# On best proximity points for set-valued contractions of Nadler type with respect to *b*-generalized pseudodistances in *b*-metric spaces

- Robert Plebaniak
^{1}Email author

**2014**:39

https://doi.org/10.1186/1687-1812-2014-39

© Plebaniak; licensee Springer. 2014

**Received: **20 November 2013

**Accepted: **28 January 2014

**Published: **14 February 2014

## Abstract

In this paper, in *b*-metric space, we introduce the concept of *b*-generalized pseudodistance which is an extension of the *b*-metric. Next, inspired by the ideas of Nadler (Pac. J. Math. 30:475-488, 1969) and Abkar and Gabeleh (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2):319-325, 2013), we define a new set-valued non-self-mapping contraction of Nadler type with respect to this *b*-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for $T:A\to {2}^{B}$. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error $inf\{d(x,y):y\in T(x)\}$, and hence the existence of a consummate approximate solution to the equation $T(x)=x$. In other words, the best proximity points theorem achieves a global optimal minimum of the map $x\to inf\{d(x;y):y\in T(x)\}$ by stipulating an approximate solution *x* of the point equation $T(x)=x$ to satisfy the condition that $inf\{d(x;y):y\in T(x)\}=dist(A;B)$. The examples which illustrate the main result given. The paper includes also the comparison of our results with those existing in the literature.

**MSC:**47H10, 54C60, 54E40, 54E35, 54E30.

## Keywords

*b*-metric spaces

*b*-generalized pseudodistancesglobal optimal minimumbest proximity pointsNadler contractionset-valued maps

## 1 Introduction

A number of authors generalize Banach’s [1] and Nadler’s [2] result and introduce the new concepts of set-valued contractions (cyclic or non-cyclic) of Banach or Nadler type, and they study the problem concerning the existence of best proximity points for such contractions; see *e.g.* Abkar and Gabeleh [3–5], Al-Thagafi and Shahzad [6], Suzuki *et al.* [7], Di Bari *et al.* [8], Sankar Raj [9], Derafshpour *et al.* [10], Sadiq Basha [11], and Włodarczyk *et al.* [12].

In 2012, Abkar and Gabeleh [13] introduced and established the following interesting and important best proximity points theorem for a set-valued non-self-mapping. First, we recall some definitions and notations.

*A*,

*B*be nonempty subsets of a metric space $(X,d)$. Then denote: $dist(A,B)=inf\{d(x,y):x\in A,y\in B\}$; ${A}_{0}=\{x\in A:d(x,y)=dist(A,B)\text{for some}y\in B\}$; ${B}_{0}=\{y\in B:d(x,y)=dist(A,B)\text{for some}x\in A\}$; $D(x,B)=inf\{d(x,y):y\in B\}$ for $x\in X$. We say that the pair $(A,B)$ has the

*P*-property if and only if

where ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$.

**Theorem 1.1** (Abkar and Gabeleh [13])

*Let* $(A,B)$ *be a pair of nonempty closed subsets of a complete metric space* $(X,d)$ *such that* ${A}_{0}\ne \mathrm{\varnothing}$ *and* $(A,B)$ *has the* *P*-*property*. *Let* $T:A\to {2}^{B}$ *be a multivalued non*-*self*-*mapping contraction*, *that is*, ${\mathrm{\exists}}_{0\u2a7d\lambda <1}{\mathrm{\forall}}_{x,y\in A}\{H(T(x),T(y))\u2a7d\lambda d(x,y)\}$. *If* $T(x)$ *is bounded and closed in* *B* *for all* $x\in A$, *and* $T({x}_{0})\subset {B}_{0}$ *for each* ${x}_{0}\in {A}_{0}$, *then* *T* *has a best proximity point in* *A*.

