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Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors

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Evolution Inclusions and Variation Inequalities for Earth Data Processing III

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 27))

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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.

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Notes

  1. 1.

    That is, V is a real reflexive separable Banach space embedded into a real Hilbert space H continuously and densely, H is identified with its conjugated space H  ∗  and V  ∗  is a dual space to V. So, we have such chain of continuous and dense embeddings: \(V \subset H \equiv {H}^{{_\ast}}\subset {V }^{{_\ast}}\) (see, e.g., [49]).

References

  1. Aubin JP, Ekeland I (1988) Applied nonlinear analysis. Mir, Moscow

    Google Scholar 

  2. Barbu V (1976) Nonlinear semigroups and differential equations in Banach spaces. Editura Acad, Bucuresti

    Google Scholar 

  3. Bearman PW, Obasaju ED (1982) An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. J Fluid Mech 119:297–321

    Google Scholar 

  4. Browder FE (1977) Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc Nat Acad Sci 74:2659–2661

    Google Scholar 

  5. Browder FE, Hess P (1972) Nonlinear mappings of monotone type in Banach spaces. J Funct Anal. doi:10.1016/0022–1236(72)90070–5

    Google Scholar 

  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)00022–6

    Google Scholar 

  7. Chikrii AA (1997) Conflict-controlled processes. Kluver Academic Publishers, Boston

    Google Scholar 

  8. Davis RW, Moore EF (1982) A numerical study of vortex shedding from rectangles. J Fluid Mech 116:475–506

    Google Scholar 

  9. Denkowski Z, Migorski S, Papageorgiou NS (2003) An introduction to nonlinear analysis. Kluwer Academic Publishers, Boston

    Google Scholar 

  10. Duvaut G, Lions JL (1980) Inequalities in mechanics and in physics. Nauka, Moskow

    Google Scholar 

  11. Gajewski H, Gr\(\mathrm{\ddot{o}}\)ger K, Zacharias K (1974) Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie, Berlin

    Google Scholar 

  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)00066-9

    Google Scholar 

  13. Hu S, Papageorgiou NS (1997) Handbook of multivalued analysis, vol. I: Theory. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  14. Hu S, Papageorgiou NS (1997) Handbook of Multivalued Analysis, vol. II: Applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  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-z

    Google Scholar 

  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/s10559-011-9359-6

    Google Scholar 

  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, 1:82–105

    Google Scholar 

  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. 4:535–581

    Google Scholar 

  19. Kasyanov PO, Mel’nik VS, Piccirillo AM (2008) Local subdifferentials and multivariational inequalities in Banach and Frechet spaces. Opuscula Mathematica 28:295–311

    Google Scholar 

  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-y

    Google Scholar 

  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):116–130

    Google Scholar 

  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.2010.05.008

    Google Scholar 

  23. Kasyanov PO, Melnik VS, Yasinsky VV (2007) Evolution inclusions and inequalities in Banach spaces with \({w}_{\lambda }\)-pseudomonotone maps. Naukova Dumka, Kiev

    Google Scholar 

  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/S0004972708000130

    Google Scholar 

  25. Kuttler K (2000) Non-degenerate implicit evolution inclusions. Electron J Differ Equat 34: 1–20

    Google Scholar 

  26. Lions JL (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier-Villars, Paris

    Google Scholar 

  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.2008.9.129

    Google Scholar 

  28. Mel’nik VS (1997) Critical points of some classes of multivalued mappings. Cybern Syst Anal. doi: 10.1007/BF02665895

    Google Scholar 

  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:1010431221039

    Google Scholar 

  30. Mel’nik VS (2006) Topological methods in the theory of operator inclusions in Banach spaces. I. Ukr Math Journ. doi: 10.1007/s11253-006-0062-0

    Google Scholar 

  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 8:787–802

    Google Scholar 

  32. Naniewicz Z, Panagiotopoulos PD (1995) Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York

    Google Scholar 

  33. Okajima A (1982) Strouhal number of rectangular cylinders. J Fluid Mech 123:379–398

    Google Scholar 

  34. Panagiotopoulos PD (1985) Inequality problems in mechanics and applications. Convex and Nonconvex Energy Functions. Birkhauser, Basel

    Google Scholar 

  35. Panagiotopoulos PD (1993) Hemivariational inequalities, applications in mechanics and engineering. Springer, Berlin

    Google Scholar 

  36. Peng Z, Liu Z (2011) A note on multivalued \({W}_{{\lambda }_{0}}\) pseudomonotone map. Appl Math Lett 24:1204–1208

    Google Scholar 

  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 44:105–132

    Google Scholar 

  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, Kyiv

    Google Scholar 

  39. Skrypnik IV (1990) Methods of investigation of nonlinear elliptic boundary problems. Nauka, Moscow

    Google Scholar 

  40. Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York

    Google Scholar 

  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 25:481–494

    Google Scholar 

  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-z

    Google Scholar 

  43. Zadoianchuk NV, Kas’yanov PO (2012) Dynamics of solutions for the class of second order autonomous evolution inclusions. Cybern Syst Anal 48(3):344–366

    Google Scholar 

  44. Zgurovsky MZ, Kasyanov PO, Melnik VS (2008) Differential-operator inclusions and variation inequalities in infinitedimensional spaces (in Russian). Naukova dumka, Kyiv

    Google Scholar 

  45. Zgurovsky MZ, Kasyanov PO, Valero J (2010) Noncoercive evolution inclusions for Sk type operators. Int J Bifurcat Chaos 20(9):2823–2834

    Google Scholar 

  46. Zgurovsky MZ, Melnik VS (2002) Ky Fan inequality and operational inclusions in Banach spaces. Cybern Syst Anal. doi: 10.1023/A:1016391328367

    Google Scholar 

  47. Zgurovsky MZ, Melnik VS (2004) Nonlinear analysis and control of physical processes and fields. Springer, Berlin

    Google Scholar 

  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. 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

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Authors and Affiliations

  1. Institute for Applied System Analysis, National Academy of Science National Technical University of Ukraine, Kyiv, Ukraine

    Mikhail Z. Zgurovsky

  2. Kyiv Polytechnic Institute Institute for Applied System Analysis, National Technical University of Ukraine, Kyiv, Ukraine

    Pavlo O. Kasyanov, Oleksiy V. Kapustyan & Nina V. Zadoianchuk

  3. Centro de Investigación Operativa, Univ. Miguel Hernández de Elche, Alicante, Spain

    José Valero

Authors
  1. Mikhail Z. Zgurovsky
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  2. Pavlo O. Kasyanov
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  3. Oleksiy V. Kapustyan
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  4. José Valero
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  5. Nina V. Zadoianchuk
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Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V. (2012). Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors. In: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Advances in Mechanics and Mathematics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28512-7_7

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