## 4 \(\alpha\)-Modified Geraghty-Pata-Suzuki-Type Proximal Contraction  
Definition 23: Let \(\mathcal{P}\) and \(\mathcal{Q}\) be nonempty subsets of a metric space \((\Omega,M_{d})\). Let \(\alpha:\Omega\times\Omega\to[0,\infty)\), then a mapping \(T:\mathcal{P}\to\mathcal{Q}\) is called an \(\alpha\)-modified Geraghty-Pata-Suzuki-type proximal contraction if for any \(u,v,\mu,y\in\mathcal{P}\), there exists \(\psi\in\Psi\), and fixed constants \(\Lambda\geq 0\), \(\lambda\geq 1,\gamma\in[0,\lambda]\) such that the following holds,  
\[\frac{1}{1+\lambda+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y)\]  
which implies  
\[\alpha(\mu,T\mu)\alpha(y,Ty)M_{d}(u,v)\leq\frac{(1-\epsilon)}{2}\beta(q(\mu,y ))q(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{ \gamma},\]  
for every \(\epsilon\in[0,1]\), where  
\[q(\mu,y)=\max\{M_{d}(\mu,y),M_{d}(\mu,T\mu),M_{d}(y,Ty)\},\]  
and  
\[M_{d}^{*}(\mu,T\mu)=M_{d}(\mu,T\mu)-M_{d}(\mathcal{P},\mathcal{Q}).\]  
Definition 24: Consider nonempty subsets \(\mathcal{P}\) and \(\mathcal{Q}\) of a metric space \((\Omega,M_{d})\). Let \(\alpha:\Omega\times\Omega\to[0,\infty)\), then a mapping \(T:\mathcal{P}\to\mathcal{Q}\) is an \(\alpha\)-Geraghty-Pata-Suzuki-type proximal contraction of type I if for any \(u,v,\mu,y\in\mathcal{P}\), there exist \(\psi\in\Psi\) and fixed constants \(\Lambda\geq 0\), \(\lambda\geq 1\), \(\gamma\in[0,\lambda]\) such that the following condition holds:  
\[\frac{1}{1+\lambda+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y)\]  
which implies  
\[\alpha(\mu,T\mu)\alpha(y,Ty)M_{d}(u,v)\leq\frac{(1-\epsilon)}{2}\beta(M_{d}( \mu,y))M_{d}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+\|\mu\|+\|y \|+\|v\|]^{\gamma},\]  
for every \(\epsilon\in[0,1]\). Remark 7: If \(q(\mu,y)=\max\{M_{d}(\mu,y),M_{d}(\mu,T\mu),M_{d}(y,Ty)\}=M_{d}(\mu,y)\), then every \(\alpha\)-modified Geraghty-Pata-Suzuki-type proximal contraction becomes an \(\alpha\)-Geraghty-Pata-Suzuki-type proximal contraction of type I.  
Theorem 4: _Let \((\mathcal{P},\mathcal{Q})\) be a pair of nonempty closed subsets of a complete metric space \((\Omega,M_{d})\), \(T:\mathcal{P}\to\mathcal{Q}\) be an \(\alpha\)-proximal admissible mapping and \(\alpha\)-modified Geraghty-Pata-Suzuki-type proximal mapping, and \(T(\mathcal{P}_{0})\subset\mathcal{Q}_{0}\), where \(\mathcal{P}_{0}\) is a nonempty and closed subset of \(\mathcal{P}\). Then, there exist elements \(\mu_{0},\mu_{1}\in\mathcal{P}_{0}\) such that \(\alpha(\mu_{0},\mu_{1})\geq 1\) and \(M_{d}(\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\). Let \(\{\mu_{n}\}\) be a sequence in \(\mathcal{P}_{0}\) such that \(\alpha(\mu_{n},\mu_{n+1})\geq 1\) and \(\lim_{n\to+\infty}\mu_{n}=\mu\in\mathcal{P}_{0}\), then \(T\) has a unique best proximity point \(\mu\in\mathcal{P}_{0}\)._  
Proof: \(\mu_{0},\mu_{1}\in\mathcal{P}_{0}\) such that \(M_{d}(\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\) and \(\alpha(\mu_{0},\mu_{1})\geq 1\). As \(T\mu_{1}\in T(\mathcal{P}_{0})\subseteq\mathcal{Q}_{0}\), there exists an element \(\mu_{2}\in\mathcal{P}_{0}\) such that \(M_{d}(\mu_{2},T\mu_{1})=M_{d}(\mathcal{P},\mathcal{Q})\). As \(T\) is \(\alpha-\)proximal admissible, it follows that \(\alpha(\mu_{1},\mu_{2})\geq 1\). In addition, we set \(\|\mu\|=M_{d}(\mu,\mu_{0})\) for all \(\mu\in\Omega\).