It is worth noticing that the map *T* in Theorem 1.1 is continuous, so it is u.s.c. on *X*, which by [[14], Theorem 6, p.112], shows that *T* is closed on *X*. In 1998, Czerwik [15] introduced of the concept of a *b*-metric space. A number of authors study the problem concerning the existence of fixed points and best proximity points in *b*-metric space; see *e.g.* Berinde [16], Boriceanu *et al.* [17, 18], Bota *et al.* [19] and many others.

In this paper, in a *b*-metric space, we introduce the concept of a *b*-generalized pseudodistance which is an extension of the *b*-metric. The idea of replacing a metric by the more general mapping is not new (see *e.g.* distances of Tataru [20], *w*-distances of Kada *et al.* [21], *τ*-distances of Suzuki [[22], Section 2] and *τ*-functions of Lin and Du [23] in metric spaces and distances of Vályi [24] in uniform spaces). Next, inspired by the ideas of Nadler [2] and Abkar and Gabeleh [13], we define a new set-valued non-self-mapping contraction of Nadler type with respect to this *b*-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for $T:A\to {2}^{B}$. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error $inf\{d(x,y):y\in T(x)\}$, and hence the existence of a consummate approximate solution to the equation $T(X)=x$. In other words, the best proximity points theorem achieves a global optimal minimum of the map $x\to inf\{d(x;y):y\in T(x)\}$ by stipulating an approximate solution *x* of the point equation $T(x)=x$ to satisfy the condition that $inf\{d(x;y):y\in T(x)\}=dist(A;B)$. Examples which illustrate the main result are given. The paper includes also the comparison of our results with those existing in the literature. This paper is a continuation of research on *b*-generalized pseudodistances in the area of *b*-metric space, which was initiated in [25].

## 2 On generalized pseudodistance

To begin, we recall the concept of *b*-metric space, which was introduced by Czerwik [15] in 1998.

**Definition 2.1** Let *X* be a nonempty subset and $s\u2a7e1$ be a given real number. A function $d:X\times X\to [0,\mathrm{\infty})$ is *b*-metric if the following three conditions are satisfied: (d1) ${\mathrm{\forall}}_{x,y\in X}\{d(x,y)=0\iff x=y\}$; (d2) ${\mathrm{\forall}}_{x,y\in X}\{d(x,y)=d(y,x)\}$; and (d3) ${\mathrm{\forall}}_{x,y,z\in X}\{d(x,z)\u2a7ds[d(x,y)+d(y,z)]\}$.

The pair $(X,d)$ is called a *b*-metric space (with constant $s\u2a7e1$). It is easy to see that each metric space is a *b*-metric space.

In the rest of the paper we assume that the *b*-metric $d:X\times X\to [0,\mathrm{\infty})$ is continuous on ${X}^{2}$. Now in *b*-metric space we introduce the concept of a *b*-generalized pseudodistance, which is an essential generalization of the *b*-metric.

**Definition 2.2** Let *X* be a *b*-metric space (with constant $s\u2a7e1$). The map $J:X\times X\to [0,\mathrm{\infty})$, is said to be a *b*-*generalized pseudodistance on* *X* if the following two conditions hold:

(J1) ${\mathrm{\forall}}_{x,y,z\in X}\{J(x,z)\u2a7ds[J(x,y)+J(y,z)]\}$; and

*X*such that

**Remark 2.1** (A) If $(X,d)$ is a *b*-metric space (with $s\u2a7e1$), then the *b*-metric $d:X\times X\to [0,\mathrm{\infty})$ is a *b*-generalized pseudodistance on *X*. However, there exists a *b*-generalized pseudodistance on *X* which is not a *b*-metric (for details see Example 4.1).

Indeed, if $J(x,y)=0$ and $J(y,x)=0$, then $J(x,x)=0$, since, by (J1), we get $J(x,x)\u2a7ds[J(x,y)+J(y,x)]=s[0+0]=0$. Now, defining $({x}_{m}=x:m\in \mathbb{N})$ and $({y}_{m}=y:m\in \mathbb{N})$, we conclude that (2.1) and (2.2) hold. Consequently, by (J2), we get (2.3), which implies $d(x,y)=0$. However, since $x\ne y$, we have $d(x,y)\ne 0$, a contradiction.

