Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Inclusions and Hemivariational Inequalities pp 3–22Cite as

Preliminaries

  • Chapter
  • First Online:

  • 1162 Accesses

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 26))

Abstract

In this chapter we present preliminary material from functional analysis which will be used subsequently. The results are stated without proofs, since they are standard and can be found in many references. For the convenience of the reader we summarize definitions and results on normed spaces, Banach spaces, duality, and weak topologies which are mostly assumed to be known as a basic material from functional analysis. We then recall some standard results on measure theory that will be applied repeatedly in this book. We assume that the reader has some familiarity with the notions of linear algebra and general topology.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier, Oxford (2003)

    MATH  Google Scholar 

  3. Ahmed, N.U., Kerbal, S.: Optimal control of nonlinear second order evolution equations. J. Appl. Math. Stoch. Anal. 6, 123–136 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amassad, A., Fabre, C., Sofonea, M.: A quasistatic viscoplastic contact problem with normal compliance and friction. IMA J. Appl. Math. 69, 463–482 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Amassad, A., Shillor, M., Sofonea, M.: A quasistatic contact problem with slip dependent coefficient of friction. Math. Methods Appl. Sci. 22, 267–284 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Amassad, A., Sofonea, M.: Analysis of a quasistatic viscoplastic problem involving Tresca friction law. Discrete Contin. Dyn. Syst. 4, 55–72 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Andersson, L.-E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. 16, 347–370 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Andersson, L.-E.: A global existence result for a quasistatic contact problem with friction. Adv. Math. Sci. Appl. 5, 249–286 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Andersson, L.-E.: A quasistatic frictional problem with a normal compliance penalization term. Nonlinear Anal. 37, 689–705 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Andersson, L.-E.: Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim. 42, 169–202 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005)

    MATH  Google Scholar 

  12. Atkinson, K., Han, W.: Theoretical numerical analysis: a functional analysis framework, 3rd edn. Texts in Applied Mathematics, vol. 39, Springer, New York (2009)

    Google Scholar 

  13. Aubin, J.P.: L’analyse non linéaire et ses motivations économiques. Masson, Paris, (1984)

    Google Scholar 

  14. Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory. Springer, Berlin, New York, Tokyo (1984)

    Book  MATH  Google Scholar 

  15. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel, Boston (1990)

    MATH  Google Scholar 

  16. Awbi, B., Rochdi, M., Sofonea, M.: Abstract evolution equations for viscoelastic frictional contact problems. Z. Angew. Math. Phys. 51, 218–235 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems. Wiley, Chichester (1984)

    MATH  Google Scholar 

  18. Barboteu, M., Fernandez, J.R., Raffat, T.: Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity. Comput. Methods Appl. Mech. Engrg. 197, 3724–3732 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Barboteu, M., Han, W., Sofonea, M.: Numerical analysis of a bilateral frictional contact problem for linearly elastic materials. IMA J. Numer. Anal. 22, 407–436 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Barboteu, M., Sofonea, M.: Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation. Appl. Math. Comput. 215, 2978–2991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Barboteu, M., Sofonea, M.: Modelling and analysis of the unilateral contact of a piezoelectric body with a conductive support. J. Math. Anal. Appl. 358, 110–124 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei, Bucharest-Noordhoff, Leyden (1976)

    Book  MATH  Google Scholar 

  23. Barbu, V.: Optimal Control of Variational Inequalities. Pitman, Boston (1984)

    MATH  Google Scholar 

  24. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. D. Reidel Publishing Company, Dordrecht (1986)

    MATH  Google Scholar 

  25. Bartosz, K.: Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear. Nonlinear Anal. 65, 546–566 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bartosz, K.: Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal. 71, 1747–1762 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Batra, R.C., Yang, J.S.: Saint-Venant’s principle in linear piezoelectricity. J. Elasticity 38, 209–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bian, W.: Existence results for second order nonlinear evolution inclusions. Indian J. Pure Appl. Math. 11, 1177–1193 (1998)

    MathSciNet  Google Scholar 

  29. Bisegna, P., Lebon, F., Maceri, F.: The unilateral frictional contact of a piezoelectric body with a rigid support. In: Martins, J.A.C., Marques, M.D.P.M. (eds.), Contact Mechanics, pp. 347–354. Kluwer, Dordrecht (2002)

    Google Scholar 

  30. Brézis, H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)

    Google Scholar 

  31. Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)

    Google Scholar 

  32. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematics Studies, North Holland, Amsterdam (1973)

    MATH  Google Scholar 

  33. Brézis, H.: Analyse fonctionnelle—Théorie et applications. Masson, Paris (1987)

    Google Scholar 

  34. Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)

