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.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier, Oxford (2003)
Ahmed, N.U., Kerbal, S.: Optimal control of nonlinear second order evolution equations. J. Appl. Math. Stoch. Anal. 6, 123–136 (1993)
Amassad, A., Fabre, C., Sofonea, M.: A quasistatic viscoplastic contact problem with normal compliance and friction. IMA J. Appl. Math. 69, 463–482 (2004)
Amassad, A., Shillor, M., Sofonea, M.: A quasistatic contact problem with slip dependent coefficient of friction. Math. Methods Appl. Sci. 22, 267–284 (1999)
Amassad, A., Sofonea, M.: Analysis of a quasistatic viscoplastic problem involving Tresca friction law. Discrete Contin. Dyn. Syst. 4, 55–72 (1998)
Andersson, L.-E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. 16, 347–370 (1991)
Andersson, L.-E.: A global existence result for a quasistatic contact problem with friction. Adv. Math. Sci. Appl. 5, 249–286 (1995)
Andersson, L.-E.: A quasistatic frictional problem with a normal compliance penalization term. Nonlinear Anal. 37, 689–705 (1999)
Andersson, L.-E.: Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim. 42, 169–202 (2000)
Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005)
Atkinson, K., Han, W.: Theoretical numerical analysis: a functional analysis framework, 3rd edn. Texts in Applied Mathematics, vol. 39, Springer, New York (2009)
Aubin, J.P.: L’analyse non linéaire et ses motivations économiques. Masson, Paris, (1984)
Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory. Springer, Berlin, New York, Tokyo (1984)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel, Boston (1990)
Awbi, B., Rochdi, M., Sofonea, M.: Abstract evolution equations for viscoelastic frictional contact problems. Z. Angew. Math. Phys. 51, 218–235 (2000)
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems. Wiley, Chichester (1984)
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)
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)
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)
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)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei, Bucharest-Noordhoff, Leyden (1976)
Barbu, V.: Optimal Control of Variational Inequalities. Pitman, Boston (1984)
Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. D. Reidel Publishing Company, Dordrecht (1986)
Bartosz, K.: Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear. Nonlinear Anal. 65, 546–566 (2006)
Bartosz, K.: Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal. 71, 1747–1762 (2009)
Batra, R.C., Yang, J.S.: Saint-Venant’s principle in linear piezoelectricity. J. Elasticity 38, 209–218 (1995)
Bian, W.: Existence results for second order nonlinear evolution inclusions. Indian J. Pure Appl. Math. 11, 1177–1193 (1998)
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)
Brézis, H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)
Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematics Studies, North Holland, Amsterdam (1973)
Brézis, H.: Analyse fonctionnelle—Théorie et applications. Masson, Paris (1987)
Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)
Browder, F., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)
Campillo, M., Dascalu, C., Ionescu, I.R.: Instability of a periodic system of faults. Geophys. J. Int. 159, 212–222 (2004)
Campillo, M., Ionescu, I.R.: Initiation of antiplane shear instability under slip dependent friction. J. Geophys. Res. 102 B9, 363–371 (1997)
Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Springer, New York (2007)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580, Springer, Berlin (1977)
Cazenave, T., Haraux, A.: Introduction aux problèmes d’évolution semi-linéaires. Ellipses, Paris (1990).