Now, we apply the *b*-generalized pseudodistance to define the ${H}^{J}$-distance of Nadler type.

**Definition 2.3**Let

*X*be a

*b*-metric space (with $s\u2a7e1$). Let the class of all nonempty closed subsets of

*X*be denoted by $Cl(X)$, and let the map $J:X\times X\to [0,\mathrm{\infty})$ be a

*b*-generalized pseudodistance on

*X*. Let ${\mathrm{\forall}}_{u\in X}{\mathrm{\forall}}_{V\in Cl(X)}\{J(u,V)={inf}_{v\in V}J(u,v)\}$. Define ${H}^{J}:Cl(X)\times Cl(X)\to [0,\mathrm{\infty})$ by

We will present now some indications that we will use later in the work.

*b*-metric space (with $s\u2a7e1$) and let $A\ne \mathrm{\varnothing}$ and $B\ne \mathrm{\varnothing}$ be subsets of

*X*and let the map $J:X\times X\to [0,\mathrm{\infty})$ be a

*b*-generalized pseudodistance on

*X*. We adopt the following denotations and definitions: ${\mathrm{\forall}}_{A,B\in Cl(X)}\{dist(A,B)=inf\{d(x,y):x\in A,y\in B\}\}$ and

**Definition 2.4**Let

*X*be a

*b*-metric space (with $s\u2a7e1$) and let the map $J:X\times X\to [0,\mathrm{\infty})$ be a

*b*-generalized pseudodistance on

*X*. Let $(A,B)$ be a pair of nonempty subset of

*X*with ${A}_{0}\ne \mathrm{\varnothing}$.

- (I)The pair $(A,B)$ is said to have the ${P}^{J}$-property if and only if$\begin{array}{c}\{[J({x}_{1},{y}_{1})=dist(A,B)]\wedge [J({x}_{2},{y}_{2})=dist(A,B)]\}\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\{J({x}_{1},{x}_{2})=J({y}_{1},{y}_{2})\},\hfill \end{array}$
where ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$.

- (II)We say that the
*b*-generalized pseudodistance*J*is associated with the pair $(A,B)$ if for any sequences $({x}_{m}:m\in \mathbb{N})$ and $({y}_{m}:m\in \mathbb{N})$ in*X*such that ${lim}_{m\to \mathrm{\infty}}{x}_{m}=x$; ${lim}_{m\to \mathrm{\infty}}{y}_{m}=y$, and${\mathrm{\forall}}_{m\in \mathbb{N}}\{J({x}_{m},{y}_{m-1})=dist(A,B)\},$

then $d(x,y)=dist(A,B)$.

**Remark 2.2**If $(X,d)$ is a

*b*-metric space (with $s\u2a7e1$), and we put $J=d$, then:

- (I)
The map

*d*is associated with each pair $(A,B)$, where $A,B\subset X$. It is an easy consequence of the continuity of*d*. - (II)
The ${P}^{d}$-property is identical with the

*P*-property. In view of this, instead of writing the ${P}^{d}$-property we will write shortly the*P*-property.

## 3 The best proximity point theorem with respect to a *b*-generalized pseudodistance

We first recall the definition of closed maps in topological spaces given in Berge [14] and Klein and Thompson [26].

**Definition 3.1** Let *L* be a topological vector space. The set-valued dynamic system $(X,T)$, *i.e.* $T:X\to {2}^{X}$ is called closed if whenever $({x}_{m}:m\in \mathbb{N})$ is a sequence in *X* converging to $x\in X$ and $({y}_{m}:m\in \mathbb{N})$ is a sequence in *X* satisfying the condition ${\mathrm{\forall}}_{m\in \mathbb{N}}\{{y}_{m}\in T({x}_{m})\}$ and converging to $y\in X$, then $y\in T(x)$.

