## 3 Optimal Coincidence Point Results  
**Definition 20**. : _A mapping \(g:\mathcal{P}\to\mathcal{P}\) satisfies the \((\alpha,\theta)_{R}\)-property if there exist functions \(\alpha,\theta:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\) such that_  
\[\alpha(g\mu,gy)\geq\theta(g\mu,gy)\text{ implies }\alpha(\mu,y)\geq\theta(\mu,y) \text{, for all }\mu,y\in\mathcal{P}.\]  
**Definition 21**. : _Let \(\mathcal{P}\) and \(\mathcal{Q}\) be two nonempty subsets of a metric space \((\Omega,M_{d})\), \(T:\mathcal{P}\to\mathcal{Q}\) and \(g:\mathcal{P}\to\mathcal{P}\) be a non-self-mapping and \(\alpha,\theta:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\), then \(T\) is said to be \(\alpha\)-\(\theta\)-generalized proximal admissible if_  
\[\left. \begin{array}{l}\alpha(\mu,y)\geq\theta(\mu,y)\\ M_{d}(gu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\\ M_{d}(gv,Ty)=M_{d}(\mathcal{P},\mathcal{Q})\end{array} \right\}\text{ implies }\alpha(gu,gv)\geq\theta(gu,gv),\]  
_for all \(\mu,y,u,v\in\mathcal{P}\)._  
**Remark 5**. : _If the mapping \(g=l_{\mathcal{P}}\), then an \(\alpha\)-\(\theta\)-generalized proximal admissible mapping becomes an \(\alpha\)-\(\theta\)-proximal admissible mapping._  
**Definition 22**. : _Let \(\mathcal{P}\) and \(\mathcal{Q}\) be nonempty subsets of a metric space \((\Omega,M_{d})\). Let \(T:\mathcal{P}\to\mathcal{Q}\) and \(g:\mathcal{P}\to\mathcal{P}\). A pair of mappings \((g,T)\) is said to be an \(\alpha\)-\(\theta\)-generalized Gergatbity-Pata-Suzuki-type proximal contraction if \(\alpha,\theta:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\), and for any \(u,v,\mu,y\) in \(\mathcal{P}\), there exist some fixed constants \(\Lambda\geq 0,\;\lambda\geq 1\), and \(\gamma\in[0,\lambda]\) such that the following holds,_  
\[\frac{1}{1+\lambda+\gamma}M_{d}^{*}(g\mu,T\mu)\leq M_{d}(g\mu,gy)\]  
_which implies_  
\[\frac{\alpha(g\mu,g\mu)}{\theta(g\mu,gy)}M_{d}(gu,gv)\leq\frac{(1-e)}{2}\beta (m_{g}(\mu,y))m_{g}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+\|\mu \|+\|y\|+\|v\|]^{\gamma},\]  
_for every \(\epsilon\in[0,1]\), and \(\psi\in\Psi\), where_  
\[m_{g}(\mu,y) = \max\{M_{d}(g\mu,gy),M_{d}^{*}(g\mu,T\mu),M_{d}^{*}(gy,Ty),\frac{ M_{d}^{*}(g\mu,T\mu)+M_{d}^{*}(gy,Ty)}{2},\] \[M_{d}^{*}(g\mu,Ty),M_{d}^{*}(gy,T\mu),\frac{M_{d}^{*}(g\mu,Ty)+M_ {d}^{*}(gy,T\mu)}{2}\},\]  
_and_  
\[M_{d}^{*}(g\mu,T\mu)=M_{d}(g\mu,T\mu)-M_{d}(\mathcal{P},\mathcal{Q}).\]  
**Remark 6**. : _If we take \(g=1_{P}\) (identity mapping over set \(\mathcal{P}\)), then the \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki-type proximal contraction becomes an \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki-type proximal contraction._  
**Theorem 3**. : _Consider an expansive mapping \(g:\mathcal{P}\rightarrow\mathcal{P}\) satisfying the \((\alpha,\theta)_{R}\)-property, and a mapping \(T:\mathcal{P}\rightarrow\mathcal{Q}\) that is \(\alpha\)-\(\theta\)-proximal admissible. Let \(\mathcal{P}_{0}\) be a nonempty closed subset of \(\mathcal{P}\), and assume \(T(\mathcal{P}_{0})\subset\mathcal{Q}_{0}\), as well as \(\mathcal{P}_{0}\subseteq g(\mathcal{P}_{0})\). If the set \(\mathcal{Q}\) is approximately compact with respect to the set \(\mathcal{P}\), if the pair \((g,T)\) satisfies the \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki-type proximal contraction and if there exists \(\mu_{0}\), \(\mu_{1}\in\mathcal{P}_{0}\) satisfying \(M_{d}(g\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\) and \(\alpha(\mu_{0},\mu_{1})\geq\theta(\mu_{0},\mu_{1})\), then the pair \((g,T)\) has a unique optimal coincidence point \(\mu\) in \(\mathcal{P}_{0}\)._  
Proof. : Since \(\mu_{0}\) is a given point in \(\mathcal{P}_{0}\), as \(T(\mathcal{P}_{0})\subseteq\mathcal{Q}_{0}\) and \(\mathcal{P}_{0}\subseteq g(\mathcal{P}_{0})\), we can choose an element \(\mu_{1}\in\mathcal{P}_{0}\) such that \(M_{d}(g\mu_{1},T\mu_{0})=M_{d}(\mathcal{P},\mathcal{Q})\) and \(\alpha(\mu_{0},\mu_{1})\geq\theta(\mu_{0},\mu_{1})\), where \(\mu_{0}\), \(\mu_{1}\in\mathcal{P}_{0}\).