Best Proximity Theory in Metrically Convex Menger PM-Spaces via Cyclic Kannan Maps

Abstract

A Takahashi convex structure is considered on Menger PM-spaces and used to investigate the existence of best proximity points for weak cyclic Kannan contractions. We then introduce a concept of a probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger PM-spaces and prove that every compact and convex pair in metrically convex Menger PM-spaces has the probabilistic proximal quasi-normal structure. By applying this geometric property, we survey the existence of a best proximity point for cyclic relatively Kannan nonexpansive maps which preserves distance. In order to provide more accurate results, we obtain the same conclusions in the framework of CAT$\left(0\right)$ spaces.

1. Introduction

The well-known Schauder’s fixed point theorem [1] states that if $\mathcal{U}$ is a nonempty, compact, and convex subset of a Banach space X, and $\mathcal{S}$ maps $\mathcal{U}$ into itself continuously, then $\mathcal{S}$ has a fixed point. Schauder’s fixed point problem is one of the most important tools in nonlinear analysis, and in particular, it has played a basic role in the development of fixed point theory and the theory of differential equations.

Now assume that $\mathcal{U},\mathcal{V}$ are nonempty subsets of a metric space $(\mathcal{M},d)$ and let $\mathcal{S}:\mathcal{U}\to \mathcal{V}$ be a non-self-mapping. For this mapping, the fixed point equation $\mathcal{S}u=u$ may not have a solution; however, it is contemplated to find an approximate solution $u\in \mathcal{U}$ for which the error $d(u,\mathcal{S}u)$ is minimal. Indeed, this is the idea behind best approximation theory.

The well-known best approximation theorem was established by Ky Fan as shown below.

Theorem 1

([2]). Let $\mathcal{U}$ be a nonempty compact convex subset of a normed linear space X, and assume that $\mathcal{S}:\mathcal{U}\to X$ is a continuous mapping. Then there is a point $u\in \mathcal{U}$ such that

$$\parallel u-\mathcal{S}u\parallel =\mathrm{dist}(\mathcal{S}u,\mathcal{U}):=inf\{\parallel \mathcal{S}u-x\parallel :x\in \mathcal{U}\}.$$

The point u in Theorem 1 is called a best approximant point of $\mathcal{S}$.

It is worth noticing that if in Theorem 1 $\mathcal{S}\left(\mathcal{U}\right)\subseteq \mathcal{U}$, then by applying Schauder’s fixed point theorem, we obtain the existence of a fixed point of the map $\mathcal{S}$, and so, Ky Fan’s best approximation theorem can be considered a generalization of Schauder’s fixed point theorem.

Motivated by Ky Fan’s best approximation theorem, the concept of best proximity points for non-self-mappings can be defined in the following way.

Definition 1.

Let $\mathcal{U}$ and $\mathcal{V}$ be nonempty subsets of a metric space $(\mathcal{M},d)$ and $\mathcal{S}:\mathcal{U}\to \mathcal{V}$ be a non-self-mapping. A point $p\in \mathcal{U}$ is called a best proximity point of $\mathcal{S}$ whenever $d(p,\mathcal{S}p)=\mathrm{dist}(\mathcal{U},\mathcal{V})$.

Note that best proximity point theorems are studied to find necessary conditions to guarantee the existence of a solution to the minimization problem

$$\begin{array}{c}\hfill \underset{u\in \mathcal{U}}{min}d(u,\mathcal{S}u).\end{array}$$

(1)

The existence and convergence of best proximity points is an interesting subject of optimization theory which was first studied in [3] for cyclic relatively nonexpansive maps and then in [4] for cyclic contractions in the framework of uniformly convex Banach spaces.

More precisely, let us take into account a mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$, where $\mathcal{U}$ and $\mathcal{V}$ are two nonempty subsets of a metric space $\left(\mathcal{M},d\right)$. The mapping $\mathcal{S}$ is referred to as cyclic if it satisfies the conditions $\mathcal{S}\left(\mathcal{U}\right)\subset \mathcal{V}$ and $\mathcal{S}\left(\mathcal{V}\right)\subset \mathcal{U}$. When $\mathcal{U}\cap \mathcal{V}=\varnothing$, it follows that a cyclic mapping cannot possess any fixed points. Under such circumstances, it becomes meaningful to investigate the presence of best proximity points (briefly, BPPs) $p\in \mathcal{U}\cup \mathcal{V}$, which are characterized by the condition

$$d(p,\mathcal{S}p)=\mathrm{dist}(\mathcal{U},\mathcal{V}):=inf\left\{d(u,v):(u,v)\in \mathcal{U}\times \mathcal{V}\right\}.$$

The significance of BPPs lies in their role as optimal solutions to the problem of best approximation between two disjoint sets.

Eldred et al. [3] demonstrated the presence of BPPs for cyclic relatively nonexpansive mappings utilizing proximal normal structure in the framework of Banach spaces. It was proved that every cyclic relatively nonexpansive mapping defined on a union of two nonempty, weakly compact and convex subsets $\mathcal{U},\mathcal{V}$ of a Banach space X has a BPP whenever the pair $(\mathcal{U},\mathcal{V})$ has the proximal normal structure. We recall that every compact and convex pair in a Banach space X possesses a proximal normal structure (see Proposition 2.2 of [3]).

A mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$ is referred to as a weak cyclic Kannan contraction map if $\mathcal{S}$ is cyclic and provides the inequality given by

$$d(\mathcal{S}u,\mathcal{S}v)\le r\left[d(u,\mathcal{S}u)+d(v,\mathcal{S}v)\right]+\left(1-2r\right)\mathrm{dist}(\mathcal{U},\mathcal{V})$$

for some $r\in (0,{\displaystyle \frac{1}{2}})$, and for all $\left(u,v\right)\in \mathcal{U}\times \mathcal{V}$ [5]. If in the above definition $r={\displaystyle \frac{1}{2}}$, then $\mathcal{S}$ is said to be relatively Kannan nonexpansive.

In [6], a BPP theorem for weak cyclic Kannan contraction mappings in Banach spaces is established. Furthermore, the existence of BPPs for relatively Kannan nonexpansive mappings which are relatively nonexpansive was investigated in [6] and after that it was extended to convex metric spaces (see Theorem 3.2 of [7]). Indeed, within the framework of convex metric spaces, in [7], a novel class of Kannan-type cyclic orbital contractions was introduced, and the existence of BPPs was examined. Subsequently, the same problem is addressed for cyclic relatively Kannan nonexpansive mappings through the employment of the concept of proximal quasi-normal structure in Banach spaces (Theorem 3.8 of [6]) and then in the convex metric setting (Theorem 4.4 of [7]).

In [8], the notions of normal structure and metrically convexity in Menger PM-space were defined and the existence of a common fixed point in such spaces obtained. The primary objective of this study is to provide the existence of BPPs for weak cyclic Kannan contractions within the framework of metrically convex Menger PM-spaces. Also, we introduce the concept of probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger PM-spaces and using this geometric property, we examine the existence of a BPP for cyclic relatively Kannan nonexpansive maps which preserves distance. These results establish a significant basis for the existence of BPPs in Menger PM-spaces through cyclic Kannan maps.

This study is organized as follows. After some standard definitions and notations are given in the theory of Menger PM-spaces in the next section, in Section 3, we demonstrate the existence of BPPs for the class of weak cyclic Kannan contractions that is defined on a union of two nonempty subsets of a metrically convex Menger PM-space. This is followed by the concept of probabilistic proximal quasi-normal structure in Section 4, and we obtain another BPP result for cyclic relatively Kannan nonexpansive mappings which preserves distance in the context of metrically convex Menger PM-spaces using this concept. Finally, in Section 5, we consider the category of CAT$\left(0\right)$ spaces as a suitable subfamily of metrically convex Menger PM-spaces which satisfy in some appropriate geometric properties and obtain similar existence results of BPPs for cyclic Kannan contractions and cyclic relatively nonexpansive maps in such spaces.

2. Preliminaries

Probabilistic metric space is a space in which the distance between two points is given by a distribution function rather than a non-negative number. The concept of this space was introduced by Menger [9] in 1942. After that, a comprehensive study was given by Scheweizer and Sklar [10] in 1983.

Let us denote by ${\mathcal{D}}^{+}$ the set of all distribution functions $f:\mathbb{R}\to [0,1]$ that is nondecreasing, left-continuous, $f\left(0\right)=0$ and ${sup}_{p\in \mathbb{R}}f\left(p\right)=1$. The space ${\mathcal{D}}^{+}$ has a maximal element ${ϵ}_0$ as

$$\begin{array}{c}\hfill {ϵ}_0\left(t\right)=\left\{\begin{array}{cc}0,\hfill & t\le 0,\hfill \\ 1,\hfill & t>0.\hfill \end{array}\right.\end{array}$$

Definition 2

([10]). A continuous triangular norm (t-norm) is an operation $\mathcal{T}:[0,1]\times [0,1]\to [0,1]$ satisfying the conditions below:

(i)

$\mathcal{T}$ is associative and commutative;

(ii)

$\mathcal{T}$ is continuous;

(iii)

$\mathcal{T}(s,1)=s$ for all $s\in [0,1]$;

(iv)

$\mathcal{T}(s_1,s_2)\le \mathcal{T}(q_1,q_2)$ whenever $s_1\le q_1$ and $s_2\le q_2$, and $s_1,s_2,q_1,q_2\in [0,1]$.

