New best proximity point results in G-metric space
Subject Areas : History and biographyA. H. Ansari 1 , A. Razani 2 , N. Hussain 3
1 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
2 - Department of Mathematics, Faculty of Science, Imam Khomeini
International University, postal code: 34149-16818, Qazvin, Iran
3 - Department of Mathematics, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia
Keywords: G-metric space, Best proximity point, Generalized proximal weakly G-contraction,
Abstract :
Best approximation results provide an approximate solution to the fixedpoint equation $Tx=x$, when the non-self mapping $T$ has no fixed point. Inparticular, a well-known best approximation theorem, due to Fan cite{5},asserts that if $K$ is a nonempty compact convex subset of a Hausdorfflocally convex topological vector space $E$ and $T:K\rightarrow E$ is acontinuous mapping, then there exists an element $x$ satisfying thecondition $d(x,Tx)=\inf \{d(y,Tx):y\in K\}$, where $d$ is a metric on $E$.Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, ArticleID 837943) introduced proximal contractive mappings and established certainbest proximity point results for these mappings in $G$-metric spaces. The aimof this paper is to introduce certain new classes of auxiliary functions andproximal contraction mappings and establish best proximity point theoremsfor such kind of mappings in $G$-metric spaces. As consequences of theseresults, we deduce certain new best proximity and fixed point results in$G$-metric spaces. Moreover, we present certain examples to illustrate theusability of the obtained results.
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