A new class of $\mathcal{S}$-contractions in complete metric spaces and $\mathcal{G_{P}}$-contractions in ordered metric spaces

Abstract

AbstractThe basic purpose of this article is to define new so-called $\mathcal{S}$ S -contractions and discuss the presence of common best proximity point theorems for such contractions in the setting of Cauchy metric spaces. We also calculate some common optimal approximate solutions of some fixed point equations when there does not exist any common fixed point. We also introduce the notions of $\mathcal{G_{P}}$ GP -functions and $\mathcal {G_{P}}$ GP -contractions with the help of $\mathcal{P}$ P -functions and prove the existence of a unique best proximity point in partially order metric spaces. We give some examples that justify the validity of our results. These results extend and unify many existing results in the literature.