Certain Interpolative Proximal Contractions, Best Proximity Point Theorems in Bipolar Metric Spaces with Applications

Abstract

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results.

1. Introduction

In the study of fractals, best proximity points can be used. Best proximity points can be used to estimate or comprehend the behavior of these fractals at various scales because fractals are frequently generated through iterated processes. In this case, optimal proximity points could be used to predict how iterated mappings will behave in the setting of the Julia set or Mandelbrot set (fractal sets constructed through iteration). Modeling complex phenomena requires the use of fractional calculus, which deals with derivatives and integrals of noninteger order. Best proximity points are often utilized to approximately solve fractional differential equations. Finding precise answers to these problems can be difficult since they sometimes require fractional-order derivatives. In certain situations, the best proximity points can offer approximations of solutions. In conclusion, the mathematical methods and applications of fixed point theorems, best proximity point theorems, fractals, and fractional calculus are connected. Together, they are frequently employed to comprehend and simulate intricate nonlinear circumstances, particularly when classical calculus is insufficient.

In 1906, the theory of metric space was introduced by Fréchet [1]. From 1906 to now, numerous generalizations of metric space have been introduced by altering the metric function. Fixed point (FP) theory is an important tool for obtaining a unique solution of different type of problems. A mapping $\mathcal{V}:\mathcal{H}\to \mathcal{H}$ has an FP if $\mathcal{V}\left(c\right)=c$. The Banach contraction principle [2] is an origin of FP theory. Even in the present time, for the benefit of human beings, researchers of several fields including computers, physics, applied mathematics, and many others are benefiting from using the Banach contraction principle. Sessa et al. [3] demonstrated some FP results and provided an application for nonlinear differential equation. In 1968, Kannan [4] gave some FP results and enhanced the era of fixed point theory. Then, in 1968, Fan [5] proved some FP results by the extension of two FP theorem of Browder. In 2018, Karapinar [6] re-examined the Kannan FP theorem with regards to interpolation. Ishtiaq et al. [7] demonstrated FP results with the help of interpolation and gave the application for fractional differential equations.

In 1997, the concept of best proximity point (BPP) and best approximity was given by Basha and Veeramani [8]. In 2011, Basha [9] initiated some BPP theorems for contractive non-self mappings. In 2012, Basha and Shahzad [10] provided the BPP theorem for generalized proximal contractions of the first and second kind in the setting of complete metric space. In 2020, Altun et al. [11] presented the concept of p-proximal contraction and p-proximal contractive non-self mappings on metric space. In 2021, Altun et al. [12] derived some BPP theorems for interpolative proximal contraction (IPC) and proved Reich–Rus–Ceric and Kannan-type proximal contraction of the first and second kind. Ishtiaq et al. [13] provided common BPP theorems for proximal contractions, Kannan-type IPC, Reich–Rus–Ciric-type IPC, and Hardy–Rogers-type IPC. Karapinar [6] initiated Kannan-type IPC and proved some FP theorems. Karapinar et al. [14] initiated Reich–Rus–Ciric-type IPC and proved some FP theorems.

Mutlu and Gurdal [15] established the notion of bipolar metric space (BMS) and discussed the basic properties and derived some FP results. Mani et al. [16] demonstrated the FP results in the context of BMS under the simulation function. Kurepa [17] presented the notion of pseudo-metric space (PMS). Khajasteh et al. [18] introduced simulation function and proved some FP results for bipolar metric space. Semet et al. [19] proved some FPs results for $\alpha -\mathsf{\Psi }$ contractive mappings in complete metric space. Gurdal et al. [20] proved some FP results for $\alpha -\mathsf{\Psi }$ contractive mappings in bipolar metric spaces. Lateef [21] proved best proximity points in $\mathcal{F}$-metric spaces. Nashine et al. [22] proved several best proximity point theorems for rational proximal contractions.

In this work, we prove some BPP results in a complete bipolar metric spaces (CBMS). Moreover, we introduce proximal contraction (PC), Reich–Rus–Ceric-type IPC, Kannan-type IPC of the first and second kind. Also, we provide some examples to illustrate the validity of our results, an application to find the solution to an integral equation, and a nonlinear fractional differential equation.

2. Preliminaries

In this section, we provide some basic definitions and results that will help to the readers to understand the main results.

Definition 1

([17]). A PMS $\left(\mathcal{H},\mathcal{V}\right)$ is a set $\mathcal{H}\ne \varnothing$ together with a non-negative real valued function $\mathcal{V}:\mathcal{H}\times \mathcal{H}\to {\mathbb{R}}^{+}$, called a PM, such that for every $c_1,c_2,p_2\in \mathcal{H}$:

1. 

$\mathcal{V}\left(c,p\right)=0\mathit{for}c,p\in \mathcal{H};$

2. 

$\mathcal{V}\left(c,p\right)=\mathcal{V}\left(p,c\right)\mathit{for}c,p\in \mathcal{H};$

3. 

$\mathcal{V}\left(c_1,p_2\right)\le \mathcal{V}\left(c_1,c_2\right)+\mathcal{V}\left(c_2,p_2\right).$

Like a metric space, points in PMS need not be distinguishable, that is, one may have $\mathcal{V}\left(c,p\right)=0$ for distinct values of $c\ne p$.

Definition 2

([15]). Let $\mathcal{H}$,$\mathcal{K}\ne \varnothing$ and $\mathcal{V}:\mathcal{H}\times \mathcal{K}\to {\mathbb{R}}^{+}$ be a function, satisfying some of the axioms:

(a1)

if $\mathcal{V}\left(c,p\right)=0$ then $c=p$, ∀$\left(c,p\right)\in \mathcal{H}\times \mathcal{K}$;

(a2)

if $c=p$ then $\mathcal{V}\left(c,p\right)=0$, ∀$\left(c,p\right)\in \mathcal{H}\times \mathcal{K}$;

(a3)

$\mathcal{V}\left(c,p\right)=\mathcal{V}\left(p,c\right)$, ∀$\left(c,p\right)\in \mathcal{H}\cap \mathcal{K}$;

(a4)

$\mathcal{V}\left(c_1,p_2\right)\le \mathcal{V}\left(c_1,p_1\right)+\mathcal{V}\left(c_2,p_1\right)+\mathcal{V}\left(c_2,p_2\right)$, ∀$c_1,c_2\in \mathcal{H}$ and $p_1,p_2\in \mathcal{K}$.

Then,

(i)

If $\left(\mathrm{a}2\right)$ and $\left(\mathrm{a}3\right)$ hold, then $\mathcal{V}$ is said to be a bipolar pseudo-semimetric (BPSM) on the pair $\left(\mathcal{H},\mathcal{K}\right)$.

(ii)

If $\mathcal{V}$ is a BPSM verifying $\left(\mathrm{a}4\right)$, it said to be a bipolar pseudo-metric (BPM).

(iii)

A BPM $\mathcal{V}$ satisfying (a1), is called a BMS.

Definition 3

([15]). Let $\left({\mathcal{H}}_1,{\mathcal{K}}_1,{\mathcal{V}}_1\right)$ and $\left({\mathcal{H}}_2,{\mathcal{K}}_2,{\mathcal{V}}_2\right)$ be a BPSM.

A map $\mathcal{J}:\left({\mathcal{H}}_1,{\mathcal{K}}_1,{\mathcal{V}}_1\right)\rightrightarrows \left({\mathcal{H}}_2,{\mathcal{K}}_2,{\mathcal{V}}_2\right)$ is called continuous at a point $c_0\in {\mathcal{H}}_1$, if for every $\xi >0$, there exists a $\delta >0$ such that whenever $p\in {\mathcal{K}}_1$ and ${\mathcal{V}}_1\left(c_0,p\right)<\delta$, ${\mathcal{V}}_2\left(\mathcal{J}\left(c_0\right),\mathcal{J}\left(p\right)\right)<\xi$. It is continuous at a point $p_0\in {\mathcal{K}}_1$ if for each $\xi >0$, there exists a $\delta >0$ such that whenever $c\in {\mathcal{H}}_1$ and ${\mathcal{V}}_1\left(c,p_0\right)<\delta$, ${\mathcal{V}}_2\left(\mathcal{J}\left(c\right),\mathcal{J}\left(p_0\right)\right)<\xi$. If $\mathcal{J}$ is continuous at every point $c\in {\mathcal{H}}_1$ and $p\in {\mathcal{K}}_1$, then it is said to be continuous.

