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

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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.

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Author notes

  1. Hongyong Cui

    Present address: Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, Sevilla, 41080, Spain

Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Hongyong Cui & Yangrong Li

  2. Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, Sevilla, 41080, Spain

    José A. Langa

  3. Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. de la Universidad s/n, Elche, 03202, Alicante, Spain

    José Valero

Authors
  1. Hongyong Cui
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  2. José A. Langa
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  3. Yangrong Li
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  4. José Valero
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Correspondence to José A. Langa.

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Cui, H., Langa, J.A., Li, Y. et al. Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship, Characterization and Robustness. Set-Valued Var. Anal 26, 493–530 (2018). https://doi.org/10.1007/s11228-016-0395-2

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  • DOI: https://doi.org/10.1007/s11228-016-0395-2

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