Proximity Point Results for Generalized p-Cyclic Reich Contractions: An Application to Solving Integral Equations

Abstract

This article studies new classes of contractions called the p-cyclic Reich contraction and p-cyclic Reich contraction pair and develops certain best proximity point results for such contractions in the setting of partial metric spaces. Furthermore, the best proximity point results for p-proximal cyclic Reich contractions of the first and second types are also discussed.

1. Introduction and Preliminaries

The Banach contraction principle [1] has been improved and generalized by many researchers for different classes of contractions in various spaces. The Banach contraction result shows that every self-mapping $\mathcal{S}$ on a complete metric space $(U,d)$ satisfies

$$\begin{array}{c}\hfill d(\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha d(\mu ,\omega ),\end{array}$$

for all $\mu ,\omega \in U$, where $\alpha \in (0,1)$ has a unique fixed point in U.

The following theorem was introduced by Reich [2] to generalize the Banach contraction principle.

Theorem 1.

Let $(U,d)$ be a complete metric space. If $\mathcal{S}:U\to U$ such that

$$\begin{array}{c}\hfill d(\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha \left[d\right(\mu ,\omega )+d(\mu ,\mathcal{S}\mu )+d(\omega ,\mathcal{S}\omega \left)\right],\end{array}$$

for all $\mu ,\omega \in U$, where $\alpha \in [0,\frac{1}{3})$, then $\mathcal{S}$ has a unique fixed point.

Popescu [3] defined p-contraction mappings in 2008 and presented a fixed-point result for such mappings, as follows:

Theorem 2

([3]). Let $(U,d)$ be a complete metric space and $\mathcal{S}:U\to U$ be a p-contraction mapping, that is,

$$\begin{array}{ccc}\hfill d(\mathcal{S}\mu ,\mathcal{S}\omega )& \le & \alpha \left[d\right(\mu ,\omega )+|d(\mu ,\mathcal{S}\mu )-d(\omega ,\mathcal{S}\omega )|],\hfill \end{array}$$

for all $\mu ,\omega \in U$, where $\alpha \in [0,1)$. Then, $\mathcal{S}$ has a unique fixed point.

In the framework of cyclic mapping, Kirk et al. [4] introduced the cyclic representation of the Banach contraction principle as follows:

Theorem 3

([4]). Let $\varnothing \ne \Re ,\aleph$ be closed subsets of a complete metric space $(U,d)$. A mapping $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is referred to as cyclic if $\mathcal{S}(\Re )\subseteq \aleph$ and $\mathcal{S}(\aleph )\subseteq \Re$. Then, $\mathcal{S}$ has a fixed point in $\Re \cap \aleph$ if there is $\alpha \in [0,1)$ such that

$$\begin{array}{c}\hfill d(\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha d(\mu ,\omega ),\end{array}$$

for all $\mu \in \Re$ and $\omega \in \aleph$.

By taking the idea of the best proximity point (BPP) into consideration, various further extensions of the Banach contraction theorem have been presented [5,6,7,8,9].

Let $\Re ,\aleph$ be two subsets of $(U,d)$. In this context, a point $\mu$ in $\Re \cup \aleph$ is called the BPP of $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ if $d(\mu ,\mathcal{S}\mu )=d(\Re ,\aleph )$ = $inf\left\{d\right(\mu ,\omega ):\mu \in \Re ,\omega \in \aleph \}$. Since $d(\mu ,\mathcal{S}\mu )>d(\Re ,\aleph )$ for all $\mu \in \Re$, then $d(\mu ,\mathcal{S}\mu )$ is at least $d(\Re ,\aleph )$.

The research on this topic has been the focus of numerous articles [10,11,12,13,14,15,16]. Note that Theorem 3 requires $\Re \cap \aleph$ to be nonempty. On the other hand, Eldred and Veeremani [17] discussed the idea of cyclic contraction mapping and obtained a result that incorporates the case $\Re \cap \aleph =\varnothing$ by combining the concepts of both cyclic mapping and the BPP.

Definition 1.

A mapping $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is called a cyclic contraction if it satisfies $\mathcal{S}(\Re )\subseteq \aleph$ and $\mathcal{S}(\aleph )\subseteq \Re$ and the inequality:

$$\begin{array}{c}\hfill d(\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha d(\mu ,\omega )+(1-\alpha )d(\Re ,\aleph ),\end{array}$$

for all $\mu \in \Re$ and $\omega \in \aleph$, where $\alpha \in (0,1)$.

Theorem 4.

Let ℜ and ℵ be two nonempty closed subsets of a metric space $(U,d)$, and let $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ be a cyclic contraction mapping. If either ℜ or ℵ is boundedly compact, then there exists $\mu \in \Re \cup \aleph$ such that $d(\mu ,\mathcal{S}\mu )=d(\Re ,\aleph ).$

Now, we recall some basic results and definitions related to the partial metric space, which are crucial to our discussion that follows.

Definition 2

([18]). A partial metric (PM) on a nonempty set U is a function $\rho :U\times U\to [0,\infty )$ that satisfies the following axioms for all $\mu ,\omega ,\nu \in U$:

(i) 

$\mu =\omega$, if and only if $\rho (\mu ,\mu )=\rho (\mu ,\omega )=\rho (\omega ,\omega )$;

(ii) 

$\rho (\mu ,\mu )\le \rho (\mu ,\omega )$;

(iii) 

$\rho (\mu ,\omega )=\rho (\omega ,\mu )$;

(iv) 

$\rho (\mu ,\omega )\le \rho (\mu ,\nu )+\rho (\nu ,\omega )-\rho (\nu ,\nu )$.

Then, the function ρ is called a partial metric on U, and the pair $(U,\rho )$ is called a partial metric space (PMS).

We know that every metric space is a PMS, but not conversely, in general. If $\rho$ is a PM on U, then

$$\begin{array}{c}\hfill d_{\rho }(\mu ,\omega )=2\rho (\mu ,\omega )-\rho (\mu ,\mu )-\rho (\omega ,\omega ),\end{array}$$

is a metric on U.

Each PM $\rho$ on U generates a $T_0$ topology ${\tau }_{\rho }$ on U, with a base of the class of open $\rho$-balls $\{{\mathcal{B}}_{\rho }(\mu ,\epsilon ),\mu \in U,\epsilon >0\}$, where ${\mathcal{B}}_{\rho }(\mu ,\epsilon )=\{\omega \in U:\rho (\mu ,\omega )<\rho (\mu ,\mu )+\epsilon \}$ for all $\mu \in U$ and $\epsilon >0$.

Definition 3

([18,19,20]). Let $(U,\rho )$ be a PMS. Then:

(i) 

A sequence $\left\{{\mu }_n\right\}$ converges to a point $\mu \in U$, if and only if $\rho (\mu ,\mu )={lim}_{n\to \infty }\rho (\mu ,{\mu }_n)$;

(ii) 

A sequence $\left\{{\mu }_n\right\}$ is called a Cauchy sequence, if ${lim}_{n,m\to \infty }\rho ({\mu }_n,{\mu }_m)$ exists and is finite;

(iii) 

$(U,\rho )$ is said to be a complete PMS if every Cauchy sequence $\left\{{\mu }_n\right\}$ in U converges to some $\mu \in U$ (with respect to ${\tau }_{\rho }$) such that

$$\begin{array}{c}\hfill \rho (\mu ,\mu )=\underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\mu }_m),\end{array}$$

(iv) 

A sequence $\left\{{\mu }_n\right\}$ in $(U,\rho )$ is called a 0-Cauchy sequence if

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\mu }_m)=0.\end{array}$$

(v) 

$(U,\rho )$ is said to be a 0-complete PMS if every 0-Cauchy sequence in U converges with respect to ${\tau }_{\rho }$ to a point $\mu \in U$ such that

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\mu }_m)=\rho (\mu ,\mu ).\end{array}$$

Lemma 1

([18,19,20]). Let $(U,\rho )$ be a PMS. Then:

(i) 

$\left\{{\mu }_n\right\}$ is a Cauchy sequence in $(U,\rho )$, if and only if it is a Cauchy sequence in the metric space $(U,d_{\rho })$.

(ii) 

$(U,\rho )$ is a complete PMS if and only if the metric space $(U,d_{\rho })$ is complete. Furthermore,

$$\underset{n\to \infty }{lim}{d}_{\rho }({\mu }_n,\mu )=0\iff \rho (\mu ,\mu )=\underset{n\to \infty }{lim}\rho ({\mu }_n,\mu )=\underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\mu }_m).$$

(iii) 

Every 0-Cauchy sequence in $(U,\rho )$ is Cauchy in $(U,d_{\rho })$.

(iv) 

If $(U,\rho )$ is complete, then it is 0-complete.

Lemma 2

([18]). Let $(U,\rho )$ be a PMS. Then:

(i) 

If $\rho (\mu ,\omega )=0$, then $\mu =\omega$. However, if $\mu =\omega$, then $\rho (\mu ,\omega )$ may not be zero;

(ii) 

If $\mu \ne \omega$, then $\rho (\mu ,\omega )>0.$

Lemma 3

([21]). Let ${\mu }_n\to \nu$ as $n\to \infty$ in a PMS $(U,\rho )$, where $\rho (\nu ,\nu )=0$. Then,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,\omega )=\rho (\nu ,\omega ),\end{array}$$

for every $\omega \in U$.

In the rest of this paper, unless stated otherwise, we consider $(U,\rho )$ as a PMS and ℜ and ℵ as its nonempty subsets.

In 2022, Sahin [16] provided the next definitions and introduced them as described below.

Definition 4.

A sequence $\left\{{\mu }_n\right\}$ in $\Re \cup \aleph$ with $\left\{{\mu }_{2n}\right\}\subseteq \Re$ and $\left\{{\mu }_{2n+1}\right\}\subseteq \aleph$ is called a cyclically Cauchy if for each $ϵ>0$, there is $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_n,{\mu }_m)<\rho (\Re ,\aleph )+ϵ,\end{array}$$

when m is odd, n is even and $n,m\ge n_0$.

If $\rho (\Re ,\aleph )=0$, then the concept of a cyclically Cauchy becomes the definition of a cyclically 0-Cauchy.

Definition 5.

A pair $(\Re ,\aleph )$ is called a cyclically 0-complete if for every cyclically Cauchy sequence $\left\{{\mu }_n\right\}$ in $\Re \cup \aleph$, either the sequence $\left\{{\mu }_{2n}\right\}$ has a convergent subsequence $\left\{{\mu }_{2n_i}\right\}$ to a point $\nu \in \Re$ with respect to ${\tau }_{\rho }$ such that

$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\rho ({\mu }_{2n_i},{\mu }_{2n_j})=\rho (\nu ,\nu )=0,\end{array}$$

(1)

or $\left\{{\mu }_{2n+1}\right\}$ has a convergent subsequence $\left\{{\mu }_{2n_i+1}\right\}$ to a point ${\nu }^{*}\in \aleph$ with respect to ${\tau }_{\rho }$ such that

$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\rho ({\mu }_{2n_i+1},{\mu }_{2n_j+1})=\rho ({\nu }^{*},{\nu }^{*})=0.\end{array}$$

Definition 6

([16]). ℜ is called a 0-boundedly compact if every bounded sequence $\left\{{\mu }_n\right\}$ has a convergent subsequence $\left\{{\mu }_{n_k}\right\}$ to a point $\nu \in \Re$ with respect to ${\tau }_{\rho }$ such that

$$\begin{array}{c}\hfill \underset{k,\ell \to \infty }{lim}\rho ({\mu }_{n_k},{\mu }_{n_{\ell }})=\rho (\nu ,\nu )=0.\end{array}$$

Remark 1

([16]). If either ℜ or ℵ is a 0-boundedly compact, then the pair $(\Re ,\aleph )$ is a cyclically 0-complete pair. If ℜ and ℵ are closed subsets of a complete PMS with $\rho (\Re ,\aleph )=0$, then $(\Re ,\aleph )$ is a cyclically 0-complete pair.

Definition 7.

Let $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ be a cyclic mapping. Then, $\mathcal{S}$ is said to be a p-cyclic contraction mapping if there is $\alpha \in [0,1)$ such that

$$\begin{array}{c}\hfill \rho (\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha \left[\rho (\mu ,\omega )+|\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{S}\omega )|\right]+(1-\alpha )\rho (\Re ,\aleph ),\end{array}$$

for all $\mu \in \Re$ and $\omega \in \aleph$, where $\rho (\Re ,\aleph )=inf\left\{\rho \right(\mu ,\omega ):\mu \in \Re and\omega \in \aleph \}.$

Then, the author proved certain BPP results for such contractions on $\Re \cup \aleph$, where $(\Re ,\aleph )$ is a cyclically 0-complete pair in a PMS (see [11,12,22,23,24]).

