Skip to main content

Advertisement

SpringerLink
Log in

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

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397, 738–756 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bensoussan, A., Lions, J.L.: Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1, 289–312 (1974/75)

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

    Book  MATH  Google Scholar 

  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)

  9. Giannessi, F., Khan, A.A.: Regularization of non-coercive quasi-variational inequalities. Control Cybern. 29, 91–110 (2000)

    MATH  MathSciNet  Google Scholar 

  10. Giannessi, F., Khan, A.A.: On the envelope of a variational inequality. Nonlinear Anal. Var. Probl. 35, 285–293 (2010)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  14. Jadamba, B., Khan, A.A., Sama, M.: Generalized solutions of quasi-variational inequalities. Optim. Lett. 6(7), 1221–1231 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  16. Kenmochi, N.: Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations. Hiroshima Math. J. 4, 229–263 (1974)

    MATH  MathSciNet  Google Scholar 

  17. Khan, A.A., Sama, M.: Optimal control of multivalued quasi-variational inequalities. Nonlinear Anal. 75(3), 1419–1428 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)

    MATH  Google Scholar 

  19. Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64, 1703–1724 (2015)

    Article  MathSciNet  Google Scholar 

  20. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (1987)

    Google Scholar 

  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)

  22. Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu, Z.: Generalized quasi-variational hemi-variational inequalities. Appl. Math. Lett. 17, 741–745 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liu, Z.: Existence results for evolution noncoercive hemivariational inequalities. J. Optim. Theory Appl. 120, 417–427 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lunsford, M.L.: Generalized variational and quasi-variational inequalities with discontinuous operators. J. Math. Anal. Appl. 214, 245–263 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 512–585 (1969)

    Article  MathSciNet  Google Scholar 

  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)

  29. Motreanu, V.V.: Existence results for constrained quasivariational inequalities. In Abstract and Applied Analysis, Art. ID 427908 (2013)

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

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Akhtar A. Khan

  2. Département de Mathématiques, Université de Perpignan, 66860, Perpignan, France

    Dumitru Motreanu

Authors
  1. Akhtar A. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dumitru Motreanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akhtar A. Khan.

Additional information

Dedicated to Prof. Franco Giannessi on his 80th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, A.A., Motreanu, D. Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities. J Optim Theory Appl 167, 1136–1161 (2015). https://doi.org/10.1007/s10957-015-0825-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0825-6

Keywords

Mathematics Subject Classification

Search

Navigation