Abstract
In this work, we define an exact generalized process and we establish some results such as monotonicity, compactness, and upper semicontinuity for the multivalued process defined by the exact generalized process. The main result is on compactness, invariance, and attraction properties of \(\omega \)-limit sets.
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References
Babin, A.V., Vishik, M.I.: Attractors of evolution equations. Studies in Applied Mathematics, vol. 25. North-Holland Publishing Co., Amsterdam (1992)
Caraballo, T., Marín-Rubio, P., Robinson, J.C.: A comparison between two theories for multi-valued semiflows and their asymptotic behaviour. Set-Valued Anal. 11, 297–322 (2003)
Carvalho, A.N., Gentile, C.B.: Asymptotic behaviour of non-linear parabolic equations with monotone principal part. J. Math. Anal. Appl. 280, 252–272 (2003)
Cholewa, J.W., Dlotko, T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000)
Chueshov, I.D.: Introduction to the Theory of Infinite-dimensional Dissipative Systems. ACTA Scientific Publishing House, Kharkiv (2002)
Hale, J.K.: Asymptotic behavior of dissipative system. mathematical surveys and monographs. in American Mathematical Society, vol. 25. Providence (1988)
Kapustyan, A.V., Melnik, V.S., Valero, J., Yasinsky, V.V.: Global Attractors for Multi-Valued Evolution Equations Without Uniqueness. Naukova Dumka, Kiev (2008)
Ladyzhenskaya, O.: Attracors for Semigroups an Evolution Equations. Lincei Lectures, vol. 25. Cambridge University Press, Cambridge (1991)
Melnik, V.S., Valero, J.: On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)
Simsen, J., Gentile, C.B.: On attractors for multivalued semigroups defined by generalized semiflows. Set-Valued Anal. 16, 105–124 (2008)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)
Caraballo, T., Kloeden, P.E., Marín-Rubio, P.: Weak pullback attractors of setvalued processes. J. Math. Anal. Appl. 288, 692–707 (2003)
Caraballo, T., Carvalho, A.N., Langa, J.A., Rivero, F.: Existence of pullback attractors for pullback asymptotically compact processes. Nonlinear Anal. 72, 1967–1976 (2010)
Carvalho, A.N., Langa, J.A., Robinson, J.C.: On the continuity of pullback attractors for evolution processes. Nonlinear Anal. 71, 1812–1824 (2009)
Cheban, D.N.: Global Attractors of Non-autonomous Dissipative Dynamical Systems. World Scientific Publishing Ltd., Singapore (2004)
Kloeden, P.E., Schmalfuss, B.: Asymptotic behaviour of non-autonomous difference inclusions. Syst. Control Lett. 33, 275–280 (1998)
Kloeden, P.E., Schmalfu\(\beta \), B.: Nonautonomous systems, cocycle attractors and variable time-step discretization. Numer. Algorithms 14, 141–152 (1997)
Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9(2), 307–341 (1997)
Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)
Kloeden, P.E.: Pullback attractors in nonautonomous difference equations. J. Differ. Equ. Appl. 6(1), 33–52 (2000)
Caraballo, T., Langa, J.A., Melnik, V.S., Valero, J.: Pullback attractors of nonautonomous and stochastic multivalued dynamical systems. Set-Valued Analysis 11, 153–201 (2003)
Melnik, V.S., Valero, J.: On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. 8, 375–403 (2000)
Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7(5), 475–502 (1997)
Ball, J.M.: On the asymptotic behavior of generalized process with applications to nonlinear evolution equations. J. Differ. Equ. 27, 224–265 (1978)
Szegö, G.P., Treccani, G.: Semigruppi di Trasformazioni Multivoche. Springer Lecture Notes in Mathematics, vol. 101 (1969)
Kloeden, P.E., Marín-Rubio, P.: Weak pullback attractors of non-autonomous difference inclusions. J. Differ. Equ. Appl. 9(5), 489–502 (2003)
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Simsen, J., Capelato, É. (2015). Some Properties for Exact Generalized Processes. In: Sadovnichiy, V., Zgurovsky, M. (eds) Continuous and Distributed Systems II. Studies in Systems, Decision and Control, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-19075-4_12
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DOI: https://doi.org/10.1007/978-3-319-19075-4_12
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