigórskiM. Sofonea and S.D. Zeng, Well-posedness of history-dependent sweeping processes, SIAM J. Math. Anal., 51 (2019), 1082-1107.  doi: 10.1137/18M1201561.

[33]

D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and its Applications, 29. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9.

[34]

Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.

[35]

P.D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410.

[36]

P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1.

[37]

P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.

[38]

M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys., 655. Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.

[39] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Notes Series, 398. Cambridge University Press, Cambridge, 2012. 
[40] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2018. 
[41]

M. SofoneaS. Migórski and A. Ochal, Two history-dependent contact problems, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 355-380.  doi: 10.1007/978-3-319-14490-0_14.

[42]

M. Sofonea, Y.-B. Xiao and S.D. Zeng, Generalized penalty method for history-dependent variational–hemivariational inequalities, submitted.

[43]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of split problems, submitted.

[44]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of a viscoplastic contact problem, Evolution Equations and Control Theory, to appear.

[45]

Y.-M. WangY.-B. XiaoX. Wang and Y.J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.

[46]

Y.-B. Xiao and M. Sofonea, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484.  doi: 10.1080/00036811.2015.1093623.

[47]

E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity, IMA J. Numer. Anal., 29 (2009), 43-71.  doi: 10.1093/imanum/drm029.

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145. 

[3]

S. Carl, V.K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics. Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[4]

S. CarlV.K. Le and D. Motreanu, Evolutionary variational-hemivariational inequalities: Existence and comparison results, J. Math. Anal. Appl., 345 (2008), 545-558.  doi: 10.1016/j.jmaa.2008.04.005.

[5]

O. ChadliQ.H. Ansari and S. Al-Homidan, Existence of solutions for nonlinear implicit differential equations: An equilibrium problem approach, Numer. Func. Anal. Optim., 37 (2016), 1385-1419.  doi: 10.1080/01630563.2016.1210164.

[6]

O. ChadliQ.H. Ansari and J.-C. Yao, Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.  doi: 10.1007/s10957-015-0707-y.

[7]

F.H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[8]

M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. angew. Math. Phys., 53 (2002), 1099-1109.  doi: 10.1007/PL00012615.

[9]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.

[11] C. EckJ. Jarušek and M. Krbeč, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, 270. Chapman/CRC Press, New York, Boca Raton, FL, 2005.  doi: 10.1201/9781420027365.
[12]

C. EckJ. Jarušek and M. Sofonea, A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction, European J. Appl. Math., 21 (2010), 229-251.  doi: 10.1017/S0956792510000045.

[13]

E.-H. Essoufi and M. Kabbaj, Existence of solutions of a dynamic Signorini's problem with nonlocal friction for viscoelastic piezoelectric materials, Bull. Math. Soc. Sc. Math. Roumanie (N.S.), 48 (2005), 181-195. 

[14]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I. Unilateral Analysis and Unilateral Mechanics, Nonconvex Optimization and its Applications, 69. Kluwer Academic Publishers, Boston, MA, 2003.

[15]

D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications. Vol. Ⅱ: Unilateral Problems, Nonconvex Optimization and its Applications, 70. Kluwer Academic Publishers, Boston, MA, 2003.

[16]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.  doi: 10.1080/02331930801951116.

[17]

J.F. HanS. Migórski and H.D. Zeng, Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response, Nonlinear Anal. Real World Appl., 28 (2016), 229-250.  doi: 10.1016/j.nonrwa.2015.10.004.

[18]

W.M. HanS. Migórski and M. Sofonea, Analysis of a general dynamic history-dependent variational-hemivariational inequality, Nonlinear Anal. Real World Appl., 36 (2017), 69-88.  doi: 10.1016/j.nonrwa.2016.12.007.

[19] W.M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002. 
[20]

J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5.

[21]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.  doi: 10.1007/BF00932654.

[22]

A. Kulig and S. Migórski, Solvability and continuous dependence results for second order nonlinear inclusion with Volterra-type operator, Nonlinear Analysis, 75 (2012), 4729-4746.  doi: 10.1016/j.na.2012.03.023.

[23]

K. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electron. J. Differ. Equ., 2004 (2004), 21 pp.

[24]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[25]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0.

[26]

S. MigórskiA. Ochal and M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 12 (2011), 3384-3396.  doi: 10.1016/j.nonrwa.2011.06.002.

[27]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[28]

S. MigórskiA. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 22 (2015), 604-618.  doi: 10.1016/j.nonrwa.2014.09.021.

[29]

S. MigórskiA. Ochal and M. Sofonea, Evolutionary inclusions and hemivariational inequalities, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 39-64.  doi: 10.1007/978-3-319-14490-0_2.

[30]

S. Migórski and J. Ogorzaly, A class of evolution variational inequalities with memory and its application to viscoelastic frictional contact problems, J. Math. Anal. Appl., 442 (2016), 685-702.  doi: 10.1016/j.jmaa.2016.04.076.

[31]

S. Migórski and J. Ogorzaly, Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics, Z. angew. Math. Phys., 68 (2017), Art. 15, 22 pp. doi: 10.1007/s00033-016-0758-4.

[32]

S. MigórskiM. Sofonea and S.D. Zeng, Well-posedness of history-dependent sweeping processes, SIAM J. Math. Anal., 51 (2019), 1082-1107.  doi: 10.1137/18M1201561.

[33]

D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and its Applications, 29. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9.

[34]

Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.

[35]

P.D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410.

[36]

P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1.

[37]

P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.

[38]

M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys., 655. Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.

[39] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Notes Series, 398. Cambridge University Press, Cambridge, 2012. 
[40] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2018. 
[41]

M. SofoneaS. Migórski and A. Ochal, Two history-dependent contact problems, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 355-380.  doi: 10.1007/978-3-319-14490-0_14.

[42]

M. Sofonea, Y.-B. Xiao and S.D. Zeng, Generalized penalty method for history-dependent variational–hemivariational inequalities, submitted.

[43]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of split problems, submitted.

[44]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of a viscoplastic contact problem, Evolution Equations and Control Theory, to appear.

[45]

Y.-M. WangY.-B. XiaoX. Wang and Y.J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.

[46]

Y.-B. Xiao and M. Sofonea, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484.  doi: 10.1080/00036811.2015.1093623.

[47]

E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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