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Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 493–530 | Cite as

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship, Characterization and Robustness

  • Hongyong Cui
  • José A. Langa
  • Yangrong Li
  • José Valero
Article

Abstract

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.

Keywords

Multi-valued dynamical systems Pullback attractor Uniform attractor Lifted invariance Lower semi-continuity Weak equi-attraction 

Mathematics Subject Classification (2010)

35B40 35B41 37L30 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Hongyong Cui
    • 1
    • 2
  • José A. Langa
    • 2
  • Yangrong Li
    • 1
  • José Valero
    • 3
  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China
  2. 2.Departamento de Ecuaciones Diferenciales Análisis NuméricoUniversidad de SevillaSevillaSpain
  3. 3.Centro de Investigación OperativaUniversidad Miguel HernándezElcheSpain

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