Definition 8

([24]). Let ℜ and ℵ be nonempty subsets of a metric space $(U,d)$. Then, $(\Re ,\aleph )$ is said to satisfy the property $UC$ if the following holds. If $\left\{{\mu }_n\right\}$ and $\left\{{\nu }_n\right\}$ are sequences in ℜ and $\left\{{\omega }_n\right\}$ is a sequence in ℵ such that

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}d({\mu }_n,{\omega }_n)& =& d(\Re ,\aleph ),\hfill \\ \hfill \underset{n\to \infty }{lim}d({\nu }_n,{\omega }_n)& =& d(\Re ,\aleph ),\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d({\mu }_n,{\nu }_n)=0.\end{array}$$

The obtained results generalize and extend certain well-known findings in metric fixed-point theory, which include results of Eldred–Veeremani [17], Popescu [3], and Sahin [16]. First, we introduce the concept of p-cyclic Reich contraction by combining the concepts of cyclic contraction and p-contraction. Then, we prove some BPP results. Next, we introduce the concept of the p-cyclic Reich contraction pair and present some BPP results in PMSs. The BPP results for p-proximal cyclic Reich contractions of the first and second types are also discussed. In addition, some examples are provided to illustrate the results. Finally, we present sufficient conditions to demonstrate the existence of the solution to integral equations.

2. Best Proximity Results of Mappings

We begin this section by introducing the p-cyclic Reich contraction mapping.

Definition 9.

Assume that $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a cyclic mapping. Then, $\mathcal{S}$ is said to be a p-cyclic Reich contraction mapping if there exists $\alpha ,\beta ,\gamma ,\delta \in {\mathbb{R}}^{+}$ with $\eta =\alpha +\beta +\gamma +\delta <1$ such that

$$\begin{array}{c}\hfill \rho (\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{S}\omega )+\delta \left|\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{S}\omega )\right|+(1-\eta )\rho (\Re ,\aleph ),\end{array}$$

for all $\mu \in \Re$ and $\omega \in \aleph$, where $\rho (\Re ,\aleph )=inf\left\{\rho \right(\mu ,\omega ):\mu \in \Re and\omega \in \aleph \}.$

Proposition 1.

Assume that $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a p-cyclic Reich contraction mapping. The sequence $\left\{{\mu }_n\right\}$ is defined by ${\mu }_{n+1}=\mathcal{S}{\mu }_n$ with the initial point $\left\{{\mu }_0\right\}\in \Re$. If there is $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}),\end{array}$$

then $\mathcal{S}$ has a BPP in $\Re \cup \aleph$.

Proof. 

Since $\mathcal{S}$ is a p-cyclic Reich contraction, there are non-negative numbers $\alpha ,\beta ,\gamma ,\delta$, where $\eta =\alpha +\beta +\gamma +\delta <1$ such that

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_{n+1})& =& \rho (\mathcal{S}{\mu }_{n-1},\mathcal{S}{\mu }_n)\hfill \\ \hfill & \le & \alpha \rho ({\mu }_{n-1},{\mu }_n)+\beta \rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})+\gamma \rho ({\mu }_n,\mathcal{S}{\mu }_n)\hfill \\ \hfill & & +\delta \left|\rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})-\rho ({\mu }_n,\mathcal{S}{\mu }_n)\right|+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

(2)

for all $n\in \mathbb{N}$. Now, if there is $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}),\end{array}$$

(3)

then, from Equations (2) and (3), we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})& =& \rho (\mathcal{S}{\mu }_{n_0},\mathcal{S}{\mu }_{n_0+1})\hfill \\ & \le & \alpha \rho ({\mu }_{n_0},{\mu }_{n_0+1})+\beta \rho ({\mu }_{n_0},\mathcal{S}{\mu }_{n_0})+\gamma \rho ({\mu }_{n_0+1},\mathcal{S}{\mu }_{n_0+1})\hfill \\ & & +\delta \left|\rho ({\mu }_{n_0},\mathcal{S}{\mu }_{n_0})-\rho ({\mu }_{n_0+1},\mathcal{S}{\mu }_{n_0+1})\right|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta )\rho ({\mu }_{n_0},{\mu }_{n_0+1})+\gamma \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})-\delta \rho ({\mu }_{n_0},{\mu }_{n_0+1})\hfill \\ & & +\delta \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta -\delta )\rho ({\mu }_{n_0},{\mu }_{n_0+1})+(\gamma +\delta )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})\hfill \\ & & +(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta -\delta )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})+(\gamma +\delta )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})\hfill \\ & & +(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta +\gamma )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& (\eta -\delta )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Since $\eta =\alpha +\beta +\gamma +\delta <1$, we obtain $\frac{1-\eta }{1-\eta +\delta }<1$. Hence,

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})\le \rho (\Re ,\aleph ).\end{array}$$

Furthermore, since $\rho (\Re ,\aleph )\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})$, we have

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})=\rho (\Re ,\aleph ),\end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )\le \rho ({\mu }_{n_0},\mathcal{S}{\mu }_{n_0})& \le & \rho ({\mu }_{n_0+1},\mathcal{S}{\mu }_{n_0+1})\hfill \\ & =& \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})=\rho (\Re ,\aleph ).\hfill \end{array}$$

Hence, ${\mu }_{n_0}$ and ${\mu }_{n_0+1}$ are the BPPs of $\mathcal{S}$. □

Remark 2.

Suppose that $\left\{{\mu }_n\right\}$ is a sequence, as in Proposition 1, such that there is $n_0\in \mathbb{N}$ satisfying

$$\rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}).$$

Then, $\mathcal{S}$ has the BPP in $\Re \cup \aleph$. Therefore, in the rest of this paper we investigate the condition $\rho ({\mu }_{n+1},{\mu }_{n+2})\le \rho ({\mu }_n,{\mu }_{n+1})$ for all n in $\mathbb{N}$.

Proposition 2.

Assume that $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a p-cyclic Reich contraction mapping. By defining the sequence $\left\{{\mu }_n\right\}$, as in Proposition 1, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

Proof. 

Since $\mathcal{S}$ is a p-cyclic Reich contraction, by Remark 2, we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_{n+1})& =& \rho (\mathcal{S}{\mu }_{n-1},\mathcal{S}{\mu }_n)\hfill \\ & \le & \alpha \rho ({\mu }_{n-1},{\mu }_n)+\beta \rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})+\gamma \rho ({\mu }_n,\mathcal{S}{\mu }_n)\hfill \\ & & +\delta \left|\rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})-\rho ({\mu }_n,\mathcal{S}{\mu }_n)\right|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{n-1},{\mu }_n)+\beta \rho ({\mu }_{n-1},{\mu }_n)+\gamma \rho ({\mu }_n,{\mu }_{n+1})\hfill \\ & & +\delta \left|\rho ({\mu }_{n-1},{\mu }_n)-\rho ({\mu }_n,{\mu }_{n+1})\right|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta )\rho ({\mu }_{n-1},{\mu }_n)+\gamma \rho ({\mu }_n,{\mu }_{n+1})+\delta \rho ({\mu }_{n-1},{\mu }_n)\hfill \\ & & -\delta \rho ({\mu }_n,{\mu }_{n+1})+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& (\alpha +\beta +\delta )\rho ({\mu }_{n-1},{\mu }_n)+(\gamma -\delta )\rho ({\mu }_n,{\mu }_{n+1})+(1-\eta )\rho (\Re ,\aleph )\hfill \\ \hfill ⟹\rho ({\mu }_n,{\mu }_{n+1})& \le & \frac{\alpha +\beta +\delta }{1+\delta -\gamma }\rho ({\mu }_{n-1},{\mu }_n)+\frac{1-\eta }{1+\delta -\gamma }\rho (\Re ,\aleph ),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_{n+1})& \le & \vartheta \rho ({\mu }_{n-1},{\mu }_n)+\xi \rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$. Therefore, we write $\vartheta =\frac{\alpha +\beta +\delta }{1+\delta -\gamma }$ and $\xi =\frac{1-\eta }{1+\delta -\gamma }$. From the last inequality, we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }_n,{\mu }_{n+1})\hfill \\ & \le & \vartheta \rho ({\mu }_{n-1},{\mu }_n)+\xi \rho (\Re ,\aleph )\hfill \\ & \le & \vartheta \left(\vartheta \rho ({\mu }_{n-2},{\mu }_{n-1})+\xi \rho (\Re ,\aleph )\right)+\xi \rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^2\rho ({\mu }_{n-2},{\mu }_{n-1})+\xi (1+\vartheta )\rho (\Re ,\aleph )\hfill \\ & & ⋮\hfill \\ & \le & {\vartheta }^n\rho ({\mu }_0,{\mu }_1)+\xi \left(1+\vartheta +{\vartheta }^2+\cdots +{\vartheta }^{n-1}\right)\rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^n\rho ({\mu }_0,{\mu }_1)+\xi \left(\frac{1-{\vartheta }^n}{1-\vartheta }\right)\rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^n\rho ({\mu }_0,{\mu }_1)+\frac{1-\eta }{1-\delta }\left(1-{\vartheta }^n\right)\rho (\Re ,\aleph )\hfill \\ & \le & {\vartheta }^n\rho ({\mu }_0,{\mu }_1)+(1-{\vartheta }^n)\rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$. Therefore, we conclude that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

(4)

□

The follow proposition is important for our main result.

Proposition 3.

Assume that $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a p-cyclic Reich contraction. Then, every sequence $\left\{{\mu }_n\right\}\subseteq \Re \cup \aleph$, as constructed in Proposition 1, is bounded.

Proof. 

Suppose that $\left\{{\mu }_n\right\}\subseteq \Re \cup \aleph$ is a sequence defined in Proposition 1. Then, by Proposition 2, the sequence $\left\{\rho ({\mu }_{2n},{\mu }_{2n+1})\right\}$ is convergent to $\rho (\Re ,\aleph )$ as $n\to \infty$; therefore, the sequence $\left\{\rho ({\mu }_{2n},{\mu }_{2n+1})\right\}$ is bounded. Then, for all $n\in \mathbb{N}$, there exists $M>0$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{2n},{\mu }_{2n+1})\le M.\end{array}$$

Since $\mathcal{S}$ is a p-cyclic Reich contraction, using Remark 2, we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_{2n+1},{\mu }_1)& \le & \rho ({\mu }_{2n+1},{\mu }_{2n+2})+\rho ({\mu }_{2n+2},{\mu }_1)\hfill \\ & \le & M+\rho (\mathcal{S}{\mu }_{2n+1},\mathcal{S}{\mu }_0)\hfill \\ & \le & M+\alpha \rho ({\mu }_{2n+1},{\mu }_0)+\beta \rho ({\mu }_{2n+1},\mathcal{S}{\mu }_{2n+1})+\gamma \rho ({\mu }_0,\mathcal{S}{\mu }_0)\hfill \\ & & +\delta |\rho ({\mu }_{2n+1},\mathcal{S}{\mu }_{2n+1})-\rho ({\mu }_0,\mathcal{S}{\mu }_0)|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& M+\alpha \rho ({\mu }_{2n+1},{\mu }_0)+\beta \rho ({\mu }_{2n+1},{\mu }_{2n+2})+\gamma \rho ({\mu }_0,{\mu }_1)\}\hfill \\ & & +\delta |\rho ({\mu }_{2n+1},{\mu }_{2n+2})-\rho ({\mu }_0,{\mu }_1)|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (1+\beta )M+\alpha \rho ({\mu }_{2n+1},{\mu }_1)+\alpha \rho ({\mu }_0,{\mu }_1)+\gamma \rho ({\mu }_0,{\mu }_1)\hfill \\ & & +\delta \{-\rho ({\mu }_{2n+1},{\mu }_{2n+2})+\rho ({\mu }_0,{\mu }_1)\}+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& (1+\beta -\delta )M+\alpha \rho ({\mu }_{2n+1},{\mu }_1)+(\alpha +\delta )\rho ({\mu }_0,{\mu }_1)+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$, which implies that

$$\rho ({\mu }_{2n+1},{\mu }_1)\le \frac{1+\beta -\delta }{1-\alpha }M+\frac{\alpha +\delta }{1-\alpha }\rho ({\mu }_1,{\mu }_0)+\frac{1-\eta }{1-\alpha }\rho (\Re ,\aleph ).$$

We take K to be defined as

$$\begin{array}{c}\hfill K=\frac{1+\beta -\delta }{1-\alpha }M+\frac{\alpha +\delta }{1-\alpha }\rho ({\mu }_1,{\mu }_0)+\frac{1-\eta }{1-\alpha }\rho (\Re ,\aleph ).\end{array}$$

Then, $\left\{{\mu }_{2n+1}\right\}$ is bounded. Moreover, we have

$$\begin{array}{c}\hfill \rho ({\mu }_{2n},{\mu }_1)\le \rho ({\mu }_{2n},{\mu }_{2n+1})+\rho ({\mu }_{2n+1},{\mu }_1)\le M+K.\end{array}$$

Hence, $\left\{{\mu }_n\right\}$ is bounded. □

Theorem 5.

