Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors

  • Mikhail Z. Zgurovsky
  • Pavlo O. Kasyanov
  • Oleksiy V. Kapustyan
  • José Valero
  • Nina V. Zadoianchuk
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 27)

Abstract

One of the most effective approaches to investigate nonlinear problems, represented by partial differential equations, inclusions and inequalities with boundary values, consists in the reduction of them into differential-operator inclusions, in infinite-dimensional spaces governed by nonlinear operators. In order to study these objects, the modern methods of nonlinear analysis have been used [7, 10, 11, 26]. Convergence of approximate solutions to an exact solution of the differential-operator equation or inclusion is frequently proved on the basis of the property of monotony or pseudomonotony of the corresponding operator.

Keywords

Weak Solution Evolution Inclusion Multivalued Operator Pullback Attractor Pseudomonotone Operator 
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.

References

  1. 1.
    Aubin JP, Ekeland I (1988) Applied nonlinear analysis. Mir, MoscowGoogle Scholar
  2. 2.
    Barbu V (1976) Nonlinear semigroups and differential equations in Banach spaces. Editura Acad, BucurestiGoogle Scholar
  3. 3.
    Bearman PW, Obasaju ED (1982) An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. J Fluid Mech 119:297–321Google Scholar
  4. 4.
    Browder FE (1977) Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc Nat Acad Sci 74:2659–2661Google Scholar
  5. 5.
    Browder FE, Hess P (1972) Nonlinear mappings of monotone type in Banach spaces. J Funct Anal. doi:10.1016/0022–1236(72)90070–5Google Scholar
  6. 6.
    Carl S, Motreanu D (2003) Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient. J Differ Equat. doi:10.1016/S0022–0396(03)00022–6Google Scholar
  7. 7.
    Chikrii AA (1997) Conflict-controlled processes. Kluver Academic Publishers, BostonGoogle Scholar
  8. 8.
    Davis RW, Moore EF (1982) A numerical study of vortex shedding from rectangles. J Fluid Mech 116:475–506Google Scholar
  9. 9.
    Denkowski Z, Migorski S, Papageorgiou NS (2003) An introduction to nonlinear analysis. Kluwer Academic Publishers, BostonGoogle Scholar
  10. 10.
    Duvaut G, Lions JL (1980) Inequalities in mechanics and in physics. Nauka, MoskowGoogle Scholar
  11. 11.
    Gajewski H, Gr\(\mathrm{\ddot{o}}\)ger K, Zacharias K (1974) Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie, BerlinGoogle Scholar
  12. 12.
    Guan Z, Karsatos AG, Skrypnik IV (2003) Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators. J Differ Equat. doi:10.1016/S0022-0396(02)00066-9Google Scholar
  13. 13.
    Hu S, Papageorgiou NS (1997) Handbook of multivalued analysis, vol. I: Theory. Kluwer Academic Publishers, DordrechtGoogle Scholar
  14. 14.
    Hu S, Papageorgiou NS (1997) Handbook of Multivalued Analysis, vol. II: Applications. Kluwer Academic Publishers, DordrechtGoogle Scholar
  15. 15.
    Kapustyan VO, Kasyanov PO, Kogut OP (2008) On solvability for one class of parameterized operator inclusions. Ukr Math Journ. doi:10.1007/s11253-009-0179-zGoogle Scholar
  16. 16.
    Kasyanov PO (2011) Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern Syst Anal 47(5):800–811. doi:10.1007/s10559-011-9359-6Google Scholar
  17. 17.
    Kasyanov PO, Melnik VS (2005) Faedo–Galerkin method differential-operator inclusions in Banach spaces with maps of \({w}_{{\lambda }_{0}}\) -pseudomonotone type. Nats Acad Sci Ukr. Kiev, Inst. Math., Prepr. Part 2, 1:82–105Google Scholar
  18. 18.
    Kasyanov PO, Melnik VS (2007) On solvabbility of differential-operator inclusions and evolution variation inequalities generated by \({w}_{{\lambda }_{0}}\) -pseudomonotone maps type. Ukr Math Bull. 4:535–581Google Scholar
  19. 19.
    Kasyanov PO, Mel’nik VS, Piccirillo AM (2008) Local subdifferentials and multivariational inequalities in Banach and Frechet spaces. Opuscula Mathematica 28:295–311Google Scholar
  20. 20.
    Kasyanov PO, Mel’nik VS, Toscano S (2006) Periodic solutions for nonlinear evolution equations with \({W}_{{\lambda }_{0}}\)-pseudomonotone maps. Nonlin Oscil. doi: 10.1007/s11072-006-0037-yGoogle Scholar
  21. 21.
    Kasyanov P, Mel’nik V, Toscano S (2009) Initial time value problem solutions for evolution inclusions with S k type operators. Syst Res Inform Tech (1):116–130Google Scholar
  22. 22.
    Kasyanov PO, Mel’nik VS, Toscano S (2010) Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued -pseudomonotone maps. J Differ Equat 249(6):1258–1287. doi:10.1016/j.jde.2010.05.008Google Scholar
  23. 23.
    Kasyanov PO, Melnik VS, Yasinsky VV (2007) Evolution inclusions and inequalities in Banach spaces with \({w}_{\lambda }\)-pseudomonotone maps. Naukova Dumka, KievGoogle Scholar
  24. 24.
