## 1 Introduction and Preliminaries
Let \(\alpha:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\) and \(T:\mathcal{P}\to\mathcal{Q}\). A mapping \(T\) is referred to as \(\alpha\)-proximal admissible if the following condition is satisfied for all \(\mu,y\in\mathcal{P}\):_  
\[\left. \begin{array}{l}\alpha(\mu,y)\geq 1\\ M_{d}(u,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\\ M_{d}(v,Ty)=M_{d}(\mathcal{P},\mathcal{Q})\end{array} \right\}\text{implies }\alpha(u,v)\geq 1.\]  
**Remark 1**. : _If we take \(\mathcal{P}=\mathcal{Q}=\Omega\), then every \(\alpha\)-proximal admissible mapping becomes an \(\alpha\)-admissible mapping._  
**Definition 10** ([12]). : _In a metric space \((\Omega,M_{d})\), let \(\mathcal{P}\) and \(\mathcal{Q}\) be nonempty subsets. An \(\alpha\)-proximal admissible mapping \(T:\mathcal{P}\to\mathcal{Q}\) is an admissible mapping:_  
1. \(\alpha\)_-Pata proximal admissible contraction of type I if_ \[\alpha(\mu,y)M_{d}(u,v)\leq\frac{(1-\epsilon)}{2}M_{d}(\mu,y)+\Lambda(\epsilon )^{\mu}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{\gamma},\]
2. \(\alpha\)_-Pata proximal admissible contraction of type II if_ \[\alpha(\mu,y)M_{d}(Tu,Tv)\leq\frac{(1-\epsilon)}{2}M_{d}(T\mu,Ty)+\Lambda( \epsilon)^{\mu}\psi(\epsilon)[1+\|T\mu\|+\|Ty\|+\|Tv\|]^{\gamma},\] _for any_ \(u,v,\mu\) _and_ \(y\) _in_ \(\mathcal{P}\)_, and constants_ \(\Lambda\geq 0\)_,_ \(\epsilon\in[0,1]\)_,_ \(\psi\in\Psi,\) _and_ \(\gamma\in[0,\mu]\)_, where_ \(\mu\geq 1\)_._  
**Lemma 1** ([12]). : _Assume that \(T:\mathcal{P}\to\mathcal{Q}\) is an \(\alpha\)-proximal admissible mapping and \(T(\mathcal{P}_{0})\subseteq\mathcal{Q}_{0}\), where \(\mathcal{P}_{0}\) is a nonempty subset of \(\mathcal{P}\). If_  
\[M_{d}(\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\text{, and }\alpha(\mu_{0}, \mu_{1})\geq 1\text{,}\]  
_for some \(\mu_{0}\) and \(\mu_{1}\) in \(\mathcal{P}_{0}\), then \(\mathcal{P}_{0}\) contains a sequence \(\{\mu_{n}\}\) such that,_  
\[M_{d}(\mu_{n+1},T\mu_{n})=M_{d}(\mathcal{P},\mathcal{Q})\text{, and }\alpha(\mu_{n},\mu_{n+1})\geq 1 \text{ for all }n\in\mathbb{N}\text{.} \tag{1}\]  
**Definition 11** ([12]). : _A sequence \(\mu_{n}\subset\mathcal{P}_{0}\) is called an \(\alpha\)-proximal admissible Picard sequence if it fulfills (1) under the following condition:_  
\[\alpha(\mu_{n},\mu_{1})\geq 1\text{, and }\alpha(\mu_{n},\mu_{0})\geq 1 \text{ for all }n\in\mathbb{N}\text{.}\]  
**Definition 12** ([12]). : _Consider the mappings \(T:\mathcal{P}\to\mathcal{Q}\), \(\alpha:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\), and \(g:\mathcal{P}\to\mathcal{P}\). We say that a pair of the mappings \((g,T)\) satisfies a generalized \(\alpha\)-Pata-proximal contractive condition if there exist constants \(\Lambda\geq 0\), \(\mu\geq 1\), and \(\gamma\in[0,\mu]\) such that_  
\[\left. \begin{array}{c}\alpha(\mu,y)\geq 1\\ M_{d}(gu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\\ M_{d}(gv,Ty)=M_{d}(\mathcal{P},\mathcal{Q})\end{array} \right\}\text{ implies }\alpha(gu,gv)\geq 1\text{,}\]  
_and_  
\[\alpha(\mu,y)M_{d}(gu,gv)\leq\tfrac{(1-\epsilon)}{2}M_{d}(\mu,y)+\Lambda( \epsilon)^{\mu}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{\gamma}\text{, for all }u,v,\mu,y\in\mathcal{P}\text{,}\]  
_for every \(\epsilon\in[0,1]\), and \(\psi\in\Psi\)._  
**Remark 2** ([12]). : _When the mapping \(g\) is the identity mapping on \(\mathcal{P}\) (denoted as \(I_{\mathcal{P}}\)), every generalized \(\alpha\)-Pata proximal contraction becomes an \(\alpha\)-Pata proximal admissible contraction of type I._  
**Definition 13** ([12]). : _A set \(\mathcal{Q}\) is said to be an approximate compact with respect to \(\mathcal{P}\) if every sequence \(\{y_{n}\}\) in \(\mathcal{Q}\) satisfying_  
\[M_{d}(\mu,y_{n})\to M_{d}(\mu,\mathcal{Q})=\inf M_{d}(\mu,b)\text{ for all }b\in\mathcal{Q}\text{,}\]  
_has a convergent subsequence._  
**Remark 3**. : _Each set is approximately compact concerning itself.