Assume that $(\Re ,\aleph )$ is a cyclically 0-complete pair and $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a p-cyclic Reich contraction, if there exist some $\alpha ,\beta ,\gamma ,\delta \in [0,1)$ such that $\eta =\alpha +\beta +\gamma +\delta <1$. Then, $\mathcal{S}$ has the BPP.

Proof. 

Let $\left\{{\mu }_0\right\}$ be an arbitrary point in $\Re \cup \aleph$, and let $\left\{{\mu }_n\right\}$ be a sequence constructed as $\left\{{\mu }_{n+1}\right\}=\mathcal{S}{\mu }_n$ for all $n\in \mathbb{S}$. If there exists $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}).\end{array}$$

Then, by Proposition 1, the mapping $\mathcal{S}$ has a BPP. Now, assume that

$$\begin{array}{c}\hfill \rho ({\mu }_{n+1},{\mu }_{n+2})\le \rho ({\mu }_n,{\mu }_{n+1}),\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$

In this case, from Proposition 2, we know that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

(5)

Now, we show that $\left\{{\mu }_n\right\}$ is a cyclically Cauchy. Let $n,m\in \mathbb{N}$ with $n\ge m$. Since $\mathcal{S}$ is a p-cyclic Reich contraction mapping, we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_m)& =& \rho (\mathcal{S}{\mu }_{n-1},\mathcal{S}{\mu }_{m-1})\hfill \\ & \le & \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+\beta \rho ({\mu }_{m-1},\mathcal{S}{\mu }_{m-1})+\gamma \rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})\hfill \\ & & +\delta |\rho ({\mu }_{m-1},\mathcal{S}{\mu }_{m-1})-\rho ({\mu }_{n-1},\mathcal{S}{\mu }_{n-1})|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+\beta \rho ({\mu }_{m-1},{\mu }_m)+\gamma \rho ({\mu }_{n-1},{\mu }_n)\hfill \\ & & +\delta \rho ({\mu }_{m-1},{\mu }_m)-\delta \rho ({\mu }_{n-1},{\mu }_n)+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+\beta \rho ({\mu }_{m-1},{\mu }_m)+\gamma \rho (\Re ,\aleph )\hfill \\ & & +\delta \rho ({\mu }_{m-1},{\mu }_m)-\delta \rho (\Re ,\aleph )+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+(\beta +\delta )\rho ({\mu }_{m-1},{\mu }_m)\hfill \\ & & +(1-\eta -\delta +\gamma )\rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$ with $n\ge m$. Assuming that $\ell =\beta +\delta$ and $q=1-\eta -\delta +\gamma$, we obtain

$$\begin{array}{c}\hfill \rho ({\mu }_n,{\mu }_m)\le \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+\ell \rho ({\mu }_{m-1},{\mu }_m)+q\rho (\Re ,\aleph ).\end{array}$$

(6)

Furthermore, using the same steps as those used for the proof of Proposition 2, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_m,{\mu }_{m+1})& \le & \vartheta \rho ({\mu }_{m-1},{\mu }_m)+\xi \rho (\Re ,\aleph ),\hfill \end{array}$$

for all $m\in \mathbb{N}$, where $\vartheta =\frac{\alpha +\beta +\delta }{1+\delta -\gamma }$ and $\xi =\frac{1-\eta }{1+\delta -\gamma }$. Then, we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }_n,{\mu }_m)\hfill \\ & \le & \rho (\mathcal{S}{\mu }_{n-1},\mathcal{S}{\mu }_{m-1})\hfill \\ & \le & \alpha \rho ({\mu }_{n-1},{\mu }_{m-1})+\ell \rho ({\mu }_{m-1},{\mu }_m)+q\rho (\Re ,\aleph )\hfill \\ & \le & \alpha \{\alpha \rho ({\mu }_{n-2},{\mu }_{m-2})+\ell \rho ({\mu }_{m-2},{\mu }_{m-1})+q\rho (\Re ,\aleph )\}+\ell \{\vartheta \rho ({\mu }_{m-2},{\mu }_{m-1})\hfill \\ & & +\xi \rho (\Re ,\aleph )\}+q\rho (\Re ,\aleph )\hfill \\ & =& {\alpha }^2\rho ({\mu }_{n-2},{\mu }_{m-2})+\ell (\alpha +\vartheta )\rho ({\mu }_{m-2},{\mu }_{m-1})+q(\alpha +1)\rho (\Re ,\aleph )+\ell \xi \rho (\Re ,\aleph )\hfill \\ & \le & {\alpha }^2\{\alpha \rho ({\mu }_{n-3},{\mu }_{m-3})+\ell \rho ({\mu }_{m-3},{\mu }_{m-2})+q\rho (\Re ,\aleph )\}+\ell (\alpha +\vartheta )\{\vartheta \rho ({\mu }_{m-3},{\mu }_{m-2})\hfill \\ & & +\xi \rho (\Re ,\aleph )\}+q(\alpha +1\left)\rho \right(\Re ,\aleph )+\ell \xi \rho (\Re ,\aleph )\hfill \\ & =& {\alpha }^3\rho ({\mu }_{n-3},{\mu }_{m-3})+\ell \{{\alpha }^2+\alpha \vartheta +{\vartheta }^2\}\rho ({\mu }_{m-3},{\mu }_{m-2})\hfill \\ & & +\{\ell (\alpha +\vartheta )\xi +\ell \xi \}\rho (\Re ,\aleph )+q(1+\alpha +{\alpha }^2)\rho (\Re ,\aleph )\hfill \\ & & ⋮\hfill \\ & \le & {\alpha }^m\rho ({\mu }_{n-m},{\mu }_0)\hfill \\ & & +\ell {\vartheta }^m\left\{1+\frac{\alpha }{\vartheta }+{\left(\frac{\alpha }{\vartheta }\right)}^2+\cdots +{\left(\frac{\alpha }{\vartheta }\right)}^m\right\}\rho ({\mu }_0,{\mu }_1)\hfill \\ & & +\xi \ell \left\{\begin{array}{c}\multicolumn{1}{c}{\left\{\alpha +{\alpha }^2+{\alpha }^3+\cdots +{\alpha }^{m-1}\right\}}\\ +\vartheta \left\{\alpha +{\alpha }^2+\cdots +{\alpha }^{m-2}\right\}\hfill \\ +{\vartheta }^2\left\{\alpha +{\alpha }^2+\cdots +{\alpha }^{m-3}\right\}\hfill \\ ⋮\hfill \\ +{\vartheta }^{m-1}\alpha +{\vartheta }^m\hfill \end{array}\right\}\rho (\Re ,\aleph )\hfill \\ & & +q(1+\alpha +{\alpha }^2+\cdots +{\alpha }^m)\rho (\Re ,\aleph )\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\alpha }^m\rho ({\mu }_0,{\mu }_{n-m})+\ell {\vartheta }^m\left\{\sum _{i=0}^m{\left(\frac{\alpha }{\vartheta }\right)}^i\right\}\rho ({\mu }_0,{\mu }_1)\hfill \\ & & +\xi \ell \left\{\sum _{i=0}^{\infty }{\alpha }^i+\vartheta \sum _{i=0}^{\infty }{\alpha }^i+\cdots +{\vartheta }^m\sum _{i=0}^{\infty }{\alpha }^i\right\}\rho (\Re ,\aleph )\hfill \\ & & +q\left\{\sum _{i=0}^{m-1}{\alpha }^i\right\}\rho (\Re ,\aleph )\hfill \\ & \le & {\alpha }^m\rho ({\mu }_0,{\mu }_{n-m})+\ell {\vartheta }^m\left\{\sum _{i=0}^m{\left(\frac{\vartheta }{\alpha }\right)}^i\right\}\rho ({\mu }_0,{\mu }_1)\hfill \\ & & +\xi \ell \frac{1}{1-\alpha }\left\{1+\vartheta +{\vartheta }^2+\cdots +{\vartheta }^m\right\}\rho (\Re ,\aleph )\hfill \\ & & +q\left\{\sum _{i=0}^{m-1}{\alpha }^i\right\}\rho (\Re ,\aleph ).\hfill \end{array}$$

Since $\left\{{\mu }_n\right\}$ is a bounded sequence, then there exists $M>0$ such that $\rho ({\mu }_0,{\mu }_{n-m})\le M$ for all $n,m\in \mathbb{N}$ with $n\ge m$, and we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }_n,{\mu }_m)\hfill \\ & \le & {\alpha }^{m}M+\ell {\vartheta }^m\left(\frac{1-{\left(\frac{\vartheta }{\alpha }\right)}^m}{1-\frac{\vartheta }{\alpha }}\right)\rho ({\mu }_0,{\mu }_1)\hfill \\ & & +\left(\frac{\xi \ell }{1-\alpha }\right)\left(\frac{1-{\vartheta }^m}{1-\vartheta }\right)\rho (\Re ,\aleph )+q\frac{1-{\alpha }^m}{1-\alpha }\rho (\Re ,\aleph ).\hfill \end{array}$$

Hence, we have

$$\begin{array}{c}\hfill \rho (\Re ,\aleph )\le \underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\mu }_m)\le \frac{1}{1-\alpha }\left(\frac{\xi \ell }{1-\vartheta }+q\right)\rho (\Re ,\aleph )\le \rho ({\mu }_n,{\mu }_m),\end{array}$$

where $\frac{1}{1-\alpha }(\frac{\xi \ell }{1-\vartheta }+q)<1$.

Since $(\Re ,\aleph )$ is a cyclically 0-complete pair, there exist subsequences $\left\{{\mu }_{2n_i}\right\}$ of $\left\{{\mu }_{2n}\right\}$ such that

$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\rho ({\mu }_{2n_i},{\mu }_{2n_j})=\underset{i\to \infty }{lim}\rho ({\mu }_{2n_i},{\mu }^{*})=\rho ({\mu }^{*},{\mu }^{*})=0,\end{array}$$

(7)

for some ${\mu }^{*}\in \Re$. Furthermore, we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }^{*},{\mu }_{2n_{i-1}})\hfill \\ & \le & \rho ({\mu }^{*},{\mu }_{2n_i})+\rho ({\mu }_{2n_i},{\mu }_{2n_{i-1}}).\hfill \end{array}$$

By taking the limit $i\to \infty$ and from inequality (5), we obtain

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}({\mu }^{*},{\mu }_{2n_i-1})=\rho (\Re ,\aleph ).\end{array}$$

Moreover, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{2n_i},\mathcal{S}{\mu }^{*})& =& \rho (\mathcal{S}{\mu }_{2n_i-1},\mathcal{S}{\mu }^{*})\hfill \\ & \le & \alpha \rho ({\mu }_{2n_i-1},{\mu }^{*})+\beta \rho ({\mu }_{2n_i-1},\mathcal{S}{\mu }_{2n_i-1})+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho ({\mu }_{2n_i-1},\mathcal{S}{\mu }_{2n_i-1})-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{2n_i-1},{\mu }^{*})+\beta \rho ({\mu }_{2n_i-1},{\mu }_{2n_i})+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho ({\mu }_{2n_i-1},{\mu }_{2n_i})-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Taking the limit $i\to \infty$ and using inequality (7) implies that

$$\begin{array}{ccc}\hfill \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& =& \underset{i\to \infty }{lim}({\mu }_{2n_i},\mathcal{S}{\mu }^{*})\hfill \\ & \le & \alpha \rho (\Re ,\aleph )+\beta \rho (\Re ,\aleph )+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho (\Re ,\aleph )-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho (\Re ,\aleph )+\beta \rho (\Re ,\aleph )+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta \rho (\Re ,\aleph )-\delta \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill (1-\gamma +\delta )\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& \le & (1-\eta +\alpha +\beta +\delta )\rho (\Re ,\aleph )\hfill \\ \hfill \Rightarrow \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& \le & \frac{1-\gamma }{1-\gamma +\delta }\rho (\Re ,\aleph )\hfill \\ & \le & \rho (\Re ,\aleph ),\hfill \end{array}$$

and then $\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})=\rho (\Re ,\aleph )$. Hence, ${\mu }^{*}$ is the BPP of $\mathcal{S}$ in ℜ. If $\left\{{\mu }_{2n+1}\right\}$ has a subsequence $\left\{{\mu }_{2n_i+1}\right\}$ such that

$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\rho ({\mu }_{2n_i+1},{\mu }_{2n_j+1})=\underset{i\to \infty }{lim}\rho ({\mu }_{2n_i+1},{\omega }^{*})=\rho ({\omega }^{*},{\omega }^{*})=0,\end{array}$$

for some ${\omega }^{*}$ in ℵ, then, in a similar way, one deduces that ${\omega }^{*}$ is the BPP of $\mathcal{S}$ in ℵ. □

Example 1.

