zbMATH — the first resource for mathematics

Relation between different types of global attractors of set-valued nonautonomous dynamical systems. (English) Zbl 1087.37016
The authors study the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems. They prove that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but they prove that every global pullback attractor of an $$\alpha$$-condensing set-valued cocycle is always a local forward attractor. The obtained results are applied to periodic and homogeneous systems. The authors give also a new criterion for the absolute asymptotic stability of nonstationary discrete linear inclusions.

MSC:
 37B55 Topological dynamics of nonautonomous systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37B25 Stability of topological dynamical systems 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
Full Text:
References:
 [1] Babin, A. V. and Vishik, M. I.: Attractors of Evolutionary Equations, Nauka, Moscow, 1989; English translation: North–Holland, Amsterdam, 1992. · Zbl 0804.58003 [2] Babin, A.: Attractors of the generalized semi-group generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math. 44(2) (1995), 207–223 (translated from Izv. Russian Acad. Sci. 58 (1994)). · Zbl 0839.35036 [3] Ball, J. M.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7(5) (1997), 475–502. · Zbl 0903.58020 [4] Ball, J. M.: Global atractors for damped semilinear wave equations, Discrete Contin. Dynam. Systems 10(1–2) (2004), 31–52. · Zbl 1056.37084 [5] Caraballo, T., Marin-Rubio, P. and Robinson, J. C.: A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal. 11(3) (2003), 297–392. · Zbl 1053.47050 [6] Cheban, D. N. and Fakeeh, D. S.: Global Attractors of Disperse Dynamical Systems, Sigma, Chişinău, 1994 (in Russian). [7] Cheban, D. N. and Fakeeh, D. S.: Global attractors of infinite-dimensional dynamical systems, III, Bull. Acad. Sci. Republic of Moldova. Mathematics 2–3(18–19) (1995), 3–13. · Zbl 1115.37362 [8] Cheban, D. N.: Global attractors of infinite-dimensional nonautonomous dynamical systems. I, Bull. Acad. Sci. Republic of Moldova. Mathematics 3(25) (1997), 42–55. · Zbl 1115.37361 [9] Cheban, D. N.: The asymptotics of solutions of infinite-dimensional homogeneous dynamical systems, Mat. Zametki 63(1) (1998), 115–126; translation in Math. Notes 63(1) (1998), 115–126. · Zbl 0927.37057 [10] Cheban, D. N. and Schmalfuss, B.: The global attractors of nonautonomous disperse dynamical systems and differential inclusions, Bull. Acad. Sci. Republic of Moldova. Mathematics 1(29) (1999), 3–22. · Zbl 1001.37012 [11] Cheban, D. N., Kloeden, P. E. and Schmalfuss, B.: Relation between pullback and global attractors of nonautonomous dynamical systems, Nonlinear Dynam. Systems Theory 2(2) (2002), 8–28. [12] Cheban, D. N.: Global Attractors of Nonautonomous Dynamical Systems, State University of Moldova, 2002 (in Russian). · Zbl 1008.34046 [13] Cheban, D. N. and Mammana, C.: Upper semicontinuity of attractors of set-valued nonautonomous dynamical systems, Internat. J. Pure Appl. Math. 4(5) (2003), 385–418. · Zbl 1152.37305 [14] Cheban, D. N.: Global Attractors of Nonautonomous Dissipative Dynamical Systems, World Scientific, Singapore, 2004 (in press). · Zbl 1098.37002 [15] Chepyzhov, V. V. and Vishik, M. I.: A Hausdorff dimension estimate for kernel sections of nonautonomous evolutions equations, Indian Univ. Math. J. 42(3) (1993), 1057–1076. · Zbl 0819.35073 [16] Chepyzhov, V. V. and Vishik, M. I.: Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. · Zbl 0986.35001 [17] Chueshov, I. D.: Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkov, 2002. · Zbl 1100.37047 [18] Fakeeh, D. S.: Levinson’s center of disperse dissipative dynamical systems, Izv. AN SSRM 3 (1990), 55–59. · Zbl 0896.54025 [19] Fakeeh, D. S.: On structure of Levinson’s center of disperse dynamical systems, Izv. Akad. Nauk SSR Moldova 1 (1991), 62–67. · Zbl 0888.54044 [20] Fakeeh, D. S.: Analogue of Levinson–Pliss’ theorem for differential inclusions, In: Mat. Issled. 124, Ştiinţa, Chişinău, 1992, pp. 100–105. [21] Fakeeh, D. S. and Cheban, D. N.: Connectedness of Levinson’s center of compact dissipative dynamical system without uniqueness, Izv. Akad. Nauk Respub. Moldova Mat. 1 (1993), 15–22. · Zbl 0842.54039 [22] Fillipov, A. F.: Differential Equations with Discontinuous Right Part, Nauka, Moscow, 1985 (in Russian). [23] Gurvits, L.: Stability of discrete linear inclusion, Linear Algebra Appl. 231 (1995), 47–85. · Zbl 0845.68067 [24] Hale, J. K.: Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. · Zbl 0642.58013 [25] Husemoller, D.: Fibre Bundles, Springer, Berlin, 1994. · Zbl 0202.22903 [26] Kloeden, P. E. and Schmalfuss, B.: Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms 14 (1997), 141–152. · Zbl 0886.65077 [27] Ladyzhenskaya, O. A.: Attractors for Semigroups and Evolution Equations, Lizioni Lincei, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0755.47049 [28] Melnik, V. S.: Multivalued semiflows and their attractors, Dokl. Akad. Nauk 343 (1995), 302–305; English translation in Dokl. Math. 52 (1995), 36–39. · Zbl 0922.54035 [29] Melnik, V. S. and Valero, J.: On ttractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111. · Zbl 0915.58063 [30] Melnik, V. S. and Valero, J.: On attractors of multi-valued semi-processes and nonautonomus evolutions inclusions, Set-Valued Anal. 8 (2000), 375–403. · Zbl 1063.35040 [31] Pilyugin, S. Yu.: Attracting sets and systems without uniqueness, Mat. Zametki 42(5) (1987), 703–711. · Zbl 0647.58026 [32] Sadovskii, B. N.: Limit compact and condensing operators, Uspekhi Mat. Nauk 27(1(163)) (1972), 81–146. · Zbl 0232.47067 [33] Sell, G. R. and You, Y.: Dynamics of Evolutionary Equations, Springer, New York, 2002. · Zbl 1254.37002 [34] Sell, G. R.: Lectures on Topological Dynamics and Differential Equations, Van Nostrand Reinhold Math. Studies 2, Van Nostrand Reinhold, London, 1971. · Zbl 0212.29202 [35] Shcherbakov, B. A.: Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972 (in Russian). · Zbl 0256.34062 [36] Shcherbakov, B. A.: Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985 (in Russian). · Zbl 0638.34046 [37] Sibirskii, K. S. and Shube, A. S.: Semidynamical Systems, Ştiinţa, Chişinău, 1987 (in Russian). [38] Zubov, V. I.: The Methods of A. M. Lyapunov and Their Application, Noordhoof, Groningen, 1964. · Zbl 0115.30204 [39] Zubov, V. I.: Stability of Motion, Vysshaya Shkola, Moscow, 1973 (in Russian). · Zbl 0273.70005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.