In this paper we introduce geometric contraction map and give a new condition for the existence and uniqueness of Best proximity point of geometric contractions. We also consider the convergence of iterates to proximity points in metric spaces.

‎Let $(C,D)$ be a nonempty pair of disjoint subsets of a metric space. ‎Main purpose of this paper is to present a range of a convergence sequence to $u\in C\cup D$ such that $d(Tu,fu)=dist(C,D)$‎, for mappings $T,f:C\cup D\to C\cup D$. ‎In fact, ‎we give a generalization of Best proximity point results for cyclic contractive mappings. ‎To this end‎, ‎we consider an example is presented to support the main result. ‎‎

This study aims to define $\perp$-proximally increasing mappings and compute some Best proximity point results regarding this mapping in the framework of new spaces on the Riemannian manifolds. The new spaces are called strongly orthogonal Riemannian metric spaces. This study aims to define $\perp$-proximally increasing mappings and compute some Best proximity point results regarding this mapping in the framework of new spaces on the Riemannian manifolds. The new spaces are called strongly orthogonal Riemannian metric spaces.

In this study, we establish some Best proximity point results for generalized cyclic contraction mappings in partially ordered metric spaces. We also prove some Best proximity point theorems by introducing the $T$-restriction property and generalized pointwise cyclic contraction mapping. Some illustrations are provided to support our results.

In this article, we introduce the notion of $F_n$-contractions $T:A^n\rightarrow B$ in standard metric spaces and explore the possibility of certain approximation results for these mappings. We prove the existence and uniqueness of $n$-tuple ($n \geq 2$) Best proximity points of $F_n$-contractions, not necessarily continuous, using the weak $P$-property in complete metric spaces. Additionally, suitable examples are presented to substantiate our main results. Moreover, we anticipate a fixed point result to prove the existence and uniqueness of the solution for a type of integral equation to elucidate our obtained theorems.