Consider the space $U=L^1\left([0,1]\right)$ with the norm

$${||f||}_{L^1}={\int }_0^1|f\left(t\right)|dt.$$

Define $\rho :U\times U\to {\mathbb{R}}^{+}$ by

$$||f\left(t\right)-{g\left(t\right)||}_{L^1}={\int }_0^1|f\left(t\right)-g\left(t\right)|dt.$$

It is clear that $(U,\rho )$ is a PMS. Suppose that $\Re =\{f\in U:-1\le f\le 0\}$ and $\aleph =\{g\in U:0\le g\le 1\}$. Then, we have $\rho (\Re ,\aleph )=0$. First, we prove that the pair $(\Re ,\aleph )$ is a cyclically 0-complete pair. Assume that $\left\{f_n\right\}$ is a cyclically Cauchy sequence in $\Re \cup \aleph$ with $\left\{f_{2n}\right\}\subseteq \Re$ and $\left\{f_{2n+1}\right\}\subseteq \aleph$. Then, we obtain

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}\rho (f_n,f_m)& =& \rho (\Re ,\aleph ).\hfill \end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{\int }_0^1|f_{2n}-f_{2m+1}|& =& 0,\hfill \end{array}$$

which implies that ${lim}_{n\to \infty }{f}_{2n}={lim}_{m\to \infty }{f}_{2m+1}=0$. Then, we have

$$\begin{array}{ccc}\hfill \underset{n,s\to \infty }{lim}\rho (f_{2n},f_{2s})& =& \underset{n\to \infty }{lim}\rho (f_n,0)=\rho (0,0)=0,\hfill \end{array}$$

that is, the sequence $\left\{f_{2n}\right\}$ has a subsequence satisfying (1). Now, we define a mapping $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ by $\mathcal{S}=\frac{-f}{4}$ then, $\mathcal{S}(\Re )\subseteq \aleph$ and $\mathcal{S}(\aleph )\subseteq \Re$; hence, $\mathcal{S}$ is a cyclic mapping. Now, we show that $\mathcal{S}$ is a p-cyclic Reich contraction mapping. Choose $\alpha =\frac{1}{4},\beta =\gamma =\frac{1}{5}$, and $\delta =\frac{1}{10}$ Then, we have the following three cases.

Case 1: Let $f\in \Re$ and $g\in \aleph$. In this case, we obtain

$$\begin{array}{ccc}& & \alpha \rho (f,g)+\beta \rho (f,\mathcal{S}f)+\gamma \rho (g,\mathcal{S}g)+\delta |\rho (f,\mathcal{S}f)-\rho (g,\mathcal{S}g)|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{5}{\int }_0^1|-f-\frac{1}{4}f|dt+\frac{1}{5}{\int }_0^1|g-\frac{-1}{4}g|dt\hfill \\ & +& \frac{1}{10}|{\int }_0^1|f-\frac{1}{4}f|dt-{\int }_0^1|g+\frac{1}{4}g|dt|+(1-\eta )\left(0\right)\hfill \\ & \ge & \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{4}{\int }_0^1|f|dt+\frac{1}{4}{\int }_0^1|g|dt\hfill \\ & +& \frac{1}{8}|{\int }_0^1|f|-|g|dt|\hfill \\ & \ge & \frac{1}{4}{\int }_0^1|f-g|dt\hfill \\ & =& \rho (\mathcal{S}f,\mathcal{S}g).\hfill \end{array}$$

Case 2: If $f,g\in \Re$, we obtain

$$\begin{array}{ccc}& & \alpha \rho (f,g)+\beta \rho (f,\mathcal{S}f)+\gamma \rho (g,\mathcal{S}g)+\delta |\rho (f,\mathcal{S}f)-\rho (g,\mathcal{S}g)|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{5}{\int }_0^1|-f-\frac{1}{4}f|dt+\frac{1}{5}{\int }_0^1|-g-\frac{1}{4}g|dt\hfill \\ & +& \frac{1}{10}|{\int }_0^1|f+\frac{1}{4}f|dt-{\int }_0^1|g+\frac{1}{4}g|dt|+(1-\eta )\left(0\right)\hfill \\ & =& \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{4}{\int }_0^1|f|dt+\frac{1}{4}{\int }_0^1|g|dt\hfill \\ & +& \frac{1}{8}|{\int }_0^1|f|-|g|dt|\hfill \\ & \ge & \frac{1}{4}{\int }_0^1|f-g|dt\hfill \\ & =& \rho (\mathcal{S}f,\mathcal{S}g).\hfill \end{array}$$

Case 3: If $f,g\in \aleph$, we have

$$\begin{array}{ccc}& & \alpha \rho (f,g)+\beta \rho (f,\mathcal{S}f)+\gamma \rho (g,\mathcal{S}g)+\delta |\rho (f,\mathcal{S}f)-\rho (g,\mathcal{S}g)|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{5}{\int }_0^1|f+\frac{1}{4}f|dt+\frac{1}{5}{\int }_0^1|g+\frac{1}{4}g|dt\hfill \\ & +& \frac{1}{10}|{\int }_0^1|f+\frac{1}{4}f|dt-{\int }_0^1|g+\frac{1}{4}g|dt|+(1-\eta )\left(0\right)\hfill \\ & =& \frac{1}{4}{\int }_0^1|f-g|dt+\frac{1}{4}{\int }_0^1|f|dt+\frac{1}{4}{\int }_0^1|g|dt\hfill \\ & +& \frac{1}{8}|{\int }_0^1|f|-|g|dt|\hfill \\ & \ge & \frac{1}{4}{\int }_0^1|f-g|dt\hfill \\ & =& \rho (\mathcal{S}f,\mathcal{S}g).\hfill \end{array}$$

Hence, $\mathcal{S}$ is a p-cyclic Reich contraction. Then, all requirements of Theorem 5 are satisfied, and $\mathcal{S}$ has the BPP $f^{*}$ in $\Re \cup \aleph$.

Corollary 1.

Assume that ℜ or ℵ is 0-boundedly complete and $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is a p-cyclic Reich contraction, where $\delta \le 2\gamma$. Then, $\mathcal{S}$ has a BPP.

Proof. 

The conclusion follows from Remark 2. □

The following corollary generalizes the results of Popescu [3] and Sahin [16].

Corollary 2.

Let $(U,\rho )$ be a 0-complete PMS and $\mathcal{S}:U\to U$ be a mapping. Assume that there are some $\alpha ,\beta ,\gamma ,\delta \in [0,1)$, with $\alpha +\beta +\gamma +\delta <1$ and $\delta \le 2\gamma$ such that

$$\begin{array}{c}\hfill \rho (\mathcal{S}\mu ,\mathcal{S}\omega )\le \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{S}\omega )+\delta \left|\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{S}\omega )\right|.\end{array}$$

(8)

Then, $\mathcal{S}$ has a fixed point.

Proof. 

If we take $\Re =\aleph =U$ and $\rho (\Re ,\aleph )=0$ with a 0-complete PMS $(U,\rho )$, from Remark 2, we have that $(\Re ,\aleph )$ is a cyclically 0-complete pair. Furthermore, from (8), $\mathcal{S}$ shows a p-cyclic contraction mapping. Thus, by Theorem 5, there exists ${\mu }^{*}\in U$ such that

$$\begin{array}{c}\hfill \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})=\rho (\Re ,\aleph )=0,\end{array}$$

which implies that ${\mu }^{*}=\mathcal{S}{\mu }^{*}$. □

3. Best Proximity Results for a Pair of Mappings

We start this section by introducing the p-cyclic Reich contraction pair mappings.

Definition 10.

Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are two mappings. Then, $(\mathcal{S},\mathcal{T})$ is said to be a p-cyclic Reich contraction pair if there exist $\alpha ,\beta ,\gamma ,\delta \in {\mathbb{R}}^{+}$ with $\eta =\alpha +\beta +\gamma +\delta <1$ such that

$$\begin{array}{c}\hfill \rho (\mathcal{S}\mu ,\mathcal{T}\omega )\le \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{T}\omega )+\delta \left|\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{T}\omega )\right|+(1-\eta )\rho (\Re ,\aleph ),\end{array}$$

for all $\mu \in \Re$ and $\omega \in \aleph$, where $\rho (\Re ,\aleph )=inf\left\{\rho \right(\mu ,\omega ):\mu \in \Re and\omega \in \aleph \}.$ Note that if $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, then $(\mathcal{T},\mathcal{S})$ is also a p-cyclic Reich contraction pair (see [12]).

Recently, Demma et al. [25] presented the BPP results in PMSs and adapted the property $UC$ in these spaces.

Definition 11.

$(\Re ,\aleph )$ is said to satisfy the property $U{C}_{\rho }$ if the following holds. If $\left\{{\mu }_n\right\}$ and $\left\{{\nu }_n\right\}$ are sequences in ℜ and $\left\{{\omega }_n\right\}$ is a sequence in ℵ such that

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\omega }_n)& =& \rho (\Re ,\aleph ),\hfill \\ \hfill \underset{n\to \infty }{lim}\rho ({\nu }_n,{\omega }_n)& =& \rho (\Re ,\aleph ),\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\nu }_n)=0.\end{array}$$

Lemma 4

([26]). Every cyclically Cauchy sequence is bounded.

The following propositions show the relation between the $U{C}_{\rho }$ property (or boundedly compact) and the notion of a cyclically 0-complete and play important roles in obtaining some corollaries from the main result.

Proposition 4.

If ℜ and ℵ are cyclically 0-complete subsets of a PMS $(U,\rho )$ and $(\Re ,\aleph )$ satisfies the $U{C}_{\rho }$ property, then $(\Re ,\aleph )$ is a cyclically 0-complete.

As a particular case, suppose that ℜ is a subset of a PMS. Then, $(\Re ,\Re )$ is cyclically 0-complete if and only if ℜ is 0-complete. As a consequence of Lemma 4, every 0-boundedly 0-compact pair in a PMS is cyclically 0-complete. Suppose $(\Re ,\aleph )$ has the property $U{C}_{\rho }$ and ℜ, ℵ are 0-complete. If $\left\{{\mu }_n\right\}$ is a cyclically Cauchy sequence in $\Re \cup \aleph$, then sequences $\left\{{\mu }_{2n}\right\}$ and $\left\{{\mu }_{2n+1}\right\}$ are 0-Cauchy and hence $(\Re ,\aleph )$ is a cyclically 0-complete. Additionally, if $\left\{{\mu }_{2n_k}\right\}$ and $\left\{{\mu }_{2m_k+1}\right\}$ are convergent subsequences of $\left\{{\mu }_{2n}\right\}$ and $\left\{{\mu }_{2n+1}\right\}$ that converge to $\nu \in \Re$ and $\omega \in \aleph$, respectively, then $\rho (\Re ,\aleph )\le \rho (\mu ,\omega )={lim}_{k\to \infty }\rho ({\mu }_{2n_k},{\mu }_{2m_k+1})=\rho (\Re ,\aleph )$. The proofs of the following results follow the technique used in Section 2.

Proposition 5.

Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are two mappings such that $(\Re ,\aleph )$ is a p-cyclic Reich contraction pair. The sequences $\left\{{\mu }_n\right\}$ and ${\mu }_{n+1}$ are constructed as ${\mu }_{n+1}=\mathcal{S}{\mu }_n,{\mu }_{n+2}=\mathcal{T}{\mu }_{n+1}$ for all $n\in \mathbb{N}$, with the initial point $\left\{{\mu }_0\right\}\in \Re$. There is $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}),\end{array}$$

Then, the mappings $\mathcal{S}$ and $\mathcal{T}$ have a BPP.

Proof. 