Next, we introduce the concepts of a set-valued non-self-closed map and a set-valued non-self-mapping contraction of Nadler type with respect to the *b*-generalized pseudodistance.

**Definition 3.2** Let *L* be a topological vector space. Let *X* be certain space and *A*, *B* be a nonempty subsets of *X*. The set-valued non-self-mapping $T:A\to {2}^{B}$ is called closed if whenever $({x}_{m}:m\in \mathbb{N})$ is a sequence in *A* converging to $x\in A$ and $({y}_{m}:m\in \mathbb{N})$ is a sequence in *B* satisfying the condition ${\mathrm{\forall}}_{m\in \mathbb{N}}\{{y}_{m}\in T({x}_{m})\}$ and converging to $y\in B$, then $y\in T(x)$.

It is worth noticing that the map *T* in Theorem 1.1 is continuous, so it is u.s.c. on *X*, which by [[14], Theorem 6, p.112], shows that *T* is closed on *X*.

**Definition 3.3**Let

*X*be a

*b*-metric space (with $s\u2a7e1$) and let the map $J:X\times X\to [0,\mathrm{\infty})$ be a

*b*-generalized pseudodistance on

*X*. Let $(A,B)$ be a pair of nonempty subsets of

*X*. The map $T:A\to {2}^{B}$ such that $T(x)\in Cl(X)$, for each $x\in X$, we call a set-valued non-self-mapping contraction of Nadler type, if the following condition holds:

It is worth noticing that if $(X,d)$ is a metric space (*i.e.* $s=1$) and we put $J=d$, then we obtain the classical Nadler condition. Now we prove two auxiliary lemmas.

**Lemma 3.1**

*Let*

*X*

*be a complete*

*b*-

*metric space*(

*with*$s\u2a7e1$).

*Let*$(A,B)$

*be a pair of nonempty closed subsets of*

*X*

*and let*$T:A\to {2}^{B}$.

*Then*

*Proof*Let $x,y\in A$, $\gamma >0$ and $w\in T(x)$ be arbitrary and fixed. Then, by the definition of infimum, there exists $v\in T(y)$ such that

Hence, by (3.3) we obtain $J(w,v)\u2a7d{H}^{J}(T(x),T(y))+\gamma $, thus (3.2) holds. □

**Lemma 3.2**

*Let*

*X*

*be a complete*

*b*-

*metric space*(

*with*$s\u2a7e1$)

*and let the sequence*$({x}_{m}:m\in \{0\}\cup \mathbb{N})$

*satisfy*

*Then* $({x}_{m}:m\in \{0\}\cup \mathbb{N})$ *is a Cauchy sequence on* *X*.

*Proof*From (3.4) we claim that

Let now ${\epsilon}_{0}>0$ be arbitrary and fixed, let ${n}_{0}({\epsilon}_{0})=max\{{n}_{2}({\epsilon}_{0}),{n}_{3}({\epsilon}_{0})\}+1$ and let $k,l\in \mathbb{N}$ be arbitrary and fixed such that $k>l>{n}_{0}$. Then $k={i}_{0}+{n}_{0}$ and $l={j}_{0}+{n}_{0}$ for some ${i}_{0},{j}_{0}\in \mathbb{N}$ such that ${i}_{0}>{j}_{0}$ and, using (d3), (3.9), and (3.10), we get $d({x}_{k},{x}_{l})=d({x}_{{i}_{0}+{n}_{0}},{x}_{{j}_{0}+{n}_{0}})\u2a7dsd({x}_{{n}_{0}},{x}_{{i}_{0}+{n}_{0}})+sd({x}_{{n}_{0}},{x}_{{j}_{0}+{n}_{0}})<s{\epsilon}_{0}/2s+s{\epsilon}_{0}/2s={\epsilon}_{0}$.