    MATH  Google Scholar 

  35. Browder, F., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  36. Campillo, M., Dascalu, C., Ionescu, I.R.: Instability of a periodic system of faults. Geophys. J. Int. 159, 212–222 (2004)

    Article  Google Scholar 

  37. Campillo, M., Ionescu, I.R.: Initiation of antiplane shear instability under slip dependent friction. J. Geophys. Res. 102 B9, 363–371 (1997)

    Google Scholar 

  38. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Springer, New York (2007)

    MATH  Google Scholar 

  39. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580, Springer, Berlin (1977)

    Google Scholar 

  40. Cazenave, T., Haraux, A.: Introduction aux problèmes d’évolution semi-linéaires. Ellipses, Paris (1990).

    MATH  Google Scholar 

  41. Chang, K.C.: Variational methods for nondifferentiable functionals and applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  42. Chau, O., Han, W., Sofonea, M.: Analysis and approximation of a viscoelastic contact problem with slip dependent friction. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 8, 153–174 (2001)

    MathSciNet  MATH  Google Scholar 

  43. Chau, O., Motreanu, D., Sofonea, M.: Quasistatic frictional problems for elastic and viscoelastic materials. Appl. Math. 47, 341–360 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ciarlet, P.G.: Mathematical Elasticity, vol. I: Three-dimensional elasticity. Studies in Mathematics and its Applications, vol. 20, North-Holland, Amsterdam (1988)

    Google Scholar 

  45. Ciulcu, C., Motreanu, D., Sofonea, M.: Analysis of an elastic contact problem with slip dependent coefficient of friction. Math. Inequal. Appl. 4, 465–479 (2001)

    MathSciNet  MATH  Google Scholar 

  46. Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  47. Clarke, F.H.: Generalized gradients of Lipschitz functionals. Adv. Math. 40, 52–67 (1981)

    Article  MATH  Google Scholar 

  48. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience, New York (1983)

    MATH  Google Scholar 

  49. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    MATH  Google Scholar 

  50. Cocu, M.: Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci. 22, 567–581 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  51. Cocu, M., Pratt, E., Raous, M.: Existence d’une solution du problème quasistatique de contact unilatéral avec frottement non local. C.R. Acad. Sci. Paris Sér. I Math. 320, 1413–1417 (1995)

    Google Scholar 

  52. Cocu, M., Pratt, E., Raous, M.: Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci. 34, 783–798 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. Cocu, M., Ricaud, J.M.: Existence results for a class of implicit evolution inequalities and application to dynamic unilateral contact problems with friction. C.R. Acad. Sci. Paris Sér. I Math. 329, 839–844 (1999)

    Google Scholar 

  54. Cocu, M., Ricaud, J.M.: Analysis of a class of implicit evolution inequalities associated to dynamic contact problems with friction. Int. J. Eng. Sci. 328, 1534–1549 (2000)

    MathSciNet  Google Scholar 

  55. Cohn, D.L., Measure Theory. Birkhäuser, Boston (1980)

    MATH  Google Scholar 

  56. Corneschi, C., Hoarau-Mantel, T.-V., Sofonea, M.: A quasistatic contact problem with slip dependent coefficient of friction for elastic materials. J. Appl. Anal. 8, 59–80 (2002)

    Article  MathSciNet  Google Scholar 

  57. Cristescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publishers, Editura Tehnicǎ, Bucharest (1982)

    MATH  Google Scholar 

  58. Deimling, K.: Differential Inclusions. Springer, New York, Heidelberg, Berlin (1980)

    Google Scholar 

  59. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  60. Demkowicz, I., Oden, J.T.: On some existence and uniqueness results in contact problems with non local friction. Nonlinear Anal. 6, 1075–1093 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Denkowski, Z., Migórski S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  62. Denkowski, Z., Migórski, S.: Hemivariational inequalities in thermoviscoelasticity. Nonlinear Anal. 63, e87–e97 (2005)

    MATH  Google Scholar 

  63. Denkowski, Z., Migórski, S.: On sensitivity of optimal solutions to control problems for hyperbolic hemivariational inequalities. In: Cagnol, J., Zolesio, J.-P. (eds.), Control and Boundary Analysis, pp. 147–156. Marcel Dekker, New York (2005)

    Google Scholar 

  64. Denkowski, Z., Migórski, S., Ochal, A.: Existence and uniqueness to a dynamic bilateral frictional contact problem in viscoelasticity. Acta Appl. Math. 94, 251–276 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Denkowski, Z., Migórski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems. Control Cybernet. 36, 611–632 (2007)

