Cosmic divergence, weak cosmic convergence, and fixed points at infinity

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Abstract

To characterize the asymptotic behavior of fixed-point iterations of non-expansive operators with no fixed points, Bauschke et al. (J Fixed Point Theory Appl 18(2):297–307, 2016) recently studied cosmic convergence and conjectured that cosmic convergence always holds. This paper presents a cosmically divergent counterexample, which disproves this conjecture. This paper also demonstrates, with a counterexample, that cosmic convergence can be weak in infinite dimensions. Finally, this paper shows positive results relating to cosmic convergence that provide an interpretation of cosmic accumulation points as fixed points at infinity.

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Correspondence to Ernest K. Ryu.

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Ryu, E.K. Cosmic divergence, weak cosmic convergence, and fixed points at infinity. J. Fixed Point Theory Appl. 20, 109 (2018). https://doi.org/10.1007/s11784-018-0592-8

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Keywords

  • Cosmic convergence
  • non-expansive mapping
  • convex optimization
  • weak convergence
  • minimal displacement vector

Mathematics Subject Classification

  • Primary 47H09
  • Secondary 90C25