Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

Abstract

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction; an $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction; and an $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.

1. Introduction and Preliminaries

The Banach–Caccioppoli fixed-point theorem is named after Stefan Banach (1892–1945) and Renato Caccioppoli (1904–1959) and was first stated by Banach [1] in 1922. The Banach contraction principle, also known as the Banach fixed-point theorem, is one of the main pillars of metric fixed-point theory. This principle states that when a mapping T is a contraction on a complete metric space $\Omega$ and maps elements from $\Omega$ back to itself, there exists a unique fixed point $\mu$ in $\Omega$. This fixed-point theorem has several applications in determining the existence of solutions of integral and differential equations. It is quite interesting to study contractive mapping cases that do not have a fixed point. It is also interesting if the contractive mapping T is a non-self-mapping, in which case it is impossible to find the fixed point such that $T\mu \ne \mu$ or $M_d(\mu ,T\mu )\ne 0$. Then, it would be interesting to approximate the fixed point to minimize the error among $\mu$ and $T\mu$ or to minimize

$$\underset{\mu \in \Omega }{min}{M}_d(\mu ,T\mu ).$$

The best proximity point result offers the necessary conditions to compute an approximate solution $\mu$, which is considered optimal as it minimizes the error $M_d(\mu ,T\mu )$ in achieving the global minimum value $M_d(\mathcal{P},\mathcal{Q})$, where $\mathcal{P}$ and $\mathcal{Q}$ are nonempty subsets of a metric space $(\Omega ,M_d)$ and $T:\mathcal{P}\to \mathcal{Q}$ is a non-self-mapping. Any point $\mu \in \mathcal{P}$ is known as the best proximity point of the non-self-mapping T if

$$M_d(\mu ,T\mu )=M_d(\mathcal{P},\mathcal{Q})=inf\{M_d(\mu ,y):\mu \in \mathcal{P},y\in \mathcal{Q}\}.$$

For more details, see the best proximity results in [2,3,4,5]. In 1969, Ky Fan [6] provided the first best approximation result. The coincidence of the best proximity point results is a generalization of the best proximity point results because it deals with two mappings, one of which is a non-self-mapping and the other is a self-mapping. Let $\mathcal{P}$ and $\mathcal{Q}$ be a nonempty subset of a metric space $(\Omega ,M_d)$. If $T:\mathcal{P}\to \mathcal{Q}$ and $g:\mathcal{P}\to \mathcal{P}$ is a self-mapping and

$$M_d(g\mu ,T\mu )=M_d(\mathcal{P},\mathcal{Q}),$$

then $\mu \in \mathcal{P}$ is referred to as the coincidence of the best proximity point of the pair of mappings $(g,T)$. The results concerning the coincidence of the best proximity points serve as a generalization of the best proximity point results and fixed-point results. This is evident when considering that if we set g as the identity mapping $I_{\mathcal{P}}$, each coincidence’s best proximity point becomes the best proximity point of the mapping T. Moreover, if the mapping T is a self-mapping, then the concept of the best proximity point reduces to the notion of a fixed point. Researchers have explored various generalizations of the Banach fixed-point theorem in different directions, leading to numerous applications in various fields. Among them are two contractive conditions presented by V. Pata [7] and T. Suzuki [8], which shall be discussed here. Recently, Karapınar et al. in [9] modified these contractive conditions and proved some fixed-point results by introducing a new type of contraction called the $\alpha -$Pata–Suzuki-type contraction. Geraghty [10] proposed another extension of the Banach contraction principle, known as the Geraghty contraction. Ayari [11] utilized this Geraghty contraction and proved the best proximity-point results for $\alpha -$proximal Geraghty non-self-mappings. Recently, Saleem et al. in [12] introduced the Pata-type best proximal contraction and proved related results in the best proximity point. The Banach space is symmetric and is related to the fixed-point problems discussed in [13]. It has a certain importance, and several researchers are working on it around the globe. Recalling that symmetry is a mapping on an object $\Omega$, preserving its underlying structure, Neugebaner [13] utilized this concept to derive various applications of a layered compression–expansion fixed-point theorem. These applications resulted in the derivation of solutions for a second-order difference equation with Dirichlet boundary conditions.

In this paper, we introduce three different types of contractive conditions: (1) an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, (2) an $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction, and (3) an $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction. The purpose of this study is to prove the existence and uniqueness of the coincidence of best proximity and the best proximity-point results for the above-mentioned proximal contractions in complete metric spaces. Several examples are given to highlight the superiority of our results. As an application, we shall derive certain recent fixed-point results as corollaries to our results.

The sets mentioned below are important in the best proximity analysis.

$$\begin{array}{c}{\mathcal{P}}_0=\{\mu \in \mathcal{P}:M_d(\mu ,y)=M_d(\mathcal{P},\mathcal{Q})\mathrm{for}\mathrm{some}y\in \mathcal{Q}\},\\ {\mathcal{Q}}_0=\{y\in \mathcal{Q}:M_d(\mu ,y)=M_d(\mathcal{P},\mathcal{Q})\mathrm{for}\mathrm{some}\mu \in \mathcal{P}\}.\end{array}$$

Definition 1 

([12]). In a metric space $(\Omega ,M_d)$, let ${\mu }_0$ be any arbitrary point. The functional defined below is called the zero of Ω

$$∥\mu ∥=M_d(\mu ,{\mu }_0),\mathrm{for}\mathrm{all}\mu \in \Omega .$$

Definition 2. 

The set Ψ comprises all continuous increasing functions $\psi :[0,1]\to [0,+\infty )$ that satisfy the condition $\psi \left(0\right)=0$ and are continuous at 0.

Definition 3 

([7]). In a metric space $(\Omega ,M_d)$, let $T:\Omega \to \Omega$ be a mapping. Then, the mapping T is a Pata contraction if the following inequality holds for all $\mu ,y\in \Omega$, and every $ϵ\in [0,1]$:

$$M_d(T\mu ,Ty)\le (1-ϵ)M_d(\mu ,y)+\Lambda {\left(ϵ\right)}^{\eta }\psi \left(ϵ\right){[1+|\mu |+|y\left|\right]}^{\gamma },$$

where $\Lambda \ge 0$, $\eta \ge 1$, and $\gamma \in [0,\eta ]$ are fixed constants, and $\psi \in \Psi$.

Definition 4 

([14]). Consider a nonempty set Ω, and let $T:\Omega \to \Omega$ be a mapping. Additionally, let $\alpha :\Omega \times \Omega \to {\mathbb{R}}^{+}$ be a function that maps the pairs of the elements from Ω to positive real numbers. We say that the mapping T is an α-admissible if the condition, $\alpha (\mu ,y)\ge 1$, then $\alpha (T\mu ,Ty)\ge 1$ for all $\mu ,y\in \Omega$ holds.

Definition 5 

([15]). Let Ω be a nonempty set, and $\alpha :\Omega \times \Omega \to [0,+\infty )$ be an auxiliary function. A self-mapping T on Ω is called an $\alpha -$orbital admissible if

$$\alpha (\mu ,T\mu )\ge 1\mathrm{implies} \alpha (T\mu ,T^2\mu )\ge 1,\mathit{for}\mathit{all}\mu \in \Omega .$$

Definition 6 

([11]). A set K represents a class of all functions $\beta :[0,+\infty )\to [0,1]$ such that $\beta \left(t_n\right)⟶1$ implies $t_n⟶0,$ where $\left\{t_n\right\}$ is bounded sequence of positive real numbers.

Definition 7 

([11]). Consider $(\Omega ,M_d)$ as a metric space, and consider a pair of nonempty subsets $(\mathcal{P},\mathcal{Q})$ of Ω. A mapping $T:\mathcal{P}\to \mathcal{Q}$ is termed a $\beta -$Geraghty mapping if there exists a constant $\beta \in K$ such that for all $\mu ,y\in \mathcal{P}$, the following inequality is satisfied:

$$M_d(T\mu ,Ty)\le \beta \left(M_d(\mu ,y)\right)M_d(\mu ,y).$$

Definition 8 

([16]). A mapping $T:\Omega \to \Omega$ is classified as an expansive mapping if, for all $\mu ,y\in \Omega$ and some $q>1$, the following inequality holds:

$$M_d(T\mu ,Ty)\ge q_d(\mu ,y).$$

Definition 9 

([17]). Consider nonempty subsets $\mathcal{P}$ and $\mathcal{Q}$ in a metric space $(\Omega ,M_d)$. Let $\alpha :\mathcal{P}\times \mathcal{P}\to [0,+\infty )$ and $T:\mathcal{P}\to \mathcal{Q}$. A mapping T is referred to as $\alpha -$proximal admissible if the following condition is satisfied for all $\mu ,y\in \mathcal{P}$:

$$\left.\begin{array}{c}\alpha (\mu ,y)\ge 1\hfill \\ M_d(u,T\mu )=M_d(\mathcal{P},\mathcal{Q})\hfill \\ M_d(v,Ty)=M_d(\mathcal{P},\mathcal{Q})\hfill \end{array}\right\}\mathrm{implies}\alpha (u,v)\ge 1.$$

Remark 1. 

If we take $\mathcal{P}=\mathcal{Q}=\Omega$, then every $\alpha -$proximal admissible mapping becomes an α-admissible mapping.

Definition 10 

([12]). In a metric space $(\Omega ,M_d)$, let $\mathcal{P}$ and $\mathcal{Q}$ be nonempty subsets. An $\alpha -$proximal admissible mapping $T:\mathcal{P}\to \mathcal{Q}$ is an admissible mapping:

1.

$\alpha -$Pata proximal admissible contraction of type I if

$$\alpha (\mu ,y)M_d(u,v)\le \frac{(1-ϵ)}{2}{M}_d(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

2.

$\alpha -$Pata proximal admissible contraction of type II if

$$\alpha (\mu ,y)M_d(Tu,Tv)\le \frac{(1-ϵ)}{2}{M}_d(T\mu ,Ty)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥T\mu ∥+∥Ty∥+∥Tv∥]}^{\gamma },$$

for any $u,v,\mu$ and $y$ in $\mathcal{P}$, and constants $\Lambda \ge 0,$ $ϵ\in [0,1],\psi \in \Psi ,$ and $\gamma \in [0,\mu ],$ where $\mu \ge 1$.

Lemma 1 

([12]). Assume that $T:\mathcal{P}\to \mathcal{Q}$ is an $\alpha -$proximal admissible mapping and $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$, where ${\mathcal{P}}_0$ is a nonempty subset of $\mathcal{P}$. If

$$M_d({\mu }_1,T{\mu }_0)=M_d(\mathcal{P},\mathcal{Q}),\mathrm{and}\alpha ({\mu }_0,{\mu }_1)\ge 1,$$

for some ${\mu }_0$ and ${\mu }_1$ in ${\mathcal{P}}_0$, then ${\mathcal{P}}_0$ contains a sequence $\left\{{\mu }_n\right\}$ such that,

$$M_d({\mu }_{n+1},T{\mu }_n)=M_d(\mathcal{P},\mathcal{Q}),\mathrm{and}\alpha ({\mu }_n,{\mu }_{n+1})\ge 1\mathit{for}\mathit{all}n\in \mathbb{N}.$$

(1)

Definition 11 

([12]). A sequence ${\mu }_n\subset {\mathcal{P}}_0$ is called an α-proximal admissible Picard sequence if it fulfills (1) under the following condition:

$$\alpha ({\mu }_n,{\mu }_1)\ge 1,\mathit{and}\alpha ({\mu }_n,{\mu }_0)\ge 1\mathit{for}\mathit{all}n\in \mathbb{N}.$$

Definition 12 

([12]). Consider the mappings $T:\mathcal{P}\to \mathcal{Q}$, $\alpha :\mathcal{P}\times \mathcal{P}\to [0,+\infty )$, and $g:\mathcal{P}\to \mathcal{P}$. We say that a pair of the mappings $(g,T)$ satisfies a generalized $\alpha -$Pata-proximal contractive condition if there exist constants $\Lambda \ge 0$, $\mu \ge 1$, and $\gamma \in [0,\mu ]$ such that

$$\left.\begin{array}{c}\alpha (\mu ,y)\ge 1\\ M_d(gu,T\mu )=M_d(\mathcal{P},\mathcal{Q})\\ M_d(gv,Ty)=M_d(\mathcal{P},\mathcal{Q})\end{array}\right\}\mathrm{implies}\alpha (gu,gv)\ge 1,$$

and

$$\alpha (\mu ,y)M_d(gu,gv)\le \frac{(1-ϵ)}{2}{M}_d(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },\mathit{for}\mathit{all}u,v,\mu ,y\in \mathcal{P},$$

for every $ϵ\in [0,1],$ and $\psi \in \Psi .$

Remark 2 

([12]). When the mapping g is the identity mapping on $\mathcal{P}$ (denoted as $I_{\mathcal{P}}$), every generalized α-Pata proximal contraction becomes an α-Pata proximal admissible contraction of type I.