    MathSciNet  MATH  Google Scholar 

  66. Denkowski, Z., Migórski, S., Papageorgiu, N.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)

    Google Scholar 

  67. Denkowski, Z., Migórski, S., Papageorgiu, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)

    Google Scholar 

  68. Diestel, J., Uhl, J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15, AMS, Providence, RI (1977)

    Google Scholar 

  69. Dinculeanu, N.: Vector Measures. Akademie, Berlin (1966)

    MATH  Google Scholar 

  70. Doghri, I.: Mechanics of Deformable Solids. Springer, Berlin (2000)

    MATH  Google Scholar 

  71. Drozdov, A.D.: Finite Elasticity and Viscoelasticity—A Course in the Nonlinear Mechanics of Solids. World Scientific, Singapore (1996)

    Book  MATH  Google Scholar 

  72. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  73. Dunford, N., Schwartz, J.: Linear Operators. Part I, Wiley, New York (1958)

    MATH  Google Scholar 

  74. Duvaut, G.: Loi de frottement non locale. J. Méc. Th. et Appl. Special issue, 73–78 (1982)

    Google Scholar 

  75. Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Methods Appl. Sci. 8, 445–468 (1998)

    Article  MATH  Google Scholar 

  76. Eck, C., Jarušek, J.: Existence results for the semicoercive static contact problem with Coulomb friction. Nonlinear Anal. 42, 961–976 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  77. Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, vol. 270, Chapman/CRC Press, New York (2005)

    Google Scholar 

  78. Eck, C., Jarušek, J., Sofonea, M.: A Dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction. European J. Appl. Math. 21, 229–251 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  79. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  80. Evans, L.C.: Partial Differential Equations. AMS Press, Providence (1999)

    Google Scholar 

  81. Fichera, G.: Problemi elastostatici con vincoli unilaterali. II. Problema di Signorini con ambique condizioni al contorno. Mem. Accad. Naz. Lincei, VIII, vol. 7, 91–140 (1964)

    MathSciNet  MATH  Google Scholar 

  82. Friedman, A.: Variational Principles and Free-boundary Problems. Wiley, New York (1982)

    MATH  Google Scholar 

  83. Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL (2006)

    Google Scholar 

  84. Gasinski, L., Smolka, M.: An existence theorem for wave-type hyperbolic hemivariational inequalities. Math. Nachr. 242, 1–12 (2002)

    Article  MathSciNet  Google Scholar 

  85. Germain, P., Muller, P.: Introduction à la mécanique des milieux continus, Masson, Paris (1980)

    MATH  Google Scholar 

  86. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    MATH  Google Scholar 

  87. Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  88. Goeleven, D., Miettinen, M., Panagiotopoulos, P.D.: Dynamic hemivariational inequalities and their applications. J. Optim. Theory Appl. 103, 567–601 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  89. Goeleven, D., Motreanu, D.: Hyperbolic hemivariational inequality and nonlinear wave equation with discontinuities. In: Gilbert, R.P. et al. (eds.) From Convexity to Nonconvexity, pp. 111–122. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  90. Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities, Theory, Methods and Applications, vol. I: Unilateral Analysis and Unilateral Mechanics. Kluwer Academic Publishers, Boston, Dordrecht, London (2003)

    Google Scholar 

  91. Goeleven, D., Motreanu, D.: Variational and Hemivariational Inequalities, Theory, Methods and Applications, vol. II: Unilateral Problems. Kluwer Academic Publishers, Boston, Dordrecht, London (2003)

    Google Scholar 

  92. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  93. Guo, X.: On existence and uniqueness of solution of hyperbolic differential inclusion with discontinuous nonlinearity. J. Math. Anal. Appl. 241, 198–213 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  94. Guo, X.: The initial boundary value problem of mixed-typed hemivariational inequality. Int. J. Math. Math. Sci. 25, 43–52 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  95. Guran, A. (ed.): Proceedings of the First International Symposium on Impact and Friction of Solids, Structures and Intelligent Machines. World Scientific, Singapore (2000)

    Google Scholar 

  96. Guran, A., Pfeiffer, F., Popp, K. (eds.): Dynamics with friction: modeling, analysis and experiment, Part I. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  97. Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)

    MATH  Google Scholar 

  98. Halmos, P.R.: Measure Theory. D. Van Nostrand Company, Princeton, Toronto, London, New York (1950)

    MATH  Google Scholar 

  99. Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)

    MATH  Google Scholar 

  100. Han, W., Sofonea, M.: Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM J. Numer. Anal. 38, 556–579 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  101. Han W., Sofonea, M.: Time-dependent variational inequalities for viscoelastic contact problems, J. Comput. Appl. Math. 136, 369–387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  102. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI-International Press, Somerville, MA (2002)

