Advertisement

On the Solvability of Inclusions with Multivalued Compact Perturbations of Bi-Mappings

  • Vy Khoi Le
Article
  • 31 Downloads

Abstract

This paper is about the solvability of the inclusion
$$A(u,u) + F(u) \ni L , $$
where X is a reflexive Banach space, LX , A is maximal monotone in one argument and continuous in the other argument in certain sense, and F is a multivalued perturbing term, which is compact in certain sense. We are interested in appropriate conditions on A and F such that this inclusion has solutions within a closed and convex set. We also prove that under such conditions the mapping uA (u, u) + F(u) is in fact generalized pseudomonotone in the sense of Browder and Hess (J. Funct. Anal. 11, 251–294, 1972).

Keywords

Inclusion Maximal monotone mapping Generalized pseudomonotone mapping Class (S)+ Multivalued mapping 

Mathematics Subject Classifications (2010)

58E35 47J20 47J25 35J87 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The author is very grateful for the referees’ careful reading of the manuscript and invaluable comments and suggestions.

References

  1. 1.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, London (1984)MATHGoogle Scholar
  2. 2.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)MATHGoogle Scholar
  3. 3.
    Bensoussan, A., Goursat, M., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C.R. Acad. Sci. Paris Sér. A-B 276, A1279–A1284 (1973)MATHGoogle Scholar
  4. 4.
    Bensoussan, A., Lions, J.-L.: Nouvelle formulation de problèmes de contrôle impulsionnel et applications. C. R. Acad. Sci. Paris Sér. A-B 276, A1189–A1192 (1973)MATHGoogle Scholar
  5. 5.
    Bensoussan, A., Lions, J.-L.: Applications of variational inequalities in stochastic control. Studies in Mathematics and Its Applications, vol. 12. North-Holland, Amsterdam (1982)Google Scholar
  6. 6.
    Bensoussan, A., Lions, J.-L.: Impulse Control and Inequalities, Quasivariational. Gauthier-Villars Montrouge (1984)Google Scholar
  7. 7.
    Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)Google Scholar
  8. 8.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics. Springer, New York (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Ceng, L.-C., Yao, J.-C.: Existence theorems for generalized set-valued mixed (quasi-) variational inequalities in Banach spaces. J. Global Optim. 55 1, 27–51 (2013)Google Scholar
  10. 10.
    Cosso, A.: Stochastic differential games involving impulse controls adn double-obstacle quasi-variational inequalities. SIAM J. Control Optim. 51 (3), 2102–2131 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland Math Studies, vol. 5. North-Holland, Amsterdam (1973)Google Scholar
  12. 12.
    Fukao, T., Kenmochi, N.: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint, Discrete Contin. Dyn. Syst. 35(6), 2523–2538 (2015)MathSciNetMATHGoogle Scholar
  13. 13.
    Jadamba, B., Khan, A.A., Sama, M.: Generalized solutions of quasi variational inequalities. Optim. Lett. 6(7), 1221–1231 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kadoya, A., Kenmochi, N., Niezgódka, M.: Quasi-variational inequalities in economic growth models with technological development. Adv. Math. Sci. Appl. 24(1), 185–214 (2014)MathSciNetMATHGoogle Scholar
  15. 15.
    Kano, R., Kenmochi, N., Murase, Y.: Existence theorems for elliptic quasi-variational inequalities in Banach spaces, Recent advances in nonlinear analysis, pp. 149–169. World Science Publisher, Hackensack (2008)Google Scholar
  16. 16.
    Kenmochi, N.: Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of differential equations, Vol. IV, pp. 203–298. Elsevier/North-Holland, Amsterdam (2007)Google Scholar
  17. 17.
    Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics Solid Mechanics and Its Applications, vol. 147. Springer, Dordrecht (2007)Google Scholar
  18. 18.
    Le, V.K.: A range existence theorem for pseudomonotone perturbations of maximal monotone operators. Proc. Amer. Math Soc. 139, 1645–1658 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Le, V.K.: On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents. J. Math. Anal. Appl. 388, 695–715 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Le, V.K.: Existence results for quasi-variational inequalities with multivalued perturbations of maximal monotone mappings, Results Math. - Online First -  10.1007/s00025-016-0547-6 (2016)
  21. 21.
    Le, V.K.: On the convergence of solutions of inclusions containing maximal monotone and generalized pseudomonotone mappings. Nonlinear Anal. 143, 64–88 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. 54(2), 371–398 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Murase, Y.: Abstract quasi-variational inequalities of elliptic type and applications, Nonlocal and abstract parabolic equations and their applications, Banach Center Publ., vol. 86, Polish Acad. Sci. Inst. Math., Warsaw (2009)Google Scholar
  24. 24.
    Murase, Y., Kano, R., Kenmochi, N.: Elliptic quasi-variational inequalities and applications. Discrete Contin. Dyn. Syst. no. Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 583–591 (2009)Google Scholar
  25. 25.
    Rockafellar, T.R.: Convex analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  26. 26.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. 2b, Nonlinear monotone operators. Springer, New York (1990)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA

Personalised recommendations