Some fundamental examples of continuous t-norms are ${\mathcal{T}}_{\pi }(s,q)=sq$ and ${\mathcal{T}}_m(s,q)=min\{s,q\}$.

Definition 3

([10]). A Menger probabilistic metric space (briefly, Menger PM-space) is a triple $(\mathcal{M},\mathcal{F},\mathcal{T})$ where $\mathcal{T}$ is a continuous t-norm, $\mathcal{M}$ is a nonempty set, and $\mathcal{F}:\mathcal{M}\times \mathcal{M}⟶{\mathcal{D}}^{+}$ is a mapping such that, if $F_{p,q}$ indicates the value of $\mathcal{F}$ at $(p,q)$, the following hold:

(1)

$F_{p,q}\left(t\right)={ϵ}_0\left(t\right)\iff p=q$, $t>0$;

(2)

$F_{p,q}=F_{q,p}$, for all $p,q\in \mathcal{M}$ (symmetry property);

(3)

$F_{p,r}(t_1+t_2)\ge \mathcal{T}\left(F_{p,q}\left(t_1\right),F_{q,r}\left(t_2\right)\right)$, for all $p,q,r\in \mathcal{M}$ and $t_1,t_2\ge 0$.

Remark 1

([11]). Every metric space is a Menger PM-space. Indeed, if $(\mathcal{M},d)$ is a metric space and $\mathcal{T}={\mathcal{T}}_m$, then by defining $F_{p,q}\left(t\right)={ϵ}_0(t-d(p,q))$ for all $p,q\in \mathcal{M}$ and $t>0$, the triple $(\mathcal{M},\mathcal{F},\mathcal{T})$ is a Menger PM-space induced by d.

It is widely acknowledged [10] that the $(\epsilon ,\mu )$-topology in a Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T})$ is given by the family of neighborhoods

$$\begin{array}{c}\hfill {\mathfrak{N}}_u=\left\{{\mathcal{N}}_u(\epsilon ,\mu ):\epsilon >0,\mu \in (0,1)\right\},\forall u\in \mathcal{M},\end{array}$$

where

$$\begin{array}{c}\hfill {\mathcal{N}}_u(\epsilon ,\mu ):=\left\{v\in X:F_{u,v}\left(\epsilon \right)>1-\mu \right\}.\end{array}$$

The closed $(\epsilon ,\mu )$-neighborhood of a point $u\in \mathcal{M}$ is given by

$$\begin{array}{c}\hfill {\mathcal{N}}_u[\epsilon ,\mu ]=\left\{v\in X:F_{u,v}\left(\epsilon \right)\ge 1-\mu \right\}.\end{array}$$

Definition 4

([12]). Let $(\mathcal{M},\mathcal{F},\mathcal{T})$ be a Menger PM-space and $\mathcal{U}\subseteq \mathcal{M}$. Define a map ${\nabla }_{\mathcal{U}}\left(t\right):(0,\infty )\to [0,1]$ with

$$\begin{array}{c}\hfill {\nabla }_{\mathcal{U}}\left(t\right):=\underset{p,q\in \mathcal{U}}{inf}\underset{\epsilon <t}{sup}{F}_{p,q}\left(\epsilon \right).\end{array}$$

The constant ${\displaystyle {\nabla }_{\mathcal{U}}:=\underset{t>0}{sup}{\nabla }_{\mathcal{U}}\left(t\right)}$ is called the probabilistic diameter of $\mathcal{U}$. In case ${\nabla }_{\mathcal{U}}=1$, it is said that $\mathcal{U}$ is probabilistic bounded (briefly p-bounded).

In 1970, Takahashi introduced a convex structure on metric spaces [13]. After that, Hadžić generalized Takahashi’s convex structure to the Menger PM-spaces as follows.

Definition 5

([14]). Let $(\mathcal{M},\mathcal{F},\mathcal{T})$ be a Menger PM-space. A map $\mathcal{W}:\mathcal{M}\times \mathcal{M}\times [0,1]\to \mathcal{M}$ is called a convex structure on $\mathcal{M}$ if for every $(p,q)\in \mathcal{M}\times \mathcal{M}$, $\mathcal{W}(p,q,0)=q,\mathcal{W}(p,q,1)=p$ and for all $p,q,r\in \mathcal{M}$, $\lambda \in (0,1)$ and $t>0$,

$$\begin{array}{c}\hfill F_{\mathcal{W}(p,q,\lambda ),r}\left(2t\right)\ge \mathcal{T}\left(F_{p,r}\left({\displaystyle \frac{t}{\lambda }}\right),F_{q,r}\left({\displaystyle \frac{t}{1-\lambda }}\right)\right).\end{array}$$

Throughout this paper we denote by $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ a Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T})$ equipped with a convex structure $\mathcal{W}:\mathcal{M}\times \mathcal{M}\times [0,1]\to \mathcal{M}$ and we call it a convex Menger PM-space.

Definition 6.

A subset $\mathcal{G}$ of a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is called a convex set if for every $p,q\in \mathcal{G}$ and $\lambda \in [0,1]$ it holds that $\mathcal{W}(p,q,\lambda )\in \mathcal{G}$.

Lemma 1

([8]). If $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is a convex Menger PM-space and ${\left\{{\mathcal{G}}_{\alpha }\right\}}_{\alpha \in \Gamma }$ is a family of convex subsets of $\mathcal{M}$, then ${\displaystyle \mathcal{G}=\bigcap _{\alpha \in \Gamma }{\mathcal{G}}_{\alpha }}$ is a convex set.

Definition 7.

A convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is said to have property (C) if every decreasing net consists of nonempty, p-bounded, closed, and convex subsets of $\mathcal{M}$ possessing a nonempty intersection.

It is worth mentioning that every compact and convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ possesses the property (C). Moreover, every reflexive Banach space can be considered a convex Menger PM-space which has the property (C) (note that in this case the convex structure is defined with $\mathcal{W}(x,y,\lambda )=(1-\lambda )x+\lambda y$, where $(x,y,\lambda )\in \mathcal{M}\times \mathcal{M}\times [0,1]$).

Definition 8

([8]). A convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is called metrically convex if for every $p,q\in \mathcal{M}$, and $\lambda \in (0,1)$, there exists a unique element $r=\mathcal{W}(p,q,\lambda )\in \mathcal{M}$ for which

$$\begin{array}{c}\hfill F_{p,q}\left({\displaystyle \frac{t}{\lambda }}\right)=F_{r,q}\left(t\right),F_{p,q}\left({\displaystyle \frac{t}{1-\lambda }}\right)=F_{r,p}\left(t\right),\forall t>0.\end{array}$$

For instance, every normed linear space can be considered a metrically convex Menger PM-space.

The following lemmas are needed to be used in the proofs of the theorems in the upcoming sections. Also, we give the condition $(♮)$ in the next lemma.

Lemma 2

(Lemma 3.7 of [8]). Let $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ be a convex Menger PM-space. Assume that for every $\lambda \in (0,1),t>0$ and $p,q,r\in \mathcal{M}$, the following condition holds:

$$F_{\mathcal{W}(p,q,\lambda ),r}\left(t\right)>min\{F_{r,p}\left(t\right),F_{r,q}\left(t\right)\}.(♮)$$

If there exists $z\in \mathcal{M}$ for which

$$F_{\mathcal{W}(p,q,\lambda ),z}\left(t\right)=min\{F_{z,p}\left(t\right),F_{z,q}\left(t\right)\},$$

for all $t>0$, then $\mathcal{W}(p,q,\lambda )\in \{p,q\}$.

Lemma 3

(Lemma 3.8 of [8]). If $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is a metrically convex Menger PM-space, then for any $p,q\in \mathcal{M}$ with $p\ne q$, there exists $\lambda \in (0,1)$ such that $\mathcal{W}(p,q,\lambda )\notin \{p,q\}$.

Lemma 4

(Lemma 4.2 of [8]). If $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is a convex Menger PM-space which satisfies the condition $(♮)$, then the closed $(\epsilon ,\mu )$-neighborhoods are convex sets.