Definition 4

([15]). Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BPSM.

(i)

A sequence $\left\{\left(c_n,s_n\right)\right\}$ on the set $\mathcal{H}\times \mathcal{K}$ is said to be a bisequence (in short, BS) on $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$.

(ii)

If both sequences $\left\{\left(c_n\right)\right\}$ and $\left\{\left(s_n\right)\right\}$ converge, then the BS $\left\{\left(c_n,s_n\right)\right\}$ is said to be convergent. If $\left\{\left(c_n\right)\right\}$ and $\left\{\left(s_n\right)\right\}$ both converge to the same point $h\in \mathcal{H}\cap \mathcal{K}$, then this BS is said to be biconvergent (in short, BC).

(iii)

A bisequence $\left(c_n,s_n\right)$ on $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is said to be Cauchy bisequence (in short, CBS), if for each $\xi >0$, there exists a number $n_0\in \mathbb{N}$, such that for all positive integers $n,m\ge n_0$, $\mathcal{V}\left(c_n,s_m\right)<\xi$.

Definition 5

([15]). A BMS is called complete if every CBS in this space is convergent.

Theorem 1

([15]). Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a complete BPSMS and a contraction $\mathcal{J}:\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)\rightrightarrows \left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$. Then, it has a unique FP.

Definition 6

([11]). Let $\left(\mathcal{H},\mathcal{V}\right)$ be an MS and $\varnothing \ne \mathcal{K}$,$\mathcal{R}\subset \mathcal{H}$. A mapping $\mathcal{J}:\mathcal{K}\to \mathcal{R}$ is said to be a proximal contraction if there exists a real number $\kappa \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{K},\mathcal{R}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{K},\mathcal{R}\right);\hfill \end{array}$$

this gives us

$$\mathcal{V}\left(c_1,c_2\right)\le \kappa \mathcal{V}\left(p_1,p_2\right)$$

for all $c_1,c_2,p_1,p_2\in \mathcal{K}$.

Definition 7

([12]). Let $\left(\mathcal{H},\mathcal{V}\right)$ be an MS and $\varnothing \ne \mathcal{K}$,$\mathcal{R}\subset \mathcal{H}$. We will consider the following subsets:

$${\mathcal{K}}_0=\left\{c\in \mathcal{K}:\mathcal{V}\left(c,p\right)=\mathcal{V}\left(\mathcal{K},\mathcal{R}\right),\mathit{for}\mathit{some}p\in \mathcal{R}\right\}\subset K,$$

$${\mathcal{R}}_0=\left\{p\in \mathcal{R}:\mathcal{V}\left(c,p\right)=\mathcal{V}\left(\mathcal{K},\mathcal{R}\right),\mathit{for}\mathit{some}c\in \mathcal{K}\right\}\subset R,$$

and

$$\mathcal{V}\left(\mathcal{K},\mathcal{R}\right)=inf\left\{\mathcal{V}\left(c,p\right):c\in \mathcal{K}\wedge p\in \mathcal{R}\right\}.$$

Definition 8

([12]). Let $\left(\mathcal{H},\mathcal{V}\right)$ be an MS and $\varnothing \ne \mathcal{K}$,$\mathcal{R}\subset \mathcal{H}$. We say that $\mathcal{R}$ is acompact with respect to $\mathcal{K}$ if each sequence $\left\{p_n\right\}$ in $\mathcal{R}$ verifying

$$\mathcal{V}\left(c,p_n\right)\to \mathcal{V}\left(c,\mathcal{R}\right)$$

for some $c\in \mathcal{K}$ has a convergent subsequence.

Definition 9

([12]). Let $\left(\mathcal{H},\mathcal{V}\right)$ be an MS and $\varnothing \ne \mathcal{K}$,$\mathcal{R}\subset \mathcal{H}$. An element $c^{*}$ in $\mathcal{K}$ is called a BPP of the mapping $\mathcal{J}:\mathcal{K}\to \mathcal{R}$, if the following equation,

$$\mathcal{V}\left(c^{*},\mathcal{J}{c}^{*}\right)=\mathcal{V}\left(\mathcal{K},\mathcal{R}\right),$$

is satisfied.

3. Main Results

In this section, we prove several BPP results by utilizing generalized interpolative contractions on BMS and provide some nontrivial examples.

Definition 10.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS, and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. We will consider the following subsets:

$${\left(\mathcal{H}\cup \mathcal{K}\right)}_0=\left\{c\in \left(\mathcal{H}\cup \mathcal{K}\right):\mathcal{V}\left(c,p\right)=\mathcal{V}\left(\left(\mathcal{H}\cup \mathcal{K}\right),\left(\mathcal{R}\cup \mathcal{S}\right)\right)\mathit{for}\mathit{some}p\in \left(\mathcal{R}\cup \mathcal{S}\right)\right\},$$

$${\left(\mathcal{R}\cup \mathcal{S}\right)}_0=\left\{p\in \left(\mathcal{R}\cup \mathcal{S}\right):\mathcal{V}\left(c,p\right)=\mathcal{V}\left(\left(\mathcal{H}\cup \mathcal{K}\right),\left(\mathcal{R}\cup \mathcal{S}\right)\right)\mathit{for}\mathit{some}c\in \left(\mathcal{H}\cup \mathcal{K}\right)\right\},$$

and

$$\mathcal{V}\left(c,p\right)=\mathcal{V}\left(\left(\mathcal{H}\cup \mathcal{K}\right),\left(\mathcal{R}\cup \mathcal{S}\right)\right)=inf\left\{\mathcal{V}\left(c,p\right):c\in \left(\mathcal{H}\cup \mathcal{K}\right)\wedge p\in \left(\mathcal{R}\cup \mathcal{S}\right)\right\}.$$

Definition 11.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$, respectively. We say that $\mathcal{R}\cup \mathcal{S}$ is acompact with respect to $\mathcal{H}\cup \mathcal{K}$ if every sequence $\left\{p_n\right\}$ in $\mathcal{R}\cup \mathcal{S}$ satisfying the following

$$\mathcal{V}\left(c,p_n\right)\to \mathcal{V}\left(c,\mathcal{R}\cup \mathcal{S}\right)$$

for some $c\in \mathcal{H}\cup \mathcal{K}$ has a convergent subsequence.

Definition 12.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$, respectively. An element $c^{*}\in$$\mathcal{R}\cap \mathcal{S}$ is called a BPP of the mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ if it satisfies the equation

$$\mathcal{V}\left(c^{*},\mathcal{J}{c}^{*}\right)=\mathcal{V}\left(\left(\mathcal{H}\cup \mathcal{K}\right),\left(\mathcal{R}\cup \mathcal{S}\right)\right).$$

Some generalized definitions are given below.

Definition 13.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$, respectively. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called proximal contraction (in short, PC) on BMS if there exists a real number $\kappa \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(c_1,c_2\right)\le \left(\mathcal{V}\left(p_1,p_2\right)\right)$$

(1)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}.$

Definition 14.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called K-PC on BMS if there exists a real number $\kappa \in [0,\frac{1}{2})$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(c_1,c_2\right)\le \kappa \left(\mathcal{V}\left(p_1,c_1\right)+\mathcal{V}\left(p_2,c_2\right)\right)$$

(2)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

Definition 15.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$, respectively. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called Reich–Rus–Ciric-type IPC on BPS if there exists a real number $\kappa \in [0,\frac{1}{3})$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(c_1,c_2\right)\le \kappa \left(\mathcal{V}\left(p_1,p_2\right)+\mathcal{V}\left(p_1,c_1\right)+\mathcal{V}\left(p_2,c_2\right)\right)$$

(3)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

In this paper, we aim to obtain some BPP results via the interpolative idea. Now, we give some definition via interpolative contraction.

