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January  2013, 12(1): 281-302. doi: 10.3934/cpaa.2013.12.281

On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence

Tomás Caraballo 1, and  David Cheban 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  June 2011 Revised  August 2011 Published  September 2011

The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so--called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper [6] to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations).
Keywords: Non-autonomous dynamical systems, almost periodic, quasi-periodic, asymptotically almost periodic solutions, convergent systems, almost automorphic, functional differential equations; evolution equations with monotone operators, cocycles, global attractor, dissipative systems, recurrent solutions, skew-product systems.
Mathematics Subject Classification: Primary: 34C11, 34C27, 34D05, 34D23, 34D45, 37C55, 7C60, 37C65, 37C70, 37C7.
Citation: Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
References:
[1]

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464.

[2]

H. Brezis, "Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert," Vol. 5 of Math. Studies, North Holland, 1973.

[3]

I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.

[4]

F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc., Ser. A 71, Indag. Math., 30 (1968), 27-35.

[5]

T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations, Journal of Differential Equations, 251 (2011), 708-727.

[6]

T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous dynamical systems with weak convergence, Communications in Pure and Applied Analysis, 11 (2012), 809-828.

[7]

T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous difference equations with weak convergence, Journal of Difference Equations and Applications, (2012), to appear.

[8]

D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528pp.

[9]

D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations, J. Dynamics and Differential Equations, 20 (2008), 669-697.

[10]

D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203 pp.

[11]

D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems," Nova Science Publishers, New York, 2010, xvii+269 p.

[12]

D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374-393.

[13]

I. D. Chueshov, "Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem. Universitetskie Lektsii po Sovremennoi Matematike," AKTA, Kharkiv, 1999. 436 pp. (in Russian) [English translation: Introduction to the theory of infinite-dimensional dissipative systems. University Lectures in Contemporary Mathematics. AKTA, Kharkiv, 1999. 436 pp.]

[14]

C. Conley, "Isolated Invariant Sets and the Morse Index," Region. Conf. Ser. Math., No. 38, 1978. Am. Math. Soc., Providence, RI.

[15]

B. P. Demidovich, "Lectures on Mathematical Theory of Stability," Moscow, Nauka, 1967. (in Russian)

[16]

A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280-284.

[17]

J. K. Hale, "Theory of Functional-Differential Equations," Springer-Verlag, New York-Heidelberg-Berlin, 1977.

[18]

J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.

[19]

N. Hassani, "Systems Dynamiques Nonautonomes Contractants et leur Applications," Theése de magister. Algerie, USTHB, 1983.

[20]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dyn. Diff. Eqns., 13 (2001), 107-131.

[21]

D. Husemoller, "Fibre Bundles," Springer-Verlag, Berlin-Heidelberg-New York, 1994.

[22]

W. A. Kirk and B. Sims, "Handbook of Metric Fixed Point Theory," Kluwer Academic Publishers, Dodrecht/Boston/London/2001, 703 pp.

[23]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis TMA, 74 (2011), 2695-2719.

[24]

B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.

[25]

J. L. Lions, "Quelques Methodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Paris, 1969.

[26]

B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 1246-1255.

[27]

G. R. Sell, "Topological Dynamics and Ordinary Differential Equations," Van Nostrand-Reinhold, London, 1971.

[28]

T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp.

[29]

V. V. Zhikov, On Stability and Unstability of Levinson's centre, Differentsial'nye Uravneniya, 8 (1972), 2167-2170.

[30]

V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations, Mat. Sbornik, 90 (1973), 214-228; English transl., Math. USSR-Sb., 19 (1974), 209-223.

show all references

References:
[1]

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464.

[2]

H. Brezis, "Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert," Vol. 5 of Math. Studies, North Holland, 1973.

[3]

I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.

[4]

F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc., Ser. A 71, Indag. Math., 30 (1968), 27-35.

[5]

T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations, Journal of Differential Equations, 251 (2011), 708-727.

[6]

T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous dynamical systems with weak convergence, Communications in Pure and Applied Analysis, 11 (2012), 809-828.

[7]

T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous difference equations with weak convergence, Journal of Difference Equations and Applications, (2012), to appear.

[8]

D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528pp.

[9]

D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations, J. Dynamics and Differential Equations, 20 (2008), 669-697.

[10]

D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203 pp.

[11]

D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems," Nova Science Publishers, New York, 2010, xvii+269 p.

[12]

D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374-393.

[13]

I. D. Chueshov, "Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem. Universitetskie Lektsii po Sovremennoi Matematike," AKTA, Kharkiv, 1999. 436 pp. (in Russian) [English translation: Introduction to the theory of infinite-dimensional dissipative systems. University Lectures in Contemporary Mathematics. AKTA, Kharkiv, 1999. 436 pp.]

[14]

C. Conley, "Isolated Invariant Sets and the Morse Index," Region. Conf. Ser. Math., No. 38, 1978. Am. Math. Soc., Providence, RI.

[15]

B. P. Demidovich, "Lectures on Mathematical Theory of Stability," Moscow, Nauka, 1967. (in Russian)

[16]

A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280-284.

[17]

J. K. Hale, "Theory of Functional-Differential Equations," Springer-Verlag, New York-Heidelberg-Berlin, 1977.

[18]

J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.

[19]

N. Hassani, "Systems Dynamiques Nonautonomes Contractants et leur Applications," Theése de magister. Algerie, USTHB, 1983.

[20]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dyn. Diff. Eqns., 13 (2001), 107-131.

[21]

D. Husemoller, "Fibre Bundles," Springer-Verlag, Berlin-Heidelberg-New York, 1994.

[22]

W. A. Kirk and B. Sims, "Handbook of Metric Fixed Point Theory," Kluwer Academic Publishers, Dodrecht/Boston/London/2001, 703 pp.

[23]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis TMA, 74 (2011), 2695-2719.

[24]

B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.

[25]

J. L. Lions, "Quelques Methodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Paris, 1969.

[26]

B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 1246-1255.

[27]

G. R. Sell, "Topological Dynamics and Ordinary Differential Equations," Van Nostrand-Reinhold, London, 1971.

[28]

T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp.

[29]

V. V. Zhikov, On Stability and Unstability of Levinson's centre, Differentsial'nye Uravneniya, 8 (1972), 2167-2170.

[30]

V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations, Mat. Sbornik, 90 (1973), 214-228; English transl., Math. USSR-Sb., 19 (1974), 209-223.

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