Let $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ be a convex Menger PM-space, and $\mathcal{U}$ and $\mathcal{V}$ be two nonempty subsets of $\mathcal{M}$. It is said that a pair $(\mathcal{U},\mathcal{V})$ in $\mathcal{M}$ provides a property if both $\mathcal{U}$ and $\mathcal{V}$ provide that property. For example, $(\mathcal{U},\mathcal{V})$ is convex if and only if $\mathcal{U}$ and $\mathcal{V}$ are convex; $({\mathcal{U}}^{\prime },{\mathcal{V}}^{\prime })\subseteq (\mathcal{U},\mathcal{V})\iff {\mathcal{U}}^{\prime }\subseteq \mathcal{U},\mathrm{and}{\mathcal{V}}^{\prime }\subseteq \mathcal{V}$. We shall also employ the following notations:

$$\begin{array}{cc}\hfill {\nabla }_{p,\mathcal{V}}\left(t\right)& :=\underset{q\in \mathcal{V}}{inf}\underset{\epsilon <t}{sup}{F}_{p,q}\left(\epsilon \right),\forall p\in \mathcal{M},\forall t>0,\hfill \\ \hfill {\nabla }_{\mathcal{U},\mathcal{V}}\left(t\right)& :=\underset{(p,q)\in \mathcal{U}\times \mathcal{V}}{inf}\underset{\epsilon <t}{sup}{F}_{p,q}\left(\epsilon \right),\forall t>0,\hfill \\ \hfill {\nabla }_{\mathcal{U},\mathcal{V}}& :=\underset{t>0}{sup}{\nabla }_{\mathcal{U},\mathcal{V}}\left(t\right),\hfill \\ \hfill {\nabla }_{\mathcal{U}}\left(t\right)& :={\nabla }_{\mathcal{U},\mathcal{U}}\left(t\right)=\underset{p\in \mathcal{U}}{inf}{\nabla }_{p,\mathcal{V}}\left(t\right),\forall t>0.\hfill \end{array}$$

The closed convex hull of the set $\mathcal{U}$ is described as follows:

$$\overline{\mathrm{con}}\left(\mathcal{U}\right):=\bigcap \left\{\mathcal{G}:\mathcal{G}\mathrm{is}\mathrm{a}\mathrm{closed}\mathrm{and}\mathrm{convex}\mathrm{subset}\mathrm{of}\mathcal{M}\mathrm{such}\mathrm{that}\mathcal{G}\supseteq \mathcal{U}\right\}.$$

The probabilistic distance (briefly p-distance) between the sets $\mathcal{U}$ and $\mathcal{V}$ is defined as

$${\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right):=\underset{(p,q)\in \mathcal{U}\times \mathcal{V}}{sup}\underset{ϵ<t}{sup}{F}_{p,q}\left(ϵ\right),\forall t>0.$$

Clearly, if $\mathcal{U}\cap \mathcal{V}\ne \varnothing$, then

$${\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)={ϵ}_0\left(t\right),\forall t>0.$$

Note that the p-distance between $p\in \mathcal{M}$ and the set $\mathcal{V}$ is indicated by ${\mathfrak{D}}_{p,\mathcal{V}}\left(t\right)$, for all $t>0$.

A pair $(p,q)\in \mathcal{U}\times \mathcal{V}$ is called proximal in $(\mathcal{U},\mathcal{V})$ if $F_{p,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$. Moreover, the probabilistic proximal pair (p-proximal pair) of $(\mathcal{U},\mathcal{V})$ is indicated by $({\mathcal{U}}_0,{\mathcal{V}}_0)$ and defined as shown below:

$${\displaystyle {\mathcal{U}}_0=\{p\in \mathcal{U}:F_{p,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\mathrm{for}\mathrm{all}t>0,\mathrm{for}\mathrm{some}q\in \mathcal{V}\},}$$

$${\displaystyle {\mathcal{V}}_0=\{q\in \mathcal{V}:F_{p,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\mathrm{for}\mathrm{all}t>0,\mathrm{for}\mathrm{some}p\in \mathcal{U}\}.}$$

It needs to be noted that the p-proximal pair $({\mathcal{U}}_0,{\mathcal{V}}_0)$ may be empty. We refer to ([15]) for some sufficient conditions to obtain the nonemptiness of p-proximal pairs.

The next lemma will be used in our coming discussions.

Lemma 5

([15]). Let $(\mathcal{U},\mathcal{V})$ be a nonempty, p-bounded, closed, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which satisfies the condition $(♮)$. If $\mathcal{M}$ has the property (C), then $({\mathcal{U}}_0,{\mathcal{V}}_0)$ is also nonempty, p-bounded, closed, and convex. Moreover, ${\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)={\mathfrak{D}}_{{\mathcal{U}}_0,{\mathcal{V}}_0}\left(t\right)$ for all $t>0$.

Definition 9.

A pair $(\mathcal{U},\mathcal{V})$ in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is said to be probabilistic proximinal (briefly, p-proximinal) provided that

$${\mathcal{U}}_0=\mathcal{U},{\mathcal{V}}_0=\mathcal{V}.$$

We also need the next lemma.

Lemma 6

([15]). If $(\mathcal{U},\mathcal{V})$ is a nonempty pair in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$, then we have

$${\nabla }_{\overline{\mathrm{con}}\left(\mathcal{U}\right),\overline{\mathrm{con}}\left(\mathcal{V}\right)}\left(t\right)={\nabla }_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0.$$

Definition 10.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty pair of subsets of a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$. A mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$ is called cyclic if $\mathcal{S}\left(\mathcal{U}\right)\subseteq \mathcal{V}$ and $\mathcal{S}\left(\mathcal{V}\right)\subseteq \mathcal{U}$. A point $p\in \mathcal{U}\cup \mathcal{V}$ is called a best proximity point (BPP for brief) for a cyclic mapping $\mathcal{S}$ if

$$F_{p,\mathcal{S}p}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0.$$

Notation: We denote by $\mathcal{B}pp\left(\mathcal{S}\right)$ for the set of all BPPs for the cyclic mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$.

To see the existence results of BPPs for non-self-maps in Menger PM-spaces, we refer to [16,17,18].

3. Weak Cyclic Kannan Contractions

In this section, we take into account the class of weak cyclic Kannan contractions and ensure the existence of BPPs in the context of metrically convex Menger PM-spaces.

Definition 11.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty pair of subsets of a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$. A mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$ is called a weak cyclic Kannan contraction if $\mathcal{S}$ is cyclic and

$$\begin{array}{c}\hfill F_{\mathcal{S}p,\mathcal{S}q}\left(t\right)\ge r\left[F_{p,\mathcal{S}p}\left(t\right)+F_{q,\mathcal{S}q}\left(t\right)\right]+(1-2r){\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)\end{array}$$

(2)

for some $r\in \left(0,{\displaystyle \frac{1}{2}}\right)$, for all $\left(p,q\right)\in \mathcal{U}\times \mathcal{V}$ and $t>0.$

The following theorem is the first existence result of BPPs.

Theorem 2.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, p-bounded, closed, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which has the property (C). Assume that $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a weak cyclic Kannan contraction. Then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Proof. 

Let Y indicate the set of all nonempty, p-bounded, closed, and convex pairs $(\mathcal{E},\mathcal{F})$ which are subsets of $(\mathcal{U},\mathcal{V})$ such that $\mathcal{S}$ is cyclic on $\mathcal{E}\cup \mathcal{F}$. This collection is nonempty since $(\mathcal{U},\mathcal{V})\in \mathrm{Y}$. Also, $\mathrm{Y}$ is partially ordered by the reverse inclusion. Since $\mathcal{M}$ possesses the property (C), every increasing chain in $\mathrm{Y}$ is bounded above. So, employing Zorn’s lemma, we derive a minimal element $({\mathcal{C}}_1,{\mathcal{C}}_2)\in \mathrm{Y}$.

