Weak convergence of orbits of nonlinear operators in reflexive Banach spaces

Dan Butnariu, Simeon Reich, Alexander J. Zaslavski

Research output: Contribution to journalArticlepeer-review


Let K be a closed convex subset of a reflexive Banach space X. We consider self-mappings of K which are bounded on bounded subsets of K and satisfy a relaxed form of nonexpansivity with respect to a given convex function f. The family of these operators is endowed with the topology of uniform convergence on bounded subsets of K. We show that "almost all" such operators T share the property that they have a fixed point zT such that, for any x ∈ K, the orbit {Tkx}k=1 converges weakly to ZT. Here the meaning of "almost all" is in the sense of Baire's categories: the collection of all those operators which do not have this property is contained in a countable union of nowhere dense sets.

Original languageEnglish
Pages (from-to)489-508
Number of pages20
JournalNumerical Functional Analysis and Optimization
Issue number5-6
StatePublished - 2003


  • Bregman distance
  • Fixed point
  • Nonexpansivity of an operator with respect to a convex function
  • Orbit
  • Reflexive Banach space
  • Total convexity
  • Uniform convexity
  • Weak convergence

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization


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