Definition 13 

([12]). A set $\mathcal{Q}$ is said to be an approximate compact with respect to $\mathcal{P}$ if every sequence $\left\{y_n\right\}$ in $\mathcal{Q}$ satisfying

$$M_d(\mu ,y_n)\to M_d(\mu ,\mathcal{Q})=inf{M}_d(\mu ,b)\mathit{for}\mathit{all}b\in \mathcal{Q},$$

has a convergent subsequence.

Remark 3. 

Each set is approximately compact concerning itself. 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 ${\mathcal{P}}_{\mathcal{0}}$ 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 Ω is called a Suzuki contraction if

$$\frac{1}{2}{M}_d(\mu ,T\mu )\le M_d(\mu ,y)\mathrm{implies}{M}_d(T\mu ,Ty)\le M_d(\mu ,y),\mathit{for}\mathit{all}\mu ,y\in \Omega .$$

Definition 15 

([9]). Consider a metric space $(\Omega ,M_d)$, and let $\Lambda \ge 0$, $\mu \ge 1$, 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 $ϵ\in [0,1]$, $\psi \in \Psi$, and for all $\mu ,y\in \Omega$, the following holds:

$$\frac{1}{2}{M}_d(\mu ,T\mu )\le M_d(\mu ,y),$$

which implies

$$\alpha (\mu ,T\mu )\alpha (y,Ty)M_d(T\mu ,Ty)\le \mathcal{P}(\mu ,y),$$

where

$$\begin{array}{ccc}\hfill \mathcal{P}(\mu ,y)& =& (1-ϵ)max\{M_d(\mu ,y),M_d(\mu ,T\mu ),M_d(y,Ty),\frac{1}{2}[M_d(\mu ,Ty)+M_d(y,T\mu )]\}+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥T\mu ∥+∥Ty∥]}^{\gamma }.\hfill \end{array}$$

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

$$\left.\begin{array}{c}\alpha (\mu ,y)\ge \theta (\mu ,y)\\ M_d(u,T\mu )=M_d(\mathcal{P},\mathcal{Q})\\ M_d(v,Ty)=M_d(\mathcal{P},\mathcal{Q})\end{array}\right\}\mathrm{implies}\alpha (u,v)\ge \theta (u,v),$$

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 $\alpha -\theta -$admissible mapping;

3.

If $\mathcal{P}=\mathcal{Q}=\Omega$ and $\theta (\mu ,y)=1$, then every $\alpha -\theta -$proximal admissible mapping becomes an α-admissible mapping.

Lemma 3 

([17]). Assuming that the mapping T is an $\alpha -\theta -$proximal admissible, and $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$, 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 $\alpha ({\mu }_0,{\mu }_1)\ge \theta ({\mu }_0,{\mu }_1)$, then there exists a sequence $\left\{{\mu }_n\right\}\subset {\mathcal{P}}_0$ such that

$$M_d({\mu }_{n+1},T{\mu }_n)=M_d(\mathcal{P},\mathcal{Q}),\mathit{and}\alpha ({\mu }_n,{\mu }_{n+1})\ge \theta ({\mu }_n,{\mu }_{n+1})\mathit{for}\mathit{all}n\in \mathbb{N}.$$

(2)

Definition 17. 

A sequence $\left\{{\mu }_n\right\}\subset {\mathcal{P}}_0$ is referred to as an $\alpha -\theta -$proximal admissible Picard sequence if it satisfies condition (2) and also fulfills $\alpha ({\mu }_n,{\mu }_1)\ge \theta ({\mu }_n,{\mu }_1)$ and $\alpha ({\mu }_n,{\mu }_0)\ge \theta ({\mu }_n,{\mu }_0)$ 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. 

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.

$\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction if there exist constants $\Lambda \ge 0$, $\mu \ge 1$ and $\gamma \in [0,\mu ]$ such that

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y),$$

implies

$$\frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m(\mu ,y)\right)m(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

2.

$\alpha -$Geraghty–Pata–Suzuki-Type proximal contraction if there exist fixed constants $\Lambda \ge 0$, $\mu \ge 1$ and $\gamma \in [0,\mu ]$ such that

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y),$$

which implies

$$\alpha (\mu ,y)M_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m(\mu ,y)\right)m(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

3.

A modified Geraghty–Pata–Suzuki proximal contraction if there exist fixed constants $\Lambda \ge 0,\mu \ge 1$ and $\gamma \in [0,\mu ]$ such that

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y)$$

which implies

$$M_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(M_d^{\ast }(\mu ,T\mu )\right)M_d^{\ast }(\mu ,T\mu )+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

for every $ϵ\in [0,1],$ where

$$\begin{array}{ccc}m(\mu ,y)\hfill & =& max\{M_d(\mu ,y),M_d^{\ast }(\mu ,T\mu ),M_d^{\ast }(y,Ty),\frac{M_d^{\ast }(\mu ,T\mu )+M_d^{\ast }(y,Ty)}{2},M_d^{\ast }(y,T\mu ),\hfill \\ & & \frac{M_d^{\ast }(\mu ,Ty)+M_d^{\ast }(y,T\mu )}{2}\},\hfill \end{array}$$

and

$$M_d^{\ast }(\mu ,y)=M_d(\mu ,y)-M_d(\mathcal{P},\mathcal{Q}),$$

for all $u,v,\mu ,y\in \mathcal{P},$ for every $ϵ\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 $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction with $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$ and ${\mathcal{P}}_0\ne \varphi .$ Then, the set ${\mathcal{P}}_0$ is $\alpha -\theta -$proximal orbital complete.

Proof. 

Let ${\mu }_0\in {\mathcal{P}}_0$ and $\left\{{\mu }_n\right\}$ be a Cauchy $\alpha -\theta -$proximal admissible Picard sequence. As $\Omega$ is complete and $\mathcal{P}$ is closed, there exists some ${\mu }_{\ast }$ in $\mathcal{P}$ such that ${lim}_{n\to +\infty }{M}_d({\mu }_n,{\mu }_{\ast })=0$ for all $n\in \mathbb{N}$. Additionally, considering that $\mu \ge 1$ and $\gamma \in [0,\mu ]$, it follows that $\frac{1}{1+\mu +\gamma }\le 1$. As T is an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, we have

$$\begin{array}{ccc}\hfill \frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_0,T{\mu }_0)& \le & M_d^{\ast }({\mu }_0,T{\mu }_0)=M_d({\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_0,{\mu }_1)\hfill \end{array}$$

Thus, the above inequality can be expressed as follows:

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_0,T{\mu }_0)\le M_d({\mu }_0,{\mu }_1),$$

which implies that

$$\begin{array}{ccc}{M}_d({\mu }_1,{\mu }_2)\hfill & \le \hfill & \frac{\alpha ({\mu }_1,{\mu }_2)}{\theta ({\mu }_1,{\mu }_2)}{M}_d({\mu }_1,{\mu }_2)\hfill \\ \hfill & \le \hfill & \frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_0,{\mu }_1)\right)m({\mu }_0,{\mu }_1)+\hfill \\ \hfill & \hfill & \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_0∥+∥{\mu }_1∥+∥{\mu }_2∥]}^{\gamma }.\hfill \end{array}$$

(3)

Taking the limit as $ϵ\to 0,$ then inequality (3) becomes

$$\begin{array}{ccc}{M}_d({\mu }_1,{\mu }_2)\hfill & \le \hfill & \frac{1}{2}\beta \left(m({\mu }_0,{\mu }_1)\right)m({\mu }_0,{\mu }_1)\hfill \\ \hfill & <\hfill & \beta \left(m({\mu }_0,{\mu }_1)\right)m({\mu }_0,{\mu }_1)\hfill \\ \hfill & \le \hfill & m({\mu }_0,{\mu }_1),\hfill \end{array}$$

(4)

where

$$\begin{array}{cc}\hfill m({\mu }_0,{\mu }_1)=& max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d({\mu }_0,T{\mu }_0)+M_d({\mu }_1,T{\mu }_1)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d({\mu }_0,T{\mu }_1)+M_d({\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ \hfill & \le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,{\mu }_2)+\hfill \\ \hfill & M_d({\mu }_2,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q}),\frac{M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)+M_d({\mu }_1,{\mu }_2)+M_d({\mu }_2,T{\mu }_1)}{2}-\hfill \\ \hfill & M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,{\mu }_2)+M_d({\mu }_2,T{\mu }_1)+M_d({\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ \hfill & \le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2),\frac{M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,{\mu }_2)}{2}\}\hfill \\ \hfill & \le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}.\hfill \end{array}$$

Consider that if

$$m({\mu }_0,{\mu }_1)\le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}=M_d({\mu }_1,{\mu }_2),$$

then from (4), we have

$$\begin{array}{c}\hfill M_d({\mu }_1,{\mu }_2)<m({\mu }_0,{\mu }_1)\le M_d({\mu }_1,{\mu }_2)\end{array}$$

which is a contradiction. Now, consider that if

$$m({\mu }_0,{\mu }_1)\le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}=M_d({\mu }_0,{\mu }_1),$$

then from inequality (4), we have

$$M_d({\mu }_1,{\mu }_2)<M_d({\mu }_0,{\mu }_1).$$

Since $T{\mu }_2\in T\left({\mathcal{P}}_0\right)\subset {\mathcal{Q}}_0,$ we can find ${\mu }_3\in {\mathcal{P}}_0$, which satisfies $M_d({\mu }_3,T{\mu }_2)=M_d(\mathcal{P},\mathcal{Q})$. Starting at point ${\mu }_0,$ continuing on the same line, we shall construct a sequence $\left\{{\mu }_n\right\}$ by ${\mu }_{n+1}\ne {\mu }_n$. Since T is an $\alpha -\theta -$Geraghty–Pata–Suzuki type proximal contraction, we have to prove that $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence. Since

$$\begin{array}{ccc}\hfill \frac{1}{1+\mu +\gamma }{M}_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})& \le & M_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})=M_d({\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_{n-1},{\mu }_n),\hfill \end{array}$$

we have

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})\le M_d({\mu }_{n-1},{\mu }_n),$$

which implies that

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_{n+1})\le \frac{\alpha ({\mu }_{n-1},{\mu }_n)}{\theta ({\mu }_{n-1},{\mu }_n)}{M}_d({\mu }_n,{\mu }_{n+1})& \le & \frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_{n-1},{\mu }_n)\right)m({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill & +& \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_{n-1}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }.\hfill \end{array}$$

(5)

Taking the limit as $ϵ\to 0$, then the above inequality becomes

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_{n+1})& \le & \frac{1}{2}\beta \left(m({\mu }_{n-1},{\mu }_n)\right)m({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

$$\begin{array}{cc}& <\beta \left(m({\mu }_{n-1},{\mu }_n)\right)m({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

(6)

$$\begin{array}{c}\le m({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

(7)

where

$$\begin{array}{ccc}\hfill m({\mu }_{n-1},{\mu }_n)& =& max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_{n-1},T{\mu }_{n-1})+M_d({\mu }_n,T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_{n-1},T{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})+M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & & \frac{M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1}),\frac{M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,{\mu }_{n+1})}{2}\}\hfill \\ & \le & max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}\hfill \end{array}$$

If

$$m({\mu }_{n-1},{\mu }_n)\le max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}=M_d({\mu }_n,{\mu }_{n+1}),$$

then inequality (7) becomes

$$\begin{array}{c}\hfill M_d({\mu }_n,{\mu }_{n+1})<m({\mu }_{n-1},{\mu }_n)\le M_d({\mu }_n,{\mu }_{n+1}),\end{array}$$

which is a contradiction if we consider

$$m({\mu }_{n-1},{\mu }_n)\le max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}=M_d({\mu }_{n-1},{\mu }_n)$$

Then, from inequality (7), we have

$$\begin{array}{c}\hfill M_d({\mu }_n,{\mu }_{n+1})<M_d({\mu }_{n-1},{\mu }_n);\end{array}$$

thus, $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence, and we have

$$M_d({\mu }_n,{\mu }_{n+1})\le M_d({\mu }_{n-1},{\mu }_n)\le \cdots \le M_d({\mu }_1,{\mu }_2)\le M_d({\mu }_0,{\mu }_1).$$

