## 1 Introduction and Preliminaries
Recalling that symmetry is a mapping on an object \(\Omega\), preserving its underlying structure, [PERSON] [13] utilized this concept to derive various applications of a layered compression-expansion fixed-point theorem. These applications resulted in the derivation of solutions for a second-order difference equation with Dirichlet boundary conditions. In this paper, we introduce three different types of contractive conditions: (1) an \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki-type proximal contraction, (2) an \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki-type proximal contraction, and (3) an \(\alpha\)-\(\theta\)-modified Geraghty-Pata-Suzuki-type proximal contraction. The purpose of this study is to prove the existence and uniqueness of the coincidence of best proximity and the best proximity-point results for the above-mentioned proximal contractions in complete metric spaces. Several examples are given to highlight the superiority of our results. As an application, we shall derive certain recent fixed-point results as corollaries to our results. The sets mentioned below are important in the best proximity analysis. \[\mathcal{P}_{0} =\{\mu\in\mathcal{P}:M_{d}(\mu,y)=M_{d}(\mathcal{P},\mathcal{Q}) \text{ for some }y\in\mathcal{Q}\},\] \[\mathcal{Q}_{0} =\{y\in\mathcal{Q}:M_{d}(\mu,y)=M_{d}(\mathcal{P},\mathcal{Q}) \text{ for some }\mu\in\mathcal{P}\}.\]  
**Definition 1** ([12]). : _In a metric space \((\Omega,M_{d})\), let \(\mu_{0}\) be any arbitrary point. The functional defined below is called the zero of \(\Omega\)_  
\[\|\mu\|=M_{d}(\mu,\mu_{0})\text{, for all }\mu\in\Omega.\]  
**Definition 2**. : _The set \(\Psi\) comprises all continuous increasing functions \(\psi:[0,1]\rightarrow[0,+\infty)\) that satisfy the condition \(\psi(0)=0\) and are continuous at \(0\)._  
**Definition 3** ([7]). : _In a metric space \((\Omega,M_{d})\), let \(T:\Omega\to\Omega\) be a mapping. Then, the mapping \(T\) is a Pata contraction if the following inequality holds for all \(\mu,y\in\Omega\), and every \(\epsilon\in[0,1]\):_  
\[M_{d}(T\mu,Ty)\leq(1-\epsilon)M_{d}(\mu,y)+\Lambda(\epsilon)^{\eta}\psi( \epsilon)[1+|\mu|+|y||^{\gamma},\]  
_where \(\Lambda\geq 0\), \(\eta\geq 1\), and \(\gamma\in[0,\eta]\) are fixed constants, and \(\psi\in\Psi\)._  
**Definition 4** ([14]). : _Consider a nonempty set \(\Omega\), and let \(T:\Omega\to\Omega\) be a mapping. Additionally, let \(\alpha:\Omega\times\Omega\to\mathbb{R}^{+}\) be a function that maps the pairs of the elements from \(\Omega\) to positive real numbers. We say that the mapping \(T\) is an \(\alpha\)-admissible if the condition, \(\alpha(\mu,y)\geq 1\), then \(\alpha(T\mu,Ty)\geq 1\) for all \(\mu,y\in\Omega\) holds._  
**Definition 5** ([15]). : _Let \(\Omega\) be a nonempty set, and \(\alpha:\Omega\times\Omega\to[0,+\infty)\) be an auxiliary function. A self-mapping \(T\) on \(\Omega\) is called an \(\alpha\)-orbital admissible if_  
\[\alpha(\mu,T\mu)\geq 1\text{ implies }\alpha(T\mu,T^{2}\mu)\geq 1,\text{ for all }\mu\in\Omega.\]  
**Definition 6** ([11]). : _A set \(K\) represents a class of all functions \(\beta:[0,+\infty)\to[0,1]\) such that \(\beta(t_{n})\longrightarrow 1\) implies \(t_{n}\longrightarrow 0\), where \(\{t_{n}\}\) is bounded sequence of positive real numbers._  
**Definition 7** ([11]). : _Consider \((\Omega,M_{d})\) as a metric space, and consider a pair of nonempty subsets \((\mathcal{P},\mathcal{Q})\) of \(\Omega\). A mapping \(T:\mathcal{P}\to\mathcal{Q}\) is termed a \(\beta\)-Geraghty mapping if there exists a constant \(\beta\in K\) such that for all \(\mu,y\in\mathcal{P}\), the following inequality is satisfied:_  
\[M_{d}(T\mu,Ty)\leq\beta(M_{d}(\mu,y))M_{d}(\mu,y).\]  
**Definition 8** ([16]). : _A mapping \(T:\Omega\to\Omega\) is classified as an expansive mapping if, for all \(\mu,y\in\Omega\) and some \(q>1\), the following inequality holds:_  
\[M_{d}(T\mu,Ty)\geq q_{d}(\mu,y).\]  
**Definition 9** ([17]). : _Consider nonempty subsets \(\mathcal{P}\) and \(\mathcal{Q}\) in a metric space \((\Omega,M_{d})\).