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Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

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  • M. De La Sen
  • Asier Ibeas

Abstract

This paper presents some results concerning the properties of distances and existence and uniqueness of best proximity points of p‐cyclic proximal, weak proximal contractions, and some of their generalizations for the non‐self‐mapping T:⋃i∈p-Ai→⋃i∈p-Bi (p≥2), where Ai and Bi, ∀i∈p-={1,2,…,p}, are nonempty subsets of X which satisfy TAi⊆Bi,∀i∈p-, such that (X, d) is a metric space. The boundedness and the convergence of the sequences of distances in the domains and in their respective image sets of the cyclic proximal and weak cyclic proximal non‐self‐mapping, and of some of their generalizations are investigated. The existence and uniqueness of the best proximity points and the properties of convergence of the iterates to such points are also addressed.

Suggested Citation

  • M. De La Sen & Asier Ibeas, 2014. "Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:914915
    DOI: 10.1155/2014/914915
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    References listed on IDEAS

    as
    1. Chiming Chen & Chingju Lin, 2012. "Best Periodic Proximity Points for Cyclic Weaker Meir‐Keeler Contractions," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. Winate Sanhan & Chirasak Mongkolkeha & Poom Kumam, 2012. "Generalized Proximal ψ‐Contraction Mappings and Best Proximity Points," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Chiming Chen & Chingju Lin, 2012. "Best Periodic Proximity Points for Cyclic Weaker Meir-Keeler Contractions," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-7, May.
    4. Winate Sanhan & Chirasak Mongkolkeha & Poom Kumam, 2012. "Generalized Proximal -Contraction Mappings and Best Proximity Points," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, October.
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