Now, we have to show that

$$\underset{n\to +\infty }{lim}{M}_d({\mu }_n,{\mu }_{n-1})=0.$$

Suppose that

$$\underset{n\to +\infty }{lim}{M}_d({\mu }_n,{\mu }_{n-1})=r\ne 0.$$

Then, inequality (6) becomes

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_{n+1})& \le & \beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)M_d({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill \frac{M_d({\mu }_n,{\mu }_{n+1})}{M_d({\mu }_n,{\mu }_{n-1})}& \le & \beta \left(M_d({\mu }_{n-1},{\mu }_n)\right),\hfill \end{array}$$

and taking THE limit as $n\to +\infty ,$ we have

$$\begin{array}{ccc}\hfill \underset{n\to +\infty }{lim}\frac{M_d({\mu }_n,{\mu }_{n+1})}{M_d({\mu }_n,{\mu }_{n-1})}& \le & \underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)\hfill \\ \hfill 1& \le & \underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)\le 1\hfill \end{array}$$

$$\underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)=1,\mathrm{implies}\underset{n\to +\infty }{lim}{M}_d({\mu }_{n-1},{\mu }_n)=0.$$

Now, we will show that the sequence $\left\{M_d({\mu }_n,{\mu }_0)\right\}$ is bounded above by constant $c.$ Since $c_n=∥{\mu }_n∥=M_d({\mu }_n,{\mu }_0),$ and

$$M_d({\mu }_n,{\mu }_{n+1})\le M_d({\mu }_{n-1},{\mu }_n)\le \cdots \le M_d({\mu }_1,{\mu }_2)\le M_d({\mu }_0,{\mu }_1)=c_1,$$

for all $n=0,1,2,\dots ,$ and $c_0,c_1\in {\mathbb{R}}^{+},$ we have

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_0)& \le & M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},{\mu }_1)+M_d({\mu }_1,{\mu }_0)\hfill \\ & \le & M_d({\mu }_0,{\mu }_1)+M_d({\mu }_{n+1},{\mu }_1)+c_1\hfill \\ & \le & 2c_1+M_d({\mu }_{n+1},{\mu }_1)\hfill \\ & \le & 2c_1+\frac{\alpha ({\mu }_{n+1},{\mu }_1)}{\theta ({\mu }_{n+1},{\mu }_1)}{M}_d({\mu }_{n+1},{\mu }_1)\hfill \\ & \le & 2c_1+\frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_n,{\mu }_0)\right)m({\mu }_n,{\mu }_0)\hfill \\ & +& \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_n∥+∥{\mu }_0∥+∥{\mu }_1∥]}^{\gamma },\hfill \end{array}$$

(8)

where

$$\begin{array}{ccc}\hfill m({\mu }_n,{\mu }_0)& =& max\{M_d({\mu }_n,{\mu }_0),M_d({\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_n,T{\mu }_n)+M_d({\mu }_0,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_0,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_n,T{\mu }_0)+M_d({\mu }_0,T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & \le & max\{M_d({\mu }_n,{\mu }_0),M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_0,{\mu }_1)+\hfill \\ & & M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)+M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_0,{\mu }_n)+M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_n,{\mu }_0)+M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)+M_d({\mu }_0,{\mu }_n)+M_d({\mu }_n,{\mu }_{n+1})}{2}+\hfill \\ & & \frac{M_d({\mu }_{n+1},T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{c_n,c_1,c_1,\frac{c_1+c_1}{2},c_n+c_1,\frac{2c_n+2c_1}{2}\}\hfill \\ & \le & max\{c_n,c_1,c_n+c_1\}\hfill \\ & \le & c_n+c_1.\hfill \end{array}$$

Then, inequality (8) can be written as

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_0)& \le & 2c_1+\frac{\left(1-ϵ\right)}{2}\left(\beta (c_n+c_1)\right)(c_n+c_1)\hfill \\ & +& \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_n∥+∥{\mu }_0∥+∥{\mu }_1∥]}^{\gamma }\hfill \\ & \le & 2c_1+\left(1-ϵ\right)\left(\beta (c_n+c_1)\right)(c_n+c_1)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+c_n+c_0+c_1]}^{\gamma }\hfill \\ & =& 2c_1+\left(1-ϵ\right)\left(\beta (c_n+c_1)\right)(c_n+c_1)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+c_n+c_0+c_1]}^{\gamma }\hfill \\ & <& 2c_1+\left(1-ϵ\right)(c_n+c_1)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+c_n+c_0+c_1]}^{\gamma }\hfill \\ & <& 2c_1+c_n+c_1-ϵc_n-ϵc_1+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+c_n+c_0+c_1]}^{\gamma }\hfill \end{array}$$

which further implies that

$$\begin{array}{ccc}\hfill ϵc_n& <& 3c_1-ϵc_1+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right)c_n^{{}^{\gamma }}{[\frac{1+c_0+c_1}{c_n}+1]}^{{}^{\gamma }}\hfill \\ & <& 3c_1+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right)c_n^{{}^{\gamma }}{[\frac{1+c_0+c_1}{c_n}+1]}^{{}^{\gamma }},\hfill \end{array}$$

as $\gamma \le \mu .$ Suppose that $s=\Lambda c_n^{{}^{\gamma }}{[\frac{1+c_0+c_1}{c_n}+1]}^{\gamma }$ and $b=3c_1,$ then the above inequality can be written as

$$ϵc_n<s{\left(ϵ\right)}^{\mu }\psi \left(ϵ\right)+b,$$

for some constants $s,b>0.$ Thus, the sequence $\left\{c_n\right\}$ is bounded. Now, if there exists a divergent subsequence $c_{n_i}$, in that case, there is a subsequence $s_{n_i}$ converging to $\Lambda$. If we select ${ϵ}_{n_i}=\frac{(1+b)}{c_{n_i}}$, then

$$1<\psi \left({ϵ}_i\right){(1+b)}^{\mu }{s}_{n_i}\to 0,$$

and this leads to a contradiction. Moving forward, our next objective is to demonstrate that $\left\{{\mu }_n\right\}$ is indeed a Cauchy sequence. To achieve this, we will show that

$$M_d({\mu }_{n+m},{\mu }_n)\le C{\omega }_n\left(\eta \right),$$

where

$$C=\underset{n\in \mathbb{N}}{sup}\Lambda {(1+3c_n)}^{\gamma }<+\infty ,\mathrm{and}{\omega }_n\left(\mu \right)={\left(\frac{\mu }{n}\right)}^{\mu }\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

Since m is a fixed natural number, define $P_n=n^{\mu }{M}_d({\mu }_{n+m},{\mu }_n),$ and

$$\begin{array}{ccc}\hfill P_{n+1}& =& {(n+1)}^{\mu }{M}_d({\mu }_{n+m+1},{\mu }_{n+1})\hfill \\ & \le & {(n+1)}^{\mu }\frac{\alpha ({\mu }_{n+m+1},{\mu }_{n+1})}{\theta ({\mu }_{n+m+1},{\mu }_{n+1})}{M}_d({\mu }_{n+m+1},{\mu }_{n+1})\hfill \\ & \le & {(n+1)}^{\mu }\frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_{n+m},{\mu }_n)\right)m({\mu }_{n+m},{\mu }_n)+\hfill \\ & & {(n+1)}^{\mu }\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_{n+m}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }\hfill \end{array}$$

(9)

If we take the limit $ϵ\to 0$, then inequality (9) becomes

$$\begin{array}{ccc}{P}_{n+1}\hfill & \le \hfill & {(n+1)}^{\mu }\frac{\beta \left(m({\mu }_{n+m},{\mu }_n)\right)}{2}m({\mu }_{n+m},{\mu }_n)\hfill \\ \hfill & <\hfill & {(n+1)}^{\mu }\beta \left(m({\mu }_{n+m},{\mu }_n)\right)m({\mu }_{n+m},{\mu }_n)\hfill \\ \hfill & \le \hfill & {(n+1)}^{\mu }m({\mu }_{n+m},{\mu }_n),\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}\hfill m({\mu }_{n+m},{\mu }_n)& =& max\{M_d({\mu }_{n+m},{\mu }_n),M_d({\mu }_{n+m},T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\frac{M_d({\mu }_{n+m},T{\mu }_{n+m})+M_d({\mu }_n,T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_n,T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\frac{M_d({\mu }_{n+m},T{\mu }_n)+M_d({\mu }_n,T{\mu }_{n+m})}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d({\mu }_{n+m},{\mu }_n),M_d({\mu }_{n+m},{\mu }_{n+m+1})+M_d({\mu }_{n+m+1},T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_{n+m},{\mu }_{n+m+1})+M_d({\mu }_{n+m+1},T{\mu }_{n+m})+M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)}{2}\hfill \\ & & -M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},{\mu }_{n+m+1})+M_d({\mu }_{n+m+1},T{\mu }_{n+m})-\hfill \\ & & M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d({\mu }_{n+m},{\mu }_{n+m+1})+M_d({\mu }_{n+m+1},{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)}{2}\hfill \\ & & +\frac{M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},{\mu }_{n+m+1})+M_d({\mu }_{n+m+1},T{\mu }_{n+m})}{2}\hfill \\ & & -M_d(\mathcal{P},\mathcal{Q})\}.\hfill \end{array}$$

Since ${lim}_{m,n\to +\infty }{M}_d({\mu }_{n+m},{\mu }_{n+m+1})\to 0,$ we get

$$m({\mu }_{n+m},{\mu }_n)\le max\{M_d({\mu }_{n+m},{\mu }_n),M_d({\mu }_{n+1},{\mu }_{n+m+1})\}.$$

If

$$m({\mu }_{n+m},{\mu }_n)\le max\{M_d({\mu }_{n+m},{\mu }_n),M_d({\mu }_{n+1},{\mu }_{n+m+1})\}=M_d({\mu }_{n+m+1},{\mu }_{n+1}),$$

then inequality (10), can be written as

$$\begin{array}{ccc}\hfill P_{n+1}& <& {(n+1)}^{\mu }{M}_d({\mu }_{n+m+1},{\mu }_{n+1})\hfill \\ & =& P_{n+1},\hfill \end{array}$$

which is a contradiction if

$$m({\mu }_{n+m},{\mu }_n)\le max\{M_d({\mu }_{n+m},{\mu }_n),M_d({\mu }_{n+1},{\mu }_{n+m+1})\}=M_d({\mu }_{n+m},{\mu }_n).$$

Choose

$$ϵ=1-{\left(\frac{n}{n+1}\right)}^{\mu }\le \frac{\mu }{n+1}.$$

then, from inequality (9), we have

$$\begin{array}{ccc}\hfill P_{n+1}& \le & {(n+1)}^{\mu }\left({\left(\frac{n}{n+1}\right)}^{\mu }\right)\beta \left(M_d({\mu }_{n+m},{\mu }_n)\right)M_d({\mu }_{n+m},{\mu }_n)+\hfill \\ & & {(n+1)}^{\mu }\Lambda {\left(\frac{\mu }{n+1}\right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right){[1+3c_n]}^{\gamma }\hfill \\ & \le & n^{\mu }{M}_d({\mu }_{n+m},{\mu }_n)+\Lambda {\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right){[1+3c_n]}^{\gamma }.\hfill \end{array}$$

Now, we have

$$\begin{array}{ccc}\hfill P_{n+1}& \le & P_n+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right)\hfill \\ & \le & P_{n-1}+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n}\right)+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right)\hfill \\ & & ⋮\hfill \\ & \le & P_0+C{\left(\mu \right)}^{\mu }[\psi \left(\frac{\mu }{1}\right)+\psi \left(\frac{\mu }{2}\right)\dots +\psi \left(\frac{\mu }{n+1}\right)].\hfill \end{array}$$

Since $P_0=0,$this gives

$$P_n\le C{\left(\mu \right)}^{\mu }\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

Since $n\ne 0,$ after division by $n^{\mu },$ we obtain that

$$\frac{P_n}{n^{\mu }}=M_d({\mu }_{n+m},{\mu }_n)\le \frac{C{\left(\mu \right)}^{\mu }}{n^{\mu }}\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

On taking the limit as $n\to +\infty ,$ then $\left\{{\mu }_n\right\}$ is a Cauchy sequence. For each $ϵ\in [0,1],$ we have

$$\begin{array}{ccc}{M}_d({\mu }_n,{\mu }_{n+1})\hfill & \le \hfill & \frac{\alpha ({\mu }_n,{\mu }_{n+1})}{\theta ({\mu }_n,{\mu }_{n+1})}{M}_d({\mu }_n,{\mu }_{n+1})\hfill \\ \hfill & \le \hfill & \frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_{n-1},{\mu }_n)\right)m({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill & +\hfill & \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_{n-1}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma },\hfill \end{array}$$

(11)

and assuming that

$$m({\mu }_{n-1},{\mu }_n)\le max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}=M_d({\mu }_{n-1},{\mu }_n),$$

then inequality (11) can be written as,

$$M_d({\mu }_n,{\mu }_{n+1})\le \frac{\left(1-ϵ\right)}{2}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)M_d({\mu }_{n-1},{\mu }_n)+K_2{\left(ϵ\right)}^{\mu }\psi \left(ϵ\right),$$

for each $ϵ\in [0,1].$ Hence, $\left\{{\mu }_n\right\}$ is a Cauchy sequence in a closed subset ${\mathcal{P}}_0$ of a complete metric space $(\Omega ,M_d).$ As we know that

$$M_d({\mu }_{n+1},T{\mu }_n)=M_d(\mathcal{P},\mathcal{Q}),$$

and every Cauchy sequence in a complete metric space is convergent, there exists some $\mu \in {\mathcal{P}}_0$ such that ${\mu }_n\to \mu$ in ${\mathcal{P}}_0\subseteq \mathcal{P},$

$$\underset{n\to +\infty }{lim}{M}_d({\mu }_n,\mu )=0.$$

Hence, ${\mathcal{P}}_0$ is $\alpha -\theta -$proximal orbital complete.    □

Theorem 1. 