    Google Scholar 

  103. Haslinger, J., Hlaváček, I., Nečas J.: Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis, vol. IV, Ciarlet, P.G., and J.-L. Lions (eds.), North-Holland, Amsterdam, pp. 313–485 (1996)

    Google Scholar 

  104. Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London (1999)

    Google Scholar 

  105. Hewitt, E., Stromberg, K.: Abstract Analysis. Springer, New York, Inc. (1965)

    Book  MATH  Google Scholar 

  106. Hild, P.: On finite element uniqueness studies for Coulomb’s frictional contact model. Int. J. Appl. Math. Comput. Sci. 12, 41–50 (2002)

    MathSciNet  MATH  Google Scholar 

  107. Hiriart-Urruty, J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms, I, II. Springer, Berlin (1993)

    Google Scholar 

  108. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  109. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I: theory. Kluwer, Dordrecht, The Netherlands (1997)

    Google Scholar 

  110. Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990)

    Google Scholar 

  111. Ionescu, I.R., Dascalu, C., Campillo, M.: Slip-weakening friction on a periodic System of faults: spectral analysis. Z. Angew. Math. Phys. 53, 980–995 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  112. Ionescu, I.R., Nguyen, Q.-L.: Dynamic contact problems with slip dependent friction in viscoelasticity. Int. J. Appl. Math. Comput. Sci. 12, 71–80 (2002)

    MathSciNet  MATH  Google Scholar 

  113. Ionescu, I.R., Nguyen, Q.-L., Wolf, S.: Slip displacement dependent friction in dynamic elasticity. Nonlinear Anal. 53, 375–390 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  114. Ionescu, I.R., Paumier, J.-C.: Friction dynamique avec coefficient dépandant de la vitesse de glissement. C.R. Acad. Sci. Paris Sér. I Math. 316, 121–125 (1993)

    Google Scholar 

  115. Ionescu, I.R., Paumier, J.-C.: On the contact problem with slip rate dependent friction in elastodynamics. Eur. J. Mech. A Solids 13, 556–568 (1994)

    MathSciNet  Google Scholar 

  116. Ionescu, I.R., Paumier, J.-C.: On the contact problem with slip displacement dependent friction in elastostatics. Int. J. Eng. Sci. 34, 471–491 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  117. Ionescu, I.R., Sofonea, M.: Functional and Numerical Methods in Viscoplasticity. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  118. Ionescu, I.R., Wolf, S.: Interaction of faults under slip dependent friction. Nonlinear eingenvalue analysis. Math. Methods Appl. Sci. 28, 77–100 (2005)

    MathSciNet  MATH  Google Scholar 

  119. Jarušek, J.: Contact problem with given time-dependent friction force in linear viscoelasticity. Comment. Math. Univ. Carolin. 31, 257–262 (1990)

    MathSciNet  MATH  Google Scholar 

  120. Jarušek, J.: Dynamic contact problems with given friction for viscoelastic bodies. Czechoslovak Math. J. 46, 475–487 (1996)

    MathSciNet  MATH  Google Scholar 

  121. Jarušek, J., Eck, C.: Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Models Methods Appl. Sci. 9, 11–34 (1999)

    Article  MATH  Google Scholar 

  122. Jarušek, J., Málek, J., Nečas, J., Šverák, V.: Variational inequality for a viscous drum vibrating in the presence of an obstacle. Rend. Mat. Appl. 12, 943–958 (1992)

    MathSciNet  MATH  Google Scholar 

  123. Jarušek, J., Sofonea, M.: On the solvability of dynamic elastic-visco-plastic contact problems. ZAMM Z. Angew. Math. Mech. 88, 3–22 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  124. Jarušek, J., Sofonea, M.: On the solvability of dynamic elastic-visco-plastic contact problems with adhesion. Ann. Acad. Rom. Sci. Ser. Math. Appl. 1, 191–214 (2009)

    MathSciNet  Google Scholar 

  125. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  126. Kalita, P.: Decay of energy of a system described by hyperbolic hemivariational inequality. Nonlinear Anal. 74, 116–1181 (2011)

    Article  MathSciNet  Google Scholar 

  127. Khludnev, A.M., Sokolowski, J.: Modelling and Control in Solid Mechanics. Birkhäuser-Verlag, Basel (1997)

    MATH  Google Scholar 

  128. Kikuchi, N., Oden, J.T.: Theory of variational inequalities with applications to problems of flow through porous media. Int. J. Eng. Sci. 18, 1173–1284 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  129. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)

