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Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

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  • Moosa Gabeleh
  • Naseer Shahzad

Abstract

Let A and B be two nonempty subsets of a Banach space X . A mapping T : is said to be cyclic relatively nonexpansive if T ( A ) and T ( B ) and for all ( ) . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X . It is shown that if ( A , B ) is a nonempty, weakly compact, and convex pair and ( A , B ) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.

Suggested Citation

  • Moosa Gabeleh & Naseer Shahzad, 2014. "Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, January.
    • Handle: RePEc:hin:jnlaaa:123613
    DOI: 10.1155/2014/123613
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