We note that $\left(\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right)\right)$ is a nonempty, p-bounded, closed, and convex pair in $\mathcal{M}$ and $\left(\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right)\right)\subseteq ({\mathcal{C}}_1,{\mathcal{C}}_2)$. Further,

$$\mathcal{S}\left(\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right)\right)\subseteq \mathcal{S}\left({\mathcal{C}}_1\right)\subseteq \overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right),$$

and also

$$\mathcal{S}\left(\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right)\right)\subseteq \mathcal{S}\left({\mathcal{C}}_2\right)\subseteq \overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right),$$

that is, $\mathcal{S}$ is cyclic on $\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right)\cup \overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right).$ It now follows from the minimality of $({\mathcal{C}}_1,{\mathcal{C}}_2)$ that

$$\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right)={\mathcal{C}}_1,\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_1\right)\right)={\mathcal{C}}_2.$$

Let $x\in {\mathcal{C}}_1$ and $t>0$ be arbitrary. If $y\in {\mathcal{C}}_2$, then by the fact that $\mathcal{S}$ is a weak cyclic Kannan contraction, we obtain

$$\begin{array}{cc}\hfill F_{\mathcal{S}x,\mathcal{S}y}\left(t\right)& \ge r\left[F_{x,\mathcal{S}x}\left(t\right)+F_{y,\mathcal{S}y}\left(t\right)\right]+(1-2r){\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)\hfill \\ \hfill & \ge 2r{\nabla }_{{\mathcal{C}}_1,{\mathcal{C}}_2}\left(t\right)+(1-2r){\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right).\hfill \end{array}$$

Set

$$1-\mu :=2r{\nabla }_{{\mathcal{C}}_1,{\mathcal{C}}_2}\left(t\right)+(1-2r){\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right).$$

Then $F_{\mathcal{S}x,\mathcal{S}y}\left(t\right)\ge 1-\mu$, and so $\mathcal{S}y\in {\mathcal{N}}_{\mathcal{S}x}[t,\mu ]$ for all $y\in {\mathcal{C}}_2$. Thereby $\mathcal{S}\left({\mathcal{C}}_2\right)\subseteq {\mathcal{N}}_{\mathcal{S}x}[t,\mu ]$ which deduces that

$${\mathcal{C}}_1=\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{C}}_2\right)\right)\subseteq {\mathcal{N}}_{\mathcal{S}x}[t,\mu ].$$

Thus for any $u\in {\mathcal{C}}_1$, we have $F_{u,\mathcal{S}x}\left(t\right)\ge 1-\mu$ and so

$${\nabla }_{{\mathcal{C}}_1,\mathcal{S}x}\left(t\right)=\underset{u\in {\mathcal{C}}_1}{inf}{F}_{u,\mathcal{S}x}\left(t\right)\ge 1-\mu ,\forall x\in {\mathcal{C}}_1,$$

which ensures that

$${\nabla }_{{\mathcal{C}}_1,\mathcal{S}\left({\mathcal{C}}_1\right)}\left(t\right)=\underset{x\in {\mathcal{C}}_1}{inf}{\nabla }_{{\mathcal{C}}_1,\mathcal{S}x}\left(t\right)\ge 1-\mu .$$

Likewise, it can be shown that ${\nabla }_{\mathcal{S}\left({\mathcal{C}}_2\right),{\mathcal{C}}_2}\left(t\right)\ge 1-\mu$. It now follows from Lemma 6 that

$$\begin{array}{cc}\hfill {\nabla }_{{\mathcal{C}}_1,{\mathcal{C}}_2}\left(t\right)& ={\nabla }_{\overline{\mathrm{con}}\mathcal{S}\left({\mathcal{C}}_2\right),{\mathcal{C}}_2}\left(t\right)\hfill \\ \hfill & ={\nabla }_{\mathcal{S}\left({\mathcal{C}}_2\right),{\mathcal{C}}_2}\left(t\right)\hfill \\ \hfill & \ge 1-\mu \hfill \\ \hfill & =2r{\nabla }_{{\mathcal{C}}_1,{\mathcal{C}}_2}\left(t\right)+(1-2r){\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right).\hfill \end{array}$$

Therefore,

$${\nabla }_{{\mathcal{C}}_1,{\mathcal{C}}_2}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0,$$

Then for all $(p,q)\in {\mathcal{C}}_1\times {\mathcal{C}}_2$,

$$F_{p,\mathcal{S}p}\left(t\right)=F_{\mathcal{S}q,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0,$$

that is, any element $(p,q)\in {\mathcal{C}}_1\times {\mathcal{C}}_2$ is a member of $\mathcal{B}pp\left(\mathcal{S}\right)$ and the proof is completed.    □

The following corollaries are derived from Theorem 2, easily.

Corollary 1.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, closed, and convex pair in a metrically convex and compact Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$. If $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a weak cyclic Kannan contraction, then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Corollary 2.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, bounded, closed, and convex pair in a reflexive Banach space X. If $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a weak cyclic Kannan contraction, then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Example 1.

Consider the Euclidean metric space $\mathbb{R}$ and let $\mathcal{U}=\mathcal{V}=\mathbb{R}$. Define a map $\mathcal{S}:\mathcal{U}\to \mathcal{U}$ with

$$\mathcal{S}u=\left\{\begin{array}{c}0,\mathit{if}u\le 2;\hfill \\ -{\displaystyle \frac{1}{2}},\mathit{if}u>2.\hfill \end{array}\right.$$

It is easy to check that $\mathcal{S}$ is a weak cyclic Kannan contraction and $p=0$ is a BPP of $\mathcal{S}$, which is a fixed point in this case.

4. Cyclic Relatively Kannan Nonexpansive Mappings

In this section, we consider the class of relatively Kannan nonexpansive mappings which preserves distance in the context of metrically convex Menger PM-spaces and establish the existence of BPPs for such maps. As an application of our main existence result, we obtain a new fixed point theorem for Kannan nonexpansive maps in metrically convex Menger PM-spaces.

4.1. Probabilistic Proximal Quasi-Normal Structure

Firstly, we need to rearrange the concept of relatively Kannan nonexpansive maps in the setting of Menger PM-spaces.

Definition 12.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty pair of subsets of a Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T})$. A cyclic mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}⟶\mathcal{U}\cup \mathcal{V}$ is said to be the following:

• Relatively Kannan nonexpansive if

  $$\begin{array}{c}\hfill F_{\mathcal{S}p,\mathcal{S}q}\left(t\right)\ge {\displaystyle \frac{1}{2}}\left[F_{p,\mathcal{S}p}\left(t\right)+F_{q,\mathcal{S}q}\left(t\right)\right]\end{array}$$

  for all $\left(p,q\right)\in \mathcal{U}\times \mathcal{V}$ and $t>0$. In the case that $\mathcal{U}=\mathcal{V}$, then we say that $\mathcal{S}$ is a Kannan nonexpansive map.
• Preserve distance if for all $\left(p,q\right)\in \mathcal{U}\times \mathcal{V}$ and $t>0$ with $F_{p,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ we have $F_{\mathcal{S}p,\mathcal{S}q}\left(t\right)=F_{p,q}\left(t\right)$.

The next lemma will be utilized in the proof of our main theorem of this section.

Lemma 7.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, p-bounded, closed, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which has the property (C) and satisfies the condition $(♮)$. Suppose that $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a cyclic mapping which preserves distance. Then there is a pair $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ that is minimal with respect to being nonempty, closed, convex pair of subsets of $(\mathcal{U},\mathcal{V})$ so that $\mathcal{S}$ is cyclic on ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ and

$${\mathfrak{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0.$$

Moreover, $({\mathcal{H}}_1,{\mathcal{H}}_2)$ is p-proximinal.

Proof. 

We note that $({\mathcal{U}}_0,{\mathcal{V}}_0)$ is a nonempty, closed, convex pair with ${\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)={\mathfrak{D}}_{{\mathcal{U}}_0,{\mathcal{V}}_0}\left(t\right)$ for all $t>0$ by Lemma 5. Also, if $p\in {\mathcal{U}}_0$, then there is $q\in {\mathcal{V}}_0$ such that $F_{p,q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$. Since $\mathcal{S}$ preserves distance $F_{\mathcal{S}p,\mathcal{S}q}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$ and so $\mathcal{S}p\in {\mathcal{V}}_0$, for all $p\in {\mathcal{U}}_0$, that is, $\mathcal{S}\left({\mathcal{U}}_0\right)\subseteq {\mathcal{V}}_0$. Similarly, $\mathcal{S}\left({\mathcal{V}}_0\right)\subseteq {\mathcal{U}}_0$ which ensures that $\mathcal{S}$ is cyclic on ${\mathcal{U}}_0\cup {\mathcal{V}}_0$.

Let $\mathrm{Y}$ indicate the set of all nonempty sets $\mathcal{H}\subseteq {\mathcal{U}}_0\cup {\mathcal{V}}_0$ such that $(\mathcal{H}\cap {\mathcal{U}}_0,\mathcal{H}\cap {\mathcal{V}}_0)$ is a nonempty, closed, and convex pair which is a p-proximinal pair, $\mathcal{S}$ is cyclic on $(\mathcal{H}\cap {\mathcal{U}}_0)\cup (\mathcal{H}\cap {\mathcal{V}}_0)$, and

$${\mathfrak{D}}_{\mathcal{H}\cap {\mathcal{U}}_0,\mathcal{H}\cap {\mathcal{V}}_0}\left(t\right)={\mathfrak{D}}_{{\mathcal{U}}_0,{\mathcal{V}}_0}\left(t\right)\left(={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)\right),\forall t>0.$$

Note that ${\mathcal{U}}_0\cup {\mathcal{V}}_0\in \mathrm{Y}$ and hence $\mathrm{Y}$ is nonempty. Suppose ${\left\{{\mathcal{G}}_{\alpha }\right\}}_{\alpha \in \Gamma }$ is a descending chain in $\mathrm{Y}$ and let ${\displaystyle \mathcal{G}=\bigcap _{\alpha \in \Gamma }{\mathcal{G}}_{\alpha }}$. Since $\mathcal{M}$ possesses the property (C), we have

$${\displaystyle \mathcal{G}\cap {\mathcal{U}}_0=\left(\bigcap _{\alpha \in \Gamma }{\mathcal{G}}_{\alpha }\right)\cap {\mathcal{U}}_0=\bigcap _{\alpha \in \Gamma }\left({\mathcal{G}}_{\alpha }\cap {\mathcal{U}}_0\right)\ne \varnothing .}$$