Definition 16.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called Reich–Rus–Ciric-type IPC of the first kind on BMS if there exists a real number $\kappa \in [0,1)$ and $\alpha ,J\in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(c_1,c_2\right)\le \kappa {\left[\mathcal{V}\left(p_1,p_2\right)\right]}^{\alpha }{\left[\mathcal{V}\left(p_1,c_1\right)\right]}^J{\left[\mathcal{V}\left(p_2,c_2\right)\right]}^{1-\alpha -J}$$

(4)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

Definition 17.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called Reich–Rus–Ciric-type IPC of the second kind on bipolar metric space if there exists a real number $\kappa \in [0,1)$ and $\alpha ,J\in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(\mathcal{J}{c}_1,\mathcal{J}{c}_2\right)\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{p}_1,\mathcal{J}{p}_2\right)\right]}^{\alpha }{\left[\mathcal{V}\left(\mathcal{J}{p}_1,\mathcal{J}{c}_1\right)\right]}^J{\left[\mathcal{V}\left(\mathcal{J}{p}_2,\mathcal{J}{c}_2\right)\right]}^{1-\alpha -J}$$

(5)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

Definition 18.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called Reich–Rus–Ciric-type IPC of the first kind on bipolar metric space if there exists a real number $\kappa \in [0,1)$ and $\alpha \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(c_1,c_2\right)\le \kappa {\left[\mathcal{V}\left(p_1,c_1\right)\right]}^{\alpha }{\left[\mathcal{V}\left(p_2,c_2\right)\right]}^{1-\alpha }$$

(6)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

Definition 19.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty subsets of $\mathcal{H}$ and $\mathcal{K}$. The mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ is called Reich–Rus–Ciric-type IPC of the second kind on BMS if there exists a real number $\kappa \in [0,1)$ and $\alpha \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$⟹\mathcal{V}\left(\mathcal{J}{c}_1,\mathcal{J}{c}_2\right)\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{p}_1,\mathcal{J}{c}_1\right)\right]}^{\alpha }{\left[\mathcal{V}\left(\mathcal{J}{p}_2,\mathcal{J}{c}_2\right)\right]}^{1-\alpha }$$

(7)

for all $c_1$, $c_2$, $p_1$, $p_2\in \mathcal{H}\cup \mathcal{K}$.

Note that if we take $\mathcal{H}\cup \mathcal{K}=\mathcal{R}\cup \mathcal{S}$, then the inequalities (4) and (6) become

$$\mathcal{V}\left(\mathcal{J}{c}_1,\mathcal{J}{c}_2\right)\le \kappa {\left[\mathcal{V}\left(c_1,c_2\right)\right]}^{\alpha }{\left[\mathcal{V}\left(c_1,\mathcal{J}{c}_1\right)\right]}^J{\left[\mathcal{V}\left(c_2,\mathcal{J}{c}_2\right)\right]}^{1-\alpha -J}$$

(8)

and

$$\mathcal{V}\left(\mathcal{J}{c}_1,\mathcal{J}{c}_2\right)\le \kappa {\left[\mathcal{V}\left(c_1,\mathcal{J}{c}_1\right)\right]}^{\alpha }{\left[\mathcal{V}\left(c_2,\mathcal{J}{c}_2\right)\right]}^{1-\alpha }$$

(9)

for all $c_1,c_2\in \left\{c\in \mathcal{H}\cup \mathcal{K}:\mathcal{V}\left(c,\mathcal{J}c\right)>0\right\}$, respectively. The mapping $\mathcal{J}$ satisfying (8) (respectively, (9)) is called Reich–Rus–Ciric (respectively, Kannan)-type IPC on BMS in the literature.

Similarly, when $\mathcal{H}\cup \mathcal{K}=\mathcal{R}\cup \mathcal{S}$, the inequalities (5) and (7) become

$$\mathcal{V}\left({\mathcal{J}}^2{c}_1,{\mathcal{J}}^2{c}_2\right)\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{p}_1,\mathcal{J}{p}_2\right)\right]}^{\alpha }{\left[\mathcal{V}\left(\mathcal{J}{p}_1,{\mathcal{J}}^2{c}_1\right)\right]}^J{\left[\mathcal{V}\left(\mathcal{J}{p}_2,{\mathcal{J}}^2{c}_2\right)\right]}^{1-\alpha -J}$$

(10)

and

$$\mathcal{V}\left({\mathcal{J}}^2{c}_1,{\mathcal{J}}^2{c}_2\right)\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{p}_1,{\mathcal{J}}^2{c}_1\right)\right]}^{\alpha }{\left[\mathcal{V}\left(\mathcal{J}{p}_2,\mathcal{J}{c}_2\right)\right]}^{1-\alpha }$$

for all $c_1,c_2\in \left\{c\in \mathcal{H}\cup \mathcal{K}:\mathcal{V}\left(\mathcal{J}c,{\mathcal{J}}^{2}c\right)>0\right\}$, respectively.

Now, we present our main results.

Theorem 2.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS, $\mathcal{R}$,$\mathcal{S}\ne \varnothing$, $\mathcal{H}\subset \mathcal{R}$, and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{R}\cup \mathcal{S}$ is acompact with respect to $\mathcal{H}\cup \mathcal{K}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a proximal contraction such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ is nonempty and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then $\mathcal{J}$ has a BPP.

Proof. 

Suppose that $c_0\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$. Since $\mathcal{J}{\left(c\right)}_0\in \mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$, then there exists $c_1\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_1,\mathcal{J}{c}_0\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

Similarly, since $\mathcal{J}\left(c_1\right)\in \mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)}_0$, there exists $c_2\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_2,\mathcal{J}{c}_1\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

Carrying on this process, we can produce a sequence $\left\{c_n\right\}$ in ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\forall n\in \mathbb{N}.$$

(11)

Thus, if there exists some $n\in \mathbb{N}$ such that $c_n=c_{n+1}$, then from (11), the point $c_n$ is a BPP of the mapping $\mathcal{J}$. On the other hand, if $c_n\ne c_{n+1}$ for all $n\in \mathbb{N}$ then

$$\mathcal{V}\left(c_n,\mathcal{J}{c}_{n-1}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

and

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\forall n\ge 1$$

then by using (1), we have

$$\mathcal{V}\left(c_{n+1},c_n\right)\le \kappa \left[\mathcal{V}\left(c_{n-1},c_n\right)\right]$$

(12)

for all $n\ge 1$. Therefore, the sequence $\left\{\mathcal{V}\left(c_{n+1},c_n\right)\right\}$ is decreasing BS of positive real numbers. Hence, it converges to some element $\gamma \ge 0$ such that ${lim}_{n\to \infty }\mathcal{V}\left(c_{n+1},c_n\right)=\gamma$. Now from (12), we have

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_{n+1},c_n\right)& \le & \kappa .\mathcal{V}\left(c_{n-1},c_n\right)\hfill \\ & \le & {\kappa }^2.\mathcal{V}\left(c_{n-2},c_{n-1}\right)\hfill \\ & & ⋮\hfill \\ & \le & {\kappa }^n.\mathcal{V}\left(c_0,c_1\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$. Supposing $n\to \infty$ in the last inequality, we have $\gamma =0$. Now, for $p,V\in \mathbb{N}$ with $p>V$, say $M=\mathcal{V}\left(c_0,c_0\right)+\mathcal{V}\left(c_0,c_1\right)$ and $k_n:=\frac{{\kappa }^n}{1-\kappa }M$, we obtain

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_{n+p},c_V\right)& \le & \mathcal{V}\left(c_{n+p},c_{V+1}\right)+\mathcal{V}\left(c_n,c_{V+1}\right)+\mathcal{V}\left(c_n,c_V\right)\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+1}\right)+{\kappa }^{n}M\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+2}\right)+\mathcal{V}\left(c_{n+1},c_{V+2}\right)+\mathcal{V}\left(c_{n+1},c_{V+1}\right)+{\kappa }^{n}M\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+2}\right)+\left({\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & & \cdots \hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{n+V}\right)+\left({\kappa }^{n+p-1}+{\kappa }^{n+p-2}+\cdots +{\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & \le & \left({\kappa }^{n+p}+{\kappa }^{n+p-1}+{\kappa }^{n+p-2}+\cdots +{\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & \le & \frac{{\kappa }^n}{1-\kappa }M=k_n\hfill \end{array}$$

and similarly $\mathcal{V}\left(c_p,c_{n+V}\right)\le k_n$.

Let $\xi >0$. Since $\kappa \in \left(0,1\right)$, there exists an $n_0\in \mathbb{N}$ such that $k_{n_0}=\frac{{\kappa }^{n}M}{1-\kappa }<\frac{\xi }{3}$. Then $\mathcal{V}\left(c_p,c_V\right)\le \mathcal{V}\left(c_p,c_{p_0}\right)+\mathcal{V}\left(c_{p_0},c_{V_0}\right)+\mathcal{V}\left(c_{p_0},c_V\right)\le 3k_{n_0}<\xi$ and $\left(c_p,c_V\right)$ is a CBS.

