## 1 Introduction and Preliminaries  
The Banach-Caccioppoli fixed-point theorem is named after [PERSON] (1892-1945) and [PERSON] (1904-1959) and was first stated by [PERSON] [1] in 1922. The Banach contraction principle, also known as the Banach fixed-point theorem, is one of the main pillars of metric fixed-point theory. This principle states that when a mapping \(T\) is a contraction on a complete metric space \(\Omega\) and maps elements from \(\Omega\) back to itself, there exists a unique fixed point \(\mu\) in \(\Omega\). This fixed-point theorem has several applications in determining the existence of solutions of integral and differential equations. It is quite interesting to study contractive mapping cases that do not have a fixed point. It is also interesting if the contractive mapping \(T\) is a non-self-mapping, in which case it is impossible to find the fixed point such that \(T\mu\
eq\mu\) or \(M_{d}(\mu,T\mu)\
eq 0\). Then, it would be interesting to approximate the fixed point to minimize the error among \(\mu\) and \(T\mu\) or to minimize  
\[\min_{\mu\in\Omega}M_{d}(\mu,T\mu).\]  
The best proximity point result offers the necessary conditions to compute an approximate solution \(\mu\), which is considered optimal as it minimizes the error \(M_{d}(\mu,T\mu)\) in achieving the global minimum value \(M_{d}(\mathcal{P},Q)\), where \(\mathcal{P}\) and \(\mathcal{Q}\) are nonempty subsets of a metric space \((\Omega,M_{d})\) and \(T:\mathcal{P}\rightarrow\mathcal{Q}\) is a non-self-mapping. Any point \(\mu\in\mathcal{P}\) is known as the best proximity point of the non-self-mapping \(T\) if  
\[M_{d}(\mu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})=\inf\{M_{d}(\mu,y):\ \mu\in \mathcal{P}\,y\in\mathcal{Q}\}.\]  
For more details, see the best proximity results in [2; 3; 4; 5]. In 1969, [PERSON] [6] provided the first best approximation result. The coincidence of the best proximity point results is a generalization of the best proximity point results because it deals with two mappings, one of which is a non-self-mapping and the other is a self-mapping. Let \(\mathcal{P}\) and \(\mathcal{Q}\) be a nonempty subset of a metric space \((\Omega,M_{d})\). If \(T:\mathcal{P}\rightarrow\mathcal{Q}\) and \(g:\mathcal{P}\rightarrow\mathcal{P}\) is a self-mapping and  
\[M_{d}(g\mu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q}),\]  
then \(\mu\in\mathcal{P}\) is referred to as the coincidence of the best proximity point of the pair of mappings \((g,T)\). The results concerning the coincidence of the best proximity points serve as a generalization of the best proximity point results and fixed-point results. This is evident when considering that if we set \(g\) as the identity mapping \(J_{P}\), each coincidence's best proximity point becomes the best proximity point of the mapping \(T\). Moreover, if the mapping \(T\) is a self-mapping, then the concept of the best proximity point reduces to the notion of a fixed point. Researchers have explored various generalizations of the Banach fixed-point theorem in different directions, leading to numerous applications in various fields. Among them are two contractive conditions presented by [PERSON] [7] and [PERSON] [8], which shall be discussed here. Recently, [PERSON] et al. in [9] modified these contractive conditions and proved some fixed-point results by introducing a new type of contraction called the \(\alpha\)-Pata-Suzuki-type contraction. [PERSON] [10] proposed another extension of the Banach contraction principle, known as the Geraghty contraction. [PERSON] [11] utilized this Geraghty contraction and proved the best proximity-point results for \(\alpha\)-proximal Geraghty non-self-mappings. Recently, [PERSON] et al. in [12] introduced the Pata-type best proximal contraction and proved related results in the best proximity point. The Banach space is symmetric and is related to the fixed-point problems discussed in [13]. It has a certain importance, and several researchers are working on it around the globe.