Also, it is evident that $\mathcal{G}\cap {\mathcal{U}}_0$ is closed and convex. Similarly, $\mathcal{G}\cap {\mathcal{V}}_0$ is nonempty, closed, and convex. It is easy to show that $\mathcal{S}$ is cyclic on $(\mathcal{G}\cap {\mathcal{U}}_0)\cup (\mathcal{G}\cap {\mathcal{V}}_0)$. We now claim that ${\mathfrak{D}}_{\mathcal{G}\cap {\mathcal{U}}_0,\mathcal{G}\cap {\mathcal{V}}_0}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$ and $(\mathcal{G}\cap {\mathcal{U}}_0,\mathcal{G}\cap {\mathcal{V}}_0)$ is a p-proximinal pair. Let $x\in \mathcal{G}\cap {\mathcal{U}}_0$. Then, for any $\alpha \in \Gamma$, $x\in {\mathcal{G}}_{\alpha }\cap {\mathcal{U}}_0$. By using the fact that $({\mathcal{G}}_{\alpha }\cap {\mathcal{U}}_0,{\mathcal{G}}_{\alpha }\cap {\mathcal{V}}_0)$ is a p-proximinal pair, there is $z\in {\mathcal{G}}_{\alpha }\cap {\mathcal{V}}_0$ such that $F_{x,z}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0.$ We claim that the element z is unique. If there is another element $\tilde{z}\in {\mathcal{V}}_0$ such that $F_{x,\tilde{z}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$, then from Lemma 3, there exists $\lambda \in (0,1)$ such that $\mathcal{W}(z,\tilde{z},\lambda )\notin \{z,\tilde{z}\}$. Since $\mathcal{M}$ satisfies the condition $(♮)$, it follows that

$$F_{W(z,\tilde{z},\lambda ),x}\left(t\right)=min\left\{F_{z,x}\left(t\right),F_{\tilde{z},x}\left(t\right)\right\}={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0.$$

Hence, by Lemma 2 we see that $\mathcal{W}(z,\tilde{z},\lambda )\in \{z,\tilde{z}\}$ which is a contradiction. Then, $(\mathcal{G}\cap {\mathcal{U}}_0,\mathcal{G}\cap {\mathcal{V}}_0)\in \mathrm{Y}$. By using Zorn’s lemma, $\mathrm{Y}$ has a minimal element, say $\mathcal{H}$. Now set ${\mathcal{H}}_1:=\mathcal{H}\cap {\mathcal{U}}_0$ and ${\mathcal{H}}_2:=\mathcal{H}\cap {\mathcal{V}}_0$ and so we obtain the result. Note that since $\mathcal{H}\in \mathrm{Y}$ is minimal, we must have that $({\mathcal{H}}_1,{\mathcal{H}}_2)$ is p-proximinal.    □

It is worth noticing that if in Lemma 7, the pair $(\mathcal{U},\mathcal{V})$ is compact, then the property (C) of $\mathcal{M}$ can be dropped.

Notation: Under the assumptions of Lemma 7, by ${\mathfrak{M}}_{\mathcal{S}}(\mathcal{U},\mathcal{V})$, we indicate the set of all nonempty, closed, convex, minimal pairs $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ such that $\mathcal{S}$ is cyclic on ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ and ${\mathfrak{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$.

Here, we introduce the following notion of probabilistic proximal quasi-normal structure in convex Menger PM-spaces to establish the existence of BPPs for cyclic relatively Kannan nonexpansive maps in metrically convex Menger PM-spaces.

Definition 13.

A convex pair $(\mathcal{U},\mathcal{V})$ in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is said to have probabilistic proximal quasi-normal structure (PPQN-structure) if for any p-bounded, closed, convex and p-proximinal pair $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ such that ${\mathcal{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ and ${\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)<{\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$, there exists a point $(p,q)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$ and $t_0>0$ such that

$$min\left\{F_{p,y}\left(t_0\right),F_{x,q}\left(t_0\right)\right\}>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t_0\right),\forall (x,y)\in {\mathcal{H}}_1\times {\mathcal{H}}_2.$$

(3)

Remark 2.

In a special case, if the convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is a Banach space, then Definition 13 leads to the proximal quasi-normal structure (PQN-structure), which was introduced in [6].

Moreover, it is worth mentioning that if in Definition 13 the inequality (3) is as follows:

$$min\left\{{\nabla }_{p,{\mathcal{H}}_2}\left(t_0\right),{\nabla }_{q,{\mathcal{H}}_1}\left(t_0\right)\right\}>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t_0\right),$$

(4)

then we say that the convex pair $(\mathcal{U},\mathcal{V})$ in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ has the probabilistic proximal normal structure (PPN-structure). The concept of (PPN-structure) was first introduced in [3] in the setting of Banach spaces, and after that it was extended by M. Gabeleh to convex metric spaces ([7]) in order to study the existence of BPPs for cyclic relatively nonexpansive maps. Notice that

$$\mathtt{PPN}\text{-}\mathrm{structure}\Rightarrow \mathtt{PPQN}\text{-}\mathrm{structure}.$$

We are in a position to present the second existence result.

Theorem 3.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, p-bounded, closed, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which has the property (C) and satisfies the condition $(♮)$. Suppose that $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a cyclic relatively Kannan nonexpansive mapping which preserves distance. If $(\mathcal{U},\mathcal{V})$ has the PPQN-structure, then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Proof. 

By using Lemma 7, we obtain that ${\mathfrak{M}}_{\mathcal{S}}(\mathcal{U},\mathcal{V})$ is nonempty. Let $({\mathcal{H}}_1,{\mathcal{H}}_2)\in {\mathfrak{M}}_{\mathcal{S}}(\mathcal{U},\mathcal{V})$. Then, $\mathcal{S}$ is cyclic on ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ and ${\mathfrak{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0.$

Let $\rho >0$ be such that $\rho \le {\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$ and consider $(\tilde{x},\tilde{y})\in {\mathcal{H}}_1\times {\mathcal{H}}_2$ such that

$$F_{\tilde{x},\tilde{y}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),F_{\tilde{x},S\tilde{x}}\left(t\right)\ge \rho \mathrm{and}{F}_{S\tilde{y},\tilde{y}}\left(t\right)\ge \rho ,$$

for all $t>0$. Set

$${\mathcal{H}}_1^{\rho }=\left\{x\in {\mathcal{H}}_1:F_{x,Sx}\left(t\right)\ge \rho \right\},{\mathcal{H}}_2^{\rho }=\left\{x\in {\mathcal{H}}_2:F_{Sx,x}\left(t\right)\ge \rho \right\}$$

and let ${\mathcal{Y}}_1^{\rho }=\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{H}}_1^{\rho }\right)\right)$ and ${\mathcal{Y}}_2^{\rho }=\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{H}}_2^{\rho }\right)\right)$. We claim that $\mathcal{S}$ is cyclic on ${\mathcal{Y}}_1^{\rho }\cup {\mathcal{Y}}_2^{\rho }$. Initially, we prove that ${\mathcal{Y}}_1^{\rho }\subseteq {\mathcal{H}}_2^{\rho }$. Let $x\in {\mathcal{Y}}_1^{\rho }$. If $F_{\mathcal{S}x,x}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$, then $x\in {\mathcal{H}}_2^{\rho }$. Say $F_{Sx,x}\left(t\right)<{\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$ and put

$$\begin{array}{c}\hfill 1-\mu :=\underset{u\in {\mathcal{H}}_1^{\rho }}{inf}{F}_{Su,Sx}\left(t\right)\end{array}$$

(5)

Note that $\mathcal{S}\left({\mathcal{H}}_1^{\rho }\right)\subseteq {\mathcal{N}}_{Sx}(t,\mu )$, that is, ${\mathcal{Y}}_1^{\rho }\subseteq {\mathcal{N}}_{Sx}(t,\mu )$. Since $x\in {\mathcal{Y}}_1^{\rho }$, we obtain $F_{\mathcal{S}x,x}\left(t\right)\ge 1-\mu$. It can be deduced from the equality (5) that for each $ϵ>0$, there exists $u\in {\mathcal{H}}_1^{\rho }$ such that $1-\mu +ϵ\ge F_{\mathcal{S}u,\mathcal{S}x}\left(t\right)$. Since $\mathcal{S}$ is relatively Kannan nonexpansive, we have

$$F_{\mathcal{S}x,x}\left(t\right)+ϵ\ge 1-\mu +ϵ\ge F_{\mathcal{S}u,\mathcal{S}x}\left(t\right)\ge {\displaystyle \frac{1}{2}}\left[F_{u,\mathcal{S}u}\left(t\right)+F_{\mathcal{S}x,x}\left(t\right)\right]\ge {\displaystyle \frac{1}{2}}{F}_{\mathcal{S}x,x}\left(t\right)+{\displaystyle \frac{1}{2}}\rho .$$

Thus, we obtain that $F_{\mathcal{S}x,x}\left(t\right)\ge \rho -2ϵ$ and so $x\in {\mathcal{H}}_2^{\rho }$. Hence, ${\mathcal{Y}}_1^{\rho }\subseteq {\mathcal{H}}_2^{\rho }.$ This shows that

$$\mathcal{S}\left({\mathcal{Y}}_1^{\rho }\right)\subseteq \mathcal{S}\left({\mathcal{H}}_2^{\rho }\right)\subseteq \overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{H}}_2^{\rho }\right)\right)={\mathcal{Y}}_2^{\rho }.$$

Similar to the above, we obtain that $\mathcal{S}\left({\mathcal{Y}}_2^{\rho }\right)\subseteq {\mathcal{Y}}_1^{\rho }$, that is, $\mathcal{S}$ is cyclic on ${\mathcal{Y}}_1^{\rho }\cup Y_2^{\rho }$.

