# American Institute of Mathematical Sciences

November  2016, 21(9): 2949-2967. doi: 10.3934/dcdsb.2016081

## Equi-attraction and continuity of attractors for skew-product semiflows

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP 3 Instituto de Ciências Matemáticas e de Computacao, Universidade de Săo Paulo-Campus de Săo Carlos, Caixa Postal 668, 13560-970 Săo Carlos SP, Brazil 4 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received  July 2015 Revised  November 2015 Published  October 2016

In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.
Citation: Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081
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