Since $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is complete., $\left(c_p,c_V\right)$ converges, thus, BC to a point $c^{*}\in \left(\mathcal{H}\cap \mathcal{K}\right)$ such that $c_n\to c^{*}$. Moreover, from (11), it can be noted that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)& \le & \mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\hfill \\ & \le & \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c_n,\mathcal{J}{c}_n\right)+\mathcal{V}\left(c_n,c_{n+1}\right)\hfill \\ & =& \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)+\mathcal{V}\left(c^{*},c_{n+1}\right)\hfill \\ & \le & 2\mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right).\hfill \end{array}$$

Therefore, $\mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\to \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)$ as $n\to \infty$. Since $\mathcal{R}\cup \mathcal{S}$ is acompact with respect to $\mathcal{H}\cup \mathcal{K}$, there exists a subsequence $\left\{\mathcal{J}{c}_{n_k}\right\}$ of $\left\{\mathcal{J}{c}_n\right\}$ such that $\mathcal{J}{c}_{n_k}\to p^{*}\in \mathcal{R}\cap \mathcal{S}$ as $k\to \infty$. Therefore, by taking $k\to \infty$ in $\mathcal{V}\left(c_{n_k+1},\mathcal{J}{c}_{n_k}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, we have $\mathcal{V}\left(c^{*},p^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, and so $c^{*}\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$. Also, since $\mathcal{J}{c}^{*}\in \mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right){\left(\mathcal{R}\cup \mathcal{S}\right)}_0$, there esits $\xi \in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

(13)

Suppose that $c^{*}\ne c_n$ for all $n\in \mathbb{N}$. Otherwise, there exists a subsequence $\left\{c_{m_k}\right\}$ of $\left\{c_n\right\}$ such that $c^{*}\ne c_{m_k}$ for all $k\in \mathbb{N}$ and so we can observe this subsequence in the following steps.

From (11), (13), and the inequality (1), we obtain that

$$\mathcal{V}\left(c_{n+1},\xi \right)\le \kappa \left[\mathcal{V}\left(c_n,c^{*}\right)\right]$$

for all $n\in \mathbb{N}$. Thus, letting $n\to \infty$, we have

$$\mathcal{V}\left(c^{*},\xi \right)=0$$

That is, $\xi =c^{*}$. From (13), the point $c^{*}$ is a BPP of the mapping $\mathcal{J}$. □

Theorem 3.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{R}$,$\mathcal{S}\ne \varnothing$, $\mathcal{H}\subset \mathcal{R}$ and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{R}\cup \mathcal{S}$ is acompact with respect to $\mathcal{H}\cup \mathcal{K}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a Reich–Rus–Ciric-type IPC of the first kind such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0\ne \varnothing$ and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then $\mathcal{J}$ has a BPP.

Proof. 

Suppose that $c_0\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$. Since $\mathcal{J}{\left(c\right)}_0\in \mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$, then there exists $c_1\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_1,\mathcal{J}{c}_0\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

Similarly, since $\mathcal{J}\left(c_1\right)\in \mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)}_0$, there exists $c_2\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_2,\mathcal{J}{c}_1\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

Carrying on this process, we can produce a sequence $\left\{c_n\right\}$ in ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\forall n\in \mathbb{N}.$$

(14)

Now, if there exists $n\in \mathbb{N}$ such that $c_n=c_{n+1}$, then from (14), the point $c_n$ is a BPP of the mapping $\mathcal{J}$. Hence, we suppose that $c_n\ne c_{n+1}$ for all $n\in \mathbb{N}$. Since

$$\mathcal{V}\left(c_n,\mathcal{J}{c}_{n-1}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

and

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

for all $n\ge 1$, then by using (4) we have

$$\mathcal{V}\left(c_{n+1},c_n\right)\le \kappa {\left[\mathcal{V}\left(c_{n-1},c_n\right)\right]}^{\alpha }{\left[\mathcal{V}\left(c_{n-1},c_n\right)\right]}^J{\left[\mathcal{V}\left(c_n,c_{n+1}\right)\right]}^{1-\alpha -J}$$

which produces that

$${\left[\mathcal{V}\left(c_{n+1},c_n\right)\right]}^{\alpha +J}\le \kappa {\left[\mathcal{V}\left(c_{n-1},c_n\right)\right]}^{\alpha +J}$$

(15)

for all $n\ge 1$. Therefore, the sequence $\left\{\mathcal{V}\left(c_{n+1},c_n\right)\right\}$ is decreasing BS of positive real numbers. Thus, there exists $\gamma \ge 0$ such that ${lim}_{n\to \infty }\mathcal{V}\left(c_{n+1},c_n\right)=\gamma$. Now, from (15) we have

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_{n+1},c_n\right)& \le & {\kappa }^{\frac{1}{\alpha +J}}.\mathcal{V}\left(c_{n-1},c_n\right)\hfill \\ & \le & {\kappa }^{\frac{2}{\alpha +J}}.\mathcal{V}\left(c_{n-2},c_{n-1}\right)\hfill \\ & & ⋮\hfill \\ & \le & {\kappa }^{\frac{n}{\alpha +J}}.\mathcal{V}\left(c_0,c_1\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$. Supposing $n\to \infty$ in the last inequality, we have $\gamma =0$. Now, for $p,V\in \mathbb{N}$ with $p>V$, say $M=\mathcal{V}\left(c_0,c_0\right)+\mathcal{V}\left(c_0,c_1\right)$ and $k_n:=\frac{{\kappa }^n}{1-\kappa }M$, we obtain

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_{n+p},c_V\right)& \le & \mathcal{V}\left(c_{n+p},c_{V+1}\right)+\mathcal{V}\left(c_n,c_{V+1}\right)+\mathcal{V}\left(c_n,c_V\right)\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+1}\right)+{\kappa }^{n}M\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+2}\right)+\mathcal{V}\left(c_{n+1},c_{V+2}\right)+\mathcal{V}\left(c_{n+1},c_{V+1}\right)+{\kappa }^{n}M\hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{V+2}\right)+\left({\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & & \cdots \hfill \\ & \le & \mathcal{V}\left(c_{n+p},c_{n+V}\right)+\left({\kappa }^{n+p-1}+{\kappa }^{n+p-2}+\cdots +{\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & \le & \left({\kappa }^{n+p}+{\kappa }^{n+p-1}+{\kappa }^{n+p-2}+\cdots +{\kappa }^{n+1}+{\kappa }^n\right)M\hfill \\ & \le & \frac{{\kappa }^n}{1-\kappa }M=k_n\hfill \end{array}$$

and, similarly, $\mathcal{V}\left(c_p,c_{n+V}\right)\le k_n$.

Let $\xi >0$. Since $\kappa \in \left(0,1\right)$, there exists an $n_0\in \mathbb{N}$ such that $k_{n_0}=\frac{{\kappa }^{n}M}{1-\kappa }<\frac{\xi }{3}$. Then $\mathcal{V}\left(c_p,c_V\right)\le \mathcal{V}\left(c_p,c_{p_0}\right)+\mathcal{V}\left(c_{p_0},c_{V_0}\right)+\mathcal{V}\left(c_{p_0},c_V\right)\le 3k_{n_0}<\xi$ and $\left(c_p,c_V\right)$ is a CBS.

Since $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is complete, $\left(c_p,c_V\right)$ converges, thus, BC, to a point $c^{*}\in \left(\mathcal{H}\cap \mathcal{K}\right)$ such that $c_n\to c^{*}$. Moreover, from (14), it can be noted that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)& \le & \mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\hfill \\ & \le & \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c_n,\mathcal{J}{c}_n\right)+\mathcal{V}\left(c_n,c_{n+1}\right)\hfill \\ & =& \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)+\mathcal{V}\left(c^{*},c_{n+1}\right)\hfill \\ & \le & 2\mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right).\hfill \end{array}$$

Therefore, $\mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\to \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)$ as $n\to \infty$. Since $\mathcal{R}\cup \mathcal{S}$ is acompact with respect to $\mathcal{H}\cup \mathcal{K}$, there exists a subsequence $\left\{\mathcal{J}{c}_{n_k}\right\}$ of $\left\{\mathcal{J}{c}_n\right\}$ such that $\mathcal{J}{c}_{n_k}\to p^{*}\in \mathcal{R}\cap \mathcal{S}$ as $k\to \infty$. Therefore, by taking $k\to \infty$ in $\mathcal{V}\left(c_{n_k+1},\mathcal{J}{c}_{n_k}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, we have $\mathcal{V}\left(c^{*},p^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, and so $c^{*}\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$. Also, since $\mathcal{J}{c}^{*}\in \mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right){\left(\mathcal{R}\cup \mathcal{S}\right)}_0$, there exists $\xi \in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

(16)

Suppose that $c^{*}\ne c_n$ for all $n\in \mathbb{N}$. Otherwise, there exists a subsequence $\left\{c_{m_k}\right\}$ of $\left\{c_n\right\}$ such that $c^{*}\ne c_{m_k}$ for all $k\in \mathbb{N}$ and so we can consider this subsequence in the following steps.