We now assert that ${\nabla }_{{\mathcal{Y}}_1^{\rho },{\mathcal{Y}}_2^{\rho }}\left(t\right)\ge \rho .$ By Lemma 6, we obtain

$$\begin{array}{cc}\hfill {\nabla }_{{\mathcal{Y}}_1^{\rho },{\mathcal{Y}}_2^{\rho }}\left(t\right)& ={\nabla }_{\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{H}}_1^{\rho }\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left({\mathcal{H}}_2^{\rho }\right)\right)}\left(t\right)={\nabla }_{\mathcal{S}\left({\mathcal{H}}_1^{\rho }\right),\mathcal{S}\left({\mathcal{H}}_2^{\rho }\right)}\left(t\right)\hfill \\ \hfill & =\underset{(x,y)\in {\mathcal{H}}_1^{\rho }\times {\mathcal{H}}_2^{\rho }}{inf}{F}_{\mathcal{S}x,\mathcal{S}y}\left(t\right)\ge \underset{(x,y)\in {\mathcal{H}}_1^{\rho }\times {\mathcal{H}}_2^{\rho }}{inf}{\displaystyle \frac{1}{2}}\left[F_{x,Sx}\left(t\right)+F_{Sy,y}\left(t\right)\right]\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}(\rho +\rho )=\rho ,\hfill \end{array}$$

for all $t>0$. Since $(\tilde{x},\tilde{y})\in {\mathcal{H}}_1^{\rho }\times {\mathcal{H}}_2^{\rho }$, $F_{\tilde{x},\tilde{y}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, we deduce that

$${\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)\ge {\mathfrak{D}}_{{\mathcal{Y}}_2^{\rho },{\mathcal{Y}}_1^{\rho }}\left(t\right)\ge F_{S\tilde{y},S\tilde{x}}\left(t\right)\ge F_{\tilde{x},\tilde{y}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),$$

that is, ${\mathfrak{D}}_{{\mathcal{Y}}_2^{\rho },{\mathcal{Y}}_1^{\rho }}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$. Say

$$\begin{array}{c}\hfill {\rho }_0:=\underset{x\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}{F}_{x,Sx}\left(t\right),\end{array}$$

(6)

for all $t>0$. Then, ${\rho }_0\le {\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$. Let $\left\{{\rho }_n\right\}$ be a non-negative sequence such that ${\rho }_n\downarrow {\rho }_0$. Hence, $\left(\left\{{\mathcal{Y}}_1^{{\rho }_n}\right\},\left\{{\mathcal{Y}}_2^{{\rho }_n}\right\}\right)$ are descending sequences of nonempty, p-bounded, closed, and convex subsets of $({\mathcal{H}}_1,{\mathcal{H}}_2).$ Since $\mathcal{M}$ has the property (C),

$${\mathcal{Y}}_1^{{\rho }_0}=\bigcap _{n=1}^{\infty }{\mathcal{Y}}_1^{{\rho }_n}\ne \varnothing ,{\mathcal{Y}}_2^{{\rho }_0}=\bigcap _{n=1}^{\infty }{\mathcal{Y}}_2^{{\rho }_n}\ne \varnothing .$$

Furthermore, it follows from the preceding argument that $S:{\mathcal{Y}}_1^{{\rho }_0}\cup {\mathcal{Y}}_2^{{\rho }_0}⟶{\mathcal{Y}}_1^{{\rho }_0}\cup {\mathcal{Y}}_2^{{\rho }_0}$ is cyclic. Moreover, since ${\mathfrak{D}}_{{\mathcal{Y}}_2^{{\rho }_n},{\mathcal{Y}}_1^{{\rho }_n}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $n\in \mathbb{N}$ and for all $t>0$, we conclude that ${\mathfrak{D}}_{{\mathcal{Y}}_2^{{\rho }_0},{\mathcal{Y}}_1^{{\rho }_0}}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$. The minimality of $({\mathcal{H}}_1,{\mathcal{H}}_2)$ now implies that ${\mathcal{Y}}_2^{{\rho }_0}={\mathcal{H}}_1$ and ${\mathcal{Y}}_1^{{\rho }_0}={\mathcal{H}}_2$. Then, $F_{x,\mathcal{S}x}\left(t\right)\ge {\rho }_0$ for all $x\in {\mathcal{H}}_1\cup {\mathcal{H}}_2$.

Assume that ${\rho }_0<{\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$. Since $(\mathcal{U},\mathcal{V})$ has the PPQN-structure, there exists $(p,q)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$ such that

$$F_{p,y}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)\ge {\rho }_0,F_{x,q}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)\ge {\rho }_0$$

for all $(x,y)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$, and $t>0$. Therefore,

$$F_{p,Sp}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)\ge {\rho }_0,F_{Sq,q}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)\ge {\rho }_0$$

This contradicts the equality (6). Hence, ${\rho }_0={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$, for all $t>0$, and so

$$F_{x,Sx}\left(t\right)=F_{y,Sy}\left(t\right)={\mathfrak{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$$

for all $(x,y)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$. This establishes the desired result.    □

The next result gives us sufficient conditions in convex Menger PM-spaces for the PPQN-structure.

Proposition 1.

Every nonempty, compact, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which satisfies the condition $(♮)$ has the PPQN-structure.

Proof. 

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, compact, and convex pair in $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$. We show that for any p-bounded, closed, convex, and p-proximinal pair $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ with ${\mathcal{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)$ for all $t>0$, there exists a point $(p,q)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$ and $t_0>0$ so that  

$$min\left\{{\nabla }_{p,{\mathcal{H}}_2}\left(t_0\right),{\nabla }_{q,{\mathcal{H}}_1}\left(t_0\right)\right\}>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t_0\right),$$

(7)

which implies that

$$min\left\{F_{p,y}\left(t_0\right),F_{x,q}\left(t_0\right)\right\}>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t_0\right),\forall (x,y)\in {\mathcal{H}}_1\times {\mathcal{H}}_2,$$

and the result will follow.

To show (7), suppose the contrary. Then there exists a p-bounded, closed, convex, and p-proximinal pair $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ with ${\mathcal{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ and ${\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)<{\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$, such that for any $(p,q)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$

$$min\left\{{\nabla }_{p,{\mathcal{H}}_2}\left(t\right),{\nabla }_{{\mathcal{H}}_1,q}\left(t\right)\right\}={\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right),\forall t>0.$$

