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  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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[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.  Google Scholar

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

[42]

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

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[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.  Google Scholar

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

[42]

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

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

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