Common best proximity point theorems for certain types of mappings
Arunachalam Murali, Krishnan Muthunagai
Received August 7, 2024. Published online March 19, 2025.
Abstract: Let $S$ and $T$ be two single-valued non-self-mappings from a nonempty set $\mathcal{P}$ to another nonempty set $\mathcal{Q}$. As they are non-self-mappings, the equations $Sx=x$ and $Tx=x$ do not have a common solution. In other words, they do not have a common fixed point. So one intends to find an element $x,$ close to $Sx$ and $Tx,$ which is called the common best proximity point. The common best proximity theorem guarantees the existence of such a best proximity point of the mappings $S$ and $T.$ In this article, we prove the existence and uniqueness of the common best proximity point for a pair of non-self-mappings for rational type contractive conditions on complex valued metric spaces. In addition, by transforming non-self-mappings into self-mappings in complex valued metric spaces, we prove the existence and uniqueness of a common best proximity point for Kannan type rational expression mappings and Chatterjea type rational expression contractive mappings. Moreover, we introduce contraction conditions involving a control function of some kind and prove the existence and uniqueness of a common best proximity point for such conditions. Our key findings extend and integrate some previously published results.
Keywords: best proximity point; fixed point; rational type contractive condition; complex valued metric space
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Affiliations: Arunachalam Murali, Krishnan Muthunagai (corresponding author), Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, Chennai - 600 127, India, e-mail: murali.a2020@vitstudent.ac.in, muthunagai@vit.ac.in