If ${\mathcal{H}}_2=\left\{v\right\}$ for some $v\in \mathcal{V}$, then by the fact that $({\mathcal{H}}_1,{\mathcal{H}}_2)$ is p-proximinal, there is $u^{\prime }\in {\mathcal{H}}_1$ such that $F_{u^{\prime },v}\left(t\right)={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)$ for all $t>0$. We now have

$${\mathcal{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)=F_{u^{\prime },v}\left(t\right)\le {\nabla }_{u^{\prime },{\mathcal{H}}_2}\left(t\right)={\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right),\forall t>0,$$

which is impossible by the assumption that ${\mathcal{D}}_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)>{\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)$ for all $t>0$. Thus ${\mathcal{H}}_2$ is not singleton. So, assume that ${\mathcal{H}}_2=\{v_1,v_2\}$. Since $\mathcal{M}$ is metrically convex, there is a number ${\eta }_0\in (0,1)$ such that $\mathcal{W}(v_1,v_2,{\eta }_0)\notin \{v_1,v_2\}$. Again, using the p-proximinality of the pair $({\mathcal{H}}_1,{\mathcal{H}}_2)$, there exist elements $u_1,u_2\in {\mathcal{H}}_1$ for which $F_{u_1,v_1}\left(t\right)={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right)=F_{u_2,v_2}\left(t\right)$ for all $t>0$. Note that if $u_1=u_2$, then by the assumption $(♮)$, we obtain

$$F_{u_1,\mathcal{W}(v_1,v_2,{\eta }_0)}\left(t\right)>min\left\{F_{u_1,v_1}\left(t\right),F_{u_2,v_2}\left(t\right)\right\}={\mathcal{D}}_{\mathcal{U},\mathcal{V}}\left(t\right),\forall t>0,$$

which is a contradiction. Thus, $u_1\ne u_2$. By using the metrically convexity of $\mathcal{M}$, there is ${\eta }_1\in (0,1)$ for which $\mathcal{W}(u_1,u_2,{\eta }_1)\notin \{u_1,u_2\}$. Considering that ${\mathcal{H}}_2$ is compact, and $F_{\mathcal{W}(u_1,u_2,{\theta }_1),•}\left(t\right)$ is continuous on the set ${\mathcal{H}}_2$, there is a point $v_3\in {\mathcal{H}}_2$ so that

$$F_{\mathcal{W}(u_1,u_2,{\eta }_1),v_3}\left(t\right)={\nabla }_{\mathcal{W}(u_1,u_2,{\eta }_1),{\mathcal{H}}_2}\left(t\right)={\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right),\forall t>0.$$

Applying the condition $(♮)$ to obtain

$${\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)=F_{\mathcal{W}(u_1,u_2,{\eta }_1),v_3}\left(t\right)>min\{F_{u_1,v_3}\left(t\right),F_{u_2,v_3}\left(t\right)\},$$

is impossible.

By the same discussion, we conclude that if ${\nabla }_{{\mathcal{H}}_1,v}\left(t\right)={\nabla }_{{\mathcal{H}}_1,{\mathcal{H}}_2}\left(t\right)$ for all $v\in {\mathcal{H}}_2$ and $t>0$, then we obtain a contradiction, and this completes the proof.    □

Corollary 3.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, compact, convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which satisfies the condition $(♮)$. Suppose that $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a cyclic relatively Kannan nonexpansive mapping which preserves distance. Then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Let us illustrate Corollary 3 with the next example.

Example 2.

Consider the Euclidean space $\mathbb{R}$ with the usual metric and usual convex structure, that is, $\mathcal{W}(u,v,\lambda )=(1-\lambda )u+\lambda v$ for any $(u,v,\lambda )\in \mathbb{R}\times \mathbb{R}\times [0,1]$. By using the Remark 1, $\mathbb{R}$ is a metrically convex Menger PM-space. Let $\mathcal{U}=[0,1]$ and $\mathcal{V}=[{\displaystyle \frac{3}{2}},2]$. Define a mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ with

$$\mathcal{S}\left(0\right)=2,\mathcal{S}\left(u\right)={\displaystyle \frac{3}{2}}\mathit{if}u\in \mathcal{U}-\left\{0\right\},\mathcal{S}\left(v\right)=1,\mathit{if}v\in \mathcal{V}.$$

It is easy to see that $\mathcal{S}$ is a cyclic relatively Kannan nonexpansive mapping which preserves distance. It now follows from Corollary 3 that $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$ Note that $\mathcal{B}pp\left(\mathcal{S}{|}_{\mathcal{U}}\right)=\left\{1\right\}$ and $\mathcal{B}pp\left(\mathcal{S}{|}_{\mathcal{V}}\right)=\left\{{\displaystyle \frac{3}{2}}\right\}$.

4.2. Application to Fixed Point Theory

Fixed point results for Kannan nonexpansive maps were studied by C. Wong in [19,20] by considering the notion of quasi-normal structure (close-to-normal structure in some of the literature) in Banach spaces. We recall that a convex subset $\mathcal{K}$ of a Banach space X has a quasi-normal structure provided that for any nonempty, bounded, closed, and convex subset $\mathcal{H}$ of $\mathcal{K}$ with $\mathrm{diam}\left(\mathcal{H}\right)>0$, there is a point $p\in \mathcal{H}$ for which

$$\parallel x-p\parallel <\mathrm{diam}\left(\mathcal{H}\right),\forall x\in \mathcal{H}.$$

It was proved in [20] that if $\mathcal{K}$ is a weakly compact and convex subset of a Banach space which has the quasi-normal structure, then every Kannan nonexpansive self-map defined on $\mathcal{K}$ has a fixed point (Wong’s fixed point theorem).

As far as we know, Wong’s fixed point theorem has not yet been studied in Menger PM-spaces. In what follows, as an application of Theorem 3, we obtain a counterpart result of Wong’s fixed point problem in convex Menger PM-spaces. For this purpose, we need the following requirements.

Definition 14.

A convex subset $\mathcal{U}$ in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is said to have a probabilistic quasi-normal structure (PQN-structure) if for any p-bounded, closed, and convex set $\mathcal{H}\subseteq \mathcal{U}$ such that ${\nabla }_{\mathcal{H}}\left(t\right)<{ϵ}_0\left(t\right)$ for all $t>0$, there exists a point $p\in \mathcal{H}$ and $t_0>0$ such that

$$F_{p,y}\left(t_0\right)>{\nabla }_{\mathcal{H}}\left(t_0\right),\forall y\in \mathcal{H}.$$

(8)

Note that Definition 14 is a special case of Definition 13 by taking $\mathcal{U}=\mathcal{V}$ in Definition 13.

Remark 3.

We mention that if the inequality (8) of Definition 14 is as below

$${\nabla }_{p,\mathcal{H}}\left(t_0\right)>{\nabla }_{\mathcal{H}}\left(t_0\right),$$

(9)

then we say that the convex set $\mathcal{U}$ in a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ has a probabilistic normal structure (PN-structure) which was first introduced in [8]. It is obvious that

$$\mathtt{PN}\text{-}\mathrm{structure}\Rightarrow \mathtt{PQN}\text{-}\mathrm{structure}.$$

We now obtain the following new fixed point theorem.

Theorem 4.

Let $\mathcal{U}$ be a nonempty, p-bounded, closed, and convex subset of a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which has the property (C) and satisfies the condition $(♮)$. Suppose that $\mathcal{S}:\mathcal{U}\to \mathcal{U}$ is a Kannan nonexpansive mapping, that is,

$$\begin{array}{c}\hfill F_{\mathcal{S}p,\mathcal{S}q}\left(t\right)\ge {\displaystyle \frac{1}{2}}\left[F_{p,\mathcal{S}p}\left(t\right)+F_{q,\mathcal{S}q}\left(t\right)\right],\forall p,q\in \mathcal{U}.\end{array}$$

If $\mathcal{U}$ has a PQN-structure, then $\mathcal{S}$ has at least one fixed point.

Proof. 

It is sufficient to consider $\mathcal{U}=\mathcal{V}$ in Theorem 3.    □

As an example of a convex subset of a convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ having (PQN-structure), we pay attention to the next proposition which is a direct conclusion of Proposition 1.

Proposition 2.

Every nonempty, compact, and convex subset in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which satisfies the condition that $(♮)$ has a PQN-structure.

The next corollary is a straightforward consequence of Theorem 4 and Proposition 2.

Corollary 4.

Let $\mathcal{U}$ be a nonempty, compact, and convex pair in a metrically convex Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ which satisfies the condition $(♮)$. Suppose that $\mathcal{S}:\mathcal{U}\to \mathcal{U}$ is a Kannan nonexpansive mapping. Then $\mathcal{S}$ admits a fixed point.

5. More Results in CAT(0) Spaces

In the last section of this paper, we present more practical results in the framework of CAT$\left(0\right)$ spaces. For this purpose, we need to recall some requirements of geodesic metric spaces.

Let $(\mathcal{M},d)$ be a metric space and $u,u$ be disjoint elements of $\mathcal{M}$. A geodesic segment from u to v in $\mathcal{M}$ is an isometry ${\gamma }_{uv}:[0,d(u,v)]\to \mathcal{M}$ for which

$${\gamma }_{uv}\left(u\right)=0,{\gamma }_{uv}\left(v\right)=d(u,v).$$

Definition 15.

A metric space $(\mathcal{M},d)$ is said to be a geodesic space if for any two distinct points $u,v\in \mathcal{M}$, there is a geodesic segment from u to v. Also, the metric space $(\mathcal{M},d)$ is called uniquely geodesic if for any $u\ne v\in \mathcal{M}$, there exists a unique geodesic segment from u to v which is denoted by $[u,v]$.

For a geodesic metric space $(\mathcal{M},d)$ and for any $u,v\in \mathcal{M}$, denote a point $p\in [u,v]$ such that $d(u,p)=\alpha d(u,v)$ by $p=(1-\alpha )u\oplus \alpha v$, where $0\le \alpha \le 1$.