From (14), (16), and the inequality (4), we obtain that

$$\mathcal{V}\left(c_{n+1},\xi \right)\le \kappa {\left[\mathcal{V}\left(c_n,c^{*}\right)\right]}^{\alpha }{\left[\mathcal{V}\left(c_n,c_{n+1}\right)\right]}^J{\left[\mathcal{V}\left(c^{*},\xi \right)\right]}^{1-\alpha -J}$$

for all $n\in \mathbb{N}$. Thus, letting $n\to \infty$, we have

$$\mathcal{V}\left(c^{*},\xi \right)=0$$

That is, $\xi =c^{*}$. From (16), the point $c^{*}$ is a BPP of the mapping $\mathcal{J}$. □

Theorem 4.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{R}$,$\mathcal{S}\ne \varnothing$, $\mathcal{H}\subset \mathcal{R}$, and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{R}\cup \mathcal{S}$ is compact with respect to $\mathcal{H}\cup \mathcal{K}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a Kannan-type IPC of the first kind such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ is nonempty and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0\subset {\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then, $\mathcal{J}$ has a best proximity point.

Proof. 

Chasing the steps in the proof of Theorems 2 and 3, we achieve the objective. □

If we take $\mathcal{H}\cup \mathcal{K}=\mathcal{R}\cup \mathcal{S}$ in Theorems 3 and 4, we obtain the following FP results:

Corollary 1.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{H}\cup \mathcal{K}$ be a Reich–Rus–Ciric-type IC. Then, $\mathcal{J}$ has a unique FP.

Proof. 

It is easy to show on the lines of the Theorem in [14]. □

Corollary 2.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{H}\cup \mathcal{K}$ be an Kannan-type IC. Then, $\mathcal{J}$ has a unique FP.

Proof. 

It is immediate from Theorem in [6]. □

Theorem 5.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{R}$ and $\mathcal{S}$ be nonempty, $\mathcal{H}\subset \mathcal{R}$, and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{H}\cup \mathcal{K}$ is approximately compact with respect to $\mathcal{R}\cup \mathcal{S}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a Reich–Rus–Ciric-type IPC of the second kind such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ is nonempty and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then, $\mathcal{J}$ has a BPP.

Proof. 

Proceeding as in Theorem 3, it is possible to find a sequence $\left\{c_n\right\}$ in ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

(17)

for all $n\in \mathbb{N}$. Now if there exists $n\in \mathbb{N}$ such that $c_n=c_{n+1}$, then from (14), the point $c_n$ is a BPP of the mapping $\mathcal{J}$. Hence, we suppose that $c_n\ne c_{n+1}$ for all $n\in \mathbb{N}$. Since

$$\mathcal{V}\left(c_n,\mathcal{J}{c}_{n-1}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

and

$$\mathcal{V}\left(c_{n+1},\mathcal{J}{c}_n\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$$

for all $n\ge 1$, then by using (5) we have

$$\mathcal{V}\left(\mathcal{J}{c}_{n+1},\mathcal{J}{c}_n\right)\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{c}_{n-1},\mathcal{J}{c}_n\right)\right]}^{\alpha }{\left[\mathcal{V}\left(\mathcal{J}{c}_{n-1},\mathcal{J}{c}_n\right)\right]}^J{\left[\mathcal{V}\left(\mathcal{J}{c}_n,\mathcal{J}{c}_{n+1}\right)\right]}^{1-\alpha -J}$$

which produces that

$${\left[\mathcal{V}\left(\mathcal{J}{c}_{n+1},\mathcal{J}{c}_n\right)\right]}^{\alpha +J}\le \kappa {\left[\mathcal{V}\left(\mathcal{J}{c}_{n-1},\mathcal{J}{c}_n\right)\right]}^{\alpha +J}$$

for all $n\ge 1$. Eventually, $\left\{\mathcal{J}{c}_n\right\}$ is a CBS in $\mathcal{R}\cup \mathcal{S}$. Since $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is complete, $\left(c_p,c_V\right)$ converges, thus, BC, to a point $p^{*}\in \left(\mathcal{H}\cap \mathcal{K}\right)$ such that $c_n\to p^{*}$. Moreover, from (17), it can be noted that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(p^{*},\mathcal{R}\cup \mathcal{S}\right)& \le & \mathcal{V}\left(p^{*},c_{n+1}\right)\hfill \\ & \le & \mathcal{V}\left(p^{*},\mathcal{J}{c}_n\right)+\mathcal{V}\left(c_n,c_{n+1}\right)+\mathcal{V}\left(c_n,\mathcal{J}{c}_n\right)\hfill \\ & =& \mathcal{V}\left(p^{*},\mathcal{J}{c}_n\right)+\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)+\mathcal{V}\left(p^{*},\mathcal{J}{c}_n\right)\hfill \\ & \le & 2\mathcal{V}\left(p^{*},\mathcal{J}{c}_n\right)+\mathcal{V}\left(p^{*},\mathcal{H}\cup \mathcal{K}\right).\hfill \end{array}$$

Therefore, $\mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\to \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)$ as $n\to \infty$. Since $\mathcal{H}\cup \mathcal{K}$ is Acompact with respect to $\mathcal{R}\cup \mathcal{S}$, there exists a subsequence $\left\{c_{n_k}\right\}$ of $\left\{c_n\right\}$ such that $c_{n_k}\to c^{*}\in \mathcal{H}\cap \mathcal{K}$ as $k\to \infty$. Therefore, by taking into account the continuity of $\mathcal{J}$, we have from (17)

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)=\underset{k\to \infty }{lim}\mathcal{V}\left(c_{n_k+1},\mathcal{J}{c}_{n_k}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

Thus, the point $c^{*}$ is a BPP of the mapping $\mathcal{J}$ □

Using the similar technique of Theorem 5, we can obtain the following theorem:

Theorem 6.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS and $\mathcal{R}$,$\mathcal{S}\ne \varnothing$, and $\mathcal{H}\subset \mathcal{R}$ and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{H}\cup \mathcal{K}$ is acompact with respect to $\mathcal{R}\cup \mathcal{S}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a Kannan-type IPC of the second kind such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ is nonempty and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then, $\mathcal{J}$ has a BPP.

Now, we present some illustrative examples.

Example 1.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS. Define the BMS by $\mathcal{V}\left(c_1,c_2\right)=\mid c_1-c_2\mid$. Let $\mathcal{H}=\left[0,8\right]$, $\mathcal{K}=\left[0,10\right]$ and $\mathcal{R}=\left[0,12\right],\mathcal{S}=\left[0,15\right]$, where $\mathcal{H}\subset \mathcal{R}$ and $\mathcal{K}\subset \mathcal{S}$. Define the mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ by $\mathcal{J}\left(c\right)=\frac{c}{3}$. Thus, $\mathcal{H}\cup \mathcal{K}\subset \mathcal{R}\cup \mathcal{S}$ and $\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)=0$, ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0\subset \mathcal{H}\cup \mathcal{K}$ and ${\left(\mathcal{R}\cup \mathcal{S}\right)}_0\subset \mathcal{R}\cup \mathcal{S}$. Then, clearly, $\mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right)\subset {\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. This shows that $\mathcal{J}$ is a proximal contraction. Also, we can easily see that the other conditions of Theorem 2 hold. Then, $\mathcal{J}$ has a BPP which is 0.

