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Best Proximity Points for Some Classes of Proximal Contractions

Author

Listed:
  • Maryam A. Alghamdi
  • Naseer Shahzad
  • Francesca Vetro

Abstract

Given a self‐mapping g : A → A and a non‐self‐mapping T : A → B, the aim of this work is to provide sufficient conditions for the existence of a unique point x ∈ A, called g‐best proximity point, which satisfies d(gx, Tx) = d(A, B). In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function x → d(gx, Tx), thereby getting an optimal approximate solution to the equation Tx = gx. An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach′s contraction principle to the case of non‐self‐mappings.

Suggested Citation

  • Maryam A. Alghamdi & Naseer Shahzad & Francesca Vetro, 2013. "Best Proximity Points for Some Classes of Proximal Contractions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:713252
    DOI: 10.1155/2013/713252
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    References listed on IDEAS

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    1. Moosa Gabeleh & Naseer Shahzad, 2013. "Existence and Convergence Theorems of Best Proximity Points," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-6, June.
    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. S. Sadiq Basha, 2011. "Best proximity points: global optimal approximate solutions," Journal of Global Optimization, Springer, vol. 49(1), pages 15-21, January.
    4. Moosa Gabeleh & Naseer Shahzad, 2013. "Existence and Convergence Theorems of Best Proximity Points," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    5. 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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    Cited by:

    1. Poom Kumam & Antonio Francisco Roldán López de Hierro, 2014. "On Existence and Uniqueness of g‐Best Proximity Points under (φ, θ, α, g)‐Contractivity Conditions and Consequences," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. M. Abbas & A. Hussain & P. Kumam, 2015. "A Coincidence Best Proximity Point Problem in G‐Metric Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2015(1).

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