Assume that $\left\{{\mu }_n\right\}$ and $\left\{{\mu }_{n+1}\right\}$ are arbitrarily defined by ${\mu }_{n+1}=\mathcal{S}{\mu }_n,{\mu }_{n+2}=\mathcal{T}{\mu }_{n+1}$, with the initial point ${\mu }_0\in \Re$. Since $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\mu }_{n+2})& =& \rho (\mathcal{S}{\mu }_n,\mathcal{T}{\mu }_{n+1})\hfill \\ \hfill & \le & \alpha \rho ({\mu }_n,{\mu }_{n+1})+\beta \rho ({\mu }_n,\mathcal{S}{\mu }_n)+\gamma \rho ({\mu }_{n+1},\mathcal{T}{\mu }_{n+1})\hfill \\ \hfill & & +\delta \left|\rho ({\mu }_n,\mathcal{S}{\mu }_n)-\rho ({\mu }_{n+1},\mathcal{T}{\mu }_{n+1})\right|+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

(9)

for all $n\in \mathbb{N}$. Now, if there is $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}),\end{array}$$

(10)

as proven in Proposition 1, from Equations (9) and (10), we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})& =& \rho (\mathcal{S}{\mu }_{n_0},\mathcal{T}{\mu }_{n_0+1})\hfill \\ & \le & (\eta -\delta )\rho ({\mu }_{n_0+1},{\mu }_{n_0+2})+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Since $\eta =\alpha +\beta +\gamma +\delta <1$, we have $\frac{1-\eta }{1-\eta +\delta }\le 1$. Hence,

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})\le \rho (\Re ,\aleph ).\end{array}$$

Then, we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }_{n_0},\mathcal{S}{\mu }_{n_0})=\rho ({\mu }_{n_0},{\mu }_{n_0+1})\hfill \\ & \le & \rho ({\mu }_{n_0+1},{\mu }_{n_0+2})=\rho ({\mu }_{n_0+1},\mathcal{T}{\mu }_{n_0+1})\hfill \\ & =& \rho (\Re ,\aleph ).\hfill \end{array}$$

Hence, we have ${\mu }_{n_0}$ as the BPP of $\mathcal{S}$, and ${\mu }_{n_0+1}$ as the BPP of $\mathcal{T}$. □

Remark 3.

Suppose that $\left\{{\mu }_n\right\}$ is a sequence, as in Proposition 4, such that there is $n_0\in \mathbb{N}$ satisfying

$$\rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}).$$

Then, the p-cyclic Reich contraction pair $(\mathcal{S},\mathcal{T})$ has a BPP. Therefore, we investigate the condition $\rho ({\mu }_{n+1},{\mu }_{n+2})\le \rho ({\mu }_n,{\mu }_{n+1})$ for all n in $\mathbb{N}$ in the following two propositions.

Proposition 6.

Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are two mappings such that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, and let ${\mu }_0\in \Re$. Then, for the sequence that is defined in Proposition 4, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

Proof. 

Since $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, by Remark 3 and following the proof of Proposition 2, we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\mu }_{n+2})& =& \rho (\mathcal{S}{\mu }_n,\mathcal{T}{\mu }_{n+1})\hfill \\ & \le & \vartheta \rho ({\mu }_n,{\mu }_{n+1})+\xi \rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$. We write $\vartheta =\frac{\alpha +\beta +\delta }{1+\delta -\gamma }$ and $\xi =\frac{1-\eta }{1+\delta -\gamma }$. From the last inequality, we obtain

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }_{n+1},{\mu }_{n+2})\hfill \\ & \le & \vartheta \rho ({\mu }_n,{\mu }_{n+1})+\xi \rho (\Re ,\aleph )\hfill \\ & \le & \vartheta \left(\vartheta \rho ({\mu }_{n-1},{\mu }_n)+\xi \rho (\Re ,\aleph )\right)+\xi \rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^2\rho ({\mu }_{n-1},{\mu }_n)+\xi (1+\vartheta )\rho (\Re ,\aleph )\hfill \\ & & ⋮\hfill \\ & \le & {\vartheta }^{n+1}\rho ({\mu }_0,{\mu }_1)+\xi \left(1+\vartheta +{\vartheta }^2+\cdots +{\vartheta }^n\right)\rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^{n+1}\rho ({\mu }_0,{\mu }_1)+\xi \left(\frac{1-{\vartheta }^{n+1}}{1-\vartheta }\right)\rho (\Re ,\aleph )\hfill \\ & =& {\vartheta }^{n+1}\rho ({\mu }_0,{\mu }_1)+\frac{1-\eta }{1-\delta }\left(1-{\vartheta }^{n+1}\right)\rho (\Re ,\aleph )\hfill \\ & \le & {\vartheta }^{n+1}\rho ({\mu }_0,{\mu }_1)+(1-{\vartheta }^{n+1})\rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$. Therefore, we conclude that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_{n+1},{\mu }_{n+2})=\rho (\Re ,\aleph ).\end{array}$$

Similarly, we can show that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

Hence, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

□

Proposition 7.

Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are two mappings such that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, and let ${\mu }_0\in \Re$. Then, the sequence $\left\{{\mu }_n\right\}$, constructed as in Proposition 5, is bounded.

Proof. 

The proof of Proposition 7 follows the same lines as the proof of Proposition 3. Hence, we have that the sequence $\left\{{\mu }_n\right\}$ is bounded. □

Theorem 6.

Assume that $(\Re ,\aleph )$ is a cyclically 0-complete pair. Let $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ be a p-cyclic Reich contraction pair. If there exist $\alpha ,\beta ,\gamma ,\delta \in [0,1)$ such that $\eta =\alpha +\beta +\gamma +\delta <1$ and $\delta \le 2\gamma$, then $\mathcal{S}$ and $\mathcal{T}$ have the BPPs.

Proof. 

Let $\left\{{\mu }_0\right\}$ be an arbitrary point in ℜ, and let $\left\{{\mu }_n\right\}$ be a sequence constructed as $\left\{{\mu }_{n+1}\right\}=\mathcal{S}{\mu }_n$, $\left\{{\mu }_{n+2}\right\}=\mathcal{T}{\mu }_{n+1}$ for all $n\in \mathbb{S}$. If there exists $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho ({\mu }_{n_0},{\mu }_{n_0+1})\le \rho ({\mu }_{n_0+1},{\mu }_{n_0+2}),\end{array}$$

then, by Proposition 5, the mappings $\mathcal{S}$ and $\mathcal{T}$ have the BPPs. Now, assume that

$$\begin{array}{c}\hfill \rho ({\mu }_{n+1},{\mu }_{n+2})\le \rho ({\mu }_n,{\mu }_{n+1}),\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$

In this case, from Proposition 6, we know that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\rho ({\mu }_n,{\mu }_{n+1})=\rho (\Re ,\aleph ).\end{array}$$

Now, we show that $\left\{{\mu }_n\right\}$ is a cyclically Cauchy. For this, it is enough to prove that ${lim}_{n,m\to \infty }\rho ({\mu }_n,{\mu }_{m+1})=\rho (\Re ,\aleph )$. Let $n\ge m$. Since $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+2},{\mu }_{m+1})& =& \rho (\mathcal{S}{\mu }_{n+1},\mathcal{T}{\mu }_m)\hfill \\ & \le & \alpha \rho ({\mu }_{n+1},{\mu }_m)+\beta \rho ({\mu }_m,\mathcal{S}{\mu }_m)+\gamma \rho ({\mu }_{n+1},\mathcal{S}{\mu }_{n+1})\hfill \\ & & +\delta |\rho ({\mu }_m,\mathcal{S}{\mu }_m)-\rho ({\mu }_{n+1},\mathcal{S}{\mu }_{n+1})|+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

for all $n\in \mathbb{N}$. Now, we follow the same lines as in the proof of Theorem 5, and deduce that

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}\rho ({\mu }_{n+2},{\mu }_{m+1})=\rho (\Re ,\aleph ).\end{array}$$

Since $(\Re ,\aleph )$ is a cyclically 0-complete pair, there exist the subsequences $\left\{{\mu }_{2n_i}\right\}$ of $\left\{{\mu }_{2n}\right\}$ and $\left\{{\mu }_{2n_j+1}\right\}$ of $\left\{{\mu }_{2n+1}\right\}$ such that $\left\{{\mu }_{2n_i}\right\}\to {\mu }^{*}\in \Re$ and $\left\{{\mu }_{2n_j+1}\right\}\to {\omega }^{*}\in \aleph$. Moreover

$$\begin{array}{ccc}\hfill \rho (\Re ,\aleph )& \le & \rho ({\mu }^{*},{\mu }_{2n_{i-1}})\hfill \\ & \le & \rho ({\mu }^{*},{\mu }_{2n_i})+\rho ({\mu }_{2n_i},{\mu }_{2n_{i-1}}).\hfill \end{array}$$

Since ${lim}_{i\to \infty }\rho ({\mu }_{2n_i},{\mu }_{2n_i-1})=\rho (\Re ,\aleph )$ implies that ${lim}_{i\to \infty }\rho ({\mu }^{*},{\mu }_{2n_i-1})=\rho (\Re ,\aleph )$, additionally, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{2n_i},\mathcal{S}{\mu }^{*})& =& \rho (\mathcal{T}{\mu }_{2n_i-1},\mathcal{S}{\mu }^{*})\hfill \\ & \le & \alpha \rho ({\mu }_{2n_i-1},{\mu }^{*})+\beta \rho ({\mu }_{2n_i-1},\mathcal{T}{\mu }_{2n_i-1})+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho ({\mu }_{2n_i-1},\mathcal{T}{\mu }_{2n_i-1})-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho ({\mu }_{2n_i-1},{\mu }^{*})+\beta \rho ({\mu }_{2n_i-1},{\mu }_{2n_i})+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho ({\mu }_{2n_i-1},{\mu }_{2n_i})-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Taking the limit $i\to \infty$ in the last inequality, we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& =& \underset{i\to \infty }{lim}({\mu }_{2n_i},\mathcal{S}{\mu }^{*})\hfill \\ & \le & \alpha \rho (\Re ,\aleph )+\beta \rho (\Re ,\aleph )+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta |\rho (\Re ,\aleph )-\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \alpha \rho (\Re ,\aleph )+\beta \rho (\Re ,\aleph )+\gamma \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\hfill \\ & & +\delta \rho (\Re ,\aleph )-\delta \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})+(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill (1-\eta +\delta )\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& \le & (1-\eta +\alpha +\beta +\delta )\rho (\Re ,\aleph )\hfill \\ \hfill \Rightarrow \rho ({\mu }^{*},\mathcal{S}{\mu }^{*})& \le & \frac{1-\delta }{1-\delta +\eta }\rho (\Re ,\aleph )\hfill \\ & \le & \rho (\Re ,\aleph ).\hfill \end{array}$$

So, $\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})=\rho (\Re ,\aleph ).$ Therefore, we have $\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})\le \rho (\Re ,\aleph )$ so $\rho ({\mu }^{*},\mathcal{S}{\mu }^{*})=\rho (\Re ,\aleph ).$ Hence, ${\mu }^{*}$ is the BPP of $\mathcal{S}$. Similarly, it can be shown that ${\omega }^{*}$ is the BPP of $\mathcal{T}$. □

By taking into account Proposition 4, we can obtain the following corollaries.

Corollary 3.

Let $(U,\rho )$ be 0-complete and $\Re ,\aleph$ be closed subsets such that $(\Re ,\aleph )$ satisfies the $U{C}_{\rho }$ property. Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are mappings such that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair. Then, $\mathcal{S}$ and $\mathcal{T}$ have the BPPs.

Proof. 

By Proposition 4, we deduce that $(\Re ,\aleph )$ is a 0-complete partial pair. Thus, from Theorem 6, we deduce that $\mathcal{S}$ and $\mathcal{T}$ have the BPPs. □

Corollary 4.

Let $\Re ,\aleph$ be closed subsets of $(U,\rho )$. Assume that $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ are two mappings such that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair. If ℜ and ℵ are 0-boundedly compact, then the mappings $\mathcal{S}$ and $\mathcal{T}$ have the BPPs.

Proof. 

From Remark 1, we know that every 0-boundedly compact subset pair $(\Re ,\aleph )$ of a partial space is cyclically 0-complete. By using Theorem 6, $\mathcal{S}$ and $\mathcal{T}$ have the BPPs. □

Example 2.