Hence, we conclude that ${\mathrm{\forall}}_{\epsilon >0}{\mathrm{\exists}}_{{n}_{0}={n}_{0}(\epsilon )\in \mathbb{N}}{\mathrm{\forall}}_{k,l\in \mathbb{N},k>l>{n}_{0}}\{d({x}_{k},{x}_{l})<\epsilon \}$. Thus the sequence $({x}_{m}:m\in \{0\}\cup \mathbb{N})$ is Cauchy. □

Next we present the main result of the paper.

**Theorem 3.1** *Let* *X* *be a complete* *b*-*metric space* (*with* $s\u2a7e1$) *and let the map* $J:X\times X\to [0,\mathrm{\infty})$ *be a* *b*-*generalized pseudodistance on* *X*. *Let* $(A,B)$ *be a pair of nonempty closed subsets of* *X* *with* ${A}_{0}\ne \mathrm{\varnothing}$ *and such that* $(A,B)$ *has the* ${P}^{J}$-*property and* *J* *is associated with* $(A,B)$. *Let* $T:A\to {2}^{B}$ *be a closed set*-*valued non*-*self*-*mapping contraction of Nadler type*. *If* $T(x)$ *is bounded and closed in* *B* *for all* $x\in A$, *and* $T(x)\subset {B}_{0}$ *for each* $x\in {A}_{0}$, *then* *T* *has a best proximity point in* *A*.

*Proof* To begin, we observe that by assumptions of Theorem 3.1 and by Lemma 3.1, the property (3.2) holds. The proof will be broken into four steps.

*We can construct the sequences*$({w}^{m}:m\in \{0\}\cup \mathbb{N})$

*and*$({v}^{m}:m\in \{0\}\cup \mathbb{N})$

*such that*

*and*

*and*

Consequently, the property (3.15) holds.

Next, by (3.15) we obtain ${lim}_{n\to \mathrm{\infty}}{sup}_{m>n}J({v}^{n},{v}^{m})=0$. Then the properties (3.11)-(3.17) hold.

Step 2. *We can show that the sequence* $({w}^{m}:m\in \{0\}\cup \mathbb{N})$ *is Cauchy*.

Indeed, it is an easy consequence of (3.16) and Lemma 3.2.

Step 3. *We can show that the sequence* $({v}^{m}:m\in \{0\}\cup \mathbb{N})$ *is Cauchy*.

Indeed, it follows by Step 1 and by a similar argumentation as in Step 2.

*There exists a best proximity point*,

*i.e.*

*there exists*${w}_{0}\in A$

*such that*

*X*is a complete space, there exist ${w}_{0},{v}_{0}\in X$ such that ${lim}_{m\to \mathrm{\infty}}{w}^{m}={w}_{0}$ and ${lim}_{m\to \mathrm{\infty}}{v}^{m}={v}_{0}$, respectively. Now, since

*A*and

*B*are closed (we recall that ${\mathrm{\forall}}_{m\in \{0\}\cup \mathbb{N}}\{{w}^{m}\in A\wedge {v}^{m}\in B\}$), thus ${w}_{0}\in A$ and ${v}_{0}\in B$. Finally, since by (3.12) we have ${\mathrm{\forall}}_{m\in \{0\}\cup \mathbb{N}}\{{v}^{m}\in T({w}^{m})\}$, by closedness of

*T*, we have

*J*and $(A,B)$ are associated, so by Definition 2.4(II), we conclude that

Finally, (3.27) and (3.28), give $inf\{d({w}_{0},z):z\in T({w}_{0})\}=dist(A,B)$. □

## 4 Examples illustrating Theorem 3.1 and some comparisons

Now, we will present some examples illustrating the concepts having been introduced so far. We will show a fundamental difference between Theorem 1.1 and Theorem 3.1. The examples will show that Theorem 3.1 is an essential generalization of Theorem 1.1. First, we present an example of *J*, a generalized pseudodistance.