    Kasyanov PO, Melnik VS, Valero J (2008) On the method of approximation for evolutionary inclusions of pseudo monotone type. Bull Aust Math Soc. doi:10.1017/S0004972708000130Google Scholar
  25. 25.
    Kuttler K (2000) Non-degenerate implicit evolution inclusions. Electron J Differ Equat 34: 1–20Google Scholar
  26. 26.
    Lions JL (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier-Villars, ParisGoogle Scholar
  27. 27.
    Liu Z, Mig\(\mathrm{\acute{o}}\)rski S (2008) Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete Continuous Dyn Syst Ser B. doi:10.3934/dcdsb.2008.9.129Google Scholar
  28. 28.
    Mel’nik VS (1997) Critical points of some classes of multivalued mappings. Cybern Syst Anal. doi: 10.1007/BF02665895Google Scholar
  29. 29.
    Mel’nik VS (2000) Multivariational inequalities and operator inclusions in banach spaces with mappings of the class (S) + . Ukr Mat Zh. doi: 10.1023/A:1010431221039Google Scholar
  30. 30.
    Mel’nik VS (2006) Topological methods in the theory of operator inclusions in Banach spaces. I. Ukr Math Journ. doi: 10.1007/s11253-006-0062-0Google Scholar
  31. 31.
    Minewitsch S, Franke R, Rodi W (1994) Numerical investigation of laminar vortex-shedding flow past a square cylinder oscillating in line with the mean flow. J Fluid Struct 8:787–802Google Scholar
  32. 32.
    Naniewicz Z, Panagiotopoulos PD (1995) Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New YorkGoogle Scholar
  33. 33.
    Okajima A (1982) Strouhal number of rectangular cylinders. J Fluid Mech 123:379–398Google Scholar
  34. 34.
    Panagiotopoulos PD (1985) Inequality problems in mechanics and applications. Convex and Nonconvex Energy Functions. Birkhauser, BaselGoogle Scholar
  35. 35.
    Panagiotopoulos PD (1993) Hemivariational inequalities, applications in mechanics and engineering. Springer, BerlinGoogle Scholar
  36. 36.
    Peng Z, Liu Z (2011) A note on multivalued \({W}_{{\lambda }_{0}}\) pseudomonotone map. Appl Math Lett 24:1204–1208Google Scholar
  37. 37.
    Perestyuk MO, Kasyanov PO, Zadoyanchuk NV (2008) On Faedo–Galerkin method for evolution inclusions with \({w}_{{\lambda }_{0}}\) -pseudomonotone maps. Memoir Differ Equat Math Phys 44:105–132Google Scholar
  38. 38.
    Perestyuk NA, Plotnikov VA, Samoilenko AM, Skrypnik NV (2007) Impulse differential equations with multivalued and discontinuous raght-hand side. Institute of mathematics NAS of Ukraine, KyivGoogle Scholar
  39. 39.
    Skrypnik IV (1990) Methods of investigation of nonlinear elliptic boundary problems. Nauka, MoscowGoogle Scholar
  40. 40.
    Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New YorkGoogle Scholar
  41. 41.
    Vickery BJ (1966) Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. J Fluid Mech 25:481–494Google Scholar
  42. 42.
    Zadoyanchuk NV, Kas’yanov PO (2009) FaedoGalerkin method for second-order evolution inclusions with \({W}_{\lambda }\)-pseudomonotone mappings. Ukrainian Math J. doi:10.1007/s11253-009-0207-zGoogle Scholar
  43. 43.
    Zadoianchuk NV, Kas’yanov PO (2012) Dynamics of solutions for the class of second order autonomous evolution inclusions. Cybern Syst Anal 48(3):344–366Google Scholar
  44. 44.
    Zgurovsky MZ, Kasyanov PO, Melnik VS (2008) Differential-operator inclusions and variation inequalities in infinitedimensional spaces (in Russian). Naukova dumka, KyivGoogle Scholar
  45. 45.
    Zgurovsky MZ, Kasyanov PO, Valero J (2010) Noncoercive evolution inclusions for Sk type operators. Int J Bifurcat Chaos 20(9):2823–2834Google Scholar
  46. 46.
    Zgurovsky MZ, Melnik VS (2002) Ky Fan inequality and operational inclusions in Banach spaces. Cybern Syst Anal. doi: 10.1023/A:1016391328367Google Scholar
  47. 47.
    Zgurovsky MZ, Melnik VS (2004) Nonlinear analysis and control of physical processes and fields. Springer, BerlinGoogle Scholar
  48. 48.
    Zgurovsky MZ, Mel’nik VS, Kasyanov PO (2010) Evolution inclusions and variation inequalities for earth data processing I. Springer, Heidelberg. doi: 10.1007/978-3-642-13837-9
  49. 49.
    Zgurovsky MZ, Mel’nik VS, Kasyanov PO (2010) Evolution inclusions and variation inequalities for earth data processing II. Springer, Heidelberg. doi: 10.1007/978-3-642-13878-2

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mikhail Z. Zgurovsky
    • 1
  • Pavlo O. Kasyanov
    • 2
  • Oleksiy V. Kapustyan
    • 2
  • José Valero
    • 3
  • Nina V. Zadoianchuk
    • 2
  1. 1.Institute for Applied System AnalysisNational Academy of Science National Technical University of UkraineKyivUkraine
  2. 2.Kyiv Polytechnic Institute Institute for Applied System AnalysisNational Technical University of UkraineKyivUkraine
  3. 3.Centro de Investigación OperativaUniv. Miguel Hernández de ElcheAlicanteSpain

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