Consider the set $U=\{({\mu }_1,{\mu }_2,\cdots ,{\mu }_n,\cdots )\subseteq \Re :{\mu }_1\ge 0and{sup}_{i\in \mathbb{N}}|{\mu }_i|<\infty \}$, and define a function $\rho :U\times U\to \mathbb{R}$ by

$$\rho (\mu ,\omega )=max\{{\mu }_1,{\omega }_1\}+\underset{i\ge 2}{sup}|{\mu }_i-{\omega }_i|,$$

where $\mu =({\mu }_1,{\mu }_2,\cdots ),\omega =({\omega }_1,{\omega }_2,\cdots )\in U.$

It is clear that $(U,\rho )$ is a PMS. Define the following subsets:

$$\begin{array}{ccc}\hfill \Re & =& {\Re }_1\cup {\Re }_1=\{(a,1,\cdots ,1,\cdots ):a\in [0,1]\}\cup \{(a,-1,\cdots ,-1,\cdots ):a\in (-1,0]\},\hfill \\ \hfill \aleph & =& \left\{\right(b,0,\cdots ,0,\cdots ):b\in [0,1\left]\right\}\cup \left\{\right(b,2,\cdots ,2,\cdots ):b\in [0,1\left]\right\}.\hfill \end{array}$$

Then, $\rho (\Re ,\aleph )=1.$ It is clear that ℜ is not boundedly compact. Additionally, the pairs $(\Re ,\aleph )$ and $(\aleph ,\Re )$ do not satisfy the $U{C}_{\rho }$ property. Actually, if we choose the sequences ${\mu }_n=(\frac{1}{2n},1,\cdots ,1,\cdots )$ and ${\nu }_n=(\frac{-1}{2n},-1,\cdots ,-1,\cdots )$ in ℜ and ${\omega }_n=(0,0,\cdots ,0,\cdots )\in \aleph$ for all $n\in \mathbb{N},$ we have

$$\begin{array}{c}\hfill \rho ({\mu }_n,{\nu }_n)\to \rho (\Re ,\aleph )and\rho ({\omega }_n,{\nu }_n)\to \rho (\Re ,\aleph ),\end{array}$$

but $\rho ({\mu }_n,{\omega }_n)\to 2$. That is, $(\Re ,\aleph )$ does not satisfy the $U{C}_{\rho }$ property. Moreover, if we take the sequences ${\mu }_n=(\frac{1}{2n},2,\cdots ,2,\cdots )$ and ${\nu }_n=(\frac{1}{2n},0,\cdots ,0,\cdots )$ in ℵ and ${\omega }_n=(\frac{1}{2n},1,\cdots ,1,\cdots )$ in ℜ for all $n\in \mathbb{N},$ we obtain

$$\begin{array}{c}\hfill \rho ({\mu }_n,{\nu }_n)\to \rho (\aleph ,\Re )and\rho ({\omega }_n,{\nu }_n)\to \rho (\aleph ,\Re ),\end{array}$$

but $\rho ({\mu }_n,{\omega }_n)\to 2$. That is, $(\aleph ,\Re )$ does not satisfy the $U{C}_{\rho }$ property.

Next, we show that the pair $(\Re ,\aleph )$ is a cyclically 0-complete pair. Indeed, let $\left\{{\mu }_n\right\}$ be a cyclically Cauchy sequence in $\Re \cup \aleph$. Then, $\left\{{\mu }_{2n}\right\}\subset \Re$, $\left\{{\mu }_{2n+1}\right\}\subset \aleph ,$ and for every $\epsilon >0,$ there exists an $N\in \mathbb{N}$ such that

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_m)& <& \rho (\Re ,\aleph )+\epsilon ,\hfill \end{array}$$

(11)

when n is even, m is odd, and $n,m\ge \mathbb{N}$. Now, we show that $\left\{{\mu }_{2n}\right\}$ and $\left\{{\mu }_{2n+1}\right\}$ have convergent subsequences in ℜ and ℵ, respectively. Since ℵ is a compact subset of U, $\left\{{\mu }_{2n+1}\right\}$ has a subsequence. Now, if $\left\{{\mu }_{2n}\right\}$ has a subsequence in ${\Re }_1$, we find that $\left\{{\mu }_{2n}\right\}$ has a convergent subsequence because ${\Re }_1$ is a compact subset of U. However, conversely, suppose that the sequence $\left\{{\mu }_{2n}\right\}$ does not have a subsequence in ${\Re }_1.$ We find that there exists $n_0$ in $\mathbb{N}$ such that $\left\{{\mu }_{2n}\right\}\in {\Re }_2$ for all $n_0\le n$. In this case, it follows from (11) that we have the sequence ${\mu }_{2n}\to (0,-1)\in {\Re }_2$ as $n\to \infty$.

Now, we define two mappings $\mathcal{S}:\Re \to \aleph$ and $\mathcal{T}:\aleph \to \Re$ by

$$\mathcal{S}\left(\mu \right)=\left\{\begin{array}{cc}\left(\frac{a}{4},0,\cdots ,0,\cdots \right)\hfill & forall\mu =(a,1,\cdots ,1,\cdots )\in {\Re }_1\hfill \\ (0,0,\cdots ,0,\cdots )\hfill & forall\mu =(a,-1,\cdots ,-1,\cdots )\in {\Re }_2,\hfill \end{array}\right.$$

and

$$\mathcal{T}\left(\omega \right)=\left\{\begin{array}{cc}\left(0,-1,\cdots ,-1,\cdots \right)\hfill & forall\omega =(b,2,\cdots ,2,\cdots )\hfill \\ (\frac{b}{4},1,\cdots ,1,\cdots )\hfill & forall\mu =(b,0,\cdots ,0,\cdots ),\hfill \end{array}\right.$$

for all $\mu \in \Re$, $\omega \in \aleph$. It can be shown that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair. Then, we have the following four conditions: Choose $\alpha =\frac{1}{2},\beta =\frac{1}{4},\gamma =\frac{1}{8}$, and $\delta =\frac{1}{32}$. Then, $1-\eta =\frac{3}{32}$. Now, we show that $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair. Then, we have the following:

Case 1: Let $\mu =(a,1,\cdots ,1,\cdots )\in {\Re }_1$ and $\omega =(b,2,\cdots ,2,\cdots )\in \aleph$. We obtain

$$\begin{array}{ccc}& & \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{T}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{T}\omega )|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{2}\left(1+1\right)+\frac{1}{4}\left(1+1\right)+\frac{1}{8}\left(1+3\right)+\frac{1}{32}|2-4|+\frac{3}{32}\rho (\Re ,\aleph )\hfill \\ & =& 1+\frac{1}{2}+\frac{1}{2}+\frac{2}{32}+\frac{3}{32}\hfill \\ & >& 1+\frac{1}{4}\hfill \\ & =& \rho (\mathcal{S}\mu ,\mathcal{T}\omega ).\hfill \end{array}$$

Case 2: Let $\mu =(a,1,\cdots ,1,\cdots )\in {\Re }_1$ and $\omega =(b,0,\cdots ,0,\cdots )\in \aleph$. In this case, we have

$$\begin{array}{ccc}& & \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{T}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{T}\omega )|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{2}\left(1+1\right)+\frac{1}{4}\left(1+1\right)+\frac{1}{8}\left(1+1\right)+\frac{3}{32}\rho (\Re ,\aleph )\hfill \\ & =& 1+\frac{1}{2}+\frac{1}{4}+\frac{3}{32}\hfill \\ & >& 1+\frac{1}{4}\hfill \\ & =& \rho (\mathcal{S}\mu ,\mathcal{T}\omega ).\hfill \end{array}$$

Case 3: Let $\mu =(a,-1,\cdots ,-1,\cdots )\in {\Re }_2$ and $\omega =(b,2,\cdots ,2,\cdots )\in \aleph$. We have

$$\begin{array}{ccc}& & \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{T}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{T}\omega )|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{2}\left(1+3\right)+\frac{1}{4}+\frac{1}{8}\left(1+3\right)+\frac{1}{32}|1-4|+\frac{3}{32}\rho (\Re ,\aleph )\hfill \\ & =& 2+\frac{1}{4}+\frac{1}{2}+\frac{6}{32}\hfill \\ & >& 1=\rho (\mathcal{S}\mu ,\mathcal{T}\omega ).\hfill \end{array}$$

Case 4: Let $\mu =(a,-1,\cdots ,-1,\cdots )\in {\Re }_2$ and $\omega =(b,0,\cdots ,0,\cdots )\in \aleph$. We obtain

$$\begin{array}{ccc}& & \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{T}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{T}\omega )|+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & =& \frac{1}{2}\left(1+1\right)+\frac{1}{4}+\frac{1}{8}\left(1+1\right)+\frac{1}{32}|1-2|+\frac{3}{32}\rho (\Re ,\aleph )\hfill \\ & =& 1+\frac{1}{4}+\frac{1}{4}+\frac{1}{32}+\frac{3}{32}\hfill \\ & >& 1+\frac{1}{4}\hfill \\ & =& \rho (\mathcal{S}\mu ,\mathcal{T}\omega ).\hfill \end{array}$$

Hence, $(\mathcal{S},\mathcal{T})$ is a p-cyclic Reich contraction pair. Hence, since all the hypotheses of Theorem 6 are satisfied, $\mathcal{S}$ and $\mathcal{T}$ have the BPPs in ℜ and ℵ, respectively.

4. Best Proximity Point Results for p-Proximal Cyclic Reich Contraction

We introduce new notions of the p-proximal cyclic Reich contraction of the first and second types. We then present some BPP results for these classes of contractions.

Definition 12.

A mapping $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is said to be a p-proximal cyclic Reich contraction of the first type if there exist non-negative constants $\alpha ,\beta ,\gamma ,\delta$ such that $\eta =\alpha +\beta +\gamma +\delta <1$, $\gamma \le \beta \le \alpha +\delta$, and for all $\mu ,x,\omega ,y\in \Re \cup \aleph$, the conditions

$$\begin{array}{c}\hfill \rho (x,\mathcal{S}\mu )=\rho (y,\mathcal{S}\omega )=\rho (\Re ,\aleph ),\end{array}$$

imply that

$$\begin{array}{ccc}\hfill \rho (x,y)& \le & \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,x)+\gamma \rho (\omega ,y)\hfill \\ & & +\delta \left|\rho (\mu ,x)-\rho (\omega ,y)\right|+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Example 3.

Let $U=\mathbb{R}$ and $\rho (\mu ,\omega )=|\mu -\omega |$ be the Euclidean metrics on U. Let $\Re =[0,\frac{\pi }{2}]$ and $\aleph =[\frac{-\pi }{2},0]$ be subsets of U. Clearly, $\rho (\Re ,\aleph )=0$. We define $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ by $\mathcal{S}\mu =\frac{11}{9}sin\mu$. Therefore, $\mathcal{S}(\Re )\subset \aleph$ and $\mathcal{S}(\aleph )\subset \Re .$

So, if $\rho (\mu ,\mathcal{S}\mu )=\rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph )=0,$ then $x=\mathcal{S}\mu =\frac{-1}{9}sin\mu$ and $y=\mathcal{S}\omega =\frac{-1}{9}sin\omega$.

Therefore, we have $\rho (x,y)=\frac{-1}{9}sin\mu +\frac{-1}{9}sin\omega \le \frac{1}{9}|\mu -\omega |.$ So, $\mathcal{S}$ is a p-proximal cyclic Reich contraction of the first type on $\Re \cup \aleph$ with $\alpha =\frac{1}{9}$.

Definition 13.

A mapping $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ is said to be a p-proximal cyclic Reich contraction of the second type if there exist non-negative real constants $\alpha ,\beta ,\gamma ,\delta$ such that $\eta =\alpha +\beta +\gamma +\delta <1,$$\gamma \le \beta \le \alpha +\delta$, and for all $\mu ,x,\omega ,y\in \Re \cup \aleph$,

$$\begin{array}{c}\hfill \rho (\mathcal{S}x,\mathcal{S}\mu )=\rho (\mathcal{S}y,\mathcal{S}\omega )=\rho (\Re ,\aleph ),\end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \rho (\mathcal{S}x,\mathcal{S}y)& \le & \alpha \rho (\mathcal{S}\mu ,\mathcal{S}\omega )+\beta \rho (\mathcal{S}\mu ,\mathcal{S}x)+\gamma \rho (\mathcal{S}\omega ,\mathcal{S}y)\hfill \\ & & +\delta \left|\rho (\mathcal{S}\mu ,\mathcal{S}x)-\rho (\mathcal{S}\omega ,\mathcal{S}y)\right|+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

To introduce our new results, we need several specific concepts related to a cycling mapping $\mathcal{S}$ in the setting of PMSs.

Two subsets of ℜ and ℵ are of paramount importance, namely

$$\begin{array}{c}\hfill {\Re }_0=\{\mu \in \Re :\rho (\mu ,\omega )=\rho (\Re ,\aleph )\mathrm{for}\mathrm{some}\omega \in \aleph \},\end{array}$$

$$\begin{array}{c}\hfill {\aleph }_0=\{\omega \in \aleph :\rho (\mu ,\omega )=\rho (\Re ,\aleph )\mathrm{for}\mathrm{some}\mu \in \Re \},\end{array}$$

where

$$\begin{array}{c}\hfill \rho (\Re ,\aleph ):=inf\left\{\rho \right(\mu ,\omega ):\mu \in \Re ,\omega \in \aleph \}.\end{array}$$

Definition 14.

ℜ is said to be approximately compact with respect to ℵ if every sequence $\left\{{\mu }_n\right\}$ in ℜ, satisfying the condition $\rho (\omega ,{\mu }_n)\to \rho (\omega ,\Re )$ for some $\omega \in \aleph$, has a convergent subsequence.

Theorem 7.

