# American Institute of Mathematical Sciences

## Forward omega limit sets of nonautonomous dynamical systems

 School of Mathematics and Statistics, and, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Dedicated to Professor Jürgen Scheurle on his 65th birthday

Received  September 2017 Revised  May 2018 Published  April 2019

Fund Project: HC was partially funded by China Postdoctoral Science Foundation 2017M612430. PEK and MY were partially supported by the Chinese NSF grant 11571125

The forward $\omega$-limit set $\omega_{\mathcal{B}}$ of a nonautonomous dynamical system $\varphi$ with a positively invariant absorbing family $\mathcal{B}$ $=$ $\{ B(t), t \in \mathbb{R}\}$ of closed and bounded subsets of a Banach space $X$ which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if $\varphi$ is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a $\varphi$-invariant family $\mathcal{A}$ $=$ $\left\{A(t), t \in \mathbb{R}\right\}$ in $\mathcal{B}$ of nonempty compact subsets of $X$ to be a forward attractor is generalised to this context.

Citation: Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020065
##### References:
 [1] M. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008. [2] T. Caraballo, J. A. Langa, V. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal, 11 (2003), 153-201. doi: 10.1023/A:1022902802385. [3] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of infinite dimensional nonautonomous dynamical systems, Springer, New York, 2013. [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. [5] H. Cui and P. E. Kloeden, Forward random attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186. [6] H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, (in press). [7] H. Cui and J. A. Langa, Uniform attractors for non-autonommous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268. doi: 10.1016/j.jde.2017.03.018. [8] P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. doi: 10.1016/0022-0396(75)90021-2. [9] P. E. Kloeden, Asymptotic invariance and the discretisation of nonautonomous forward attracting sets, J. Comput. Dynamics, 3 (2016), 179-189. doi: 10.3934/jcd.2016009. [10] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176. [12] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550. [13] V. Lakshmikantham and S. Leela, Asymptotic self-invariant sets and conditional stability, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, Puerto Rico 1965, Academic Press, New York, 1967, 363-373. [14] J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976. [15] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations., Cambridge University Press, Cambridge, 1992.

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##### References:
 [1] M. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008. [2] T. Caraballo, J. A. Langa, V. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal, 11 (2003), 153-201. doi: 10.1023/A:1022902802385. [3] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of infinite dimensional nonautonomous dynamical systems, Springer, New York, 2013. [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. [5] H. Cui and P. E. Kloeden, Forward random attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186. [6] H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, (in press). [7] H. Cui and J. A. Langa, Uniform attractors for non-autonommous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268. doi: 10.1016/j.jde.2017.03.018. [8] P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105. doi: 10.1016/0022-0396(75)90021-2. [9] P. E. Kloeden, Asymptotic invariance and the discretisation of nonautonomous forward attracting sets, J. Comput. Dynamics, 3 (2016), 179-189. doi: 10.3934/jcd.2016009. [10] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176. [12] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550. [13] V. Lakshmikantham and S. Leela, Asymptotic self-invariant sets and conditional stability, in Proc. Inter. Symp. Diff. Equations and Dynamical Systems, Puerto Rico 1965, Academic Press, New York, 1967, 363-373. [14] J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976. [15] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations., Cambridge University Press, Cambridge, 1992.
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