## 5 Application to Fixed-Point Theory  
This section is devoted to the fixed-point theory for generalized and modified \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki-type proximal mappings. If \(\mathcal{P}=\mathcal{Q}=\Omega\), then the best proximity point can be reduced to a fixed point; thus, the following contractive conditions are formed. **Definition 25**. : _A mapping \(T:\Omega\to\Omega\) is called a:_  
1. _An_ \(\alpha\)_-_\(\theta\)_-Geraghty-Pata-Suzuki contraction if_ \[\frac{1}{1+\lambda+\gamma}M_{d}(\mu,T\mu)\leq M_{d}(\mu,y)\] _implies_ \[\tfrac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(T\mu,Ty)\leq\tfrac{(1-\epsilon)}{2} \beta(m_{1}(\mu,y))m_{1}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+ \|\mu\|+\|y\|+\|v\|]^{\gamma};\]
2. _An_ \(\alpha\)_-_\(\theta\)_-generalized Geraghty-Pata-Suzuki contraction if_ \[\frac{1}{1+\lambda+\gamma}M_{d}(g\mu,Ty)\leq M_{d}(g\mu,gy)\] _implies_ \[\tfrac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(T\mu,Ty)\leq\tfrac{(1-\epsilon)}{2} \beta(m_{1}(\mu,y))m_{1}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+ \|\mu\|+\|y\|+\|v\|]^{\gamma};\]
3. _An_ \(\alpha\)_-_\(\theta\)_-modified Geraghty-Pata-Suzuki contraction if_ \[\frac{1}{1+\lambda+\gamma}M_{d}(\mu,T\mu)\leq M_{d}(\mu,y)\] _implies_ \[\tfrac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(T\mu,Ty)\leq\tfrac{(1-\epsilon)}{2} \beta(m_{2}(\mu,y))m_{2}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+ \|\mu\|+\|y\|+\|v\|]^{\gamma};\]
4. _An_ \(\alpha\)_-_\(\theta\)_-modified generalized Geraghty-Pata-Suzuki contraction if_ \[\frac{1}{1+\lambda+\gamma}M_{d}(g\mu,Ty)\leq M_{d}(g\mu,gy)\] _implies_ \[\tfrac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(T\mu,Ty)\leq\tfrac{(1-\epsilon)}{2} \beta(m_{2}(\mu,y))m_{2}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+ \|\mu\|+\|y\|+\|v\|]^{\gamma},\] _for every_ \(\epsilon\in[0,1]\)_, where_ \(\alpha,\theta:\Omega\times\Omega\to[0,+\infty)\)_, and_ \(\alpha(\mu,y)\geq\theta(\mu,y)\)_,_ \(\lambda\geq 1\) _and_ \(\gamma\in[0,\lambda]\)_, for every_ \(\epsilon\in[0,1]\)_, and_ \(\psi\in\Psi\)_, where_ \[m_{1}(\mu,y) = \max\{M_{d}(\mu,y),M_{d}(\mu,T\mu),M_{d}(y,Ty),\frac{M_{d}(\mu,T \mu)+M_{d}(y,Ty)}{2},\] \[M_{d}(y,T\mu),\frac{M_{d}(\mu,Ty)+M_{d}(y,T\mu)}{2}\},\] _and_ \[\tfrac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(T\mu,Ty)\leq\tfrac{(1-\epsilon)}{2} \beta(m_{1}(\mu,y))m_{1}(\mu,y)+\Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+ \|\mu\|+\|y\|+\|v\|]^{\gamma};\]0 _for all_ \(\mu,y\in\mathcal{P}\)  
**Theorem 5**: _If there exists some \(\mu_{0}\) with \(\alpha(\mu_{0},T\mu_{0})\geq\theta(\mu_{0},T\mu_{0})\), then the mapping \(T:\Omega\to\Omega\) which satisfies the \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki contractive condition on a complete metric space \((\Omega,M_{d})\) has a unique fixed point._  
By applying Theorem (1) to the case when \(\mathcal{P}=\mathcal{Q}=\Omega\), then the \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki-type proximal contraction becomes simply the \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki contraction. We can then find a point \(\mu\) which satisfies \(M_{d}(\mu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\). However, in this scenario, \(\mathcal{P}=\mathcal{Q}=\Omega\), and we have \(M_{d}(\mu,T\mu)=0\); thus, there exists a fixed point \(\mu\) for the \(\alpha\)-\(\theta\)-Geraghty-Pata-Suzuki contraction of the mapping \(T\). Theorem 6: _If there exists some \(\mu_{0}\) with \(\alpha(\mu_{0},T\mu_{0})\geq\theta(\mu_{0},T\mu_{0}),\ g:\Omega\to\Omega,\) and \(T:\Omega\to\Omega\) satisfies the \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki contractive condition on a complete metric space, \((\Omega,M_{d})\) has a unique coincidence point._  
By applying Theorem (3) to the case when \(\mathcal{P}=\mathcal{Q}=\Omega\), then the \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki-type proximal contraction becomes the \(\alpha\)-\(\theta\)-generalized Geraghty-Pata-Suzuki contraction. We can find a point \(\mu\) which satisfies \(M_{d}(g\mu,T\mu)=M_{d}(\mathcal{P},\mathcal{Q})\).