Let $(\Re ,\aleph )$ be a proximately complete pair in a PMS U and ${\Re }_0\ne \varnothing$. Let $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ be a mapping satisfying the following conditions:

(i) 

$\mathcal{S}\left({\Re }_0\right)\subseteq {\aleph }_0$, $\mathcal{S}\left({\aleph }_0\right)\subseteq {\Re }_0$,

(ii) 

$\mathcal{S}$ is a continuous p-proximal cyclic Reich contraction of the first type.

Then, there exists $(\mu ,\omega )\in \Re \times \aleph$ such that

$$\begin{array}{c}\hfill \rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph ),\rho (\omega ,\mathcal{S}\omega )=\rho (\Re ,\aleph )and\rho (\mu ,\omega )=\rho (\Re ,\aleph ).\end{array}$$

Proof. 

From the assumption ${\Re }_0\ne \varnothing$, there exists ${\mu }_0\in {\Re }_0$. Since $\mathcal{S}\left({\Re }_0\right)\subseteq {\aleph }_0$, it follows that $\mathcal{S}\left({\mu }_0\right)\in {\aleph }_0$. So, there exists ${\mu }_1\in {\Re }_0$ such that $\rho ({\mu }_1,\mathcal{S}{\mu }_0)=\rho (\Re ,\aleph )$. By continuing this process, we obtain a sequence $\left\{{\mu }_n\right\}$ in ℜ such that $\rho ({\mu }_{n+1},\mathcal{S}{\mu }_0)=\rho (\Re ,\aleph )$. On the other hand, since ${\mu }_0\in {\Re }_0$, there exists ${\omega }_0\in {\aleph }_0$ such that $\rho ({\mu }_0,{\omega }_0)=\rho (\Re ,\aleph )$, and since $\mathcal{S}\left({\aleph }_0\right)\subseteq {\Re }_0$, it follows that $\mathcal{S}\left({\omega }_0\right)\subseteq {\aleph }_0$. So, there exists ${\omega }_0\in {\aleph }_0$ such that $\rho ({\omega }_0,\mathcal{S}{\omega }_0)=\rho (\Re ,\aleph )$.

By continuing this process, we obtain a sequence $\left\{{\omega }_n\right\}$ in ℵ such that $\rho ({\omega }_{n+1},\mathcal{S}{\omega }_n)=\rho (\Re ,\aleph )$.

Now, we define

$${\nu }_n=\left\{\begin{array}{cc}{\mu }_r,\hfill & n=2r,\hfill \\ {\omega }_r,\hfill & n=2r+1.\hfill \end{array}\right.$$

We will prove that $\left\{{\nu }_n\right\}$ is a cyclically Cauchy in $\Re \cup \aleph$. To do this, we need to verify that ${lim}_{m,n\to \infty }\rho ({\nu }_{2n},{\nu }_{2m+1})=\rho (\Re ,\aleph )$, or equivalently,

$$\begin{array}{c}\hfill \underset{m,n\to \infty }{lim}\rho ({\nu }_n,{\nu }_m)=\rho (\Re ,\aleph ).\end{array}$$

Let $m\le n$. Since $\rho ({\mu }_{n+1},\mathcal{S}{\mu }_n)=\rho ({\omega }_{m+1},\mathcal{S}{\omega }_m)=\rho (\Re ,\aleph )$ and $\mathcal{S}$ is a p-proximal cyclic Reich contraction of the first type on $\Re \cup \aleph$, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & \alpha \rho ({\mu }_n,{\omega }_m)+\beta \rho ({\mu }_n,{\mu }_{n+1})+\gamma \rho ({\omega }_m,{\omega }_{m+1})\hfill \\ & & +\delta \left|\rho ({\mu }_n,{\mu }_{n+1})-\rho ({\omega }_m,{\omega }_{m+1})\right|+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

By using the inequality

$$\begin{array}{ccc}\hfill \rho ({\mu }_n,{\mu }_{n+1})& \le & \rho ({\mu }_n,{\omega }_m)+\rho ({\omega }_m,{\omega }_{m+1})+\rho ({\omega }_{m+1},{\mu }_{n+1}),\hfill \end{array}$$

we obtain

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & (\alpha +\beta )\rho ({\mu }_n,{\omega }_m)+(\beta +\gamma )\rho ({\omega }_m,{\omega }_{m+1})+\beta \rho ({\omega }_{m+1},{\mu }_{n+1})\hfill \\ & & +\delta \left|\rho ({\mu }_n,{\omega }_m)+\rho ({\omega }_{m+1},{\mu }_{n+1})\right|+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph )\hfill \\ & \le & (\alpha +\beta +\delta )\rho ({\mu }_n,{\omega }_m)+(\beta +\gamma )\rho ({\omega }_m,{\omega }_{m+1})\hfill \\ & & +(\beta +\delta )\rho ({\omega }_{m+1},{\mu }_{n+1})+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph ),\hfill \end{array}$$

and hence

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & \frac{\alpha +\beta +\delta }{1-\beta -\delta }\rho ({\mu }_n,{\omega }_m)+\frac{\beta +\gamma }{1-\beta -\delta }\rho ({\omega }_m,{\omega }_{m+1})+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Since $\gamma \le \alpha +\delta$, it follows that

$$\begin{array}{c}\hfill \frac{\beta +\gamma }{1-\beta -\delta }\le \frac{\alpha +\beta +\delta }{1-\beta -\delta }=K.\end{array}$$

So, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & K\rho ({\mu }_n,{\omega }_m)+K\rho ({\omega }_m,{\omega }_{m+1})+(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

(12)

Also, note that

$$\begin{array}{c}\hfill \rho ({\omega }_m,\mathcal{S}{\omega }_{m-1})=\rho ({\omega }_{m+1},\mathcal{S}{\omega }_{m-1})=\rho (\Re ,\aleph ),\end{array}$$

and that $\mathcal{S}$ is a cyclic generalized proximal contraction of the first type, so that a manipulation yields

$$\begin{array}{ccc}\hfill \rho ({\omega }_m,{\omega }_{m+1})& =& \alpha \rho ({\omega }_{m-1},{\omega }_m)+\beta \rho ({\omega }_{m-1},{\omega }_m)+\gamma \rho ({\omega }_m,{\omega }_{m+1})\hfill \\ & & +\delta |\rho ({\omega }_{m-1},{\omega }_m)-\rho ({\omega }_m,{\omega }_{m+1})|+(1-\beta -\delta )(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \rho ({\omega }_m,{\omega }_{m+1})& =& \frac{\alpha +\beta +\delta }{1-\gamma -\delta }\rho ({\omega }_{m-1},{\omega }_m)+\gamma \rho ({\omega }_m,{\omega }_{m+1})+\frac{1-\beta -\delta }{1-\gamma -\delta }(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Since $\gamma \le \beta$, we have

$$\begin{array}{c}\hfill \rho ({\omega }_m,{\omega }_{m+1})=\frac{\alpha +\beta +\delta }{1-\beta -\delta }\rho ({\omega }_{m-1},{\omega }_m)+(1-\eta )\rho (\Re ,\aleph ),\end{array}$$

from which it follows that

$$\begin{array}{c}\hfill \rho ({\omega }_m,{\omega }_{m+1})=K\rho ({\omega }_{m-1},{\omega }_m)+(1-\eta )\rho (\Re ,\aleph ).\end{array}$$

(13)

It now follows from Equations (12) and (13) that

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & K\left[K\rho ({\mu }_{n-1},{\omega }_{m-1})+K\rho ({\omega }_{m-1},{\omega }_m)+(1-\eta )\rho (\Re ,\aleph )\right]\hfill \\ & & +K\left[K\rho ({\mu }_{m-1},{\omega }_m)+(1-\eta )\rho (\Re ,\aleph )\right]+(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & =K^2\rho ({\mu }_{n-1},{\omega }_{m-1})+2K^2\rho ({\omega }_{m-1},{\omega }_m)+2K(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & \le K^2\left[K\rho ({\mu }_{n-2},{\omega }_{m-2})+K\rho ({\omega }_{m-2},{\omega }_{m-1})+(1-\eta )\rho (\Re ,\aleph )\right]\hfill \\ & & +2K^2\left[K\rho ({\omega }_{m-2},{\omega }_{m-1})+(1-\eta )\rho (\Re ,\aleph )\right]+2K(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & \le K^3\rho ({\mu }_{n-2},{\omega }_{m-2})+3K^3\rho ({\omega }_{m-2},{\omega }_{m-1})+K(3K+2)(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

Now, we assume that the above relation holds for $r<m$, and we will show that it holds for $r=m$. To this end, we note that

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & K^{r+1}\rho ({\mu }_{n-r},{\omega }_{m-r+1})+(r+1)K^{r+1}\rho ({\omega }_{m-r},{\omega }_{m-r+1})\hfill \\ & & +K((r+1)K^r+r)(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

So, for $r=m$, we have

$$\begin{array}{ccc}\hfill \rho ({\mu }_{n+1},{\omega }_{m+1})& \le & K^{m+1}\rho ({\mu }_{n-m},{\omega }_0)+(m+1)K^m\rho ({\omega }_0,{\omega }_1)\hfill \\ & & +K((m+1)K^m+m)(1-\eta )\rho (\Re ,\aleph ).\hfill \end{array}$$

We now prove that the sequence $\left\{{\mu }_n\right\}$ is bounded. Suppose, conversely, that it is not bounded, so for

$$M=\frac{2K}{1-K}\rho ({\omega }_1,{\omega }_0)+\frac{K(1-\eta )}{1-K}\rho (\Re ,\aleph )),$$

there exists a natural number n such that $\rho ({\omega }_n,{\omega }_1)\le M$ and $\rho ({\omega }_{n+1},{\omega }_1)>M$. Now, we have

$$\begin{array}{ccc}\hfill M& <& \rho ({\mu }_{n+1},{\omega }_1)\hfill \\ & & \le K\rho ({\mu }_n,{\omega }_0)+K\rho ({\omega }_1,{\omega }_0)\hfill \\ & & +K(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & \le K\left[\rho ({\mu }_n,{\omega }_1)+\rho ({\mu }_1,{\omega }_0)\right]+K\rho ({\omega }_1,{\omega }_0)\hfill \\ & & +K(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & \le KM+2K\rho ({\omega }_1,{\omega }_0)+K(1-\eta )\rho (\Re ,\aleph )\hfill \\ & & =KM+(1-K)M=M,\hfill \end{array}$$

which is a contradiction. This argument shows that the sequence $\left\{{\mu }_n\right\}$ is bounded, so that

$$\begin{array}{ccc}\hfill & & \underset{m,n\to \infty }{lim}\rho ({\mu }_{n+1},{\omega }_{m+1})\hfill \\ & & \le \underset{m,n\to \infty }{lim}[K^{m+1}\rho ({\mu }_{n-m},{\omega }_0)+(m+1)K^m\rho ({\omega }_0,{\omega }_1)\hfill \\ & & +K((m+1)K^m+m)(1-\eta )\rho (\Re ,\aleph )]=\rho (\Re ,\aleph ).\hfill \end{array}$$

(14)

Therefore, $\left\{{\nu }_n\right\}$ is a cyclically Cauchy sequence in $\Re \cup \aleph$. From our assumption, $(\Re ,\aleph )$ is a proximally complete pair in U so that the sequences $\left\{{\mu }_n\right\}$ and $\left\{{\omega }_m\right\}$ have convergent subsequences in ℜ and ℵ, respectively, that is, there exist the subsequences $\left\{{\mu }_{n_r}\right\}$ and $\left\{{\omega }_{m_s}\right\}$ of $\left\{{\mu }_n\right\}$ and $\left\{{\omega }_m\right\}$ and $(\mu ,\omega )\in (\Re \times \aleph )$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\mu }_{n_r}=\mu ,\mathrm{and}\underset{s\to \infty }{lim}{\omega }_{m_s}=\omega .\end{array}$$

(15)

Since $\mathcal{S}$ is continuous, it follows that

$$\begin{array}{c}\hfill \rho (\mu ,\mathcal{S}\mu )=\underset{r\to \infty }{lim}\rho ({\mu }_{n_{r+1}},\mathcal{S}\mu )=\rho (\Re ,\aleph ),\end{array}$$

and

$$\begin{array}{c}\hfill \rho (\omega ,\mathcal{S}\omega )=\underset{r\to \infty }{lim}\rho ({\omega }_{m_{s+1}},\mathcal{S}{\omega }_{m_s})=\rho (\Re ,\aleph ).\end{array}$$

Finally, from Equations (14) and (15), we obtain

$$\begin{array}{c}\hfill \rho (\mu ,\omega )=\underset{r,s\to \infty }{lim}\rho ({\mu }_{n_r},{\omega }_{m_s})=\rho (\Re ,\aleph ).\end{array}$$

□

Example 4.