Example 2.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS. Define the bipolar metric space by $\mathcal{V}\left(c_1,c_2\right)=\mid c_1-c_2\mid$. Let $\mathcal{H}=\left[0,8\right]$, $\mathcal{K}=\left[0,10\right]$ and $\mathcal{R}=\left[0,12\right],\mathcal{S}=\left[0,15\right]$, where $\mathcal{H}\subset \mathcal{R}$ and $\mathcal{K}\subset \mathcal{S}$. Define the mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ by $\mathcal{J}\left(c\right)=c$. Thus, $\mathcal{H}\cup \mathcal{K}\subset \mathcal{R}\cup \mathcal{S}$ and $\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)=0$, ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0\subset \mathcal{H}\cup \mathcal{K}$ and ${\left(\mathcal{R}\cup \mathcal{S}\right)}_0\subset \mathcal{R}\cup \mathcal{S}$. Then, clearly, $\mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right){\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. This shows that $\mathcal{J}$ is a Kannan-type IPC of the first kind. Also, we can easily see that the other conditions of Theorem 4 hold. Then, $\mathcal{J}$ has a best proximity point which is 0. On the other hand, consider $c_1=2$, $p_1=4$, $c_2=3$, $p_2=9\in \mathcal{H}\cup \mathcal{K}$ and $\kappa =0.1$.

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& 0=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right),\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& 0=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,c_2\right)& \le & \kappa \left(\mathcal{V}\left(p_1,p_2\right)\right)\hfill \\ \hfill 1& \le & \left(0.1\right)\left(5\right)\hfill \\ \hfill 1& \le & 0.5,\hfill \end{array}$$

which is a contradiction. Hence, $\mathcal{J}$ is not a proximal contraction. Also, for K-proximal contraction,

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,c_2\right)& \le & \kappa \left(\mathcal{V}\left(p_1,c_1\right)+\mathcal{V}\left(p_2,c_2\right)\right)\hfill \\ \hfill 1& \le & \left(0.1\right)\left(2+6\right)\hfill \\ \hfill 1& \le & 0.8,\hfill \end{array}$$

which is a contradiction. That is, $\mathcal{J}$ is not a K-proximal contraction of the first kind.

Example 3.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a BMS. Define the bipolar metric space by $\mathcal{V}\left(c_1,c_2\right)=\mid c_1-c_2\mid$. Let $\mathcal{H}=\left[0,8\right]$, $\mathcal{K}=\left[0,10\right]$ and $\mathcal{R}=\left[0,12\right],\mathcal{S}=\left[0,15\right]$ where, $\mathcal{H}\subset \mathcal{R}$ and $\mathcal{K}\subset \mathcal{S}$. Define the mapping $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ by $\mathcal{J}\left(c\right)=\frac{c}{2}$. Thus, $\mathcal{H}\cup \mathcal{K}\subset \mathcal{R}\cup \mathcal{S}$ and $\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)=0$, ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0\subset \mathcal{H}\cup \mathcal{K}$ and ${\left(\mathcal{R}\cup \mathcal{S}\right)}_0\subset \mathcal{R}\cup \mathcal{S}$. Then, clearly, $\mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right)\subset {\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. This shows that $\mathcal{J}$ is a Reich–Rus–Ciric-type IPC of the first kind. Also, we can easily see that the other conditions of Theorem 3 hold. Then, $\mathcal{J}$ has a BPP which is 0. On the other hand, consider $c_1=2$, $p_1=4$, $c_2=3$, $p_2=6\in \mathcal{H}\cup \mathcal{K}$ and $\kappa =0.1$.

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& 0=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right),\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& 0=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,c_2\right)& \le & \kappa \left(\mathcal{V}\left(p_1,p_2\right)+\mathcal{V}\left(p_1,c_1\right)+\mathcal{V}\left(p_2,c_2\right)\right)\hfill \\ \hfill 1& \le & \left(0.1\right)\left(4+2+3\right)\hfill \\ \hfill 1& \le & 0.9,\hfill \end{array}$$

which is a contradiction. Hence, $\mathcal{J}$ is not a Reich–Rus–Ciric-type IPC of the first kind.

4. Application

Now, we examine the existence and unique solution to an integral equation as an application for proximal contraction.

Theorem 7.

Let us consider an integral equation

$$\mathcal{U}\left(l\right)=h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(c\right)\right)dc,l\in \mathcal{H}\cup \mathcal{K},$$

where $\mathcal{H}\cup \mathcal{K}$ is a Lebesgue-measurable set. Suppose:

(1) 

There is a continuous function $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ and $\kappa \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$\mid \mathcal{V}\left(c_1,c_2\right)\mid \le \kappa \mid \mathcal{V}\left(p_1,p_2\right)\mid$$

for $c_1,c_2,p_1,p_2\in \mathcal{H}\cup \mathcal{K}$;

(2) 

$\Vert \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c\right)dc\Vert \le 1$, i.e., ${sup}_{l\in \mathcal{H}\cup \mathcal{K}}\mid \mathcal{V}\left(l,c\right)\mid dc\le 1$.

Then, the integral equation has a unique solution in $L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)$.

Proof. 

Let $c\in \left(L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)\right)$ be a normed linear space, where $\mathcal{H},\mathcal{K}$ are Lebesgue-measurable sets and $m\left(\mathcal{H}\cup \mathcal{K}\right)<\infty$.

Consider $V:\mathcal{H}\times \mathcal{K}\to [0,\infty )$ to be defined by $\mathcal{V}\left(c_1,c_2\right)={\Vert c_1-c_2\Vert }_{\infty }$ for all $\left(c_1,c_2\right)\in \mathcal{H}\times \mathcal{K}$. Then, $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is a CBMS.

Define the mapping $\mathcal{J}:L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)\to L^{\infty }\left(\mathcal{R}\right)\cup L^{\infty }\left(\mathcal{S}\right)$ by

$$\mathcal{J}\left(\mathcal{U}\left(l\right)\right)=h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p\right)\right)dc,l\in \mathcal{H}\cup \mathcal{K}$$

Now, we have

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,c_2\right)& =& \Vert \mathcal{J}\left(\mathcal{U}\left(c_1\right)\right)-\mathcal{J}\left(\mathcal{U}\left(c_2\right)\right)\Vert \hfill \\ & =& \mid h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p_1\right)\right)dc-\left(h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p_2\right)\right)dc\right)\mid \hfill \\ & \le & \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c,\mathcal{U}\left(p_1\right)\right)-\mathcal{V}\left(l,c,\mathcal{U}\left(p_2\right)\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \underset{l\in \mathcal{H}\cup \mathcal{K}}{sup}\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \hfill \\ & =& \kappa \mathcal{V}\left(p_1,p_2\right).\hfill \end{array}$$

Hence, all conditions of Theorem 2 hold. That is, the integral equation has a unique solution. □

Theorem 8.

Let $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ be a CBMS, $\mathcal{R}$ and $\mathcal{S}$ be nonempty, $\mathcal{H}\subset \mathcal{R}$, and $\mathcal{K}\subset \mathcal{S}$ such that $\mathcal{R}\cup \mathcal{S}$ is approximately compact with respect to $\mathcal{H}\cup \mathcal{K}$. Let $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ be a proximal contraction such that ${\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ is nonempty and $\mathcal{J}{\left(\mathcal{H}\cup \mathcal{K}\right)}_0{\left(\mathcal{R}\cup \mathcal{S}\right)}_0$. Then, $\mathcal{J}$ has a best proximity point.

Proof. 