Chang, K.C.: Variational methods for nondifferentiable functionals and applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)
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)
Chau, O., Motreanu, D., Sofonea, M.: Quasistatic frictional problems for elastic and viscoelastic materials. Appl. Math. 47, 341–360 (2002)
Ciarlet, P.G.: Mathematical Elasticity, vol. I: Three-dimensional elasticity. Studies in Mathematics and its Applications, vol. 20, North-Holland, Amsterdam (1988)
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)
Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262 (1975)
Clarke, F.H.: Generalized gradients of Lipschitz functionals. Adv. Math. 40, 52–67 (1981)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience, New York (1983)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Cocu, M.: Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci. 22, 567–581 (1984)
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)
Cocu, M., Pratt, E., Raous, M.: Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci. 34, 783–798 (1996)
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)
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)
Cohn, D.L., Measure Theory. Birkhäuser, Boston (1980)
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)
Cristescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publishers, Editura Tehnicǎ, Bucharest (1982)
Deimling, K.: Differential Inclusions. Springer, New York, Heidelberg, Berlin (1980)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Demkowicz, I., Oden, J.T.: On some existence and uniqueness results in contact problems with non local friction. Nonlinear Anal. 6, 1075–1093 (1982)
Denkowski, Z., Migórski S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005)
Denkowski, Z., Migórski, S.: Hemivariational inequalities in thermoviscoelasticity. Nonlinear Anal. 63, e87–e97 (2005)
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)
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)
Denkowski, Z., Migórski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems. Control Cybernet. 36, 611–632 (2007)
Denkowski, Z., Migórski, S., Papageorgiu, N.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)
Denkowski, Z., Migórski, S., Papageorgiu, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)
Diestel, J., Uhl, J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15, AMS, Providence, RI (1977)
Dinculeanu, N.: Vector Measures. Akademie, Berlin (1966)
Doghri, I.: Mechanics of Deformable Solids. Springer, Berlin (2000)
Drozdov, A.D.: Finite Elasticity and Viscoelasticity—A Course in the Nonlinear Mechanics of Solids. World Scientific, Singapore (1996)
Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Dunford, N., Schwartz, J.: Linear Operators. Part I, Wiley, New York (1958)
Duvaut, G.: Loi de frottement non locale. J. Méc. Th. et Appl. Special issue, 73–78 (1982)
Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Methods Appl. Sci. 8, 445–468 (1998)
Eck, C., Jarušek, J.: Existence results for the semicoercive static contact problem with Coulomb friction. Nonlinear Anal. 42, 961–976 (2000)
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)
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)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Evans, L.C.: Partial Differential Equations. AMS Press, Providence (1999)
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)
Friedman, A.: Variational Principles and Free-boundary Problems. Wiley, New York (1982)
Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL (2006)
Gasinski, L., Smolka, M.: An existence theorem for wave-type hyperbolic hemivariational inequalities. Math. Nachr. 242, 1–12 (2002)
Germain, P., Muller, P.: Introduction à la mécanique des milieux continus, Masson, Paris (1980)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)
Goeleven, D., Miettinen, M., Panagiotopoulos, P.D.: Dynamic hemivariational inequalities and their applications. J. Optim. Theory Appl. 103, 567–601 (1999)
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)
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)
Goeleven, D., Motreanu, D.: Variational and Hemivariational Inequalities, Theory, Methods and Applications, vol. II: Unilateral Problems. Kluwer Academic Publishers, Boston, Dordrecht, London (2003)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)
Guo, X.: On existence and uniqueness of solution of hyperbolic differential inclusion with discontinuous nonlinearity. J. Math. Anal. Appl. 241, 198–213 (2000)
Guo, X.: The initial boundary value problem of mixed-typed hemivariational inequality. Int. J. Math. Math. Sci. 25, 43–52 (2001)
Guran, A. (ed.): Proceedings of the First International Symposium on Impact and Friction of Solids, Structures and Intelligent Machines. World Scientific, Singapore (2000)
Guran, A., Pfeiffer, F., Popp, K. (eds.): Dynamics with friction: modeling, analysis and experiment, Part I. World Scientific, Singapore (1996)
Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)
Halmos, P.R.: Measure Theory. D. Van Nostrand Company, Princeton, Toronto, London, New York (1950)
Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)
