Document Type: Research Paper

Authors

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

Abstract

Best approximation results provide an approximate solution to the fixed point equation $Tx=x$, when the non-self mapping $T$ has no fixed point. In particular, a well-known best approximation theorem, due to Fan cite{5}, asserts that if $K$ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space $E$ and $T:K\rightarrow E$ is a continuous mapping, then there exists an element $x$ satisfying the condition $d(x,Tx)=\inf \{d(y,Tx):y\in K\}$, where $d$ is a metric on $E$. Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, Article ID 837943) introduced proximal contractive mappings and established certain best proximity point results for these mappings in $G$-metric spaces. The aim of this paper is to introduce certain new classes of auxiliary functions and proximal contraction mappings and establish best proximity point theorems for such kind of mappings in $G$-metric spaces. As consequences of these results, we deduce certain new best proximity and fixed point results in $G$-metric spaces. Moreover, we present certain examples to illustrate the usability of the obtained results.

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Main Subjects

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