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

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Abstract

This paper is about the solvability of the inclusion

$$A(u,u) + F(u) \ni L , $$

where X is a reflexive Banach space, L ∈ X∗, 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 u ↦ A (u, u) + F(u) is in fact generalized pseudomonotone in the sense of Browder and Hess (J. Funct. Anal. 11, 251–294, 1972).

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References

  1. Attouch, H.: Variational Convergence for Functions and Operators. Pitman, London (1984)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

  6. Bensoussan, A., Lions, J.-L.: Impulse Control and Inequalities, Quasivariational. Gauthier-Villars Montrouge (1984)

  7. Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)

  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)

    Book  MATH  Google Scholar 

  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)

  10. Cosso, A.: Stochastic differential games involving impulse controls adn double-obstacle quasi-variational inequalities. SIAM J. Control Optim. 51 (3), 2102–2131 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  17. Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics Solid Mechanics and Its Applications, vol. 147. Springer, Dordrecht (2007)

  18. Le, V.K.: A range existence theorem for pseudomonotone perturbations of maximal monotone operators. Proc. Amer. Math Soc. 139, 1645–1658 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Le, V.K.: On the convergence of solutions of inclusions containing maximal monotone and generalized pseudomonotone mappings. Nonlinear Anal. 143, 64–88 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  25. Rockafellar, T.R.: Convex analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  26. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. 2b, Nonlinear monotone operators. Springer, New York (1990)

    Book  MATH  Google Scholar 

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Acknowledgments

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

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  1. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Vy Khoi Le

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  1. Vy Khoi Le
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Le, V.K. On the Solvability of Inclusions with Multivalued Compact Perturbations of Bi-Mappings. Set-Valued Var. Anal 27, 129–149 (2019). https://doi.org/10.1007/s11228-017-0431-x

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  • DOI: https://doi.org/10.1007/s11228-017-0431-x

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