Han, W., Sofonea, M.: Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM J. Numer. Anal. 38, 556–579 (2000)
Han W., Sofonea, M.: Time-dependent variational inequalities for viscoelastic contact problems, J. Comput. Appl. Math. 136, 369–387 (2001)
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)
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)
Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London (1999)
Hewitt, E., Stromberg, K.: Abstract Analysis. Springer, New York, Inc. (1965)
Hild, P.: On finite element uniqueness studies for Coulomb’s frictional contact model. Int. J. Appl. Math. Comput. Sci. 12, 41–50 (2002)
Hiriart-Urruty, J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms, I, II. Springer, Berlin (1993)
Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1988)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I: theory. Kluwer, Dordrecht, The Netherlands (1997)
Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990)
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)
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)
Ionescu, I.R., Nguyen, Q.-L., Wolf, S.: Slip displacement dependent friction in dynamic elasticity. Nonlinear Anal. 53, 375–390 (2003)
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)
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)
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)
Ionescu, I.R., Sofonea, M.: Functional and Numerical Methods in Viscoplasticity. Oxford University Press, Oxford (1993)
Ionescu, I.R., Wolf, S.: Interaction of faults under slip dependent friction. Nonlinear eingenvalue analysis. Math. Methods Appl. Sci. 28, 77–100 (2005)
Jarušek, J.: Contact problem with given time-dependent friction force in linear viscoelasticity. Comment. Math. Univ. Carolin. 31, 257–262 (1990)
Jarušek, J.: Dynamic contact problems with given friction for viscoelastic bodies. Czechoslovak Math. J. 46, 475–487 (1996)
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)
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)
Jarušek, J., Sofonea, M.: On the solvability of dynamic elastic-visco-plastic contact problems. ZAMM Z. Angew. Math. Mech. 88, 3–22 (2008)
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)
Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)
Kalita, P.: Decay of energy of a system described by hyperbolic hemivariational inequality. Nonlinear Anal. 74, 116–1181 (2011)
Khludnev, A.M., Sokolowski, J.: Modelling and Control in Solid Mechanics. Birkhäuser-Verlag, Basel (1997)
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)
Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
Kim, J.U.: A boundary thin obstacle problem for a wave equation. Comm. Part. Differ. Equat. 14, 1011–1026 (1989)
Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Classics in Applied Mathematics, vol. 31, SIAM, Philadelphia (2000)
Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer, Dordrecht, The Netherlands (1991)
Klarbring, A., Mikelič, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)
Klarbring, A., Mikelič, A., Shillor, M.: On friction problems with normal compliance. Nonlinear Anal. 13, 935–955 (1989)
Komura, Y.: Nonlinear semigroups in Hilbert Spaces. J. Math. Soc. Japan 19, 493–507 (1967)
Kufner, A., John, O., Fučik, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)
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)
Kuttler, K.L.: Dynamic friction contact problem with general normal and friction laws. Nonlinear Anal. 28, 559–575 (1997)
Kuttler, K.: Topics in Analysis. Private communication (2006)
Kuttler, K.L., Shillor, M.: Set-valued pseudomonotone maps and degenerate evolution inclusions. Commun. Contemp. Math. 1, 87–123 (1999)
Kuttler, K.L., Shillor, M.: Dynamic bilateral contact with discontinuous friction coefficient. Nonlinear Anal. 45, 309–327 (2001)
Kuttler, K.L., Shillor, M.: Dynamic contact with normal compliance, wear and discontinuous friction coefficient. SIAM J. Math. Anal. 34, 1–27 (2002)
Larsen, R.: Functional Analysis. Marcel Dekker, New York (1973)
Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)
Lemaître, J., Chaboche, J.-L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)
Lerguet, Z., Shillor, M., Sofonea, M.: A frictional contact problem for an electro-viscoelastic body. Electron. J. Differential Equations 170, 1–16 (2007)
Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Anal. 71, 1414–1424 (2009)
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)
Lions, J.-L.: Quelques méthodes de resolution des problémes aux limites non linéaires. Dunod, Paris (1969)
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)
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)
Liu, Z., Migórski, S., Ochal, A.: Homogenization of boundary hemivariational inequalities in linear elasticity. J. Math. Anal. Appl. 340, 1347–1361 (2008)
Maceri, F., Bisegna, P.: The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Model. 28, 19–28 (1998)
Malvern, L.E.: Introduction to the Mechanics of a Continuum Medium. Princeton-Hall, NJ (1969)
Martins, J.A.C., Marques, M.D.P.M. (eds.): Contact Mechanics. Kluwer, Dordrecht (2002)
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)
