## 1 Introduction and Preliminaries
Furthermore, when set \(\mathcal{P}\) is compact, and set \(\mathcal{Q}\) is approximately compact with respect to \(\mathcal{P}\), it follows that both \(\mathcal{P}_{0}\) and \(\mathcal{Q}_{0}\) are nonempty._  
**Lemma 2** ([18]). : _Consider a metric space \((\Omega,M_{d})\) with nonempty closed subsets \(\mathcal{P}\) and \(\mathcal{Q}\). Assume that \(\mathcal{P}_{0}\) is also a nonempty subset and \(\mathcal{Q}\) is approximately compact with respect to \(\mathcal{P}\). Then, \(\mathcal{P}_{0}\) is closed._  
**Definition 14** ([8]). : _Let \((\Omega,M_{d})\) be a metric space. A self-mapping \(T\) on \(\Omega\) is called a Suzuki contraction if_  
\[\frac{1}{2}M_{d}(\mu,T\mu)\leq M_{d}(\mu,y)\text{ implies }M_{d}(T\mu,Ty)\leq M_{d}(\mu,y) \text{, for all }\mu,y\in\Omega\text{.}\]  
**Definition 15** ([9]). : _Consider a metric space \((\Omega,M_{d})\), and let \(\Lambda\geq 0\), \(\mu\geq 1\), and \(\gamma\in[0,\mu]\) be constants. An \(\alpha-\)orbital admissible mapping \(T:\Omega\to\Omega\) is an \(\alpha\)-Pata-Suzuki contraction if for every \(\epsilon\in[0,1]\), \(\psi\in\Psi\), and for all \(\mu,y\in\Omega\), the following holds:_  
\[\frac{1}{2}M_{d}(\mu,T\mu)\leq M_{d}(\mu,y),\]  
_which implies_  
\[\alpha(\mu,T\mu)\alpha(y,Ty)M_{d}(T\mu,Ty)\leq\mathcal{P}(\mu,y),\]  
_where_  
\[\mathcal{P}(\mu,y) = (1-\epsilon)\max\{M_{d}(\mu,y),M_{d}(\mu,T\mu),M_{d}(y,Ty),\frac{ 1}{2}[M_{d}(\mu,Ty)+M_{d}(y,T\mu)]\}+\] \[\Lambda(\epsilon)^{\mu}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|T\mu\|+ \|Ty\|]^{\gamma}.\]  
**2. Main Results**  
**Definition 16** ([17]). : _Let \(\mathcal{P}\) and \(\mathcal{Q}\) be nonempty subsets of a metric space \((\Omega,M_{d})\), \(T:\mathcal{P}\to\mathcal{Q}\) be a mapping and \(\alpha,\theta:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\). Then, the mapping \(T\) is said to be an \(\alpha\)-\(\theta\)-proximal admissible mapping if_  
\[\left. \begin{array}{c}\alpha(\mu,y)\geq\theta(\mu,y)\\ M_{d}(u,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\\ M_{d}(v,Ty)=M_{d}(\mathcal{P},\mathcal{Q})\end{array} \right\}\text{implies }\alpha(u,v)\geq\theta(u,v),\]  
_for all \(\mu,y,u,v\in\mathcal{P}\)._  
**Remark 4**. : _Note that,_  
1. _If_ \(\theta(\mu,y)=1\) _for all_ \(\mu,y\in\mathcal{P}\)_, then every_ \(\alpha\)_-_\(\theta\)_-proximal admissible mapping becomes an_ \(\alpha\)_-proximal admissible mapping;_
2. _If_ \(\mathcal{P}=\mathcal{Q}=\Omega\)_, then every_ \(\alpha\)_-_\(\theta\)_-proximal admissible mapping becomes an_ \(\alpha\)_-admissible mapping;_
3. _If_ \(\mathcal{P}=\mathcal{Q}=\Omega\) _and_ \(\theta(\mu,y)=1\)_, then every_ \(\alpha\)_-_\(\theta\)_-proximal admissible mapping becomes an_ \(\alpha\)_-admissible mapping._  
**Lemma 3** ([17]). : _Assuming that the mapping \(T\) is an \(\alpha\)-\(\theta\)-proximal admissible, and \(T(\mathcal{P}_{0})\subseteq\mathcal{Q}_{0}\), where \(\mathcal{P}_{0}\) is a nonempty subset of \(\mathcal{P}\). If there exist \(\mu_{0}\) and \(\mu_{1}\) in \(\mathcal{P}_{0}\) which satisfy \(M_{d}(\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\) and \(\alpha(\mu_{0},\mu_{1})\geq\theta(\mu_{0},\mu_{1})\), then there exists a sequence \(\{\mu_{n}\}\subset\mathcal{P}_{0}\) such that_  
\[M_{d}(\mu_{n+1},T\mu_{n})=M_{d}(\mathcal{P},\mathcal{Q}),\text{ and }\alpha(\mu_{n},\mu_{n+1})\geq\theta(\mu_{n},\mu_{n+1})\text{ for all }n\in\mathbb{N}. \tag{2}\]  
**Definition 17**. : _A sequence \(\{\mu_{n}\}\subset\mathcal{P}_{0}\) is referred to as an \(\alpha\)-\(\theta\)-proximal admissible Picard sequence if it satisfies condition (2) and also fulfills \(\alpha(\mu_{n},\mu_{1})\geq\theta(\mu_{n},\mu_{1})\) and \(\alpha(\mu_{n},\mu_{0})\geq\theta(\mu_{n},\mu_{0})\) for all \(n\in\mathbb{N}\)._  
**Definition 18**. : _A set \(\mathcal{P}_{0}\) is said to be \(\alpha\)-\(\theta\)-proximal orbital complete if and only if every Cauchy \(\alpha\)-\(\theta\)-proximal admissible Picard sequence in \(\mathcal{P}_{0}\) converges in \(\mathcal{P}_{0}\)._  
**Definition 19**.