Let $\mathcal{P}$ and $\mathcal{Q}$ be nonempty subsets of a complete metric space $(\Omega ,M_d)$. Furthermore, assume that the subset $\mathcal{P}$ is closed and we have a continuous mapping $T:\mathcal{P}\to \mathcal{Q}$, which is an $\alpha -\theta -$proximal admissible and $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction. Additionally, let $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$, where ${\mathcal{P}}_0\ne \varphi$. If $\mathcal{Q}$ is approximately compact with respect to $\mathcal{P}$, then μ is a unique best proximity point for T in ${\mathcal{P}}_0$.

Proof. 

Consider ${\mu }_0$ in ${\mathcal{P}}_0$, from Lemma (4), the sequence $\left\{{\mu }_n\right\}$ is a Cauchy $\alpha -\theta -$proximal admissible Picard sequence in ${\mathcal{P}}_0$. As a result, we can derive a sequence $\left\{{\mu }_n\right\}$ in ${\mathcal{P}}_0$ such that

$$\left.\begin{array}{c}\alpha ({\mu }_n,{\mu }_{n+1})\ge \theta ({\mu }_n,{\mu }_{n+1})\\ M_d({\mu }_n,T{\mu }_{n-1})=M_d(\mathcal{P},\mathcal{Q})\\ M_d({\mu }_{n+1},T{\mu }_n)=M_d(\mathcal{P},\mathcal{Q})\end{array}\right\}\mathrm{implies}\alpha ({\mu }_{n-1},{\mu }_n)\ge \theta ({\mu }_{n-1},{\mu }_n),$$

for all $n\in \mathbb{N}.$ Given that T is an $\alpha -\theta -$proximal admissible mapping, and it is also an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, we can apply reasoning similar to that used in the proof of Lemma (4). Consequently, we can deduce that the sequence $\left\{{\mu }_n\right\}\in {\mathcal{P}}_0$ is a Cauchy $\alpha -\theta -$proximal admissible Picard sequence. Since $\Omega$ is complete and $\mathcal{P}$ is closed, there exists some $\mu$ in $\mathcal{P}$ such that ${lim}_{n\to +\infty }{M}_d({\mu }_n,\mu )=0$. Since every Cauchy sequence in a complete metric space converges, there exists a $\mu$ in ${\mathcal{P}}_0$ such that ${\mu }_n\to \mu$ within ${\mathcal{P}}_0$. Furthermore, the mapping T is continuous, so we have $T{\mu }_n\to T\mu$. Now, based on the above inequality, we have:

$$M_d(\mathcal{P},\mathcal{Q})=\underset{n\to +\infty }{lim}{M}_d({\mu }_n,T{\mu }_n)=M_d(\mu ,T\mu ).$$

Thus, we can deduce that $\mu$ represents the $\alpha -\theta -$Geraghty–Pata–Suzuki-type best proximity point of the mapping T. Now, to show the uniqueness, we assume there exists $y\ne \mu$ ($M_d(\mu ,y)>0$) in $\mathcal{P}$ such that the mapping T is an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, so

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d^{\ast }(\mu ,T\mu )=M_d(\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q})=0<M_d(\mu ,y).$$

Thus, the above inequality can be written as,

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y),$$

which implies

$$\begin{array}{ccc}\hfill M_d(\mu ,y)& \le & \frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(\mu ,y)\hfill \\ & \le & \frac{\left(1-ϵ\right)}{2}\beta \left(m(\mu ,y)\right)m(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥y∥]}^{\gamma }\hfill \end{array}$$

(12)

Taking the limit $ϵ\to 0$, then inequality (12) becomes

$$\begin{array}{ccc}\hfill M_d(\mu ,y)& \le & \frac{\beta \left(m\right(\mu ,y\left)\right)}{2}m(\mu ,y)\hfill \end{array}$$

$$\begin{array}{ccc}& <& \beta \left(m\right(\mu ,y\left)\right)m(\mu ,y)\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& \le & m(\mu ,y),\hfill \end{array}$$

(14)

where

$$\begin{array}{ccc}\hfill m(\mu ,y)& =& max\{M_d(\mu ,y),M_d(\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}),M_d(y,Ty)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(\mu ,T\mu )+M_d(y,Ty)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d(y,\mu )+M_d(\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(\mu ,y)+M_d(y,Ty)+M_d(y,\mu )+M_d(\mu ,T\mu )}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d(\mu ,y),M_d(y,\mu ),M_d(y,\mu )\}=M_d(\mu ,y).\hfill \end{array}$$

Then, inequality (14) becomes

$$\begin{array}{ccc}\hfill M_d(\mu ,y)& <& M_d(\mu ,y),\hfill \end{array}$$

which is a contradiction; therefore, $\mu =y,$ which shows that $\mu$ is the unique best proximity point of mapping $T.$    □

Example 1. 

Consider the usual metric space $\Omega =\mathbb{R}$. Let

$$\mathcal{P}=\{\frac{1}{3},\frac{3}{3},\frac{5}{3},\frac{7}{3}\}\mathrm{and}\mathcal{Q}=\{\frac{2}{3},\frac{4}{3},\frac{6}{3},\frac{8}{3}\},$$

be two closed subsets of $\Omega ,$where

$$M_d(\mathcal{P},\mathcal{Q})=\frac{1}{3},{\mathcal{P}}_0=\mathcal{P},\mathrm{and}{\mathcal{Q}}_0=\mathcal{Q}.$$

Define a mapping $T:\mathcal{P}\to \mathcal{Q}$ as

$$T\mu =\left\{\begin{array}{c}\frac{2}{3};\mathrm{when}\mu \in \{\frac{1}{3},\frac{7}{3}\},\\ \frac{8}{3};\mathrm{when}\mu \in \{\frac{3}{3},\frac{5}{3}\}.\end{array}\right.$$

Obviously, $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0.$Further, suppose that $\alpha (\mu ,y)=2M_d(\mu ,y)$ and$\theta (\mu ,y)=M_d(\mu ,y).$ If we take$\mu =\frac{1}{3},y=\frac{3}{3}\in {\mathcal{P}}_0,$then $\alpha (\mu ,y)=2\left(\frac{2}{3}\right)>\frac{2}{3}=\theta (\mu ,y),$ and we have

$$\left.\begin{array}{c}\alpha (\mu ,y)>\theta (\mu ,y)\\ M_d(\frac{3}{3},T\frac{1}{3})=M_d(\mathcal{P},\mathcal{Q})=M_d(\frac{7}{3},T\frac{3}{3})\end{array}\right\}\mathrm{implies}\alpha (\frac{3}{3},\frac{7}{3})>\theta (\frac{3}{3},\frac{7}{3}).$$

Define $\psi \left(s\right)=s{e}^s$ for all $s\ge 0.$ If $\mu =2$ and $\gamma \in [0,2],$ assuming that $\lambda =\gamma$, taking $ϵ=\frac{1}{10}$ and verifying that $T$satisfies the conditions of an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, it is evident the mapping T satisfies the following condition:

$$\frac{\alpha (u,v)}{\theta (u,v)}{M}_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m(\mu ,y)\right)m(\mu ,y)+\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma }$$

(15)

for all $u,v,\mu$ and $y\in {\mathcal{P}}_0.$ For this setting and the following simple calculation, we have

$$\begin{array}{ccc}\hfill m(\frac{1}{3},\frac{3}{3})& =& max\{M_d(\frac{1}{3},\frac{3}{3}),M_d(\frac{1}{3},\frac{2}{3})-\frac{1}{3},M_d(\frac{3}{3},\frac{8}{3})-\frac{1}{3},\frac{,M_d(\frac{1}{3},\frac{2}{3})+M_d(\frac{3}{3},\frac{8}{3})}{2}\hfill \\ & & -\frac{1}{3},M_d(\frac{3}{3},\frac{2}{3})-\frac{1}{3},\frac{M_d(\frac{1}{3},\frac{8}{3})+M_d(\frac{3}{3},\frac{2}{3})}{2}-\frac{1}{3}\}=\frac{4}{3}.\hfill \end{array}$$

Define

$$\beta \left(u\right)=\left\{\begin{array}{c}1,\mathrm{if}u=0,\\ {\displaystyle \frac{ln(1+u)+1}{u}},\mathrm{if}u>0.\end{array}\right.$$

With simple calculation steps, one can verify that inequality (15) holds for $\mu =\frac{1}{3}$ and $y=\frac{3}{3},$ and the mapping T satisfies an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction for $76.25768383\le \Lambda .$

• If $\mu =\frac{1}{3}$ and $y=\frac{5}{3}$, then$T\mu =\frac{2}{3}$ and $Ty=\frac{8}{3}$, and by taking $u=\frac{3}{3}$ and $v=\frac{7}{3},$ then inequality (15) holds.
• If $\mu =\frac{1}{3}$ and $y=\frac{7}{3}$, then$T\mu =\frac{2}{3}$ and $Ty=\frac{2}{3},$ and by taking $u=\frac{3}{3}$ and $v=\frac{1}{3},$ then inequality (15) holds.
• If $\mu =\frac{3}{3}$ and$y=\frac{7}{3}$, then$T\mu =\frac{8}{3}$ and $Ty=\frac{2}{3},$ and by taking $u=\frac{7}{3}$ and $v=\frac{1}{3},$ then inequality (15) holds.
• If $\mu =\frac{3}{3}$ and $y=\frac{5}{3}$, then$T\mu =\frac{8}{3}$ and $Ty=\frac{8}{3},$ and by taking $u=\frac{7}{3}$ and $v=\frac{7}{3},$ then inequality (15) holds.
• If $\mu =\frac{7}{3}$ and $y=\frac{5}{3}$, then$T\mu =\frac{2}{3}$ and $Ty=\frac{8}{3},$and by taking$u=\frac{3}{3}$ and $v=\frac{7}{3},$ then inequality (15) holds.

Since the mapping T satisfies an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction for every pair of $\mu ,y\in {\mathcal{P}}_0,$ there exists some $u,v\in {\mathcal{P}}_0$ for $\Lambda =164,$ and mapping T satisfies the conditions of Theorem (1), and $\mu =\frac{1}{3}$ is the unique best proximity point of the mapping $T.$

Lemma 5. 

Let $\mathcal{P}$ and $\mathcal{Q}$ be two nonempty subsets of a complete metric space $(\Omega ,M_d),$ and $T:\mathcal{P}\to \mathcal{Q}$ be a continuous $\alpha -$Geraghty–Pata–Suzuki-type proximal contraction with $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$ and ${\mathcal{P}}_0\ne \varphi .$ Then, ${\mathcal{P}}_0$ is $\alpha -$proximal orbital complete.

Proof. 

By taking $\theta (\mu ,y)=1$ for all $\mu ,y\in \mathcal{P}$ in an $\alpha -\theta -$proximal admissible, we have an $\alpha -$proximal admissible. Continuing on the same line of proof as Lemma (4), then ${\mathcal{P}}_0$ is $\alpha -$proximal orbital complete.    □

Corollary 1. 

Consider a continuous mapping $T:\mathcal{P}\to \mathcal{Q}$, which is both an $\alpha -$proximal admissible mapping and an $\alpha -$Geraghty–Pata–Suzuki-type proximal contraction, with ${\mathcal{A}}_0\ne \varphi$ and $T\left({\mathcal{A}}_0\right)\subseteq \mathcal{Q}0$. If the set $\mathcal{Q}$ is approximately compact with respect to the set $\mathcal{P}$, then the mapping T has a unique best proximity point μ in ${\mathcal{A}}_0$.