    Book  MATH  Google Scholar 

  130. Kim, J.U.: A boundary thin obstacle problem for a wave equation. Comm. Part. Differ. Equat. 14, 1011–1026 (1989)

    Article  MATH  Google Scholar 

  131. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Classics in Applied Mathematics, vol. 31, SIAM, Philadelphia (2000)

    Google Scholar 

  132. Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer, Dordrecht, The Netherlands (1991)

    Google Scholar 

  133. Klarbring, A., Mikelič, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)

    Article  MATH  Google Scholar 

  134. Klarbring, A., Mikelič, A., Shillor, M.: On friction problems with normal compliance. Nonlinear Anal. 13, 935–955 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  135. Komura, Y.: Nonlinear semigroups in Hilbert Spaces. J. Math. Soc. Japan 19, 493–507 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  136. Kufner, A., John, O., Fučik, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)

    Google Scholar 

  137. Kulig, A.: Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction. Nonlinear Anal. Real World Appl. 9, 1741–1755 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  138. Kuttler, K.L.: Dynamic friction contact problem with general normal and friction laws. Nonlinear Anal. 28, 559–575 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  139. Kuttler, K.: Topics in Analysis. Private communication (2006)

    Google Scholar 

  140. Kuttler, K.L., Shillor, M.: Set-valued pseudomonotone maps and degenerate evolution inclusions. Commun. Contemp. Math. 1, 87–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  141. Kuttler, K.L., Shillor, M.: Dynamic bilateral contact with discontinuous friction coefficient. Nonlinear Anal. 45, 309–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  142. Kuttler, K.L., Shillor, M.: Dynamic contact with normal compliance, wear and discontinuous friction coefficient. SIAM J. Math. Anal. 34, 1–27 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  143. Larsen, R.: Functional Analysis. Marcel Dekker, New York (1973)

    MATH  Google Scholar 

  144. Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)

    MATH  Google Scholar 

  145. Lemaître, J., Chaboche, J.-L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  146. Lerguet, Z., Shillor, M., Sofonea, M.: A frictional contact problem for an electro-viscoelastic body. Electron. J. Differential Equations 170, 1–16 (2007)

    MathSciNet  Google Scholar 

  147. Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Anal. 71, 1414–1424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  148. Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with normal damped response and friction. J. Math. Anal. Appl. 373, 726–738 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  149. Lions, J.-L.: Quelques méthodes de resolution des problémes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  150. Lions, J.-L., Magenes, E.: Problèmes aux limites non-homogènes I. Dunod, Paris (1968); English translation: Non-homogeneous boundary value problems and applications, vol. I. Springer, New York, Heidelberg (1972)

    Google Scholar 

  151. Liu, Z., Migórski, S.: Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 9, 129–143 (2008)

    MathSciNet  MATH  Google Scholar 

  152. Liu, Z., Migórski, S., Ochal, A.: Homogenization of boundary hemivariational inequalities in linear elasticity. J. Math. Anal. Appl. 340, 1347–1361 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  153. Maceri, F., Bisegna, P.: The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Model. 28, 19–28 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  154. Malvern, L.E.: Introduction to the Mechanics of a Continuum Medium. Princeton-Hall, NJ (1969)

    Google Scholar 

  155. Martins, J.A.C., Marques, M.D.P.M. (eds.): Contact Mechanics. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  156. Martins, J.A.C., Oden, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11, 407–428 (1987)

    Article  MathSciNet  Google Scholar 

  157. Matysiak, S.J. (ed.): Contact Mechanics, Special issue of J. Theoret. Appl. Mech. 39(3) (2001)

    Google Scholar 

  158. Maugis, D.: Contact, Adhesion and Rupture of Elastic Solids. Springer, Berlin, Heidelberg (2000)

    MATH  Google Scholar 

  159. Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  160. Migórski, S.: Existence, variational and optimal control problems for nonlinear second order evolution inclusions. Dynam. Systems Appl. 4, 513–528 (1995)

    MathSciNet  MATH  Google Scholar 

  161. Migórski, S.: Existence and relaxation results for nonlinear second order evolution inclusions. Discuss. Math. Differ. Incl. Control Optim. 15, 129–148 (1995)

    MATH  Google Scholar 

  162. Migórski, S.: Evolution hemivariational inequalities in infinite dimension and their control. Nonlinear Anal. 47, 101–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  163. Migórski, S.: Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Probl. Sci. Eng. 11, 229–242 (2003)

    Article  Google Scholar 

  164. Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)