**Example 4.1**Let

*X*be a

*b*-metric space (with constant $s=2$) where

*b*-metric $d:X\times X\to [0,\mathrm{\infty})$ is of the form $d(x,y)={|x-y|}^{2}$, $x,y\in X$. Let the closed set $E\subset X$, containing at least two different points, be arbitrary and fixed. Let $c>0$ such that $c>\delta (E)$, where $\delta (E)=sup\{d(x,y):x,y\in X\}$ be arbitrary and fixed. Define the map $J:X\times X\to [0,\mathrm{\infty})$ as follows:

The map *J* is a *b*-generalized pseudodistance on *X*. Indeed, it is worth noticing that the condition (J1) does not hold only if some ${x}_{0},{y}_{0},{z}_{0}\in X$ such that $J({x}_{0},{z}_{0})>s[J({x}_{0},{y}_{0})+J({y}_{0},{z}_{0})]$ exists. This inequality is equivalent to $c>s[d({x}_{0},{y}_{0})+d({y}_{0},{z}_{0})]$ where $J({x}_{0},{z}_{0})=c$, $J({x}_{0},{y}_{0})=d({x}_{0},{y}_{0})$ and $J({y}_{0},{z}_{0})=d({y}_{0},{z}_{0})$. However, by (4.1), $J({x}_{0},{z}_{0})=c$ shows that there exists $v\in \{{x}_{0},{z}_{0}\}$ such that $v\notin E$; $J({x}_{0},{y}_{0})=d({x}_{0},{y}_{0})$ gives $\{{x}_{0},{y}_{0}\}\subset E$; $J({y}_{0},{z}_{0})=d({y}_{0},{z}_{0})$ gives $\{{y}_{0},{z}_{0}\}\subset E$. This is impossible. Therefore, ${\mathrm{\forall}}_{x,y,z\in X}\{J(x,y)\u2a7ds[J(x,z)+J(z,y)]\}$, *i.e.* the condition (J1) holds.

*X*satisfy (2.1) and (2.2). Then, in particular, (2.2) yields

Therefore, the sequences $({x}_{m}:m\in \mathbb{N})$ and $({y}_{m}:m\in \mathbb{N})$ satisfy (2.3). Consequently, the property (J2) holds.

The next example illustrates Theorem 3.1.

**Example 4.2**Let

*X*be a

*b*-metric space (with constant $s=2$), where $X=[0,3]$ and $d(x,y)={|x-y|}^{2}$, $x,y\in X$. Let $A=[0,1]$ and $B=[2,3]$. Let $E=[0,\frac{1}{4}]\cup [1,3]$ and let the map $J:X\times X\to [0,\mathrm{\infty})$ be defined as follows:

*E*is closed set and $\delta (E)=9<10$, by Example 4.1 we see that the map

*J*is the

*b*-generalized pseudodistance on

*X*. Moreover, it is easy to verify that ${A}_{0}=\{1\}$ and ${B}_{0}=\{2\}$. Indeed, $dist(A,B)=1$, thus

and, by (4.4), $\{x,y\}\cap E=\{x,y\}$, so $J(x,y)=d(x,y)$, $y\in [2,3]$ and $x\in [0,1/4]\cup \{1\}$. Consequently ${B}_{0}=\{2\}$.

We observe the following.

(I) *We can show that the pair* $(A,B)$ *has the* ${P}^{J}$-*property*.

*E*, by (4.4) we have

(II) *We can show that the map* *J* *is associated with* $(A,B)$.

*X*, such that ${lim}_{m\to \mathrm{\infty}}{x}_{m}=x$, ${lim}_{m\to \mathrm{\infty}}{y}_{m}=y$ and

*d*, we have $d(x,y)=dist(A,B)$.

- (III)
*It is easy to see that**T**is a closed map on**X*. - (IV)
*We can show that**T**is a set*-*valued non*-*self*-*mapping contraction of Nadler type with respect**J*(*for*$\lambda =1/2$;*as a reminder*:*we have*$s=2$).