Proof. 

By taking $\theta (\mu ,y)=1$ in the proof of Theorem (1), we have the desired result.    □

Theorem 2. 

Let $\mathcal{P}$ and $\mathcal{Q}$ be nonempty subsets of a complete metric space $(\Omega ,M_d),$ and suppose that $T:\mathcal{P}\to \mathcal{Q}$ is a modified Geraghty–Pata–Suzuki proximal contraction satisfying $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0.$ If $\mathcal{Q}$ is approximately compact with respect to $\mathcal{P},$then $T$has a unique best proximity point $\mu$ in ${\mathcal{P}}_0.$

Proof. 

By taking $m(\mu ,y)=M_d(\mu ,y),$ and $\alpha (\mu ,y)=\theta (\mu ,y)$ or $\alpha (\mu ,y)=\theta (\mu ,y)=1,$ in the proof of Theorem (1).    □

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

$$\alpha (g\mu ,gy)\ge \theta (g\mu ,gy)\mathrm{implies}\alpha (\mu ,y)\ge \theta (\mu ,y),\mathrm{for}\mathrm{all}\mu ,y\in \mathcal{P}.$$

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

$$\left.\begin{array}{c}\alpha (\mu ,y)\ge \theta (\mu ,y)\\ M_d(gu,T\mu )=M_d(\mathcal{P},\mathcal{Q})\\ M_d(gv,Ty)=M_d(\mathcal{P},\mathcal{Q})\end{array}\right\}\mathrm{implies}\alpha (gu,gv)\ge \theta (gu,gv),$$

for all $\mu ,y,u,v\in \mathcal{P}.$

Remark 5. 

If the mapping $g=I_{\mathcal{P}}$, 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 Geraghty–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 $\Lambda \ge 0,\lambda \ge 1,$and $\gamma \in [0,\lambda ]$ such that the following holds,

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(g\mu ,T\mu )\le M_d(g\mu ,gy)$$

which implies

$$\frac{\alpha (g\mu ,g\mu )}{\theta (g\mu ,gy)}{M}_d(gu,gv)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_g(\mu ,y)\right)m_g(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

for every $ϵ\in [0,1],$ and $\psi \in \Psi ,$ where

$$\begin{array}{ccc}\hfill m_g(\mu ,y)& =& max\{M_d(g\mu ,gy),M_d^{\ast }(g\mu ,T\mu ),M_d^{\ast }(gy,Ty),\frac{M_d^{\ast }(g\mu ,T\mu )+M_d^{\ast }(gy,Ty)}{2},\hfill \\ & & M_d^{\ast }(g\mu ,Ty),M_d^{\ast }(gy,T\mu ),\frac{M_d^{\ast }(g\mu ,Ty)+M_d^{\ast }(gy,T\mu )}{2}\},\hfill \end{array}$$

and

$$M_d^{\ast }(g\mu ,T\mu )=M_d(g\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}).$$

Remark 6. 

If we take $g=I_{\mathcal{P}}$ (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 $g:\mathcal{P}\to \mathcal{P}$ satisfying the ${(\alpha ,\theta )}_R-$property, and a mapping $T:\mathcal{P}\to \mathcal{Q}$ that is $\alpha -\theta -$proximal admissible. Let ${\mathcal{P}}_0$ be a nonempty closed subset of $\mathcal{P}$, and assume $T\left({\mathcal{P}}_0\right)\subset {\mathcal{Q}}_0$, as well as ${\mathcal{P}}_0\subseteq g\left({\mathcal{P}}_0\right)$. 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 $\alpha ({\mu }_0,{\mu }_1)\ge \theta ({\mu }_0,{\mu }_1)$, then the pair $(g,T)$ has a unique optimal coincidence point μ in ${\mathcal{P}}_0$.

Proof. 

Since ${\mu }_0$ is a given point in ${\mathcal{P}}_0,$ as $T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$ and ${\mathcal{P}}_0\subseteq g\left({\mathcal{P}}_0\right),$ 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 $\alpha ({\mu }_0,{\mu }_1)\ge \theta ({\mu }_0,{\mu }_1),$ where ${\mu }_0,{\mu }_1\in {\mathcal{P}}_0.$ Furthermore, since $T{\mu }_1\in T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0$and $g{\mu }_2\in {\mathcal{P}}_0\subseteq g\left({\mathcal{P}}_0\right)$, it follows that there exists an element ${\mu }_2\in {\mathcal{P}}_0$such that $M_d(g{\mu }_2,T{\mu }_1)=M_d(\mathcal{P},\mathcal{Q}).$ As the mapping T is $\alpha -\theta -$proximal admissible, then $\alpha (g{\mu }_1,g{\mu }_2)\ge \theta (g{\mu }_1,g{\mu }_2).$ Further, the mapping g satisfies the ${(\alpha ,\theta )}_R-$property, which implies $\alpha ({\mu }_1,{\mu }_2)\ge \theta ({\mu }_1,{\mu }_2),$where $g{\mu }_1,g{\mu }_2\in {\mathcal{P}}_0$. Since $\lambda \ge 1,$and $\gamma \in [0,\lambda ]$ and $\frac{1}{1+\lambda +\gamma }\le 1,$ we have

$$\begin{array}{ccc}\hfill \frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(g{\mu }_0,T{\mu }_0)& \le & M_d^{\ast }(g{\mu }_0,T{\mu }_0)=M_d(g{\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d(g{\mu }_0,g{\mu }_1),\hfill \end{array}$$

so the above inequality can be written as,

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(g{\mu }_0,T{\mu }_0)\le M_d(g{\mu }_0,g{\mu }_1).$$

This implies

$$\begin{array}{ccc}\hfill M_d(g{\mu }_1,g{\mu }_2)& \le & \frac{\alpha (g{\mu }_0,g{\mu }_1)}{\theta (g{\mu }_0,g{\mu }_1)}{M}_d(g{\mu }_1,g{\mu }_2)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_g({\mu }_0,{\mu }_1)\right)m_g({\mu }_0,{\mu }_1)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_0∥+∥{\mu }_1∥+∥{\mu }_2∥]}^{\gamma },\hfill \end{array}$$

(16)

and by taking the limit $ϵ\to 0$ in inequality (16), we have

$$\begin{array}{c}\hfill M_d(g{\mu }_1,g{\mu }_2)\le \frac{\beta \left(m_g({\mu }_0,{\mu }_1)\right)}{2}{m}_g({\mu }_0,{\mu }_1)<m_g({\mu }_0,{\mu }_1)\end{array}$$

(17)

where

$$\begin{array}{cc}\hfill m_g({\mu }_0,{\mu }_1)=& max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_1,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d(g{\mu }_0,T{\mu }_0)+M_d(g{\mu }_1,T{\mu }_1)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d(g{\mu }_0,T{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ \hfill \le & max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)+M_d(g{\mu }_1,g{\mu }_2)+M_d(g{\mu }_2,T{\mu }_1)}{2}\hfill \\ \hfill -& M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_1,g{\mu }_2)+M_d(g{\mu }_2,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ \hfill & \frac{M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,g{\mu }_2)+M_d(g{\mu }_2,T{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ \hfill \le & max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_1,g{\mu }_2),\frac{M_d({\mu }_0,{\mu }_1)+M_d(g{\mu }_1,g{\mu }_2)}{2}\}\hfill \\ \hfill \le & max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_1,g{\mu }_2)\}.\hfill \end{array}$$

If

$$m_g({\mu }_0,{\mu }_1)\le max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_1,g{\mu }_2)\}=M_d(g{\mu }_1,g{\mu }_2),$$

then from inequality (17), we have

$$\begin{array}{c}\hfill M_d(g{\mu }_1,g{\mu }_2)<m_g({\mu }_0,{\mu }_1)\le M_d(g{\mu }_1,g{\mu }_2),\end{array}$$

which is a contradiction. Now, if

$$m_g({\mu }_0,{\mu }_1)\le max\{M_d(g{\mu }_0,g{\mu }_1),M_d(g{\mu }_1,g{\mu }_2)\}=M_d(g{\mu }_0,g{\mu }_1),$$

then from inequality (17), we have

$$\begin{array}{c}\hfill M_d(g{\mu }_1,g{\mu }_2)<M_d(g{\mu }_0,g{\mu }_1).\end{array}$$

Since mapping g satisfies the ${(\alpha ,\theta )}_R-$property, we have

$$\begin{array}{c}\hfill M_d({\mu }_1,{\mu }_2)<M_d({\mu }_0,{\mu }_1),\end{array}$$

which shows that $\left\{M_d({\mu }_1,{\mu }_2)\right\}$ is a decreasing sequence. Starting at the point ${\mu }_0,$we can construct sequence $\left\{{\mu }_n\right\}$by ${\mu }_{n+1}\ne {\mu }_n$. As mapping T is an $\alpha -\theta -$proximal admissible mapping, we have to prove that $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence. Since the mapping T is an $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction, we have

$$\begin{array}{ccc}\hfill \frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(g{\mu }_{n-1},T{\mu }_{n-1})& \le & M_d^{\ast }(g{\mu }_{n-1},T{\mu }_{n-1})\hfill \\ & =& M_d(g{\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & \le & M_d(g{\mu }_{n-1},g{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d(g{\mu }_{n-1},g{\mu }_n),\hfill \end{array}$$

which can be written as,

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(g{\mu }_{n-1},T{\mu }_{n-1})\le M_d(g{\mu }_{n-1},g{\mu }_n),$$

for all $n\in \mathbb{N}.$Since the mapping T is an $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction, then

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,g{\mu }_{n+1})& \le & \frac{\alpha (g{\mu }_{n-1},g{\mu }_n)}{\theta (g{\mu }_{n-1},g{\mu }_n)}{M}_d(g{\mu }_n,g{\mu }_{n+1})\hfill \\ & \le & \frac{\left(1-ϵ\right)}{2}\beta \left(m_g({\mu }_{n-1},{\mu }_n)\right)m_g({\mu }_{n-1},{\mu }_n)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_{n-1}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }.\hfill \end{array}$$

(18)

By taking the limit $ϵ\to 0$, then inequality (18) becomes

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,g{\mu }_{n+1})& \le & \frac{\beta \left(m_g({\mu }_{n-1},{\mu }_n)\right)}{2}{m}_g({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

$$\begin{array}{ccc}& <& \beta \left(m_g({\mu }_{n-1},{\mu }_n)\right)m_g({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& <& m_g({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

(20)

where

$$\begin{array}{ccc}\hfill m_g({\mu }_{n-1},{\mu }_n)& =& max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_n,T{\mu }_n)-\hfill \\ & & M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n-1},T{\mu }_{n-1})+M_d(g{\mu }_n,T{\mu }_n)}{2}-\hfill \\ & & M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n-1},T{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n-1})}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_{n-1},g{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n-1},g{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n-1})+M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)}{2}\hfill \\ & & -M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n-1},g{\mu }_n)+M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n-1})}{2}\hfill \\ & & -M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_n,g{\mu }_{n+1}),\frac{M_d({\mu }_{n-1},{\mu }_n)+M_d(g{\mu }_n,g{\mu }_{n+1})}{2}\}\hfill \\ & \le & max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_n,g{\mu }_{n+1})\}.\hfill \end{array}$$

If

$$m_g({\mu }_{n-1},{\mu }_n)\le max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_n,g{\mu }_{n+1})\}=M_d(g{\mu }_n,g{\mu }_{n+1}),$$

then inequality (20) becomes

$$\begin{array}{c}\hfill M_d(g{\mu }_n,g{\mu }_{n+1})<m_g({\mu }_{n-1},{\mu }_n)\le M_d(g{\mu }_n,g{\mu }_{n+1}),\end{array}$$

which is a contradiction. Now, if

$$m_g({\mu }_{n-1},{\mu }_n)\le max\{M_d(g{\mu }_{n-1},g{\mu }_n),M_d(g{\mu }_n,g{\mu }_{n+1})\}=M_d(g{\mu }_n,g{\mu }_{n-1}),$$

then inequality (20) becomes

$$\begin{array}{c}\hfill M_d(g{\mu }_n,g{\mu }_{n+1})<m_g({\mu }_{n-1},{\mu }_n)\le M_d(g{\mu }_n,g{\mu }_{n-1}).\end{array}$$

Since the mapping g satisfies the ${(\alpha ,\theta )}_R-$property, we have

$$M_d({\mu }_n,{\mu }_{n+1})<M_d({\mu }_{n-1},{\mu }_n)$$

(21)