Proceeding Theorem 2

$$\mathcal{V}\left(c_{n+1},c_n\right)\le \kappa \left[\mathcal{V}\left(c_{n-1},c_n\right)\right]$$

(18)

for all $n\ge 1$. Therefore, the sequence $\left\{\mathcal{V}\left(c_{n+1},c_n\right)\right\}$ is decreasing BS of positive real numbers. Thus, there exists $\gamma \ge 0$ such that ${lim}_{n\to \infty }\mathcal{V}\left(c_{n+1},c_n\right)=\gamma$. Assume that $\gamma \ne 0$. Let $L_{n+1}=\mathcal{V}\left(c_{n+1},c_n\right)$ and $L_n=\mathcal{V}\left(c_{n-1},c_n\right)$, then ${lim}_{n\to +\infty }{L}_n={lim}_{n\to +\infty }{L}_n=\gamma >0$ and $L_{n+1}<L_n$, for all $n\in \mathbb{N}$. Therefore,

$$0\le \underset{n\to +\infty }{lim}sup\xi \left(L_{n+1},L_n\right)<0,$$

which is a contradiction. Thus,

$$\underset{n\to +\infty }{lim}\mathcal{V}\left(c_{n+1},c_n\right)=0.$$

Now, we show that $\left(\left\{c_{n+1}\right\},\left\{c_n\right\}\right)$ is a CBS. On the contrary, assume that $\left(\left\{c_{n+1}\right\},\left\{c_n\right\}\right)$ is not a CBS. Then, there exists an $\xi >0$ for which we can find a subsequence $\left\{\mathcal{J}{c}_{n_k}\right\}$ of $\left\{\mathcal{J}{c}_n\right\}$ such that $c_{n_k+1}>c_{n_k}>c$

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)\ge \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

(19)

Suppose that $c_{n_k+1}$ is the least integer exceeding $c_{n_k}$ satisfying the inequality (19). Then,

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)<\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

(20)

Suppose that $c^{*}\ne c_n$ for all $n\in \mathbb{N}$. Otherwise, there exists a subsequence $\left\{c_{m_k}\right\}$ of $\left\{c_n\right\}$ such that $c^{*}\ne c_{m_k}$ for all $k\in \mathbb{N}$ and so we can consider this subsequence in the following steps.

From (19), (20) and the inequality (1), we obtain that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)& \le & \mathcal{V}\left(c_n,c^{*}\right)\hfill \\ & \le & \mathcal{V}\left(c_n,c_{m_k}\right)+\mathcal{V}\left(c_{m_k},c_n\right)+\mathcal{V}\left(c_n,c^{*}\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$. Thus, letting $n\to \infty$, we have

$$\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)<0.$$

Therefore, $\left(\left\{c_{n+1}\right\},\left\{c_n\right\}\right)$ is a CBS. Since $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is complete, $\left(c_p,c_V\right)$ converges, thus, BC, to a point $c^{*}\in \left(\mathcal{H}\cap \mathcal{K}\right)$ such that $c_n\to c^{*}$. Moreover, from (18), it can be noted that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)& \le & \mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\hfill \\ & \le & \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c_n,\mathcal{J}{c}_n\right)+\mathcal{V}\left(c_n,c_{n+1}\right)\hfill \\ & =& \mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)+\mathcal{V}\left(c^{*},c_{n+1}\right)\hfill \\ & \le & 2\mathcal{V}\left(c^{*},c_{n+1}\right)+\mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right).\hfill \end{array}$$

Therefore, $\mathcal{V}\left(c^{*},\mathcal{J}{c}_n\right)\to \mathcal{V}\left(c^{*},\mathcal{R}\cup \mathcal{S}\right)$ as $n\to \infty$. Since $\mathcal{R}\cup \mathcal{S}$ is a compact with respect to $\mathcal{H}\cup \mathcal{K}$, there exists a subsequence $\left\{\mathcal{J}{c}_{n_k}\right\}$ of $\left\{\mathcal{J}{c}_n\right\}$ such that $\mathcal{J}{c}_{n_k}\to p^{*}\in \mathcal{R}\cap \mathcal{S}$ as $k\to \infty$. Therefore, by taking $k\to \infty$ in $\mathcal{V}\left(c_{n_k+1},\mathcal{J}{c}_{n_k}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, we have $\mathcal{V}\left(c^{*},p^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)$, and so $c^{*}\in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$. Also, since $\mathcal{J}{c}^{*}\in \mathcal{J}\left({\left(\mathcal{H}\cup \mathcal{K}\right)}_0\right){\left(\mathcal{R}\cup \mathcal{S}\right)}_0$, there exists $\xi \in {\left(\mathcal{H}\cup \mathcal{K}\right)}_0$ such that

$$\mathcal{V}\left(\xi ,\mathcal{J}{c}^{*}\right)=\mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right).$$

(21)

Suppose that $c^{*}\ne c_n$ for all $n\in \mathbb{N}$. Otherwise, there exists a subsequence $\left\{c_{m_k}\right\}$ of $\left\{c_n\right\}$ such that $c^{*}\ne c_{m_k}$ for all $k\in \mathbb{N}$ and so we can consider this subsequence in the following steps.

From (18), (21) and the inequality (1), we obtain that

$$\mathcal{V}\left(c_{n+1},\xi \right)\le \kappa \left[\mathcal{V}\left(c_n,c^{*}\right)\right]$$

for all $n\in \mathbb{N}$. Thus, letting $n\to \infty$, we have

$$\mathcal{V}\left(c^{*},\xi \right)=0$$

That is, $\xi =c^{*}$. From (21), the point $c^{*}$ is a BPP of the mapping $\mathcal{J}$. That is, $\xi =c^{*}$. From (21), the point $c^{*}$ is a BPP of the mapping $\mathcal{J}$. □

Theorem 9.

Let us consider an integral equation,

$$\mathcal{U}\left(l\right)=h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(c\right)\right)dc,l\in \mathcal{H}\cup \mathcal{K},$$

where $\mathcal{H}\cup \mathcal{K}$ is a Lebesgue-measurable set. Suppose:

(1) 

There is a continuous function $\mathcal{J}:\mathcal{H}\cup \mathcal{K}\to \mathcal{R}\cup \mathcal{S}$ and $\kappa \in \left(0,1\right)$ such that

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,\mathcal{J}{p}_1\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \\ \hfill \mathcal{V}\left(c_2,\mathcal{J}{p}_2\right)& =& \mathcal{V}\left(\mathcal{H}\cup \mathcal{K},\mathcal{R}\cup \mathcal{S}\right)\hfill \end{array}$$

$$\mid \mathcal{V}\left(c_1,c_2\right)\mid \le \kappa \mid \mathcal{V}\left(p_1,p_2\right)\mid$$

for $c_1,c_2,p_1,p_2\in \mathcal{H}\cup \mathcal{K}$;

(2) 

$\Vert \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c\right)dc\Vert \le 1$, i.e., ${sup}_{l\in \mathcal{H}\cup \mathcal{K}}\mid \mathcal{V}\left(l,c\right)\mid dc\le 1$.

Then, the integral equation has a unique solution in $L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)$.

Proof. 

Let $c\in \left(L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)\right)$ be a normed linear space, where $\mathcal{H},\mathcal{K}$ are Lebesgue-measurable sets and $m\left(\mathcal{H}\cup \mathcal{K}\right)<\infty$.

Consider $V:\mathcal{H}\times \mathcal{K}\to [0,\infty )$ to be defined by $\mathcal{V}\left(c_1,c_2\right)={\Vert c_1-c_2\Vert }_{\infty }$ for all $\left(c_1,c_2\right)\in \mathcal{H}\times \mathcal{K}$. Then, $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is a CBMS.

Define the mapping $\mathcal{J}:L^{\infty }\left(\mathcal{H}\right)\cup L^{\infty }\left(\mathcal{K}\right)\to L^{\infty }\left(\mathcal{R}\right)\cup L^{\infty }\left(\mathcal{S}\right)$ by

$$\mathcal{J}\left(\mathcal{U}\left(l\right)\right)=h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p\right)\right)dc,l\in \mathcal{H}\cup \mathcal{K}$$

Now, we have

$$\begin{array}{ccc}\hfill \mathcal{V}\left(c_1,c_2\right)& =& \Vert \mathcal{J}\left(\mathcal{U}\left(c_1\right)\right)-\mathcal{J}\left(\mathcal{U}\left(c_2\right)\right)\Vert \hfill \\ & =& \mid h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p_1\right)\right)dc-\left(h\left(l\right)+\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mathcal{V}\left(l,c,\mathcal{U}\left(p_2\right)\right)dc\right)\mid \hfill \\ & \le & \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c,\mathcal{U}\left(p_1\right)\right)-\mathcal{V}\left(l,c,\mathcal{U}\left(p_2\right)\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \underset{l\in \mathcal{H}\cup \mathcal{K}}{sup}\underset{\mathcal{H}\cup \mathcal{K}}{\int }\mid \mathcal{V}\left(l,c\right)\mid dc\hfill \\ & \le & \kappa \Vert \left(\mathcal{U}\left(p_1\right)-\mathcal{U}\left(p_2\right)\right)\Vert \hfill \\ & =& \kappa \mathcal{V}\left(p_1,p_2\right),\hfill \end{array}$$

which implies that

$$\kappa \mathcal{V}\left(p_1,p_2\right)-\mathcal{V}\left(c_1,c_2\right)\ge 0.$$

We consider the simulation function as $\xi \left(c_1,c_2\right)=\kappa c_2-c_1$. Then,

$$\xi \left(\mathcal{V}\left(p_1,p_2\right),\mathcal{V}\left(c_1,c_2\right)\right)\ge 0.$$

