Variational, Hemivariational and Variational-Hemivariational Inequalities: Existence Results

  • D. Motreanu
  • V. Rădulescu
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 67)


The celebrated Hartman-Stampacchia theorem (see [6], Lemma 3.1, or [9], Theorem I.3.1) asserts that if V is a finite dimensional Banach space, KV is non-empty, compact and convex, A : KV* is continuous, then there exists uK such that, for every vK,
$$\langle Au,v - u\rangle \geqslant 0.$$


Banach Space Convex Subset Inequality Problem Nonempty Closed Convex Subset Lower Semicontinuous Function 
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  1. [1]
    H. Brézis, Analyse fonctionnelle. Théorie et applications, Masson, 1992.Google Scholar
  2. [2]
    F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal. 11 (1972), 251.-294.Google Scholar
  3. [3]
    F. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.zbMATHGoogle Scholar
  4. [4]
    J. Dugundji and A. Granas, KKM-maps and variational inequalities, Ann. Scuola Norm. Sup. Pisa 5 (1978), 679–682.MathSciNetzbMATHGoogle Scholar
  5. [5]
    G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.zbMATHGoogle Scholar
  6. [6]
    G. J. Hartman and G. Stampacchia, On some nonlinear elliptic differential equations, Acta Math. 15 (1966), 271–310.MathSciNetCrossRefGoogle Scholar
  7. [7]
    R. B. Holmes, Geometric Functional Analysis and its Applications, SpringerVerlag-New York, 1975.Google Scholar
  8. [9]
    D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities, Academic Press, New York, 1980.zbMATHGoogle Scholar
  9. [10]
    B. Knaster, K. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Mat. 14 (1929), 132–137.zbMATHGoogle Scholar
  10. [11]
    U. Mosco, Implicit variational problems and quasi-variational inequalities, in Nonlinear Operators and the Calculus of Variations (J.P. Gossez, E.J. Lami Dozo, J. Mawhin and L. Waelbroeck, Eds.), Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1976, pp. 83–156.Google Scholar
  11. [12]
    D. Motreanu and P. D. Panagiotopoulos, Double eigenvalue problems for hemivariational inequalities, Arch. Rat. Mech. Analysis 140 (1997), 225–251.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [13]
    D. Motreanu and V. Râdulescu, Existence results for inequality problems with lack of convexity, Numer. Funct. Anal. Optimiz. 21 (2000), 869–884.zbMATHCrossRefGoogle Scholar
  13. [14]
    Z. Naniewicz, Hemivariational Inequalities with functionals which are not locally Lipschitz, Nonlinear Analysis, T.M.A., 25 (1995), No. 12, pp. 1307–1320.MathSciNetzbMATHGoogle Scholar
  14. [15]
    Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.Google Scholar
  15. [16]
    P. D. Panagiotopoulos, Hemivariational Inequalities: Applications to Mechanics and Engineering, Springer-Verlag, New-York/Boston/Berlin, 1993.zbMATHCrossRefGoogle Scholar
  16. [17]
    P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functionals, Birkhduser-Verlag, Basel, 1985.CrossRefGoogle Scholar
  17. [18]
    D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoff International Publishers, The Netherlands, 1978.zbMATHGoogle Scholar
  18. [19]
    W. Prager, Problems of network flow, Zeitschrift für Angewandte Mathematik und Physik (Z.A.M.P.) 16 (1965), 185–193.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [20]
    H. H. Schaefer, Topological Vector Spaces, Macmillan Series in Advances Mathematics and Theoretical Physics, New York, 1966.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • D. Motreanu
    • 1
  • V. Rădulescu
    • 2
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsUniversity of CraiovaCraiovaRomania

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