Thus, $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence, and hence we have

$$M_d({\mu }_n,{\mu }_{n+1})\le M_d({\mu }_{n-1},{\mu }_n)\le \dots \le M_d({\mu }_1,{\mu }_2)\le M_d({\mu }_0,{\mu }_1).$$

Now, we have to show that

$$\underset{n\to +\infty }{lim}{M}_d({\mu }_n,{\mu }_{n-1})=0.$$

Consider that ${lim}_{n\to +\infty }{M}_d({\mu }_n,{\mu }_{n-1})=r\ne 0.$ Then, inequality (20) can be written as

$$\begin{array}{ccc}\hfill M_d({\mu }_n,{\mu }_{n+1})& \le & \beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)M_d({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill \frac{M_d({\mu }_n,{\mu }_{n+1})}{M_d({\mu }_n,{\mu }_{n-1})}& \le & \beta \left(M_d({\mu }_{n-1},{\mu }_n)\right).\hfill \end{array}$$

Taking the limit as $n\to +\infty ,$ we have

$$\begin{array}{ccc}\hfill \underset{n\to +\infty }{lim}\frac{M_d({\mu }_n,{\mu }_{n+1})}{M_d({\mu }_n,{\mu }_{n-1})}& \le & \underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)\hfill \\ \hfill 1& \le & \underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)\le 1\hfill \end{array}$$

$$\underset{n\to +\infty }{lim}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)=1,\mathrm{implies}\underset{n\to +\infty }{lim}{M}_d({\mu }_{n-1},{\mu }_n)=0.$$

Further, we have

$$M_d(g{\mu }_n,g{\mu }_{n-1})\le \beta \left(M_d({\mu }_{n-1},{\mu }_{n-2})\right)M_d({\mu }_{n-1},{\mu }_{n-2})+K{\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right)$$

and taking the limit as $ϵ\to 0,$ then the above inequality can be written as

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,g{\mu }_{n-1})& \le & \beta \left(M_d({\mu }_{n-1},{\mu }_{n-2})\right)M_d({\mu }_{n-1},{\mu }_{n-2})\hfill \\ & \le & M_d({\mu }_{n-1},{\mu }_{n-2}),\hfill \end{array}$$

which can aldo be written as

$$M_d(g{\mu }_n,g{\mu }_{n-1})\le M_d({\mu }_{n-1},{\mu }_{n-2}).$$

Since mapping g is an expensive mapping, the above inequality can be written as

$$M_d({\mu }_n,{\mu }_{n-1})\le M_d(g{\mu }_n,g{\mu }_{n-1})\le M_d({\mu }_{n-1},{\mu }_{n-2}),$$

which shows that the sequence $\left\{M_d({\mu }_n,{\mu }_{n-1})\right\}$ is a decreasing sequence. Now, we will show that the sequence $\left\{M_d(g{\mu }_n,{\mu }_0)\right\}$ is bounded above by the constant $\frac{C}{2}=M_d({\mu }_0,g{\mu }_1)$, which is the case when $n=1.$ Suppose that $M_d(g{\mu }_n,{\mu }_0)\le C,$ where $M_d(g{\mu }_{n-1},{\mu }_0)\le \frac{C}{2}.$ As

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,{\mu }_0)& \le & M_d(g{\mu }_n,g{\mu }_{n-1})+M_d(g{\mu }_{n-1},{\mu }_0)\hfill \\ & \le & M_d({\mu }_{n-1},{\mu }_{n-2})+M_d(g{\mu }_{n-1},{\mu }_0)\hfill \\ & & ⋮\hfill \\ & \le & M_d({\mu }_1,{\mu }_0)+\frac{C}{2}\hfill \\ & \le & \frac{C}{2}+\frac{C}{2}=C,\hfill \end{array}$$

for all $n=0,1,2,\dots ,$ and $C_0,C_1\in {\mathbb{R}}^{+},$ since $C_0=∥g{\mu }_0∥$, and $C_1=∥g{\mu }_1∥,$ we have

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,{\mu }_0)& \le & M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},g{\mu }_1)+M_d(g{\mu }_1,{\mu }_0)\hfill \\ & \le & M_d({\mu }_0,{\mu }_1)+M_d(g{\mu }_1,{\mu }_0)+M_d(g{\mu }_{n+1},g{\mu }_1)\hfill \\ & \le & \frac{C_1}{2}+C_0+M_d(g{\mu }_{n+1},g{\mu }_1)\hfill \\ & \le & \frac{C_1}{2}+C_0+\frac{\alpha ({\mu }_n,{\mu }_0)}{\theta ({\mu }_n,{\mu }_0)}{M}_d(g{\mu }_{n+1},g{\mu }_1),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,{\mu }_0)& \le & \frac{C_1}{2}+C_0+\frac{\left(1-ϵ\right)}{2}\beta \left(m_g({\mu }_n,{\mu }_0)\right)m_g({\mu }_n,{\mu }_0)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_n∥+∥{\mu }_0∥+∥{\mu }_1∥]}^{\gamma },\hfill \end{array}$$

(22)

where

$$\begin{array}{ccc}\hfill m_g({\mu }_n,{\mu }_0)& =& max\{M_d(g{\mu }_n,g{\mu }_0),M_d(g{\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_n,T{\mu }_n)+M_d(g{\mu }_0,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d(g{\mu }_0,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_n,T{\mu }_0)+M_d(g{\mu }_0,T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d(g{\mu }_n,g{\mu }_0),M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)+M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_0,g{\mu }_n)+M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{\begin{array}{c}{M}_d(g{\mu }_n,g{\mu }_0)+M_d(g{\mu }_0,g{\mu }_1)+M_d(g{\mu }_1,T{\mu }_0)+M_d(g{\mu }_0,g{\mu }_n)+\\ M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)\end{array}}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{C_n,C_1,C_1,\frac{C_1+C_1}{2},C_n+C_1,\frac{2C_n+2C_1}{2}\}\hfill \\ & \le & max\{C_n,C_1,C_n+C_1\}\hfill \\ & \le & C_n+C_1.\hfill \end{array}$$

Then, inequality (22) can be written as

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,{\mu }_0)& \le & \frac{C_1}{2}+C_0+\frac{\left(1-ϵ\right)}{2}\left(\beta (C_n+C_1)\right)(C_n+C_1)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_n∥+∥{\mu }_0∥+∥{\mu }_1∥]}^{\gamma }\hfill \\ & \le & \frac{C_1}{2}+C_0+\left(1-ϵ\right)\left(\beta (C_n+C_1)\right)(C_n+C_1)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+C_n+C_0+C_1]}^{\gamma }\hfill \\ & <& \frac{C_1}{2}+C_0+\left(1-ϵ\right)(C_n+C_1)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+C_n+C_0+C_1]}^{\gamma }\hfill \\ & =& \frac{C_1}{2}+C_0+C_n+C_1-ϵC_n-ϵC_1+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+C_n+C_0+C_1]}^{\gamma }.\hfill \end{array}$$

After simplification, we have

$$\begin{array}{ccc}\hfill ϵC_n& <& \frac{3}{2}{C}_1-ϵC_1+C_0+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right)C_n^{\lambda }[\frac{1+C_0+C_1}{C_n}+1]\hfill \\ & <& \frac{3}{2}{C}_1+C_0+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right)C_n^{\lambda }[\frac{1+C_0+C_1}{C_n}+1],\hfill \end{array}$$

as $\gamma \le \lambda$. Suppose that $s=\Lambda {[\frac{1+C_0+C_1}{C_n}+1]}^{\gamma }$and $b=\frac{3}{2}{C}_1+C_0,$then

$$ϵC_n<s{\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right)+b,$$

for some constants$s,b>0.$ Thus, the sequence $\left\{C_n\right\}$is bounded if there is a divergent subsequence $\left\{C_{n_i}\right\}.$ We have a subsequence $\left\{s_{n_i}\right\}$ such that $s_{n_i}\to \Lambda .$ If we choose ${ϵ}_{n_i}=\frac{(1+b)}{C_{n_i}},$ then

$$1<\psi \left({ϵ}_i\right){(1+b)}^{\lambda }{s}_{n_i}\to 0,$$

which is a contradiction. Our next step is to prove that $\left\{g{\mu }_n\right\}$ is a Cauchy sequence. For this, we prove that

$$M_d(g{\mu }_{n+m},g{\mu }_n)\le C{\omega }_n\left(\eta \right),$$

where

$$C=\underset{n\in \mathbb{N}}{sup}\Lambda {(1+3c_n)}^{\gamma }<+\infty ,\mathrm{and}{\omega }_n\left(\mu \right)={\left(\frac{\mu }{n}\right)}^{\mu }\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

Let m be fixed, define$P_n=n^{\mu }{M}_d(g{\mu }_{n+m},g{\mu }_n),$and

$$\begin{array}{ccc}{P}_{n+1}\hfill & =\hfill & {(n+1)}^{\mu }{M}_d(g{\mu }_{n+m+1},g{\mu }_{n+1})\hfill \\ \hfill & \le \hfill & {(n+1)}^{\mu }\frac{\alpha (g{\mu }_{n+m+1},g{\mu }_{n+1})}{\theta (g{\mu }_{n+m+1},g{\mu }_{n+1})}{M}_d(g{\mu }_{n+m+1},g{\mu }_{n+1})\hfill \\ \hfill & \le \hfill & {(n+1)}^{\mu }\frac{\left(1-ϵ\right)}{2}\beta \left(m_g({\mu }_{n+m},{\mu }_n)\right)m_g({\mu }_{n+m},{\mu }_n)+\hfill \\ \hfill & \hfill & {(n+1)}^{\mu }\Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_{n+m}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }\hfill \end{array}$$

(23)

Taking $ϵ\to 0$, then inequality (23) becomes

$$\begin{array}{ccc}{P}_{n+1}\hfill & \le \hfill & {(n+1)}^{\mu }\frac{\beta \left(m_g({\mu }_{n+m},{\mu }_n)\right)}{2}{m}_g({\mu }_{n+m},{\mu }_n)\hfill \\ \hfill & <\hfill & {(n+1)}^{\mu }\beta \left(m_g({\mu }_{n+m},{\mu }_n)\right)m_g({\mu }_{n+m},{\mu }_n)\hfill \\ \hfill & \le \hfill & {(n+1)}^{\mu }{m}_g({\mu }_{n+m},{\mu }_n)\hfill \end{array}$$

(24)

where

$$\begin{array}{ccc}\hfill m_g({\mu }_{n+m},{\mu }_n)& =& max\{M_d(g{\mu }_{n+m},g{\mu }_n),M_d(g{\mu }_{n+m},T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n+m},T{\mu }_{n+m})+M_d(g{\mu }_n,T{\mu }_n)}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n+m},T{\mu }_n)+M_d(g{\mu }_n,T{\mu }_{n+m})}{2}-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d(g{\mu }_{n+m},g{\mu }_n),M_d(g{\mu }_{n+m},g{\mu }_{n+m+1})+M_d(g{\mu }_{n+m+1},T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{\begin{array}{c}{M}_d(g{\mu }_{n+m},g{\mu }_{n+m+1})+M_d(g{\mu }_{n+m+1},T{\mu }_{n+m})+\\ M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)\end{array}}{2}-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},g{\mu }_{n+m+1})+M_d(g{\mu }_{n+m+1},T{\mu }_{n+m})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g{\mu }_{n+m},g{\mu }_{n+m+1})+M_d(g{\mu }_{n+m+1},g{\mu }_{n+1})+M_d(g{\mu }_{n+1},T{\mu }_n)}{2}\hfill \\ & & +\frac{M_d(g{\mu }_n,g{\mu }_{n+1})+M_d(g{\mu }_{n+1},g{\mu }_{n+m+1})+M_d(g{\mu }_{n+m+1},T{\mu }_{n+m})}{2}-\hfill \\ & & M_d(\mathcal{P},\mathcal{Q})\}.\hfill \end{array}$$

Taking ${lim}_{m,n\to +\infty }{M}_d({\mu }_{n+m},{\mu }_{n+m+1})\to 0,$ then we have

$$m_g({\mu }_{n+m},{\mu }_n)\le max\{M_d(g{\mu }_{n+m},g{\mu }_n),M_d(g{\mu }_{n+1},g{\mu }_{n+m+1})\}.$$

