IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v54y2012i2p367-373.html
   My bibliography  Save this article

Common best proximity points: global minimization of multi-objective functions

Author

Listed:
  • S. Sadiq Basha

Abstract

Given non-empty subsets A and B of a metric space, let $${S{:}A{\longrightarrow} B}$$ and $${T {:}A{\longrightarrow} B}$$ be non-self mappings. Due to the fact that S and T are non-self mappings, the equations Sx=x and Tx=x are likely to have no common solution, known as a common fixed point of the mappings S and T. Consequently, when there is no common solution, it is speculated to determine an element x that is in close proximity to Sx and Tx in the sense that d(x, Sx) and d(x, Tx) are minimum. As a matter of fact, common best proximity point theorems inspect the existence of such optimal approximate solutions, called common best proximity points, to the equations Sx=x and Tx=x in the case that there is no common solution. It is highlighted that the real valued functions $${x{\longrightarrow}d(x, Sx)}$$ and $${x{\longrightarrow}d(x, Tx)}$$ assess the degree of the error involved for any common approximate solution of the equations Sx=x and Tx=x. Considering the fact that, given any element x in A, the distance between x and Sx, and the distance between x and Tx are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions $${x{\longrightarrow}d(x, Sx)}$$ and $${x{\longrightarrow}d(x, Tx)}$$ by imposing a common approximate solution of the equations Sx=x and Tx=x to satisfy the constraint that d(x, Sx)=d(x, Tx)=d(A, B). The purpose of this article is to derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations in the event there is no common solution. Copyright Springer Science+Business Media, LLC. 2012

Suggested Citation

  • S. Sadiq Basha, 2012. "Common best proximity points: global minimization of multi-objective functions," Journal of Global Optimization, Springer, vol. 54(2), pages 367-373, October.
  • Handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:367-373
    DOI: 10.1007/s10898-011-9760-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-011-9760-8
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10898-011-9760-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Manuel De la Sen & Mujahid Abbas & Naeem Saleem, 2017. "On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces," Mathematics, MDPI, vol. 5(2), pages 1-20, April.
    2. Chayut Kongban & Poom Kumam & Somayya Komal & Kanokwan Sitthithakerngkiet, 2018. "On p -Common Best Proximity Point Results for S -Weakly Contraction in Complete Metric Spaces," Mathematics, MDPI, vol. 6(11), pages 1-11, November.
    3. Hussain, Nawab & Kutbi, M.A. & Salimi, Peyman, 2020. "Global optimal solutions for proximal fuzzy contractions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
    4. Calogero Vetro & Peyman Salimi, 2013. "Best proximity point results in non-Archimedean fuzzy metric spaces," Fuzzy Information and Engineering, Springer, vol. 5(4), pages 417-429, December.
    5. Watchareepan Atiponrat & Anchalee Khemphet & Wipawinee Chaiwino & Teeranush Suebcharoen & Phakdi Charoensawan, 2024. "Common Best Proximity Point Theorems for Generalized Dominating with Graphs and Applications in Differential Equations," Mathematics, MDPI, vol. 12(2), pages 1-21, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:367-373. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.