    MathSciNet  MATH  Google Scholar 

  165. Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  166. Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31, 505–533 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  167. Migórski, S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 6, 1339–1356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  168. Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  169. Migórski, S.: A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction. Discrete Contin. Dyn. Syst. Ser. S 1, 117–126 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  170. Migórski, S.: Evolution hemivariational inequalities with spplications. In: Gao, D.Y., Motreanu, D. (eds.), Handbook of Nonconvex Analysis and Applications, pp. 409–473. International Press, Boston (2010)

    Google Scholar 

  171. Migórski, S., Ochal, A.: Inverse coefficient problem for elliptic hemivariational inequality. In: Gao, D.Y., Ogden, R.W., Stavroulakis, G.E. (eds.), Nonsmooth/Nonconvex Mechanics: Modeling, Analysis and Numerical Methods, pp. 247–262. Kluwer Academic Publishers, Dordrecht, Boston, London (2001)

    Google Scholar 

  172. Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip-dependent friction. Nonlinear Anal. 61, 135–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  173. Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  174. Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elasticity 83, 247–275 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  175. Migórski, S., Ochal, A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  176. Migórski, S.,Ochal, A.: Nonlinear impulsive evolution inclusions of second order. Dynam. Systems Appl. 16, 155–174 (2007)

    MathSciNet  MATH  Google Scholar 

  177. Migórski, S., Ochal, A.: Vanishing viscosity for hemivariational inequality modeling dynamic problems in elasticity. Nonlinear Anal. 66, 1840–1852 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  178. Migórski, S., Ochal, A.: Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. 69, 495–509 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  179. Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  180. Migórski, S., Ochal, A.: Nonconvex inequality models for contact problems of nonsmooth mechanics. In: Kuczma, M., Wilmanski, K. (eds.) Computer Methods in Mechanics: Computational Contact Mechanics, Advanced Structured Materials, vol. 1, pp. 43–58. Springer, Berlin, Heidelberg (2010)

    Google Scholar 

  181. Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  182. Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. 70, 3738–3748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  183. Migórski, S., Ochal, A., Sofonea, M.: Modeling and analysis of an antiplane piezoelectric contact problem. Math. Models Methods Appl. Sci. 19, 1295–1324 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  184. Migórski, S., Ochal, A., Sofonea, M.: Variational analysis of static frictional contact problems for electro-elastic materials. Math. Nachr. 283, 1314–1335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  185. Migórski, S., Ochal, A., Sofonea, M.: A dynamic frictional contact problem for piezoelectric materials. J. Math. Anal. Appl. 361, 161–176 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  186. Migórski, S., Ochal, A., Sofonea, M.: Weak solvability of antiplane frictional contact problems for elastic cylinders. Nonlinear Anal. Real World Appl. 11, 172–183 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  187. Migórski, S., Ochal, A., Sofonea, M.: Analysis of a dynamic contact problem for electro-viscoelastic cylinders. Nonlinear Anal. 73, 1221–1238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  188. Migórski, S., Ochal, A., Sofonea, M.: Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete Contin. Dyn. Syst. Ser. B 15, 687–705 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  189. Migórski, S., Ochal, A., Sofonea, M.: History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 12, 3384–3396 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  190. Mordukhovich, B.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin, Heidelberg (2006)

    Google Scholar 

  191. Mordukhovich, B.: Variational Analysis and Generalized Differentiation II: Applications. Springer, Berlin, Heidelberg (2006)

    Google Scholar 

  192. Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London (1999)

    Google Scholar 

  193. Motreanu, D., Sofonea, M.: Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials. Abstr. Appl. Anal. 4, 255–279 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  194. Motreanu, D., Sofonea, M.: Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv. Math. Sci. Appl. 10, 103–118 (2000)

    MathSciNet  MATH  Google Scholar 

  195. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)

    Google Scholar 

  196. Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Academia, Praha (1967)

    MATH  Google Scholar 

  197. Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York (1981)

    Google Scholar 

  198. Nguyen, Q.S.: Stability and Nonlinear Solid Mechanics. Wiley, Chichester (2000)

    Google Scholar 

  199. Ochal, A.: Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60, 1369–1391 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  200. Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Engrg. 52, 527–634 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  201. Oden, J.T., Pires, E.B.: Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. J. Appl. Mech. 50, 67–76 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  202. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Boston (1985)

    Book  MATH  Google Scholar 

  203. Panagiotopoulos, P.D.: Nonconvex problems of semipermeable media and related topics. ZAMM Z. Angew. Math. Mech. 65, 29–36 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  204. Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. 16, 209–231 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  205. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  206. Panagiotopoulos, P.D.: Modelling of nonconvex nonsmooth energy problems: dynamic hemivariational inequalities with impact effects. J. Comput. Appl. Math. 63, 123–138 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  207. Panagiotopoulos, P.D.: Hemivariational inequalities and fan-variational inequalities. New applications and results. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 43, 159–191 (1995)