If

$$m_g({\mu }_{n+m},{\mu }_n)\le max\{M_d(g{\mu }_{n+m},g{\mu }_n),M_d(g{\mu }_{n+1},g{\mu }_{n+m+1})\}=M_d(g{\mu }_{n+m+1},g{\mu }_{n+1}),$$

then inequality (24) can be written as

$$\begin{array}{c}\hfill P_{n+1}<{(n+1)}^{\mu }{m}_g({\mu }_{n+m},{\mu }_n)\le M_d(g{\mu }_{n+m+1},g{\mu }_{n+1})=P_{n+1}\end{array}$$

which is a contradiction; hence,

$$m_g({\mu }_{n+m},{\mu }_n)\le max\{M_d(g{\mu }_{n+m},g{\mu }_n),M_d(g{\mu }_{n+1},g{\mu }_{n+m+1})\}=M_d(g{\mu }_{n+m},g{\mu }_n).$$

Choose

$$ϵ=1-{\left(\frac{n}{n+1}\right)}^{\mu }\le \frac{\mu }{n+1}.$$

Then, from (23),

$$\begin{array}{ccc}\hfill P_{n+1}& \le & {(n+1)}^{\mu }\left({\left(\frac{n}{n+1}\right)}^{\mu }\right)\beta \left(M_d(g{\mu }_{n+m},g{\mu }_n)\right)M_d(g{\mu }_{n+m},g{\mu }_n)+\hfill \\ & & {(n+1)}^{\mu }\Lambda {\left(\frac{\mu }{n+1}\right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right){[1+3c_n]}^{\gamma }\hfill \\ & \le & n^{\mu }{M}_d(g{\mu }_{n+m},g{\mu }_n)+\Lambda {\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right){[1+3c_n]}^{\gamma },\hfill \end{array}$$

and we get

$$\begin{array}{ccc}\hfill P_{n+1}& \le & P_n+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right)\hfill \\ & \le & P_{n-1}+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n}\right)+C{\left(\mu \right)}^{\mu }\psi \left(\frac{\mu }{n+1}\right)\hfill \\ & & ⋮\hfill \\ & \le & P_0+C{\left(\mu \right)}^{\mu }[\psi \left(\frac{\mu }{1}\right)+\psi \left(\frac{\mu }{2}\right)\dots +\psi \left(\frac{\mu }{n+1}\right)].\hfill \end{array}$$

Since $P_0=0,$this gives

$$P_n=C{\left(\mu \right)}^{\mu }\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

After division by $n^{\mu },$ we obtain that

$$\frac{p_n}{n^{\mu }}=M_d(g{\mu }_{n+m},g{\mu }_n)\le \frac{C{\left(\mu \right)}^{\mu }}{n^{\mu }}\sum _{k=1}^n\psi \left(\frac{\mu }{k}\right).$$

On taking the limit as $n\to +\infty ,$ we can show that $\left\{g{\mu }_n\right\}$ is a Cauchy sequence. For each $ϵ\in [0,1]$, from inequality (18), we have

$$\begin{array}{ccc}\hfill M_d(g{\mu }_n,g{\mu }_{n+1})& \le & \frac{\alpha (g{\mu }_{n-1},g{\mu }_n)}{\theta (g{\mu }_{n-1},g{\mu }_n)}{M}_d(g{\mu }_n,g{\mu }_{n+1})\hfill \\ & \le & \frac{\left(1-ϵ\right)}{2}\beta \left(m({\mu }_{n-1},{\mu }_n)\right)m({\mu }_{n-1},{\mu }_n)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥{\mu }_{n-1}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }.\hfill \end{array}$$

Since

$$m({\mu }_{n-1},{\mu }_n)\le M_d({\mu }_{n-1},{\mu }_n),$$

and the mapping g satisfies the ${(\alpha ,\theta )}_R-$property, then we have

$$M_d(g{\mu }_n,g{\mu }_{n+1})\le \frac{\left(1-ϵ\right)}{2}\beta \left(M_d({\mu }_{n-1},{\mu }_n)\right)M_d({\mu }_{n-1},{\mu }_n)+K_2{\left(ϵ\right)}^{\mu }\psi \left(ϵ\right),$$

for each $ϵ\in [0,1];$ hence, $\left\{g{\mu }_n\right\}\subseteq {\mathcal{P}}_0$is a Cauchy sequence in a complete metric space $(\Omega ,M_d),$ and every Cauchy sequence in a complete metric space is convergent. Since ${\mathcal{P}}_0$ is closed, there exists some $g\mu \in {\mathcal{P}}_0\subseteq g\left({\mathcal{P}}_0\right)$ such that $g{\mu }_n\to g\mu$in ${\mathcal{P}}_0.$

$$\underset{n\to +\infty }{lim}{M}_d(g{\mu }_n,g\mu )=0,$$

and the mapping T is continuous, so $T{\mu }_n\to T\mu$. Now, from the above inequality, we have,

$$M_d(\mathcal{P},\mathcal{Q})=\underset{n\to +\infty }{lim}{M}_d(g{\mu }_n,T{\mu }_n)=M_d(g\mu ,T\mu ),$$

which shows that $\mu$ is the coincidence best proximity point of the pair of mappings $(g,T)$.

To show the uniqueness of the coincidence best proximity point, assume on the contrary that there exists another point $y\ne \mu$ in ${\mathcal{P}}_0$ such that it satisfies an $\alpha -\theta -$proximal admissible Picard sequence. Since the pair of mappings $(g,T)$ satisfies the $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction, then

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(g\mu ,T\mu )\le M_d^{\ast }(g\mu ,T\mu )=M_d(g\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q})=0<M_d(g\mu ,gy),$$

as $\mu \ne y$ and $0<M_d(g\mu ,gy);$ thus, the above inequality can be written as,

$$\frac{1}{1+\mu +\gamma }{M}_d^{\ast }(g\mu ,T\mu )\le M_d(g\mu ,gy),$$

which implies

$$\begin{array}{ccc}\hfill M_d(g\mu ,gy)\le \frac{\alpha (g\mu ,gy)}{\theta (g\mu ,gy)}{M}_d(g\mu ,gy)& \le & \frac{\left(1-ϵ\right)}{2}\beta \left(m_g(\mu ,y)\right)m_g(\mu ,y)+\hfill \\ & & \Lambda {\left(ϵ\right)}^{\mu }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥y∥]}^{\gamma },\hfill \end{array}$$

(25)

where

$$\begin{array}{ccc}\hfill m_g(\mu ,y)& =& max\{M_d(\mu ,y),M_d(g\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}),M_d(gy,Ty)-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g\mu ,T\mu )+M_d(gy,Ty)}{2}-M_d(\mathcal{P},\mathcal{Q}),M_d(gy,g\mu )+M_d(g\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & \frac{M_d(g\mu ,gy)+M_d(gy,Ty)+M_d(gy,g\mu )+M_d(g\mu ,T\mu )}{2}-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & M_d(\mu ,y).\hfill \end{array}$$

Taking the limit $ϵ\to 0$ and mapping g is an expansive mapping, then inequality (25) will become

$$\begin{array}{ccc}\hfill M_d(\mu ,y)\le M_d(g\mu ,gy)& \le & \frac{\beta \left(m_g(\mu ,y)\right)}{2}{m}_g(\mu ,y)\hfill \\ & <& \beta \left(m_g(\mu ,y)\right)m_g(\mu ,y)\hfill \\ & \le & m_g(\mu ,y)\hfill \end{array}$$

$$M_d(\mu ,y)<M_d(\mu ,y),$$

which is a contradiction. Therefore, $\mu =y,$ which shows the uniqueness of the coincidence best proximity point of the pair of the mappings $(g,T).$    □

Corollary 2. 

Let $\mathcal{P}\mathrm{and}\mathcal{Q}$ be nonempty subsets of a complete metric space $(\Omega ,M_d)$, and $T:\mathcal{P}\to \mathcal{Q}$ be an $\alpha -\theta -$proximal admissible mapping where ${\mathcal{P}}_0$ is a nonempty and closed subset of set $\mathcal{P}$, $T\left({\mathcal{P}}_0\right)\subset {\mathcal{Q}}_0$ and ${\mathcal{P}}_0\subseteq g\left({\mathcal{P}}_0\right)$ for any $t>0.$If the set $\mathcal{Q}$is approximately compact with respect to the set $\mathcal{P},$ and mapping T satisfies $\alpha -\theta -$Geraghty Pata Suzuki type proximal contraction. If there exists ${\mu }_0,{\mu }_1\in {\mathcal{P}}_0$ satisfying $M_d({\mu }_1,T{\mu }_0)=M_d(\mathcal{P},\mathcal{Q})$ and $\alpha ({\mu }_0,{\mu }_1)\ge \theta ({\mu }_0,{\mu }_1)$, then the mapping T has a unique best proximity point μ in ${\mathcal{P}}_0$.

Proof. 

By taking $g=I_{\mathcal{P}}$(identity mapping over set $\mathcal{P})$ in Theorem (3), then we have the desired result.    □

4. α–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 $\alpha :\Omega$ × $\Omega \to [0,\infty )$, 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 $\Lambda \ge 0,\lambda \ge 1,\gamma \in [0,\lambda ]$ such that the following holds,

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y)$$

which implies

$$\alpha (\mu ,T\mu )\alpha (y,Ty)M_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(q(\mu ,y)\right)q(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

for every $ϵ\in [0,1],$ where

$$q(\mu ,y)=max\{M_d(\mu ,y),M_d(\mu ,T\mu ),M_d(y,Ty)\},$$

and

$$M_d^{\ast }(\mu ,T\mu )=M_d(\mu ,T\mu )-M_d(\mathcal{P},\mathcal{Q}).$$

Definition 24. 

Consider nonempty subsets $\mathcal{P}$ and $\mathcal{Q}$ of a metric space $(\Omega ,M_d)$. Let $\alpha :\Omega$×$\Omega \to [0,\infty )$, 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 $\Lambda \ge 0$, $\lambda \ge 1$, $\gamma \in [0,\lambda ]$ such that the following condition holds:

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }(\mu ,T\mu )\le M_d(\mu ,y)$$

which implies

$$\alpha (\mu ,T\mu )\alpha (y,Ty)M_d(u,v)\le \frac{\left(1-ϵ\right)}{2}\beta \left(M_d(\mu ,y)\right)M_d(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

for every $ϵ\in [0,1].$

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 $T\left({\mathcal{P}}_0\right)\subset {\mathcal{Q}}_0$, 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 $\alpha ({\mu }_0,{\mu }_1)\ge 1$ and $M_d({\mu }_1,T{\mu }_0)=M_d(\mathcal{P},\mathcal{Q}).$ Let $\left\{{\mu }_n\right\}$ be a sequence in ${\mathcal{P}}_0$ such that $\alpha ({\mu }_n,{\mu }_{n+1})\ge 1$ 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 $\alpha ({\mu }_0,{\mu }_1)\ge 1.$ As $T{\mu }_1\in T\left({\mathcal{P}}_0\right)\subseteq {\mathcal{Q}}_0,$ 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 $\alpha \left({\mu }_{1,}{\mu }_2\right)\ge 1.$ In addition, we set $∥\mu ∥=M_d(\mu ,{\mu }_0)$ for all $\mu \in \Omega$. Since $\lambda \ge 1$, $\gamma \in [0,\lambda ]$ and $\frac{1}{1+\lambda +\gamma }\le 1,$ we have

$$\begin{array}{ccc}\hfill \frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_0,T{\mu }_0)& \le & M_d^{\ast }({\mu }_0,T{\mu }_0)=M_d({\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_0,{\mu }_1),\hfill \end{array}$$

and the above inequality can be written as,

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_0,T{\mu }_0)\le M_d({\mu }_0,{\mu }_1),$$

which implies

$$\begin{array}{ccc}{M}_d({\mu }_1,{\mu }_2)\hfill & \le \hfill & \alpha ({\mu }_0,T{\mu }_0)\alpha ({\mu }_1,T{\mu }_1)M_d({\mu }_1,{\mu }_2)\hfill \\ \hfill & \le \hfill & \frac{\left(1-ϵ\right)}{2}\beta \left(q({\mu }_0,{\mu }_1)\right)q({\mu }_0,{\mu }_1)+\hfill \\ \hfill & \hfill & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_0∥+∥{\mu }_1∥+∥{\mu }_2∥]}^{\gamma }.\hfill \end{array}$$

(26)

Taking the limit $ϵ\to 0$, then inequality (26) becomes

$$\begin{array}{ccc}{M}_d({\mu }_1,{\mu }_2)\hfill & \le \hfill & \frac{\beta \left(q({\mu }_0,{\mu }_1)\right)}{2}q({\mu }_0,{\mu }_1)\hfill \\ \hfill & <\hfill & \beta \left(q({\mu }_0,{\mu }_1)\right)q({\mu }_0,{\mu }_1)\hfill \\ \hfill & \le \hfill & q({\mu }_0,{\mu }_1)\hfill \end{array}$$

