Advanced Search
MyIDEAS: Login to save this article or follow this journal

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

Contents:

Author Info

  • S. Sadiq Basha

    ()

Registered author(s):

    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

    Download Info

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
    File URL: http://hdl.handle.net/10.1007/s10898-011-9760-8
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

    Bibliographic Info

    Article provided by Springer in its journal Journal of Global Optimization.

    Volume (Year): 54 (2012)
    Issue (Month): 2 (October)
    Pages: 367-373

    as in new window
    Handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:367-373

    Contact details of provider:
    Web page: http://www.springer.com/business/operations+research/journal/10898

    Order Information:
    Web: http://link.springer.de/orders.htm

    Related research

    Keywords: Optimal approximate solution; Common best proximity point; Common fixed point; Proximally commuting mappings;

    References

    No references listed on IDEAS
    You can help add them by filling out this form.

    Citations

    Lists

    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

    Statistics

    Access and download statistics

    Corrections

    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.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Guenther Eichhorn) or (Christopher F Baum).

    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.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with 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 profile, as there may be some citations waiting for confirmation.

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