Matysiak, S.J. (ed.): Contact Mechanics, Special issue of J. Theoret. Appl. Mech. 39(3) (2001)
Maugis, D.: Contact, Adhesion and Rupture of Elastic Solids. Springer, Berlin, Heidelberg (2000)
Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985)
Migórski, S.: Existence, variational and optimal control problems for nonlinear second order evolution inclusions. Dynam. Systems Appl. 4, 513–528 (1995)
Migórski, S.: Existence and relaxation results for nonlinear second order evolution inclusions. Discuss. Math. Differ. Incl. Control Optim. 15, 129–148 (1995)
Migórski, S.: Evolution hemivariational inequalities in infinite dimension and their control. Nonlinear Anal. 47, 101–112 (2001)
Migórski, S.: Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Probl. Sci. Eng. 11, 229–242 (2003)
Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)
Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005)
Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31, 505–533 (2005)
Migórski, S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 6, 1339–1356 (2006)
Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006)
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)
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)
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)
Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip-dependent friction. Nonlinear Anal. 61, 135–161 (2005)
Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)
Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elasticity 83, 247–275 (2006)
Migórski, S., Ochal, A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006)
Migórski, S.,Ochal, A.: Nonlinear impulsive evolution inclusions of second order. Dynam. Systems Appl. 16, 155–174 (2007)
Migórski, S., Ochal, A.: Vanishing viscosity for hemivariational inequality modeling dynamic problems in elasticity. Nonlinear Anal. 66, 1840–1852 (2007)
Migórski, S., Ochal, A.: Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. 69, 495–509 (2008)
Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)
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)
Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008)
Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. 70, 3738–3748 (2009)
Migórski, S., Ochal, A., Sofonea, M.: Modeling and analysis of an antiplane piezoelectric contact problem. Math. Models Methods Appl. Sci. 19, 1295–1324 (2009)
Migórski, S., Ochal, A., Sofonea, M.: Variational analysis of static frictional contact problems for electro-elastic materials. Math. Nachr. 283, 1314–1335 (2010)
Migórski, S., Ochal, A., Sofonea, M.: A dynamic frictional contact problem for piezoelectric materials. J. Math. Anal. Appl. 361, 161–176 (2010)
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)
Migórski, S., Ochal, A., Sofonea, M.: Analysis of a dynamic contact problem for electro-viscoelastic cylinders. Nonlinear Anal. 73, 1221–1238 (2010)
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)
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)
Mordukhovich, B.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin, Heidelberg (2006)
Mordukhovich, B.: Variational Analysis and Generalized Differentiation II: Applications. Springer, Berlin, Heidelberg (2006)
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)
Motreanu, D., Sofonea, M.: Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials. Abstr. Appl. Anal. 4, 255–279 (1999)
Motreanu, D., Sofonea, M.: Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv. Math. Sci. Appl. 10, 103–118 (2000)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)
Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Academia, Praha (1967)
Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York (1981)
Nguyen, Q.S.: Stability and Nonlinear Solid Mechanics. Wiley, Chichester (2000)
Ochal, A.: Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60, 1369–1391 (2005)
Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Engrg. 52, 527–634 (1985)
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)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Boston (1985)
Panagiotopoulos, P.D.: Nonconvex problems of semipermeable media and related topics. ZAMM Z. Angew. Math. Mech. 65, 29–36 (1985)
Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. 16, 209–231 (1991)
Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)
Panagiotopoulos, P.D.: Modelling of nonconvex nonsmooth energy problems: dynamic hemivariational inequalities with impact effects. J. Comput. Appl. Math. 63, 123–138 (1995)
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)
Panagiotopoulos, P.D., Pop, G.: On a type of hyperbolic variational-hemivariational inequalities. J. Appl. Anal. 5, 95–112 (1999)
Papageorgiou, N.S.: Existence of solutions for second-order evolution inclusions. Bull. Math. Soc. Sci. Math. Roumanie 37, 93–107 (1993)
Papageorgiou, N.S., Yannakakis, N.: Second order nonlinear evolution inclusions I: existence and relaxation results. Acta Mat. Sin. (Engl. Ser.) 21, 977–996 (2005)
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)
Park, J.Y., Ha, T.G.: Existence of antiperiodic solutions for hemivariational inequalities. Nonlinear Anal. 68, 747–767 (2008)
