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Journal of Optimization Theory and Applications

, Volume 167, Issue 3, pp 1136–1161 | Cite as

Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities

  • Akhtar A. Khan
  • Dumitru Motreanu
Article

Abstract

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge’s fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

Keywords

Quasi-variational inequalities Variational inequalities Hemivariational inequalities Monotone Pseudo-monotone Generalized pseudo-monotone Coercivity  Asymptotic recessivity 

Mathematics Subject Classification

49J20 90C51 90C30 

References

  1. 1.
    Alber, Y.I., Butnariu, D., Ryazantseva, I.: Regularization of monotone variational inequalities with Mosco approximations of the constraint sets. Set Valued Anal. 13, 265–290 (2005)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alber, Y.I., Notik, A.I.: Perturbed unstable variational inequalities with unbounded operators on approximately given sets. Set Valued Anal. 1, 393–402 (1993)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397, 738–756 (2013)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)MATHGoogle Scholar
  5. 5.
    Barrett, J.W., Prigozhin, L.: Lakes and rivers in the landscape: a quasi-variational inequality approach. Interfaces Free Bound. 16(2), 269–296 (2014)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bensoussan, A., Lions, J.L.: Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1, 289–312 (1974/75)Google Scholar
  7. 7.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer, New York (2007)MATHCrossRefGoogle Scholar
  8. 8.
    Giannessi, F.: Separation of sets and gap functions for quasi-variational inequalities. Variational Inequalities and Network Equilibrium Problems (Erice, 1994), pp. 101–121. Plenum, New York (1995)Google Scholar
  9. 9.
    Giannessi, F., Khan, A.A.: Regularization of non-coercive quasi-variational inequalities. Control Cybern. 29, 91–110 (2000)MATHMathSciNetGoogle Scholar
  10. 10.
    Giannessi, F., Khan, A.A.: On the envelope of a variational inequality. Nonlinear Anal. Var. Probl. 35, 285–293 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. I. Unilateral Analysis and Unilateral Mechanics, Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston (2003)Google Scholar
  12. 12.
    Goeleven, D., Motreanu, D.: Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. II. Unilateral Problems. Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston (2003)Google Scholar
  13. 13.
    Jadamba, B., Khan, A.A., Raciti, F., Rouhani, D.B.: Generalized solutions of multi-valued monotone quasi-variational inequalities. Optim. optim. control 39, 227–240 (2010)CrossRefGoogle Scholar
  14. 14.
    Jadamba, B., Khan, A.A., Sama, M.: Generalized solutions of quasi-variational inequalities. Optim. Lett. 6(7), 1221–1231 (2012)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kano, R., Kenmochi, N., Murase, Y.: Existence theorems for elliptic quasi-variational inequalities in Banach spaces. In Recent Advances in Nonlinear Analysis, pp. 149–169. World Scientific Publishing, New Jersey (2008)Google Scholar
  16. 16.
    Kenmochi, N.: Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations. Hiroshima Math. J. 4, 229–263 (1974)MATHMathSciNetGoogle Scholar
  17. 17.
    Khan, A.A., Sama, M.: Optimal control of multivalued quasi-variational inequalities. Nonlinear Anal. 75(3), 1419–1428 (2012)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)MATHGoogle Scholar
  19. 19.
    Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64, 1703–1724 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (1987)Google Scholar
  21. 21.
    Kluge, R.: On some parameter determination problems and quasi-variational inequalities, Theory of nonlinear operators. In: Proc. Fifth Internat. Summer School, Central Inst. Math. Mech. Acad. Sci. GDR, Berlin, 1977, pp. 129-139. Akademie-Verlag, Berlin (1978)Google Scholar
  22. 22.
    Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics. Springer, Dordrecht (2007)MATHCrossRefGoogle Scholar
  23. 23.
    Lenzen, F., Lellmann, J., Becker, F., Schnorr, C.: Solving quasi-variational inequalities for image restoration with adaptive constraint sets. SIAM J. Imaging Sci. 7(4), 2139–2174 (2014)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, Z.: Generalized quasi-variational hemi-variational inequalities. Appl. Math. Lett. 17, 741–745 (2004)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu, Z.: Existence results for evolution noncoercive hemivariational inequalities. J. Optim. Theory Appl. 120, 417–427 (2004)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Lunsford, M.L.: Generalized variational and quasi-variational inequalities with discontinuous operators. J. Math. Anal. Appl. 214, 245–263 (1997)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 512–585 (1969)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mosco, U.: Implicit variational problems and quasi-variational inequalities. In: Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)Google Scholar
  29. 29.
    Motreanu, V.V.: Existence results for constrained quasivariational inequalities. In Abstract and Applied Analysis, Art. ID 427908 (2013)Google Scholar
  30. 30.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Center for Applied and Computational Mathematics, School of Mathematical SciencesRochester Institute of TechnologyRochesterUSA
  2. 2.Département de MathématiquesUniversité de PerpignanPerpignanFrance

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