## 1 Introduction and Preliminaries
: _Consider nonempty subsets \(\mathcal{P}\) and \(\mathcal{Q}\) in a metric space \((\Omega,M_{d})\), and mappings \(\alpha,\theta:\mathcal{P}\times\mathcal{P}\to[0,+\infty)\). A mapping \(T:\mathcal{P}\to\mathcal{Q}\) is an:_  
1. \(\alpha\)_-_\(\theta\)_-Gerghty-Pata-Suzuki-type proximal contraction if there exist constants_ \(\Lambda\geq 0\)_,_ \(\mu\geq 1\) _and_ \(\gamma\in[0,\mu]\) _such that_ \[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\]implies_  
\[\frac{\alpha(\mu,y)}{\theta(\mu,y)}M_{d}(u,v)\leq\frac{(1-e)}{2}\beta(m(\mu,y))m( \mu,y)+\Lambda(\epsilon)^{\mu}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{\gamma},\]
2. \(\alpha-\)_Gergatly-Pata-Suzuki-Type proximal contraction if there exist fixed constants_ \(\Lambda\geq 0\)_,_ \(\mu\geq 1\) _and_ \(\gamma\in[0,\mu]\) _such that_ \[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\] _which implies_ \[\alpha(\mu,y)M_{d}(u,v)\leq\frac{(1-e)}{2}\beta(m(\mu,y))m(\mu,y)+\Lambda( \epsilon)^{\mu}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{\gamma},\]
3. _A modified Geraghty-Pata-Suzuki proximal contraction if there exist fixed constants_ \(\Lambda\geq 0\)_,_ \(\mu\geq 1\) _and_ \(\gamma\in[0,\mu]\) _such that_ \[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y)\] _which implies_ \[M_{d}(u,v)\leq\frac{(1-e)}{2}\beta(M_{d}^{*}(\mu,T\mu))M_{d}^{*}(\mu,T\mu)+ \Lambda(\epsilon)^{\lambda}\psi(\epsilon)[1+\|\mu\|+\|y\|+\|v\|]^{\gamma},\] _for every_ \(\epsilon\in[0,1]\)_, where_ \[m(\mu,y) = \max\{M_{d}(\mu,y),M_{d}^{*}(\mu,T\mu),M_{d}^{*}(y,Ty),\frac{M_{d }^{*}(\mu,T\mu)+M_{d}^{*}(y,Ty)}{2},M_{d}^{*}(y,T\mu),\] \[\frac{M_{d}^{*}(\mu,Ty)+M_{d}^{*}(y,T\mu)}{2}\},\] _and_ \[M_{d}^{*}(\mu,y)=M_{d}(\mu,y)-M_{d}(\mathcal{P},\mathcal{Q}),\] _for all_ \(u,v,\mu,y\in\mathcal{P}\)_, for every_ \(\epsilon\in[0,1]\)_,_ \(\beta\in K\)_, and_ \(\psi\in\Psi\)_._  
**Lemma 4**: _Let \(\mathcal{P}\) and \(\mathcal{Q}\) be nonempty subsets of a complete metric space \((\Omega,M_{d})\); furthermore, the set \(\mathcal{P}\) is closed, and \(T:\mathcal{P}\to\mathcal{Q}\) is a continuous \(\alpha\)-\(\theta\)-Gergatly-Pata-Suzuki-type proximal contraction with \(T(\mathcal{P}_{0})\subseteq\mathcal{Q}_{0}\) and \(\mathcal{P}_{0}\
eq\phi\). Then, the set \(\mathcal{P}_{0}\) is \(\alpha\)-\(\theta\)-proximal orbital complete._  
Let \(\mu_{0}\in\mathcal{P}_{0}\) and \(\{\mu_{n}\}\) be a Cauchy \(\alpha\)-\(\theta\)-proximal admissible Picard sequence. As \(\Omega\) is complete and \(\mathcal{P}\) is closed, there exists some \(\mu_{*}\) in \(\mathcal{P}\) such that \(\lim_{n\to+\infty}M_{d}(\mu_{n},\mu_{*})=0\) for all \(n\in\mathbb{N}\). Additionally, considering that \(\mu\geq 1\) and \(\gamma\in[0,\mu]\), it follows that \(\frac{1}{1+\mu+\gamma}\leq 1\). As \(T\) is an \(\alpha\)-\(\theta\)-Gergatby-Pata-Suzuki-type proximal contraction, we have  
\[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\]0 \[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\]1 \[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\]2  
Thus, the above inequality can be expressed as follows:  
\[\frac{1}{1+\mu+\gamma}M_{d}^{*}(\mu,T\mu)\leq M_{d}(\mu,y),\]3which implies that  
\[ \begin{array}{rcl}M_{d}(\mu_{1},\mu_{2})&\leq&\frac{a(\mu_{1},\mu_{2})}{\theta( \mu_{1},\mu_{2})}M_{d}(\mu_{1},\mu_{2})\\ &\leq&\frac{(1-\epsilon)}{2}\beta(m(\mu_{0},\mu_{1}))m(\mu_{0},\mu_{1})+\\ &&\Lambda(\epsilon)^{\mu}\psi(\epsilon)[1+\|\mu_{0}\|+\|\mu_{1}\|+\|\mu_{2}\| ]^{\gamma}.\end{array}  \tag{3}\]  
Taking the limit as \(\epsilon\to 0\), then inequality (3) becomes  
\[ \begin{array}{rcl}M_{d}(\mu_{1},\mu_{2})&\leq&\frac{1}{2}\beta(m(\mu_{0},\mu _{1}))m(\mu_{0},\mu_{1})\\ &<&\beta(m(\mu_{0},\mu_{1}))m(\mu_{0},\mu_{1})\\ &\leq&m(\mu_{0},\mu_{1}),\end{array}  \tag{4}\]  
where  
\[