Advertisement

Existence and Uniqueness Results

  • Anca Capatina
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 31)

Abstract

This chapter deals with existence and uniqueness results for variational and quasi-variational inequalities. With the intention of focusing the differences among the proofs of results, we first consider elliptic variational inequalities of the first and second kind with linear and continuous operators in Hilbert space or monotone and hemicontinuous operators in Banach space. Next, we deal with elliptic quasi-variational inequalities involving monotone and hemicontinuous or potential operators. The last section concerns the study of a class of evolutionary quasi-variational inequalities. The results presented here will be applied, in the last part of the book, in the study of frictional contact problems.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York/San Francisco/London (1975)zbMATHGoogle Scholar
  2. 2.
    Andersson, L.E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. Theory Methods Appl. 16, 347–369 (1991)CrossRefzbMATHGoogle Scholar
  3. 3.
    Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund. Math. 3, 133–181 (1922)zbMATHGoogle Scholar
  4. 4.
    Bensoussan, A., Lions, J.L.: Applications des Inéquations Variationnelles et Contrôle Stochastique. Dunod, Paris (1978)Google Scholar
  5. 5.
    Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)zbMATHGoogle Scholar
  6. 6.
    Browder, F.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Capatina, A.: Quasi-variational inequalities for potential operators. St. Cerc. Mat. 46, 1 (1994)Google Scholar
  8. 8.
    Capatina, A., Cocou, M., Raous, M.: A class of implicit variational inequalities and applications to frictional contact. Math. Meth. Appl. Sci. 32, 1804–1827 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Céa, J.: Optimisation, Théorie et Algorithmes. Dunod, Paris (1971)zbMATHGoogle Scholar
  11. 11.
    Cocu, M., Pratt, E., Raous, M.: Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci. 34, 783–798 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dinca, G.: Sur la monotonie d’àprés Minty-Browder de l’opérateur de la théorie de plasticité. C. R. Acad. Sci. Paris 269, 535–538 (1969)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dincă, G.: Monotone Operators in the Theory of Plasticity (in romanian). Romanian Academy, Bucharest (1972)Google Scholar
  14. 14.
    Dincă, G.: Variational Methods and Applications (in romanian). Tech. Publ. House, Bucharest (1980)Google Scholar
  15. 15.
    Dincă, G., Roşca, I.: Une méthode variationnelle pour l’étude des opérateurs non-linéaires à differentiellle K-positivement définie. C. R. Acad. Sci. Paris 286, 25–28 (1978)zbMATHGoogle Scholar
  16. 16.
    Dunford, L., Schwartz, L.: Linear Operators, Part I. Interscience, New York (1958)zbMATHGoogle Scholar
  17. 17.
    Ekeland, I., Temam, R.: Analyse Convexe et Problèmes Variationnells. Dunod, Gauthier-Villars, Paris (1974)Google Scholar
  18. 18.
    Friedrichs, K.: On the boundary-value problem of the theory of elasticity and Korn’s inequality. Ann. Math. 48, 441–471 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  20. 20.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. In: AMS/IP Studies in Advanced Mathematics, vol. 30, (2002)Google Scholar
  21. 21.
    Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential equations. Acta. Math. 115, 271–310 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ionescu, I., Rosca, I., Sofonea, M.: A variational method for nonlinear multivalued operators. Nonlinear Anal. TMA 9, 259–273 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kantorovici, I.V., Akilov, G.P.: Functional Analysis (in romanian). Sci. and Tech. Publ. House, Bucharest (1986)Google Scholar
  24. 24.
    Fan, K.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121–126 (1952)CrossRefzbMATHGoogle Scholar
  25. 25.
    Langenbach, A.: Variationsmethoden in der nichtlinearen Elastizitats und Plastizitatstheorie. Wiss, Z. Humboldt Univ. Berlin, Mat. Nat. Reihe IX, (1959/1960)Google Scholar
  26. 26.
    Martiniuk, A.E.: On some generalization of variational method. Dokl. Akad. Nauk. S.S.S.R. 1222–1225 (1959)Google Scholar
  27. 27.
    Minty, G.J.: On the generalization of a direct method of the calculus of variations. Bull. Am. Math. Soc. 73, 314–321 (1967)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Moreau, J.J.: Fonctionnelles convexes. Sém. sur les équations aux dérivées partielles, Collége de France (1967)Google Scholar
  30. 30.
    Mosco, U.: Implicit variational problems and quasi-variational inequalities. Lect. Note Math. 543, 83–156 (1975)CrossRefGoogle Scholar
  31. 31.
    Oden, J.T., Kikuchi, N.: Theory of variational inequalities with applications to problems of flow through porous media. Int. J. Eng. Sci. 18, 1171–1284 (1980)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Oden, J.T., Pires, E.: Contact problems in elastostatic with non-local friction laws. TICOM Report 81–82, Austin, Texas (1981)Google Scholar
  33. 33.
    Petryshyn, W.V.: On a class of K-p.d. and non K-p.d. operators and operators equation. J. Math. Anal. Appl. 10, 1–24 (1965)Google Scholar
  34. 34.
    Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38, 1517–1533 (2001)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sofonea, M.: Une méthode variationnelle pour une classe d’inéquations non-linéaires dans les espaces de Hilbert. Bull. Soc. Sci. Math. Roumaine 30, 47–55 (1986)MathSciNetGoogle Scholar
  36. 36.
    Sofonea, M., Matei, A.: Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems. Advances in Mechanics, vol. 18. Springer, New York (2009)Google Scholar
  37. 37.
    Stampacchia, G.: Variational inequalities: Theory and application of monotone operators. In: Proceedings of a NATO Advanced Study Institute Venice, Italy (1968)Google Scholar
  38. 38.
    Stampaccia, G.: Variational inequalities. Actes Congres intern. Math. 2, 877–883 (1970)Google Scholar
  39. 39.
    Tartar, L.: Inéquations quasi variationnelles abstraites. C. R. Acad. Sci. Paris 278, 1193–1196 (1974)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Tartar, L.: Variational Methods and Monotonicity. Mathematics Research Center, University of Winsconsin (1975)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Anca Capatina
    • 1
  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

Personalised recommendations