    MathSciNet  MATH  Google Scholar 

  208. Panagiotopoulos, P.D., Pop, G.: On a type of hyperbolic variational-hemivariational inequalities. J. Appl. Anal. 5, 95–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  209. Papageorgiou, N.S.: Existence of solutions for second-order evolution inclusions. Bull. Math. Soc. Sci. Math. Roumanie 37, 93–107 (1993)

    MathSciNet  MATH  Google Scholar 

  210. Papageorgiou, N.S., Yannakakis, N.: Second order nonlinear evolution inclusions I: existence and relaxation results. Acta Mat. Sin. (Engl. Ser.) 21, 977–996 (2005)

    Google Scholar 

  211. Papageorgiou, N.S., Yannakakis, N.: Second order nonlinear evolution inclusions II: structure of the solution set. Acta Mat. Sin. (Engl. Ser.) 22, 195–206 (2006)

    Google Scholar 

  212. Park, J.Y., Ha, T.G.: Existence of antiperiodic solutions for hemivariational inequalities. Nonlinear Anal. 68, 747–767 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  213. Park, S.H., Park, J.Y., Jeong, J.M.: Boundary stabilization of hyperbolic hemivariational inequalities Acta Appl. Math. 104, 139–150 (2008)

    MathSciNet  MATH  Google Scholar 

  214. Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Sijthoff and Noordhoff International Publishers, Alpen aan den Rijn (1978)

    MATH  Google Scholar 

  215. Patron, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon & Breach, London (1988)

    Google Scholar 

  216. Pipkin, A.C.: Lectures in Viscoelasticity Theory. Applied Mathematical Sciences, vol. 7, George Allen & Unwin Ltd. London, Springer, New York (1972)

    Google Scholar 

  217. Rabinowicz, E.: Friction and Wear of Materials, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  218. Raous, M., Jean, M., Moreau, J.J. (eds.): Contact Mechanics. Plenum Press, New York (1995)

    Google Scholar 

  219. Rocca, R.: Existence of a solution for a quasistatic contact problem with local friction. C.R. Acad. Sci. Paris Sér. I Math. 328, 1253–1258 (1999)

    Google Scholar 

  220. Rocca, R., Cocu, M.: Existence and approximation of a solution to quasistatic problem with local friction. Int. J. Eng. Sci. 39, 1233–1255 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  221. Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elasticity 51, 105–126 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  222. Rockafellar, T.R.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  223. Rodríguez–Aros, A.D., Sofonea, M., Viaño, J.M.: A class of evolutionary variational inequalities with Volterra-type integral term. Math. Models Methods Appl. Sci. 14, 555–577 (2004)

    Google Scholar 

  224. Rodríguez–Aros, A.D., Sofonea, M., Viaño, J.M.: Numerical approximation of a viscoelastic frictional contact problem. C.R. Acad. Sci. Paris, Sér. II Méc. 334, 279–284 (2006)

    Google Scholar 

  225. Rodríguez–Aros, A.D., Sofonea, M., Viaño, J.M.: Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory. Numer. Math. 198, 327–358 (2007)

    Google Scholar 

  226. Royden, H.L.: Real Analysis. The Macmillan Company, New York, Collier-Macmillan Ltd, London (1963)

    MATH  Google Scholar 

  227. Schatzman, M.: A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle. J. Math. Anal. Appl. 73, 138–191 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  228. Scholz, C.H.: The Mechanics of Earthquakes and Faulting. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  229. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1951)

    MATH  Google Scholar 

  230. Shillor, M. (ed.): Recent advances in contact mechanics, Special issue of Math. Comput. Mode. 28 (4–8) (1998)

    Google Scholar 

  231. Shillor, M.: Quasistatic problems in contact mechanics. Int. J. Appl. Math. Comput. Sci. 11, 189–204 (2001)

    MathSciNet  MATH  Google Scholar 

  232. Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38, 1517–1533 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  233. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655, Springer, Berlin (2004)

    Google Scholar 

  234. Showalter, R.: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. vol. 49, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI (1997)

    Google Scholar 

  235. Signorini, A.: Sopra alcune questioni di elastostatica. Atti della Società Italiana per il Progresso delle Scienze (1933)

    Google Scholar 

  236. Simon, J.: Compact sets in the space L p(0; T; B). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  237. Sofonea, M., El Essoufi, H.: A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9, 229–242 (2004)

    Google Scholar 

  238. Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Pure and Applied Mathematics, vol. 276, Chapman-Hall/CRC Press, New York (2006)

    Google Scholar 

  239. Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Advances in Mechanics and Mathematics, vol. 18, Springer, New York (2009)

