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Paper   IPM / M / 14839
School of Mathematics
  Title:   A characterization of proximal normal structure via proximal diametral sequences
  Author(s):  Moosa Gabeleh
  Status:   Published
  Journal: J. Fixed Point Theory Appl.
  Year:  2017
  Pages:   DOI: 10.1007/s11784-017-0460-y
  Supported by:  IPM
We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.

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