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Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

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Abstract

This paper is about the existence of solutions of the following abstract quasi-variational inequality:

$$\left\{\begin{array}{ll}\langle {\mathcal{A}}(u) + {\mathcal F} (u), v-u\rangle + J_u(v) -J_u(u) \ge 0,&\quad \forall v\in X,\\ u \in D(J_u).&\end{array}\right.$$

where X is a reflexive Banach space, \({{\mathcal{A}}}\) is a maximal monotone mapping, \({{\mathcal F}}\) is a multivalued mapping which is compact in certain sense, and J u is a convex and lower semicontinuous functional depending on u. We are interested in solvability conditions for the above inequality that extends those for a class of boundary value problems containing elliptic operators with multivalued coefficients.

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References

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

    MATH  Google Scholar 

  2. 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 

  3. 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 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Carl S., Le V.K.: Elliptic inequalities with multi-valued operators: existence, comparison and related variational–hemivariational type inequalities. Nonlinear Anal. 121, 130–152 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ceng L.-C., Yao J.-C.: Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces. J. Glob. Optim. 55(1), 27–51 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Mathematics and its Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)

  11. 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 

  12. Joly, J.-L., Mosco, U.: À propos de l’existence et de la régularité des solutions de certaines inéquations quasi-variationnelles. J. Funct. Anal. 34, 107–137 (1979)

  13. 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)

  14. 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 Scientific Publishing, Hackensack (2008)

  15. Kenmochi, N.: Monotonicity and Compactness Methods for Nonlinear Variational Inequalities. Handbook of Differential Equations, vol. IV, pp. 203–298. Elsevier/North-Holland, Amsterdam (2007)

  16. Kourogenis N.C., Papageorgiou N.S.: Nonlinear hemivariational inequalities of second order using the method of upper–lower solutions. Proc. Am. Math. Soc. 131, 2359–2369 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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.: Existence of positive solutions of variational inequalities by a subsolution–supersolution approach. J. Math. Anal. Appl. 252, 65–90 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Le V.K.: Subsolution–supersolution method in variational inequalities. Nonlinear Anal. 45, 775–800 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Le V.K.: Sub-supersolution method in variational inequalities with multivalued operators given by integrals. Rocky Mt. J. Math. 41, 535–553 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Le V.K.: Variational inequalities with general multivalued lower order terms given by integrals. Adv. Nonlinear Stud. 11, 1–24 (2011)

    Article  MathSciNet  Google Scholar 

  23. 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 

  24. Le V.K.: On variational and quasi-variational inequalities with multivalued lower order terms and convex functionals. Nonlinear Anal. 94, 12–31 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

  26. Loc N.H., Schmitt K.: Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities. Nonlinear Anal. 75, 4664–4671 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Loc N.H., Schmitt K.: Nonlinear elliptic Dirichlet and no-flux boundary value problems. Ann. Univ. Buchar. Math. Ser. 3(LXI), 201–217 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Moreira D.R., Teixeira E.V.: Weak convergence under nonlinearities. An. Acad. Brasil. Ciênc. 75, 9–19 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Moreira, D.R., Teixeira, E.V.: On the behavior of weak convergence under nonlinearities and applications. Proc. Am. Math. Soc. 133, 1647–1656 (2005) (electronic)

  30. Murase, Y.: Abstract Quasi-Variational Inequalities of Elliptic Type and Applications. Nonlocal and Abstract Parabolic Equations and Their Applications, vol. 86, pp. 235–246. Banach Center Publications/Polish Academy of Sciences Institute of Mathematics, Warsaw (2009)

  31. Tartar L.: Inéquations quasi variationnelles abstraites. C. R. Acad. Sci. Paris Sér. A 278, 1193–1196 (1974)

    MATH  Google Scholar 

  32. Zeidler E.: Nonlinear Functional Analysis and its Applications, vol. 3: Variational Methods and Optimization. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

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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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Le, V.K. Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings. Results Math 71, 423–453 (2017). https://doi.org/10.1007/s00025-016-0547-6

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  • DOI: https://doi.org/10.1007/s00025-016-0547-6

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