(27)

where

$$\begin{array}{ccc}\hfill q({\mu }_0,{\mu }_1)& =& max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_0,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_0,{\mu }_1)+M_d({\mu }_1,T{\mu }_0)-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_1,{\mu }_2)\hfill \\ & +& M_d({\mu }_2,T{\mu }_1)-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}.\hfill \end{array}$$

Assume that

$$q({\mu }_0,{\mu }_1)\le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}=M_d({\mu }_1,{\mu }_2),$$

then inequality (27) can be written as

$$\begin{array}{c}\hfill M_d({\mu }_1,{\mu }_2)<q({\mu }_0,{\mu }_1)\le M_d({\mu }_1,{\mu }_2)\end{array}$$

which is a contradiction. Now, if we have

$$q({\mu }_0,{\mu }_1)\le max\{M_d({\mu }_0,{\mu }_1),M_d({\mu }_1,{\mu }_2)\}=M_d({\mu }_0,{\mu }_1),$$

then inequality (27) can be written as

$$\begin{array}{ccc}\hfill M_d({\mu }_1,{\mu }_2)& \le & \frac{\beta \left(M_d({\mu }_0,{\mu }_1)\right)}{2}{M}_d({\mu }_0,{\mu }_1)\hfill \\ & <& \beta \left(M_d({\mu }_0,{\mu }_1)\right)M_d({\mu }_0,{\mu }_1)\hfill \\ & \le & M_d({\mu }_0,{\mu }_1).\hfill \end{array}$$

Starting at this point ${\mu }_0,$we shall construct a sequence $\left\{{\mu }_n\right\}$such that ${\mu }_{n+1}\ne {\mu }_n$. Continuing on the same lines, as T is an $\alpha -$modified Geraghty–Pata–Suzuki-type proximal mapping, we have to prove that $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence. Since

$$\begin{array}{ccc}\hfill \frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})& \le & M_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})\hfill \\ & =& M_d({\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & \le & M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q})\hfill \\ & =& M_d({\mu }_{n-1},{\mu }_n),\hfill \end{array}$$

the above inequality can be written as

$$\frac{1}{1+\lambda +\gamma }{M}_d^{\ast }({\mu }_{n-1},T{\mu }_{n-1})\le M_d({\mu }_{n-1},{\mu }_n),$$

for all $n\in \mathbb{N}.$Since $T$is an $\alpha -$modified Geraghty–Pata–Suzuki-type proximal mapping,

$$\begin{array}{ccc}{M}_d({\mu }_n,{\mu }_{n+1})\le \frac{\alpha ({\mu }_n,{\mu }_{n+1})}{\theta ({\mu }_n,{\mu }_{n+1})}{M}_d({\mu }_n,{\mu }_{n+1})\hfill & \le \hfill & \frac{\left(1-ϵ\right)}{2}\beta \left(q({\mu }_{n-1},{\mu }_n)\right)q({\mu }_{n-1},{\mu }_n)+\hfill \\ \hfill & \hfill & \Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥{\mu }_{n-1}∥+∥{\mu }_n∥+∥{\mu }_{n+1}∥]}^{\gamma }.\hfill \end{array}$$

(28)

By taking the limit $ϵ\to 0,$ then inequality (28) becomes

$$\begin{array}{ccc}{M}_d({\mu }_n,{\mu }_{n+1})\hfill & \le \hfill & \frac{\beta \left(q({\mu }_{n-1},{\mu }_n)\right)}{2}q({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill & <\hfill & \beta \left(q({\mu }_{n-1},{\mu }_n)\right)q({\mu }_{n-1},{\mu }_n)\hfill \\ \hfill & \le \hfill & q({\mu }_{n-1},{\mu }_n)\hfill \end{array}$$

(29)

where

$$\begin{array}{ccc}\hfill q({\mu }_{n-1},{\mu }_n)& =& max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_{n-1},T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),M_d({\mu }_n,T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q})\},\hfill \\ & \le & max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_{n-1},{\mu }_n)+M_d({\mu }_n,T{\mu }_{n-1})-M_d(\mathcal{P},\mathcal{Q}),\hfill \\ & & M_d({\mu }_n,{\mu }_{n+1})+M_d({\mu }_{n+1},T{\mu }_n)-M_d(\mathcal{P},\mathcal{Q})\}\hfill \\ & \le & max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}.\hfill \end{array}$$

If we consider

$$q({\mu }_{n-1},{\mu }_n)\le max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}=M_d({\mu }_n,{\mu }_{n+1}),$$

then inequality (29) can be written as,

$$\begin{array}{c}\hfill M_d({\mu }_n,{\mu }_{n+1})<\beta \left(q({\mu }_{n-1},{\mu }_n)\right)q({\mu }_{n-1},{\mu }_n)\le M_d({\mu }_n,{\mu }_{n+1}),\end{array}$$

which is a contradiction. Now consider

$$q({\mu }_{n-1},{\mu }_n)\le max\{M_d({\mu }_{n-1},{\mu }_n),M_d({\mu }_n,{\mu }_{n+1})\}=M_d({\mu }_{n-1},{\mu }_n),$$

then inequality (29) can be written as

$$\begin{array}{c}\hfill M_d({\mu }_n,{\mu }_{n+1})<q({\mu }_{n-1},{\mu }_n)\le M_d({\mu }_{n-1},{\mu }_n)\end{array}$$

Thus, $\left\{M_d({\mu }_n,{\mu }_{n+1})\right\}$ is a decreasing sequence; hence,

$$M_d({\mu }_n,{\mu }_{n+1})\le M_d({\mu }_{n-1},{\mu }_n)\le \dots \le M_d({\mu }_1,{\mu }_2)\le M_d({\mu }_0,{\mu }_1).$$

By taking $\theta (\mu ,y)=1$, for all $\mu ,y\in \mathcal{P},$ and continuing on the same line as in the proof of Theorem (1), we can prove that the mapping T possesses a unique best proximity point for every $ϵ\in [0,1].$    □

Corollary 3. 

Let $(\mathcal{P},\mathcal{Q})$ be a pair of nonempty closed subsets of a complete metric space $(\Omega ,M_d)$, and the mapping $T:\mathcal{P}\to \mathcal{Q}$ be an $\alpha -$proximal admissible mapping that satisfies an$\alpha -$Geraghty–Pata–Suzuki-type proximal contraction of type I on$\mathcal{P}.$If there exist elements ${\mu }_0,{\mu }_1\in {\mathcal{P}}_0$such that $\alpha ({\mu }_0,{\mu }_1)\ge 1$ and $M_d({\mu }_1,T{\mu }_0)=M_d(\mathcal{P},\mathcal{Q}),$then the mapping T possess a unique best proximity point.

Proof. 

By choosing $q(\mu ,y)=M_d(\mu ,y)$ in Theorem (4), and following on the same line of proof, we have the desired result.    □

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

$$\frac{1}{1+\lambda +\gamma }{M}_d(\mu ,T\mu )\le M_d(\mu ,y)$$

implies

$$\frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(T\mu ,Ty)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_1(\mu ,y)\right)m_1(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma };$$

2.

An $\alpha -\theta -$generalized Geraghty–Pata–Suzuki contraction if

$$\frac{1}{1+\lambda +\gamma }{M}_d(g\mu ,T\mu )\le M_d(g\mu ,gy)$$

implies

$$\frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(T\mu ,Ty)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_1(\mu ,y)\right)m_1(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma };$$

3.

An $\alpha -\theta -$modified Geraghty–Pata–Suzuki contraction if

$$\frac{1}{1+\lambda +\gamma }{M}_d(\mu ,T\mu )\le M_d(\mu ,y)$$

implies

$$\frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(T\mu ,Ty)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_2(\mu ,y)\right)m_2(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma };$$

4.

An $\alpha -\theta -$modified generalized Geraghty–Pata–Suzuki contraction if

$$\frac{1}{1+\lambda +\gamma }{M}_d(g\mu ,T\mu )\le M_d(g\mu ,gy)$$

implies

$$\frac{\alpha (\mu ,y)}{\theta (\mu ,y)}{M}_d(T\mu ,Ty)\le \frac{\left(1-ϵ\right)}{2}\beta \left(m_2(\mu ,y)\right)m_2(\mu ,y)+\Lambda {\left(ϵ\right)}^{\lambda }\psi \left(ϵ\right){[1+∥\mu ∥+∥y∥+∥v∥]}^{\gamma },$$

for every $ϵ\in [0,1],$where $\alpha ,\theta :\Omega \times \Omega \to [0,+\infty ),$and $\alpha (\mu ,y)\ge \theta (\mu ,y),$ $\lambda \ge 1$and $\gamma \in [0,\lambda ]$, for every $ϵ\in [0,1],$ and $\psi \in \Psi ,$ where

$$\begin{array}{ccc}\hfill m_1(\mu ,y)& =& max\{M_d(\mu ,y),M_d(\mu ,T\mu ),M_d(y,Ty),\frac{M_d(\mu ,T\mu )+M_d(y,Ty)}{2},\hfill \\ & & M_d(y,T\mu ),\frac{M_d(\mu ,Ty)+M_d(y,T\mu )}{2}\},\hfill \end{array}$$

and

$$m_2(\mu ,y)=max\{M_d(\mu ,y),M_d(\mu ,T\mu ),M_d(y,Ty)\},$$

for all $\mu ,y\in \mathcal{P}.$

Theorem 5. 

If there exists some ${\mu }_0$with $\alpha ({\mu }_0,T{\mu }_0)\ge \theta ({\mu }_0,T{\mu }_0),$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.

Proof. 

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 $\alpha ({\mu }_0,T{\mu }_0)\ge \theta ({\mu }_0,T{\mu }_0),g:\Omega \to \Omega ,$ 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.

Proof. 

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 $M_d(g\mu ,T\mu )=M_d(\mathcal{P},\mathcal{Q})$. However, in the scenario $\mathcal{P}=\mathcal{Q}=\Omega$, we have $M_d(g\mu ,T\mu )=0$, which makes $\mu$ a coincidence point of the pair of mappings $(g,T)$ satisfying the $\alpha -\theta -$generalized Geraghty–Pata–Suzuki contraction condition.    □

Theorem 7. 

If there exists ${\mu }_0$ with $\alpha ({\mu }_0,T{\mu }_0)\ge \theta ({\mu }_0,T{\mu }_0)$, then the mapping $T:\Omega \to \Omega$ which satisfies the $\alpha -\theta -$modified Geraghty–Pata–Suzuki contractive condition on a complete metric space $(\Omega ,M_d)$ has a unique fixed point.

Proof. 

By applying Theorem (4) to the case where $\mathcal{P}=\mathcal{Q}=\Omega$, the $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction becomes the $\alpha -\theta -$Geraghty–Pata–Suzuki contraction. We can find a point $\mu$ that satisfies $M_d(\mu ,T\mu )=M_d(\mathcal{P},\mathcal{Q})$, but in the case of a self-mapping where $\mathcal{P}=\mathcal{Q}=\Omega$, we have $M_d(\mu ,T\mu )=0$. This implies that there exists a fixed point $\mu$ for the $\alpha -\theta -$Geraghty–Pata–Suzuki contraction of the mapping T.    □

Theorem 8. 

If there exists ${\mu }_0$with $\alpha ({\mu }_0,T{\mu }_0)\ge \theta ({\mu }_0,T{\mu }_0),$the mappings $g:\Omega \to \Omega ,$ and $T:\Omega \to \Omega$ which satisfy the $\alpha -\theta -$modified generalized Geraghty–Pata–Suzuki contractive condition on a complete metric space $(\Omega ,M_d)$ have a unique coincidence point.

Proof. 

By applying Theorem (4) to the case where $\mathcal{P}=\mathcal{Q}=\Omega$, then the $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction becomes the $\alpha -\theta -$Geraghty–Pata–Suzuki contraction. We can find a point $\mu$ that satisfies $M_d(g\mu ,T\mu )=M_d(\mathcal{P},\mathcal{Q})$. However, for the case of a self-mapping where $\mathcal{P}=\mathcal{Q}=\Omega$, we have $M_d(g\mu ,T\mu )=0$, which makes $\mu$ a fixed point of the $\alpha -\theta -$modified generalized Geraghty–Pata–Suzuki contraction of the pair of mappings $(g,T)$.    □

6. Conclusions

In this study, we introduce three types of contractions called the $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction, the $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction, and the $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction. The coincidence best proximity point and best proximity point results have been proven by using the multi-valued mapping in the framework of a complete metric space. We provided several remarks, corollaries and results which show that obtained results are proper generalizations of the results discussed in [9,11,12,19]. In the future, one can extend these results to incomplete spaces.