A subset $\mathcal{U}$ of a geodesic metric space $(\mathcal{M},d)$ is said to be convex provided that $\mathcal{U}$ includes every geodesic segment joining any two points of its points.

Definition 16

([21]). A geodesic space $(\mathcal{M},d)$ is said to be reflexive if for every decreasing chain $\left\{C_i\right\}\subseteq X$ such that $C_i$ is nonempty, bounded, closed, and convex for all i, we have that ${\displaystyle \bigcap _i{C}_i\ne \varnothing }$.

Clearly, reflexive Banach spaces can be considered a subclass of reflexive geodesic metric spaces.

Definition 17

([22]). A geodesic space $(\mathcal{M},d)$ is strictly convex if for every $r>0$, $a,x$ and $y\in \mathcal{M}$ with $d(x,a)\le r$, $d(y,a)\le r$ and $x\ne y$, it is the case that $d(a,p)<r,$ where $p\in [x,y]-\{x,y\}$.

Uniformly convex Banach spaces and more general Busemann convex spaces are strictly convex (see [23] for more details).

A very important subclass of geodesic metric spaces is CAT$\left(0\right)$ spaces. Here, we present an equivalent and more useful notion of CAT$\left(0\right)$ spaces which was established by Bruhat and Tits in [24].

Definition 18.

A geodesic space $(\mathcal{M},d)$ is said to be a CAT$\left(0\right)$ space if for any three points $u,v_1,v_2\in \mathcal{M}$ and noting that $v_0$ is the metric midpoint of $v_1,v_2$, i.e., $d(v_1,v_2)=d(v_1,v_0)+d(v_0,v_2)$, then we have

$$d{(u,v_0)}^2\le {\displaystyle \frac{1}{2}}d{(u,v_1)}^2+{\displaystyle \frac{1}{2}}d{(u,v_2)}^2-{\displaystyle \frac{1}{4}}d{(v_1,v_2)}^2.$$

(10)

The inequality (10) is well-known as the (CN)-inequality.

Hilbert spaces are a subfamily of CAT$\left(0\right)$ spaces. Obviously, CAT$\left(0\right)$ spaces are uniquely geodesic and strictly convex. Furthermore, it was proved in [25] that complete CAT$\left(0\right)$ spaces are reflexive.

For more information about the existence and convergence of BPPs in complete CAT$\left(0\right)$ spaces for various classes of non-self-maps, we refer to [26,27,28].

Remark 4.

It is worth noticing that if $(\mathcal{M},d)$ is a CAT$\left(0\right)$ space and we define

$$\mathcal{W}(u,v;\lambda )=\lambda u\oplus \alpha v,\forall u,v\in \mathcal{M},\forall \lambda \in [0,1],$$

then $(\mathcal{M},d;\mathcal{W})$ is a convex metric space in the sense of Takahashi ([13]). Now if we consider the Menger PM-space $(\mathcal{M},\mathcal{F},\mathcal{T})$ induced by the metric d as in Remark 1, then $(\mathcal{M},\mathcal{F},\mathcal{T},\mathcal{W})$ is a convex Menger PM-space which is metrically convex and has the property (C). Furthermore, since any CAT$\left(0\right)$ space is strictly convex, it satisfies the condition $(♮)$.

We now conclude the following corollaries.

Theorem 5.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, bounded, closed, and convex pair in a complete CAT$\left(0\right)$ space $(\mathcal{M},d)$ and let $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ be a weak cyclic Kannan contraction. Then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Proof. 

The proof is obtained by considering Theorem 2 and Remark 4.    □

The proof of the next result follows similar patterns to those considered in [29] in Proposition 3.5 and we omit the details.

Proposition 3.

Every nonempty and convex pair $(\mathcal{U},\mathcal{V})$ in a complete CAT$\left(0\right)$ space $(\mathcal{M},d)$ has a proximal quasi-normal structure (PQN-structure), that is, for any nonempty, bounded, closed, convex and proximinal pair $({\mathcal{H}}_1,{\mathcal{H}}_2)\subseteq (\mathcal{U},\mathcal{V})$ with $\mathrm{dist}({\mathcal{H}}_1,{\mathcal{H}}_2)=\mathrm{dist}(\mathcal{U},\mathcal{V})$ and $\mathrm{dist}({\mathcal{H}}_1,{\mathcal{H}}_2)<\delta ({\mathcal{H}}_1,{\mathcal{H}}_2):=sup\{d(u,v):(u,v)\in {\mathcal{H}}_1\times {\mathcal{H}}_2\}$, there is a point $(p,q)\in {\mathcal{H}}_1\times {\mathcal{H}}_2$ for which

$$max\{d(p,v),d(u,q)\}<\delta ({\mathcal{H}}_1,{\mathcal{H}}_2),\forall (u,v)\in {\mathcal{H}}_1\times {\mathcal{H}}_2.$$

Theorem 6.

Let $(\mathcal{U},\mathcal{V})$ be a nonempty, bounded, closed, and convex pair in a complete CAT$\left(0\right)$ space $(\mathcal{M},d)$. Assume that $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ is a cyclic relatively Kannan nonexpansive mapping which preserves distance, that is, $d(\mathcal{S}u,\mathcal{S}v)=\mathrm{dist}(\mathcal{U},\mathcal{V})$ if $d(u,v)=\mathrm{dist}(\mathcal{U},\mathcal{V})$, and otherwise,

$$d(\mathcal{S}u,\mathcal{S}v)\le {\displaystyle \frac{1}{2}}\left(d(u,\mathcal{S}u)+d(v,\mathcal{S}v)\right),\forall (u,v)\in \mathcal{U}\times \mathcal{V}.$$

Then $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$

Proof. 

The result follows by invoking Proposition 3, Remark 4, and Theorem 3. □

Example 3.

Consider the Euclidean space $\mathcal{M}={\mathbb{R}}^2$ equipped with the river metric $d_{\mathrm{riv}}$ defined with

$$d_{\mathrm{riv}}((x_1,y_1),(x_2,y_2))=\left\{\begin{array}{c}|y_1-y_2|,\mathit{if}{x}_1=x_2,\hfill \\ |x_1-x_2|+|y_1|+|y_2|,\mathit{if}{x}_1\ne x_2.\hfill \end{array}\right.$$

It is well known that $({\mathbb{R}}^2,d_{\mathrm{riv}})$ is a complete CAT$\left(0\right)$ metric space (see [30] for more information). Let

$$\mathcal{U}=\left\{(0,x):0\le x\le 1\right\},\mathcal{V}=\left\{(1,y):0\le y\le 1\right\}.$$

and define a cyclic mapping $\mathcal{S}:\mathcal{U}\cup \mathcal{V}\to \mathcal{U}\cup \mathcal{V}$ with

$$\mathcal{S}(0,x)=(1,x^2),\mathcal{S}(1,x)=(0,x^2),\forall x\in [0,1].$$

Then for any $x,y\in [0,1]$, we have

$$\begin{array}{cc}\hfill d_{\mathrm{riv}}\left(\mathcal{S}(0,x),\mathcal{S}(1,y)\right)& =d_{\mathrm{riv}}\left((1,x^2),(0,y^2)\right)=1+x^2+y^2\hfill \\ \hfill & \le 1+x+y=d_{\mathrm{riv}}\left((0,x),(1,y)\right),\hfill \end{array}$$

which implies that $\mathcal{S}$ preserves distance. Additionally,

$$\begin{array}{cc}\hfill d_{\mathrm{riv}}\left(\mathcal{S}(0,x),\mathcal{S}(1,y)\right)& =d_{\mathrm{riv}}\left((1,x^2),(0,y^2)\right)=1+x^2+y^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left(1+x+x^2+1+y+y^2\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(d_{\mathrm{riv}}\left((0,x),\mathcal{S}(0,x)\right)+d_{\mathrm{riv}}\left((1,y),\mathcal{S}(1,y)\right)\right),\hfill \end{array}$$

which deduces that $\mathcal{S}$ is a Kannan relatively nonexpansive mapping. Applying Theorem 6 to conclude that $\mathcal{B}pp\left(\mathcal{S}\right)\ne \varnothing .$ It is worth noticing that $\mathcal{B}pp\left(\mathcal{S}\right)=\left\{\right(0,1),(1,0\left)\right\}$.

6. Conclusions

In this paper, the existence of BPPs for weak cyclic Kannan contractions as well as for relatively Kannan nonexpansive mappings which preserve distance in the setting of metrically convex Menger PM-spaces and, in special cases, in CAT$\left(0\right)$ metric spaces is considered. To this end, we use the geometric concept of probabilistic proximal quasi-normal structure defined on a convex pair in a metrically convex Menger PM-space. It is interesting to ask whether we can drop the condition of symmetry property of Definition 3 and redefine the concept of nonsymmetry Menger PM-spaces. The main idea of this question goes back to the concept of asymmetric norms which induce $T_0$-quasi-metric spaces (see [31] for more details).