Indeed, let $x,y\in A$ be arbitrary and fixed. First we observe that since $T(A)\subset B=[2,3]\subset E$, by (4.4) we have ${H}^{J}(T(x),T(y))=H(T(x),T(y))\u2a7d1$, for each $x,y\in A$. We consider the following two cases.

Case 1. If $\{x,y\}\cap E\ne \{x,y\}$, then by (4.4), $J(x,y)=10$, and consequently ${H}^{J}(T(x),T(y))\u2a7d1<10/4=(1/4)\cdot 10=(\lambda /s)J(x,y)$. In consequence, $s{H}^{J}(T(x),T(y))\u2a7d\lambda J(x,y)$.

can be deduced that ${\mathrm{\forall}}_{x,y\in [0,/1/4]\cup \{1\}}\{{H}^{J}(T(x),T(y))=0\}$. Hence, $s{H}^{J}(T(x),T(y))=0\u2a7d\lambda J(x,y)$.

*T*is the set-valued non-self-mapping contraction of Nadler type with respect to

*J*.

- (V)
*We can show that*$T(x)$*is bounded and closed in**B**for all*$x\in A$.

- (VI)
*We can show that*$T(x)\subset $ ${B}_{0}$*for each*$x\in {A}_{0}$.

Indeed, by (I), we have ${A}_{0}=\{1\}$ and ${B}_{0}=\{2\}$, from which, by (4.5), we get $T(1)=\{2\}\subseteq {B}_{0}$.

All assumptions of Theorem 3.1 hold. We see that $D(1,T(1))=D(1,\{2\})=1=dist(A,B)$, *i.e.* 1 is the best proximity point of *T*.

**Remark 4.1** (I) The introduction of the concept of *b*-generalized pseudodistances is essential. If *X* and *T* are like in Example 4.2, then we can show that *T* *is not a set-valued non*-*self*-*mapping contraction of Nadler type with respect to* *d*. Indeed, suppose that *T* *is a set-valued non-self-mapping contraction of Nadler type*, *i.e.* ${\mathrm{\exists}}_{0\u2a7d\lambda <1}{\mathrm{\forall}}_{x,y\in X}\{sH(T(x),T(y))\u2a7d\lambda d(x,y)\}$. In particular, for ${x}_{0}=\frac{1}{2}$ and ${y}_{0}=1$ we have $T({x}_{0})=[5/2,3]$, $T({y}_{0})=\{2\}$ and $2=2H(T({x}_{0}),T({y}_{0}))=sH(T({x}_{0}),T({y}_{0}))\u2a7d\lambda d({x}_{0},{y}_{0})=\lambda {|1/2-1|}^{2}=\lambda \cdot 1/4<1/4$. This is absurd.

(II) If *X* is metric space ($s=1$) with metric $d(x,y)=|x-y|$, $x,y\in X$, and *T* is like in Example 4.2, then we can show that *T* *is not a set*-*valued non*-*self*-*mapping contraction of Nadler type with respect to* *d*. Indeed, suppose that *T* *is a set*-*valued non*-*self*-*mapping contraction of Nadler type*, *i.e.* ${\mathrm{\exists}}_{0\u2a7d\lambda <1}{\mathrm{\forall}}_{x,y\in X}\{H(T(x),T(y))\u2a7d\lambda d(x,y)\}$. In particular, for ${x}_{0}=\frac{1}{2}$ and ${y}_{0}=1$ we have $2=2H(T({x}_{0}),T({y}_{0}))=sH(T({x}_{0}),T({y}_{0}))\u2a7d\lambda d({x}_{0},{y}_{0})=\lambda |1/2-1|=\lambda \cdot 1/2<1/2$. This is absurd. Hence, we find that our theorem is more general than Theorem 1.1 (Abkar and Gabeleh [13]).

## Declarations

## Authors’ Affiliations

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