Park, S.H., Park, J.Y., Jeong, J.M.: Boundary stabilization of hyperbolic hemivariational inequalities Acta Appl. Math. 104, 139–150 (2008)
Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Sijthoff and Noordhoff International Publishers, Alpen aan den Rijn (1978)
Patron, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon & Breach, London (1988)
Pipkin, A.C.: Lectures in Viscoelasticity Theory. Applied Mathematical Sciences, vol. 7, George Allen & Unwin Ltd. London, Springer, New York (1972)
Rabinowicz, E.: Friction and Wear of Materials, 2nd edn. Wiley, New York (1995)
Raous, M., Jean, M., Moreau, J.J. (eds.): Contact Mechanics. Plenum Press, New York (1995)
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)
Rocca, R., Cocu, M.: Existence and approximation of a solution to quasistatic problem with local friction. Int. J. Eng. Sci. 39, 1233–1255 (2001)
Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elasticity 51, 105–126 (1998)
Rockafellar, T.R.: Convex Analysis. Princeton University Press, Princeton (1970)
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)
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)
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)
Royden, H.L.: Real Analysis. The Macmillan Company, New York, Collier-Macmillan Ltd, London (1963)
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)
Scholz, C.H.: The Mechanics of Earthquakes and Faulting. Cambridge University Press, Cambridge (1990)
Schwartz, L.: Théorie des Distributions. Hermann, Paris (1951)
Shillor, M. (ed.): Recent advances in contact mechanics, Special issue of Math. Comput. Mode. 28 (4–8) (1998)
Shillor, M.: Quasistatic problems in contact mechanics. Int. J. Appl. Math. Comput. Sci. 11, 189–204 (2001)
Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38, 1517–1533 (2000)
Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655, Springer, Berlin (2004)
Showalter, R.: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. vol. 49, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI (1997)
Signorini, A.: Sopra alcune questioni di elastostatica. Atti della Società Italiana per il Progresso delle Scienze (1933)
Simon, J.: Compact sets in the space L p(0; T; B). Ann. Mat. Pura Appl. 146, 65–96 (1987)
Sofonea, M., El Essoufi, H.: A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9, 229–242 (2004)
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)
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)
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)
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)
Strömberg, N.: Thermomechanical modelling of tribological systems. Ph.Thesis, D., no. 497, Linköping University, Sweden (1997)
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)
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)
Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)
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)
Telega, J.J.: Quasi-static Signorini’s contact problem with friction and duality. Int. Ser. Numer. Math. 101, 199–214 (1991)
Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics. Cambridge University Press, Cambridge (2001)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Truesdell, C. (ed.): Mechanics of Solids, vol. III: Theory of Viscoelasticity, Plasticity, Elastic Waves and Elastic Stability. Springer, Berlin (1973)
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)
Wloka, J.: Partielle Differentialgleichungen, Teubner, B.G., Stuttgart (1982) English translation: Partial Differential Equations. Cambridge University Press, Cambridge (1987)
Wriggers, P.: Computational Contact Mechanics. Wiley, Chichester (2002)
Wriggers, P., Nackenhorst, U. (eds.): Analysis and simulation of contact problems. Lecture Notes in Applied and Computational Mechanics, vol. 27, Springer, Berlin (2006)
Wriggers, P., Panagiotopoulos, P.D. (eds.): New Developments in Contact Problems. Springer, Wien, New York (1999)
Xiao, Y., Huang, N.: Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities. J. Global Optim. 45, 371–388 (2009)
Yang, S.S.: An Introduction to the Theory of Piezoelectricity. Springer, New York (2005)
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)
Yosida, K.: Functional Analysis, 5th edn. Springer, Berlin (1978)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. I: Fixed-point Theorems, Springer, New York (1985)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization. Springer, New York (1986)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. IV: Applications to Mathematical Physics. Springer, New York (1988)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators. Springer, New York (1990)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990)
Zeidler, E.: Applied Functional Analysis: Main Principles and Their Applications. Springer, New York (1995)
Zeidler, E.: Applied Functional Analysis: Applications of Mathematical Physics. Springer, New York (1995)
Author information
Authors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-1-4614-4232-5_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4231-8
Online ISBN: 978-1-4614-4232-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)