Journal of Mathematical Sciences

, Volume 127, Issue 5, pp 2284–2314 | Cite as

Solvability of Unilateral Parabolic Problems and Variational Inequalities

  • O. V. Solonukha


Sufficient conditions are given for the existence of solutions of strong parabolic variational inequalities with nonlinear and multi-valued operators. Pseudomonotone operators are used. A new multi-valued analogue of the acute-angle lemma is used for the localization. The theory can be applied to problems with distributed parameters. We also consider conditions for the existence of strong generalized solutions of parabolic equations with unilateral boundary conditions with applications to control problems.


Boundary Condition Generalize Solution Variational Inequality Parabolic Equation Parabolic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Arkhipova and N. N. Uraltseva, “Regularity of solutions of a problem for elliptic and parabolic equations with bilateral constraints on the boundary,” Tr. Mat. Inst. Steklov., 179, 5–22 (1987).Google Scholar
  2. 2.
    J. P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer, New York (1984).Google Scholar
  3. 3.
    V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston (1995).Google Scholar
  4. 4.
    N. Bourbaki, Topological Vector Spaces [Russian translation], Inostr. Lit., Moscow (1959).Google Scholar
  5. 5.
    H. Brezis, “Equations et inequations non lineaires dans les espaces vectoriels en dualite,” Ann. Inst. Fourier, 18, 115–175 (1968).Google Scholar
  6. 6.
    F. E. Browder, “The fixed point theory of multivalued mappings in topological vector spaces,” Math. Ann., 177, 89–113 (1968).CrossRefGoogle Scholar
  7. 7.
    F. E. Browder and P. Hess, “Nonlinear mappings of monotone type in Banach spaces,” J. Funct. Anal., 11, No. 2, 251–294 (1972).CrossRefGoogle Scholar
  8. 8.
    S.-S. Chang, Yu-Q. Chen, and B. S. Lee, “Some existence theorems for differential inclusions in Hilbert spaces,” Bull. Austral. Math. Soc., 54, 317–327 (1996).Google Scholar
  9. 9.
    F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York (1998).Google Scholar
  10. 10.
    B. Cornet, “Existence of slow solutions for a class of differential inclusions,” J. Math. Anal. Appl., 96, 130–147 (1983).CrossRefGoogle Scholar
  11. 11.
    H. Debrunner and F. Flor, “Ein Erweiterungssatz fur monotone Mengen,” Arch. Math. 15, 445–447 (1964).CrossRefGoogle Scholar
  12. 12.
    Yu. A. Dubinskii, “Nonlinear elliptic and parabolic equations,” in: Itogi Nauki i Tekhn., Sovr. Probl. Mat., 9, All-Union Institute for Scientific and Technical Information, Moscow (1976), pp. 5–130.Google Scholar
  13. 13.
    G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics [Russian translation], Nauka, Moscow (1980).Google Scholar
  14. 14.
    M. Fuchs, “Existence of solutions of nonlinear degenerated systems of parabolic variational inequalities,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, 221, 243–252 (1995).Google Scholar
  15. 15.
    Kh. Gaevsky, K. Greger, and K. Zakharias, Nonlinear Operator Equations and Operator-Differential Equations [Russian translation], Mir, Moscow (1978).Google Scholar
  16. 16.
    Z. Guan, A. G. Kartsatos, and I. V. Skrypnik, “Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators,” J. Differ. Equat., 188, 332–351 (2003).CrossRefGoogle Scholar
  17. 17.
    P. Hartman and G. Stampacchia, “On some nonlinear elliptic differential functional equations,” Acta Math., 115, 271–310 (1966).Google Scholar
  18. 18.
    G. I. Laptev, “The first boundary-value problem for quasilinear second-order elliptic equations with double degeneration,” Differ. Uravn., 30, No.6, 1057–1068 (1994).Google Scholar
  19. 19.
    J.-L. Lions, Some Methods of Solution of Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).Google Scholar
  20. 20.
    J.-L. Lions and G. Stampacchia, “Variational inequalities,” Comm. Pure Appl. Math., XX, 493–519 (1967).Google Scholar
  21. 21.
    V. S. Mel’nik, “Multi-valued inequalities and operator inclusions in Banach spaces with operators of class (S +),” Ukr. Mat. Zh., 52, No.11, 1513–1523 (2000).CrossRefGoogle Scholar
  22. 22.
    V. S. Mel’nik and O. V. Solonukha, “On stationary variational inequalities with multi-valued operators,” Kibernet. Sistem. Anal., No. 3, 74–89 (1997).Google Scholar
  23. 23.
    V. S. Mel’nik and A. N. Vakulenko, “Topological method in the theory of operator inclusions with densely defined mappings in Banach spaces,” Nonlinear Boundary Value Problems, 11, 132–145 (2001).Google Scholar
  24. 24.
    V. S. Mel’nik and M. Z. Zgurovskii, Nonlinear Analysis and Control of Infinite-Dimensional Systems [in Russian], Naukova Dumka, Kiev (1999).Google Scholar
  25. 25.
    R. T. Rockafellar, “Local boundedness of nonlinear monotone operators,” Michigan Math. J., 16, 397–407 (1969).CrossRefGoogle Scholar
  26. 26.
    R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, New York (1998).Google Scholar
  27. 27.
    I. V. Skrypnik, Methods for the Investigation of Nonlinear Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow (1990).Google Scholar
  28. 28.
    O. V. Solonoukha, “On the stationary variational inequalities with generalized pseudomonotone operators,” Methods of Functional Analysis and Topology, 3, No.4, 81–95 (1977).Google Scholar
  29. 29.
    O. V. Solonukha, “On solvability of monotone type problems with non-coercive set-valued operators,” Methods of Functional Analysis and Topology, 6, No.2, 66–72 (2000).Google Scholar
  30. 30.
    O. V. Solonukha, “On the existence of solutions of parabolic equations with monotone operators,” Naukovi Visti NTUU “KPI”,” No. 6, 137–146 (2001).Google Scholar
  31. 31.
    O. V. Solonukha, “On surjectivity of noncoercive mappings: existence of solutions of equations, inclusions, and variational inequalities,” Dopovidi NAN Ukraini, No. 3, 31–37 (2003).Google Scholar
  32. 32.
    O. V. Solonoukha, “On the existence of solutions of operator-differential inclusions and nonstationary variational inequalities,” Dopovidi NAN Ukraini, No. 4, 25–31 (2003).Google Scholar
  33. 33.
    N. N. Uraltseva, “Existence of strong solutions of quasilinear parabolic equations with unilateral conditions on the boundary,” Vestnik Leningrad Univ., 13, 89–98 (1977).Google Scholar
  34. 34.
    M. Z. Zgurovskii and V. S. Melnik, “Ki Fang’s inequality and operator inclusions in Banach spaces,” Kibernet. Sistem. Anal., No. 2, 70–85 (2002).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • O. V. Solonukha

There are no affiliations available

Personalised recommendations