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Keywords:
best proximity point; fixed point; rational type contractive condition; complex valued metric space
best proximity point; fixed point; rational type contractive condition; complex valued metric space
Summary:
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.
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