Therefore, all conditions of Theorem 8 hold. Hence, the integral equation has a unique solution. □

5. Application to Nonlinear Fractional Differential Equation

In this section, we apply Theorem 2 to examine the existence and uniqueness of a solution of nonlinear fractional differential equation given by

$$D_c^{\alpha }c\left(\partial \right)=\mathcal{J}\left(\partial ,c\left(\partial \right)\right)\partial \in \left(0,1\right),\alpha \in (1,2],$$

with boundary conditions

$$c\left(0\right)=0,c^{{}^{\prime }}\left(0\right)=Ic\left(\partial \right)\partial \in \left(0,1\right).$$

where $D_c^{\alpha }$ means a Caputo-fractional derivative of order $\alpha$, given by

$$D_c^{\alpha }\mathcal{J}\left(\partial \right)={\frac{1}{\mathsf{\Gamma }\left(n-\alpha \right)}}_0^{\partial }{\left(\partial -\varpi \right)}^{n-\alpha -1}{\mathcal{J}}^n\left(\varpi \right)d\varpi \left(n-1<\alpha <n,n=\left[\alpha \right]+1\right),$$

and $\mathcal{J}:\left[0,1\right]\times \mathbb{R}\to {\mathbb{R}}^{+}$ is a continuous function. We assume that $\xi =C\left(\left[0,1\right],\mathbb{R}\right)$ into $\mathbb{R}$ with supremum $\mid c\mid ={sup}_{\partial \in \left[0,1\right]}\mid c\left(\partial \right)\mid$.

The Riemann–Liouville fractional integral of order $\alpha$ is provided by

$$I^{\alpha }\mathcal{J}\left(c\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }\left(\partial -\varpi \right)\mathcal{J}\left(\varpi \right)d\varpi \left(\alpha >0\right)$$

Firstly, we examine the simple form for a nonlinear fractional differential equation before finding the existence of a solution. For this, assume the below fractional differential equation:

$$D_c^{\alpha }c\left(\partial \right)=\mathcal{J}\left(\partial ,c\left(\partial \right)\right)\partial \in \left(0,1\right),\alpha \in (1,2],$$

(22)

where

• $\mathcal{J}:\left[0,1\right]\times \mathbb{R}\to {\mathbb{R}}^{+}$ is a continuous function,
• $c\left(\partial \right):\left[0,1\right]\to \mathbb{R}$ is continuous,

And verifying the below condition

$$\mid \mathcal{J}\left(\partial ,c_1\right)-\mathcal{J}\left(\partial ,c_2\right)\mid \le \pi K\mid c_1-c_2\mid ,$$

for all $\partial \in \left[0,1\right]$ and K is a constant with $\pi K<1$, where

$$\pi =\frac{1}{\mathsf{\Gamma }\left(\alpha +1\right)}+\frac{2c_2^{\alpha +1}\mathsf{\Gamma }\left(\alpha \right)}{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha +1\right)}.$$

Then, Equation (22) has a unique solution.

Proof. 

Let

$$\mathcal{V}\left(c_1,c_2\right)=\mid c_1-c_2{\mid }^p\mathrm{for}\mathrm{all}{c}_1,c_2\in \xi .$$

Let $\mid c_1-c_2\mid ={sup}_{\partial \in \left[0,1\right]}\mid c_1\left(\partial \right)-c_2\left(\partial \right)\mid$ for all $c_1,c_2\in \xi$. □

Then, $\left(\mathcal{H},\mathcal{K},\mathcal{V}\right)$ is a complete BMS. We define a mapping $\xi :\xi \to \xi$ by

$$\begin{array}{ccc}\hfill \xi c\left(\partial \right)& =& \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mathcal{J}\left(\varpi ,c\left(\partial \right)\right)d\varpi \hfill \\ & & +\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}\mathcal{J}\left(m,c\left(m\right)\right)dm\right)d\varpi \hfill \end{array}$$

(23)

for all $\partial \in \left[0,1\right]$. Equation (22) has a unique solution $c\in \xi$ if and only if $c\left(\partial \right)=\xi c\left(\partial \right)$ for all $\partial \in \left[0,1\right]$. Now,

$$\mathcal{V}\left(c_1,c_2\right)=\mid c_1-c_2{\mid }^p$$

(24)

$$\mid \xi c_1\left(\partial \right)-\xi c_2\left(\partial \right)\mid =\mid \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mathcal{J}\left(\varpi ,c_1\left(\partial \right)\right)d\varpi +$$

$$+\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}\mathcal{J}\left(m,c_1\left(m\right)\right)dm\right)d\varpi \mid$$

$$-\mid \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mathcal{J}\left(\varpi ,c_2\left(\partial \right)\right)d\varpi$$

$$+\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}\mathcal{J}\left(m,c_2\left(m\right)\right)dm\right)d\varpi \mid .$$

That is,

$$\mid \xi c_1\left(\partial \right)-\xi c_2\left(\partial \right)\mid =\mid \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mathcal{J}\left(\varpi ,c_1\left(\partial \right)\right)d\varpi +$$

$$+\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}\mathcal{J}\left(m,c_1\left(m\right)\right)dm\right)d\varpi$$

$$-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mathcal{J}\left(\varpi ,c_2\left(\partial \right)\right)d\varpi$$

$$-\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}\mathcal{J}\left(m,c_2\left(m\right)\right)dm\right)d\varpi \mid .$$

$$\begin{array}{ccc}& \le & \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}\mid \mathcal{J}\left(\varpi ,c_1\left(\partial \right)\right)-\mathcal{J}\left(\varpi ,c_2\left(\partial \right)\right)\mid d\varpi \hfill \\ & & +\frac{2\partial }{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }\mid \mathcal{J}\left(m,c_1\left(\partial \right)\right)-\mathcal{J}\left(m,c_2\left(\partial \right)\right)\mid dm\right)d\varpi \hfill \end{array}$$

$$\le \frac{K\mid c_1-c_2\mid }{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{\partial }{\left(\partial -\varpi \right)}^{\alpha -1}d\varpi +\frac{2K\mid c_1-c_2\mid }{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{c_2}\left({\int }_0^{\varpi }{\left(\varpi -m\right)}^{\alpha -1}dm\right)d\varpi$$

$$\le \frac{K\mid c_1-c_2\mid }{\mathsf{\Gamma }\left(\alpha +1\right)}+\frac{2c^{\alpha +1}K\mid c_1-c_2\mid \mathsf{\Gamma }\left(\alpha \right)}{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha +2\right)}$$

$$\le K\mid c_1-c_2\mid \left(\frac{1}{\mathsf{\Gamma }\left(\alpha +1\right)}+\frac{2c^{\alpha +1}\mathsf{\Gamma }\left(\alpha \right)}{\left(2-c_2^2\right)\mathsf{\Gamma }\left(\alpha +2\right)}\right)$$

$$=K\pi \mid c_1-c_2\mid .$$

By applying $K\pi <1$ and (24), we obtain

$$\begin{array}{ccc}\hfill \mathcal{V}\left(\xi c_1\left(\partial \right),\xi c_2\left(\partial \right)\right)& =& \mid \xi c_1\left(\partial \right)-\xi c_2\left(\partial \right){\mid }^p\le K\pi \mid c_1\left(\partial \right)-c_2\left(\partial \right){\mid }^p\hfill \\ & \le & k\mid p_1\left(\partial \right)-p_2\left(\partial \right)\mid =k\left(p_1\left(\partial \right)-p_2\left(\partial \right)\right).\hfill \end{array}$$

That is, all the conditions of Theorem 2 are satisfied. Hence, $\xi$ has a unique solution.

6. Conclusions

In this paper, we determined the best proximity point results using interpolative proximal contractions and simulation functions. We determined best proximity point results for proximal contraction, Reich–Rus–Ciric-type interpolative contraction, and Kannan-type proximal interpolative contraction of the first and second kind. Our results are an extension of some proven results in the literature. The derived results were supported with suitable examples and an application to find an analytical solution of an integral equation. Readers can explore extension of the results in the setting of bipolar p-metric space, complex metric space, fuzzy bipolar metric space, and many others.