Let $U=\mathbb{R}$ and $\rho (\mu ,\omega )=|\mu -\omega |$ be Euclidean metrics on U. Let $\Re =[0,\frac{\pi }{2}]$ and $\aleph =[\frac{-\pi }{2},0]$ be subsets of U. Since ℜ and ℵ are closed subsets of the complete metric space $\mathbb{R}$ and $\rho (\Re ,\aleph )=0$, it follows that $(\Re ,\aleph )$ is a proximally complete pair. It is clear that ${\Re }_0={\aleph }_0=\left\{0\right\}$. Moreover, $\mathcal{S}\left({\Re }_0\right)\subseteq {\aleph }_0$ and $\mathcal{S}\left({\aleph }_0\right)\subseteq {\Re }_0$. As in Example 3, we define $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ by $\mathcal{S}\mu =\frac{-1}{9}sin\mu$. It is easy to see that $\mathcal{S}$ is a continuous p-proximal cyclic Reich contraction of the first type on $\Re \cup \aleph$. Therefore, all the conditions of Theorem 7 are fulfilled, so there exists $(\mu ,\omega )\in \Re \times \aleph$ such that

$$\rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph )\rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph )and\rho (\mu ,\omega )=\rho (\Re ,\aleph ).$$

Since $\mathcal{S}0=0,$ it follows that $\rho (0,\mathcal{S}0)=0=\rho (\Re ,\aleph ).$ Note also that $(0,0)\in \Re \times \aleph$ and $\rho (0,0)=0=\rho (\Re ,\aleph ).$

Theorem 8.

Let $(\Re ,\aleph )$ be a cyclically complete pair in a PMS U and ${\Re }_0\ne \varnothing$. Let ℜ be approximately compact with respect to ℵ and ℵ be approximately compact with respect to ℜ. Let $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ be a mapping satisfying the following conditions:

(i) 

$\mathcal{S}\left({\Re }_0\right)\subseteq {\aleph }_0$, $\mathcal{S}\left({\aleph }_0\right)\subseteq {\Re }_0,$

(ii) 

$\mathcal{S}$ is a continuous p-proximal cyclic Reich contraction of the first type.

Then, there exists $(\mu ,\omega )\in \Re \times \aleph$ such that

$$\begin{array}{c}\hfill \rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph ),\rho (\omega ,\mathcal{S}\omega )=\rho (\Re ,\aleph )and\rho (\mu ,\omega )=\rho (\Re ,\aleph ).\end{array}$$

Proof. 

We could use the same method as in the proof of Theorem 7. We can construct the sequences $\left\{{\mu }_n\right\}$ in ℜ and $\left\{{\omega }_n\right\}$ in ℵ such that $\rho ({\mu }_{n+1},\mathcal{S}{\mu }_n)=\rho (\Re ,\aleph )$ and $\rho ({\omega }_{n+1},\mathcal{S}{\omega }_n)=\rho (\Re ,\aleph ).$

Now, define the sequence $\left\{{\nu }_n\right\}$ by

$${\nu }_n=\left\{\begin{array}{cc}{\mu }_r,\hfill & n=2r,\hfill \\ {\omega }_r,\hfill & n=2r+1.\hfill \end{array}\right.$$

It follows that $\left\{{\nu }_n\right\}$ is a cyclically Cauchy sequence in $\Re \cup \aleph$. Moreover,

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}\rho ({\mu }_n,{\omega }_m)=\rho (\Re ,\aleph ).\end{array}$$

(16)

Since from the assumption, $(\Re ,\aleph )$ is a cyclically complete pair in U, either ${\mu }_n$ or ${\omega }_m$ converges. Assume that the sequence ${\mu }_n$ converges in ℜ, so there exists $\mu \in \Re$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\mu }_n=\mu .\end{array}$$

(17)

Note also that

$$\rho (\mu ,\aleph )\le \rho ({\omega }_m,\mu )\le \rho ({\omega }_m,{\mu }_n)+\rho ({\mu }_n,\mu ),$$

from which it follows that

$$\rho (\mu ,\aleph )\le \underset{m\to \infty }{lim}\underset{n\to \infty }{lim}\rho ({\omega }_m,\mu )\le \rho (\Re ,\aleph )\le \rho (\mu ,\aleph ),$$

and

$$\rho (\mu ,\aleph )=\underset{m\to \infty }{lim}\rho ({\omega }_m,\mu ).$$

Since ℵ is approximately compact with respect to ℜ, there exists a subsequence $\left\{{\omega }_{m_s}\right\}$ of $\left\{{\omega }_m\right\}$ and an element $\omega \in \aleph$ such that

$$\underset{s\to \infty }{lim}{\omega }_{m_s}=\omega .$$

Then, we use the continuity of $\mathcal{S}$ and obtain

$$\rho (\mu ,\mathcal{S}\mu )=\underset{n\to \infty }{lim}\rho ({\mu }_{n+1},\mathcal{S}{\mu }_n)=\rho (\Re ,\aleph ),$$

and

$$\rho (\omega ,\mathcal{S}\omega )=\underset{s\to \infty }{lim}\rho ({\omega }_{m_{s+1}},\mathcal{S}{\omega }_{m_s})=\rho (\Re ,\aleph ).$$

Finally, it follows from Equations (4), (16), and (17) that

$$\rho (\mu ,\omega )=\underset{n,s\to \infty }{lim}\rho ({\mu }_n,{\omega }_{m_s})=\rho (\Re ,\aleph ).$$

□

Theorem 9.

Let $(\Re ,\aleph )$ be a proximally complete pair in a PMS U and ${\Re }_0\ne \varnothing$. Let ℜ and ℵ be approximately compact with respect to each other. Let $\mathcal{S}:\Re \cup \aleph \to \Re \cup \aleph$ be a mapping satisfying the following conditions:

(i) 

$\mathcal{S}\left({\Re }_0\right)\subseteq {\aleph }_0$, $\mathcal{S}\left({\aleph }_0\right)\subseteq {\Re }_0$,

(ii) 

$\mathcal{S}$ is a continuous p-proximal cyclic Reich contraction of the second type.

Then, there exists $(\mu ,\omega )\in \Re \times \aleph$ such that

$$\begin{array}{c}\hfill \rho (\mu ,\mathcal{S}\mu )=\rho (\Re ,\aleph ),\rho (\omega ,\mathcal{S}\omega )=\rho (\Re ,\aleph )and\rho (\mu ,\omega )=\rho (\Re ,\aleph ).\end{array}$$

Proof. 

The proof follows the same lines as the proof of Theorem 7; hence, the proof is omitted. □

Theorem 10.

If in Theorem 7, instead of the assumption that $(\Re ,\aleph )$ is proximally complete, we assume that $(\Re ,\aleph )$ is cyclically complete, the conclusion of Theorem 7 holds.

Proof. 

The proof follows the same lines as the proof of Theorem 8; hence, the proof is omitted. □

5. Application to System of Integral Equations

Consider the Volterra–Hammerstein integral equation

$$\begin{array}{c}\hfill \mu \left(t\right)=\psi \left(t\right)+\tau {\int }_0^1\hslash (t,\kappa )\Im (\kappa ,\mu \left(\kappa \right))d\kappa ,\end{array}$$

(18)

for all $t\in [0,1]$, where $\psi \left(t\right)$, $\hslash (t,\kappa )$ and $\Im (\kappa ,\mu (\kappa \left)\right)$ are real-valued functions that are measurable both in t and $\kappa$ on $[0,1]$, and $\tau$ is a real number.

Let $U=L^1([0,1],\mathbb{R})$ and $\rho (\mu ,\omega )=d(\mu ,\omega )+z$ for all $\mu ,\omega \in U$, where

$$\begin{array}{c}\hfill d(\mu ,\omega )=||\mu \left(\kappa \right)-\omega \left(\kappa \right)||={\int }_0^1|\mu \left(\kappa \right)-\omega \left(\kappa \right)|d\kappa ,\end{array}$$

and z is a s positive real number. It is easy to verify that $(U,\rho )$ is a 0-complete PMS. We consider the following operator in the next theorem:

$$\begin{array}{c}\hfill \mathcal{S}\mu \left(t\right)=\psi \left(t\right)+\tau {\int }_0^1\hslash (t,\kappa )\Im (\kappa ,\mu \left(\kappa \right))d\kappa .\end{array}$$

Theorem 11.

Assume that the following hypotheses are satisfied:

(i) 

${\int }_0^1{sup}_{0\le \kappa \le 1}|\hslash (t,\kappa )|d\kappa ={\mathcal{K}}_1<+\infty .$

(ii) 

$\Im \in L^1[0,1]$ and $|\Im (\kappa ,\mu \left(\kappa \right))-\Im (\kappa ,\omega \left(\kappa \right))|+z\le \mathcal{A}$,

where $\mathcal{A}=\alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{S}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\gamma \rho (\omega ,\mathcal{S}\omega )|$, with $\alpha ,\beta ,\gamma ,\delta \in {\mathbb{R}}^{+}$ and $\alpha +\beta +\gamma +\delta \le 1$, for all $\kappa \in [0,1]$ and $\mu ,\omega \in L^1[0,1]$.

Then, the integral Equation (18) has a positive solution for each τ with $\tau {\mathcal{K}}_1<1$.

Proof. 

We first show that $\mathcal{S}$ is an operator from U into itself. Indeed, we have

$$\begin{array}{ccc}\hfill |\mathcal{S}\mu (t)|& \le & |\psi \left(t\right)|+|\tau |{\int }_0^1|\hslash (t,\kappa )\Im (\kappa ,\mu \left(\kappa \right))|d\kappa \hfill \\ & \le & |\psi \left(t\right)|+|\tau |\underset{0\le \kappa \le 1}{sup}|\hslash (t,\kappa )|{\int }_0^1|\Im (\kappa ,\mu \left(\kappa \right))|d\kappa .\hfill \end{array}$$

From assumptions $\left(i\right)$ and $\left(ii\right)$, we have

$$\begin{array}{ccc}\hfill {\int }_0^1|\mathcal{S}\mu \left(t\right)|dt& \le & {\int }_0^1|\psi \left(t\right)|dt+|\tau |{\int }_0^1\underset{0\le \kappa \le 1}{sup}|\hslash (t,\kappa )|dt{\int }_0^1|\Im (\kappa ,\mu \left(\kappa \right))|d\kappa <+\infty .\hfill \end{array}$$

This implies that $\mathcal{S}\mu \in U$. Now, consider for all $\mu ,\omega \in U$ that

$$\begin{array}{ccc}\hfill \rho (\mathcal{S}\mu ,\mathcal{S}\omega )& =& d(\mathcal{S}\mu ,\mathcal{S}\omega )+z\hfill \\ & =& ||\mathcal{S}\mu -\mathcal{S}\omega ||+z\hfill \\ & =& {\int }_0^1|\mathcal{S}\mu -\mathcal{S}\omega |dt+z\hfill \\ & =& {\int }_0^1|\tau {\int }_0^1\hslash (t,\kappa )\Im (\kappa ,\mu \left(\kappa \right))d\kappa -\tau {\int }_0^1\hslash (t,\kappa )\Im (\kappa ,\omega \left(\kappa \right))d\kappa |dt+z\hfill \\ & =& {\int }_0^1|\tau {\int }_0^1\hslash (t,\kappa )\left[\Im (\kappa ,\mu (\kappa \left)\right)-\Im (\kappa ,\omega (\kappa \left)\right)\right]d\kappa |dt+z\hfill \\ & \le & |\tau |{\int }_0^1\underset{0\le \kappa \le 1}{sup}|\hslash (t,\kappa )dt{\int }_0^1|\Im (\kappa ,\mu \left(\kappa \right))-\Im (\kappa ,\omega \left(\kappa \right))|d\kappa +z\hfill \\ & \le & |\tau |{\mathcal{K}}_1(\mathcal{A}-z)+z\hfill \\ & \le & \mathcal{A}\hfill \\ & =& \alpha \rho (\mu ,\omega )+\beta \rho (\mu ,\mathcal{S}\mu )+\gamma \rho (\omega ,\mathcal{S}\omega )+\delta |\rho (\mu ,\mathcal{S}\mu )-\rho (\omega ,\mathcal{S}\omega )|,\hfill \end{array}$$

for all $\mu ,\omega \in U$. Hence, all conditions of Corollary 2 hold, and the integral Equation (18) has a positive solution. □

6. Conclusions

The concept of a new class of contraction, called the p-cyclic Reich contraction, for single-valued mappings was introduced and studied. Next, we presented some BPP results in partial metric spaces. Then, we introduced the concept of the p-cyclic Reich contraction pair for these mappings and provided certain BPP results in partial metric spaces. After that, we presented two new concepts—the p-proximal cyclic Reich contraction of the first and second types—with an application of the main result to the existence of the solution of integral equations. We suggest that researchers prove these results for multi-valued mappings in Banach spaces or partially ordered metric linear spaces and convex metric spaces.