    Google Scholar 

  240. Sofonea, M., Rodríguez–Aros, A.D., Viaño J.M.: A class of integro-differential variational inequalities with applications to viscoelastic contact. Math. Comput. Model. 41, 1355–1369 (2005)

    Google Scholar 

  241. Sofonea, M., Viaño, J.M. (eds.), Mathematical Modelling and Numerical Analysis in Solid Mechanics. Special issue of Int. J. Appl. Math. Comput. Sci. 12 (1) (2002)

    Google Scholar 

  242. Strömberg, N.: Thermomechanical modelling of tribological systems. Ph.Thesis, D., no. 497, Linköping University, Sweden (1997)

    Google Scholar 

  243. Strömberg, N., Johansson, P., A. Klarbring, A.: Generalized standard model for contact friction and wear. In: Raous, M., Jean, M., Moreau, J.J. (eds.) Contact Mechanics, Plenum Press, New York (1995)

    Google Scholar 

  244. Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact friction and wear. Int. J. Solids Structures 33, 1817–1836 (1996)

    Article  MATH  Google Scholar 

  245. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)

    MATH  Google Scholar 

  246. Telega, J.J.: Topics on unilateral contact problems of elasticity and inelasticity. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.), Nonsmooth Mechanics and Applications, pp. 340–461. Springer, Wien (1988)

    Google Scholar 

  247. Telega, J.J.: Quasi-static Signorini’s contact problem with friction and duality. Int. Ser. Numer. Math. 101, 199–214 (1991)

    MathSciNet  Google Scholar 

  248. Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  249. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

    Google Scholar 

  250. Truesdell, C. (ed.): Mechanics of Solids, vol. III: Theory of Viscoelasticity, Plasticity, Elastic Waves and Elastic Stability. Springer, Berlin (1973)

    Google Scholar 

  251. Wilson, W.R.D.: Modeling friction in sheet-metal forming simulation. In: Zabaras et al. (eds.), The Integration of Materials, Process and Product Design, pp. 139–147. Balkema, Rotterdam (1999)

    Google Scholar 

  252. Wloka, J.: Partielle Differentialgleichungen, Teubner, B.G., Stuttgart (1982) English translation: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  253. Wriggers, P.: Computational Contact Mechanics. Wiley, Chichester (2002)

    Google Scholar 

  254. Wriggers, P., Nackenhorst, U. (eds.): Analysis and simulation of contact problems. Lecture Notes in Applied and Computational Mechanics, vol. 27, Springer, Berlin (2006)

    Google Scholar 

  255. Wriggers, P., Panagiotopoulos, P.D. (eds.): New Developments in Contact Problems. Springer, Wien, New York (1999)

    MATH  Google Scholar 

  256. Xiao, Y., Huang, N.: Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities. J. Global Optim. 45, 371–388 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  257. Yang, S.S.: An Introduction to the Theory of Piezoelectricity. Springer, New York (2005)

    MATH  Google Scholar 

  258. Yang, J.S., Batra, J.M., Liang, X.Q.: The cylindrical bending vibration of a laminated elastic plate due to piezoelectric acutators. Smart Mater. Struct. 3, 485–493 (1994)

    Article  Google Scholar 

  259. Yosida, K.: Functional Analysis, 5th edn. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

  260. Zeidler, E.: Nonlinear Functional Analysis and its Applications. I: Fixed-point Theorems, Springer, New York (1985)

    Google Scholar 

  261. Zeidler, E.: Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization. Springer, New York (1986)

    Google Scholar 

  262. Zeidler, E.: Nonlinear Functional Analysis and its Applications. IV: Applications to Mathematical Physics. Springer, New York (1988)

    Google Scholar 

  263. Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators. Springer, New York (1990)

    Google Scholar 

  264. Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990)

    Google Scholar 

  265. Zeidler, E.: Applied Functional Analysis: Main Principles and Their Applications. Springer, New York (1995)

    MATH  Google Scholar 

  266. Zeidler, E.: Applied Functional Analysis: Applications of Mathematical Physics. Springer, New York (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science Institute of Computer Science, Jagiellonian University, Kraków, Poland

    Stanisław Migórski & Anna Ochal

  2. Laboratoire de Mathématiques et Physique (LAMPS), Université de Perpignan Via Domitia, Perpignan, France

    Mircea Sofonea

Authors
  1. Stanisław Migórski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Ochal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mircea Sofonea
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Migórski, S., Ochal, A., Sofonea, M. (2013). Preliminaries. In: Nonlinear Inclusions and Hemivariational Inequalities. Advances in Mechanics and Mathematics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4232-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation