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

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.

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.

Let 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

$$\{d(x_1,y_1)=dist(A,B)\wedge d(x_2,y_2)=dist(A,B)\}\Rightarrow d(x_1,x_2)=d(y_1,y_2),$$

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, ${\mathrm{\exists }}_{0⩽\lambda <1}{\mathrm{\forall }}_{x,y\in A}\{H(T(x),T(y))⩽\lambda d(x,y)\}$. 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 $s⩾1$ 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) ${\mathrm{\forall }}_{x,y,z\in X}\{d(x,z)⩽s[d(x,y)+d(y,z)]\}$.

The pair $(X,d)$ is called a b-metric space (with constant $s⩾1$). 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 $s⩾1$). 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) ${\mathrm{\forall }}_{x,y,z\in X}\{J(x,z)⩽s[J(x,y)+J(y,z)]\}$; and

(J2) for any sequences $(x_m:m\in \mathbb{N})$ and $(y_m:m\in \mathbb{N})$ in X such that

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m>n}{sup}J(x_n,x_m)=0$$

(2.1)

and

$$\underset{m\to \mathrm{\infty }}{lim}J(x_m,y_m)=0,$$

(2.2)

we have

$$\underset{m\to \mathrm{\infty }}{lim}d(x_m,y_m)=0.$$

(2.3)

Remark 2.1 (A) If $(X,d)$ is a b-metric space (with $s⩾1$), 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).

(B) From (J1) and (J2) it follows that if $x\ne y$, $x,y\in X$, then

$$J(x,y)>0\vee J(y,x)>0.$$

Indeed, if $J(x,y)=0$ and $J(y,x)=0$, then $J(x,x)=0$, since, by (J1), we get $J(x,x)⩽s[J(x,y)+J(y,x)]=s[0+0]=0$. 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 $s⩾1$). 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

$${\mathrm{\forall }}_{A,B\in Cl(X)}\{H^J(A,B)=max\{\underset{u\in A}{sup}J(u,B),\underset{v\in B}{sup}J(v,A)\}\}.$$

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

Let $(X,d)$ be a b-metric space (with $s⩾1$) 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

$$\begin{array}{r}{A}_0=\{x\in A:J(x,y)=dist(A,B)\text{ for some }y\in B\};\\ B_0=\{y\in B:J(x,y)=dist(A,B)\text{ for some }x\in A\}.\end{array}$$

Definition 2.4 Let X be a b-metric space (with $s⩾1$) 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

  $$\begin{array}{c}\{[J(x_1,y_1)=dist(A,B)]\wedge [J(x_2,y_2)=dist(A,B)]\}\hfill \\ \Rightarrow \{J(x_1,x_2)=J(y_1,y_2)\},\hfill \end{array}$$

  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 $s⩾1$), 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 $s⩾1$) 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:

$${\mathrm{\exists }}_{0⩽\lambda <1}{\mathrm{\forall }}_{x,y\in A}\{s{H}^J(T(x),T(y))⩽\lambda J(x,y)\}.$$

(3.1)

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 $s⩾1$). Let $(A,B)$ be a pair of nonempty closed subsets of X and let $T:A\to 2^B$. Then

$${\mathrm{\forall }}_{x,y\in A}{\mathrm{\forall }}_{\gamma >0}{\mathrm{\forall }}_{w\in T(x)}{\mathrm{\exists }}_{v\in T(y)}\{J(w,v)⩽H^J(T(x),T(y))+\gamma \}.$$

(3.2)

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

$$J(w,v)<inf\{J(w,u):u\in T(y)\}+\gamma .$$

(3.3)

Next,

$$\begin{array}{c}inf\{J(w,u):u\in T(y)\}+\gamma \hfill \\ ⩽sup\{inf\{J(z,u):u\in T(y)\}:z\in T(x)\}+\gamma \hfill \\ ⩽max\{sup\{inf\{J(z,u):u\in T(y)\}:z\in T(x)\},\hfill \\ sup\{inf\{J(u,z):z\in T(x)\}:u\in T(y)\}\}+\gamma \hfill \\ =H^J(T(x),T(y))+\gamma .\hfill \end{array}$$

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

Lemma 3.2 Let X be a complete b-metric space (with $s⩾1$) and let the sequence $(x_m:m\in \{0\}\cup \mathbb{N})$ satisfy

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m>n}{sup}J(x_n,x_m)=0.$$

(3.4)

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

Proof From (3.4) we claim that

$${\mathrm{\forall }}_{\epsilon >0}{\mathrm{\exists }}_{n_1=n_1(\epsilon )\in \mathbb{N}}{\mathrm{\forall }}_{n>n_1}\{sup\{J(x_n,x_m):m>n\}<\epsilon \}$$

and, in particular,

$${\mathrm{\forall }}_{\epsilon >0}{\mathrm{\exists }}_{n_1=n_1(\epsilon )\in \mathbb{N}}{\mathrm{\forall }}_{n>n_1}{\mathrm{\forall }}_{t\in \mathbb{N}}\{J(x_n,x_{t+n})<\epsilon \}.$$

(3.5)

Let $i_0,j_0\in \mathbb{N}$, $i_0>j_0$, be arbitrary and fixed. If we define

$$z_n=x_{i_0+n}\text{and}{u}_n=x_{j_0+n}\text{for }n\in \mathbb{N},$$

(3.6)

then (3.5) gives

$$\underset{n\to \mathrm{\infty }}{lim}J(x_n,z_n)=\underset{n\to \mathrm{\infty }}{lim}J(x_n,u_n)=0.$$

(3.7)

Therefore, by (3.4), (3.7), and (J2),

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,z_n)=\underset{n\to \mathrm{\infty }}{lim}d(x_n,u_n)=0.$$

(3.8)

From (3.8) and (3.6) we then claim that

$${\mathrm{\forall }}_{\epsilon >0}{\mathrm{\exists }}_{n_2=n_2(\epsilon )\in \mathbb{N}}{\mathrm{\forall }}_{n>n_2}\{d(x_n,x_{i_0+n})<\frac{\epsilon }{2s}\}$$

(3.9)

and

$${\mathrm{\exists }}_{n_3=n_3(\epsilon )\in \mathbb{N}}{\mathrm{\forall }}_{n>n_3}\{d(x_n,x_{j_0+n})<\frac{\epsilon }{2s}\}.$$

(3.10)

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 $d(x_k,x_l)=d(x_{i_0+n_0},x_{j_0+n_0})⩽sd(x_{n_0},x_{i_0+n_0})+sd(x_{n_0},x_{j_0+n_0})<s{\epsilon }_0/2s+s{\epsilon }_0/2s={\epsilon }_0$.

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 $s⩾1$) 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.

Step 1. We can construct the sequences $(w^m:m\in \{0\}\cup \mathbb{N})$ and $(v^m:m\in \{0\}\cup \mathbb{N})$ such that

$${\mathrm{\forall }}_{m\in \{0\}\cup \mathbb{N}}\{w^m\in A_0\wedge v^m\in B_0\},$$

(3.11)

$${\mathrm{\forall }}_{m\in \{0\}\cup \mathbb{N}}\{v^m\in T\left(w^m\right)\},$$

(3.12)

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,v^{m-1})=dist(A,B)\},$$

(3.13)

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(v^{m-1},v^m)⩽H^J(T\left(w^{m-1}\right),T\left(w^m\right))+{\left(\frac{\lambda }{s}\right)}^m\}$$

(3.14)

and

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,w^{m+1})=J(v^{m-1},v^m)\},$$

(3.15)

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m>n}{sup}J(w^n,w^m)=0,$$

(3.16)

and

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m>n}{sup}J(v^n,v^m)=0.$$

(3.17)

Indeed, since $A_0\ne \mathrm{\varnothing }$ and $T(x)\subseteq B_0$ for each $x\in A_0$, we may choose $w^0\in A_0$ and next $v^0\in T(w^0)\subseteq B_0$. By definition of $B_0$, there exists $w^1\in A$ such that

$$J(w^1,v^0)=dist(A,B).$$

(3.18)

Of course, since $v^0\in B$, by (3.18), we have $w^1\in A_0$. Next, since $T(x)\subseteq B_0$ for each $x\in A_0$, from (3.2) (for $x=w^0$, $y=w^1$, $\gamma =\lambda /s$, $w=v^0$) we conclude that there exists $v^1\in T(w^1)\subseteq B_0$ (since $w^1\in A_0$) such that

$$J(v^0,v^1)⩽H^J(T\left(w^0\right),T\left(w^1\right))+\frac{\lambda }{s}.$$

(3.19)

Next, since $v^1\in B_0$, by definition of $B_0$, there exists $w^2\in A$ such that

$$J(w^2,v^1)=dist(A,B).$$

(3.20)

Of course, since $v^1\in B$, by (3.20), we have $w^2\in A_0$. Since $T(x)\subseteq B_0$ for each $x\in A_0$, from (3.2) (for $x=w^1$, $y=w^2$, $\gamma ={(\lambda /s)}^2$, $w=v^1$) we conclude that there exists $v^2\in T(w^2)\subseteq B_0$ (since $w^2\in A_0$) such that

$$J(v^1,v^2)⩽H^J(T\left(w^1\right),T\left(w^2\right))+{\left(\frac{\lambda }{s}\right)}^2.$$

(3.21)

By (3.18)-(3.21) and by the induction, we produce sequences $(w^m:m\in \{0\}\cup \mathbb{N})$ and $(v^m:m\in \{0\}\cup \mathbb{N})$ such that:

$$\begin{array}{c}{\mathrm{\forall }}_{m\in \{0\}\cup \mathbb{N}}\{w^m\in A_0\wedge v^m\in B_0\},\hfill \\ {\mathrm{\forall }}_{m\in \{0\}\cup \mathbb{N}}\{v^m\in T\left(w^m\right)\},\hfill \\ {\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,v^{m-1})=dist(A,B)\}\hfill \end{array}$$

and

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(v^{m-1},v^m)⩽H^J(T\left(w^{m-1}\right),T\left(w^m\right))+{\left(\frac{\lambda }{s}\right)}^m\}.$$

Thus (3.11)-(3.14) hold. In particularly (3.13) gives ${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,v^{m-1})=dist(A,B)\wedge J(w^{m+1},v^m)=dist(A,B)\}$. Now, since the pair $(A,B)$ has the $P^J$-property, from the above we conclude

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,w^{m+1})=J(v^{m-1},v^m)\}.$$

Consequently, the property (3.15) holds.

We recall that the contractive condition (see (3.1)) is as follows:

$${\mathrm{\exists }}_{0⩽\lambda <1}{\mathrm{\forall }}_{x,y\in A}\{s{H}^J(T(x),T(y))⩽\lambda J(x,y)\}.$$

(3.22)

In particular, by (3.22) (for $x=w^m$, $y=w^{m+1}$, $m\in \{0\}\cup \mathbb{N}$) we obtain

$${\mathrm{\forall }}_{m\in \{0\}\cup \mathbb{N}}\{H^J(T\left(w^m\right),T\left(w^{m+1}\right))⩽\frac{\lambda }{s}J(w^m,w^{m+1})\}.$$

(3.23)

Next, by (3.15), (3.14), and (3.23) we calculate:

$$\begin{array}{c}{\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,w^{m+1})=J(v^{m-1},v^m)⩽H^J(T\left(w^{m-1}\right),T\left(w^m\right))+{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽\frac{\lambda }{s}J(w^{m-1},w^m)+{\left(\frac{\lambda }{s}\right)}^m=\frac{\lambda }{s}J(v^{m-2},v^{m-1})+{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽\frac{\lambda }{s}[H^J(T\left(w^{m-2}\right),T\left(w^{m-1}\right))+{\left(\frac{\lambda }{s}\right)}^{m-1}]+{\left(\frac{\lambda }{s}\right)}^m\hfill \\ =\frac{\lambda }{s}{H}^J(T\left(w^{m-2}\right),T\left(w^{m-1}\right))+2{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽{\left(\frac{\lambda }{s}\right)}^{2}J(w^{m-2},w^{m-1})+2{\left(\frac{\lambda }{s}\right)}^m={\left(\frac{\lambda }{s}\right)}^{2}J(v^{m-3},v^{m-2})+2{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽{\left(\frac{\lambda }{s}\right)}^2[H^J(T\left(w^{m-3}\right),T\left(w^{m-2}\right))+{\left(\frac{\lambda }{s}\right)}^{m-2}]+2{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ={\left(\frac{\lambda }{s}\right)}^2{H}^J(T\left(w^{m-3}\right),T\left(w^{m-2}\right))+3{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽{\left(\frac{\lambda }{s}\right)}^{3}J(w^{m-3},w^{m-2})+3{\left(\frac{\lambda }{s}\right)}^m\hfill \\ ⩽\cdots ⩽{\left(\frac{\lambda }{s}\right)}^{m}J(w^0,w^1)+m{\left(\frac{\lambda }{s}\right)}^m\}.\hfill \end{array}$$

Hence,

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,w^{m+1})⩽{\left(\frac{\lambda }{s}\right)}^{m}J(w^0,w^1)+m{\left(\frac{\lambda }{s}\right)}^m\}.$$

(3.24)

Now, for arbitrary and fixed $n\in \mathbb{N}$ and all $m\in \mathbb{N}$, $m>n$, by (3.24) and (d3), we have

$$\begin{array}{rcl}J(w^n,w^m)& ⩽& sJ(w^n,w^{n+1})+sJ(w^{n+1},w^m)\\ ⩽& sJ(w^n,w^{n+1})+s[sJ(w^{n+1},w^{n+2})+sJ(w^{n+2},w^m)]\\ =& sJ(w^n,w^{n+1})+s^{2}J(w^{n+1},w^{n+2})+s^{2}J(w^{n+2},w^m)\\ ⩽& \cdots ⩽\sum _{k=0}^{m-(n+1)}{s}^{k+1}J(w^{n+k},w^{n+1+k})\\ ⩽& \sum _{k=0}^{m-(n+1)}{s}^{k+1}[{\left(\frac{\lambda }{s}\right)}^{n+k}J(w^0,w^1)+(n+k){\left(\frac{\lambda }{s}\right)}^{n+k}]\\ =& \sum _{k=0}^{m-(n+1)}[\left(\frac{{\lambda }^{n+k}}{s^{n-1}}\right)J(w^0,w^1)+(n+k)\left(\frac{{\lambda }^{n+k}}{s^{n-1}}\right)]\\ =& \frac{1}{s^{n-1}}\sum _{k=0}^{m-(n+1)}[{\lambda }^{n+k}J(w^0,w^1)+(n+k){\lambda }^{n+k}].\end{array}$$

Hence

$$J(w^n,w^m)⩽\frac{1}{s^{n-1}}\sum _{k=0}^{m-(n+1)}[J(w^0,w^1)+(n+k)]{\lambda }^{n+k}.$$

(3.25)

Thus, as $n\to \mathrm{\infty }$ in (3.25), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m>n}{sup}J(w^n,w^m)=0.$$

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.

Step 4. There exists a best proximity point, i.e. there exists $w_0\in A$ such that

$$inf\{d(w_0,z):z\in T(w_0)\}=dist(A,B).$$

Indeed, by Steps 2 and 3, the sequences $(w^m:m\in \{0\}\cup \mathbb{N})$ and $(v^m:m\in \{0\}\cup \mathbb{N})$ are Cauchy and in particularly satisfy (3.12). Next, since 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

$$v_0\in T(w_0).$$

(3.26)

Next, since $w_0\in A$, $v_0\in B$ and $T(A)\subset B$, by (3.26) we have $T(w_0)\subset B$ and

$$\begin{array}{rcl}dist(A,B)& =& inf\{d(a,b):a\in A\wedge b\in B\}⩽D(w_0,B)⩽D(w_0,T(w_0))\\ =& inf\{d(w_0,z):z\in T(w_0)\}⩽d(w_0,v_0).\end{array}$$

(3.27)

We know that ${lim}_{m\to \mathrm{\infty }}{w}^m=w_0$, ${lim}_{m\to \mathrm{\infty }}{v}^m=v_0$. Moreover by (3.13)

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(w^m,v^{m-1})=dist(A,B)\}.$$

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

$$d(w_0,v_0)=dist(A,B).$$

(3.28)

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:

$$J(x,y)=\{\begin{array}{ll}d(x,y)& \text{if }\{x,y\}\cap E=\{x,y\},\\ c& \text{if }\{x,y\}\cap E\ne \{x,y\}.\end{array}$$

(4.1)

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, ${\mathrm{\forall }}_{x,y,z\in X}\{J(x,y)⩽s[J(x,z)+J(z,y)]\}$, i.e. the condition (J1) holds.

Proving that (J2) holds, we assume that the sequences $(x_m:m\in \mathbb{N})$ and $(y_m:m\in \mathbb{N})$ in X satisfy (2.1) and (2.2). Then, in particular, (2.2) yields

$${\mathrm{\forall }}_{0<\epsilon <c}{\mathrm{\exists }}_{m_0=m_0(\epsilon )\in \mathbb{N}}{\mathrm{\forall }}_{m⩾m_0}\{J(x_m,y_m)<\epsilon \}.$$

(4.2)

By (4.2) and (4.1), since $\epsilon <c$, we conclude that

$${\mathrm{\forall }}_{m⩾m_0}\{E\cap \{x_m,y_m\}=\{x_m,y_m\}\}.$$

(4.3)

From (4.3), (4.1), and (4.2), we get

$${\mathrm{\forall }}_{0<\epsilon <c}{\mathrm{\exists }}_{m_0\in \mathbb{N}}{\mathrm{\forall }}_{m⩾m_0}\{d(x_m,y_m)<\epsilon \}.$$

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:

$$J(x,y)=\{\begin{array}{ll}d(x,y)& \text{if }\{x,y\}\cap E=\{x,y\},\\ 10& \text{if }\{x,y\}\cap E\ne \{x,y\}.\end{array}$$

(4.4)

Of course, since 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

$$A_0=\{x\in A=[0,1]:J(x,y)=dist(A,B)=1\text{ for some }y\in B=[2,3]\},$$

and by (4.4) $\{x,y\}\cap E=\{x,y\}$, so $J(x,y)=d(x,y)$, $x\in [0,1/4]\cup \{1\}$ and $y\in [2,3]$. Consequently $A_0=\{1\}$. Similarly,

$$B_0=\{y\in B=[2,3]:J(x,y)=dist(A,B)=1\text{ for some }x\in A=[0,1]\},$$

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 $B_0=\{2\}$.

Let $T:A\to 2^B$ be given by the formula

$$T(x)=\{\begin{array}{ll}\{2\}\cup [\frac{11}{4},3]& \text{for }x\in [0,\frac{1}{4}],\\ [\frac{11}{4},3]& \text{for }x\in (\frac{1}{4},\frac{1}{2}),\\ [\frac{5}{2},3]& \text{for }x\in [\frac{1}{2},\frac{3}{4}),\\ [\frac{9}{4},3]& \text{for }x\in [\frac{3}{4},\frac{7}{8}),\\ \{2\}\cup [\frac{9}{4},3]& \text{for }x=\frac{7}{8},\\ \{2\}& \text{for }x\in (\frac{7}{8},1],\end{array}x\in X.$$

(4.5)

We observe the following.

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

Indeed, as we have previously calculated $A_0=\{1\}$ and $B_0=\{2\}$. This gives the following result: for each $x_1,x_2\in A_0$ and $y_1,y_2\in B_0$, such that $J(x_1,y_1)=dist(A,B)=1$ and $J(x_2,y_2)=dist(A,B)=1$, since $A_0$ and $B_0$ are included in E, by (4.4) we have

$$J(x_1,x_2)=d(x_1,x_2)=d(1,1)=0=d(2,2)=d(y_1,y_2)=J(y_1,y_2).$$

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

Indeed, let the 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

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{J(x_m,y_{m-1})=dist(A,B)\},$$

(4.6)

be arbitrary and fixed. Then, since $dist(A,B)=1<10$, by (4.6) and (4.4), we have

$${\mathrm{\forall }}_{m\in \mathbb{N}}\{d(x_m,y_{m-1})=J(x_m,y_{m-1})=dist(A,B)\}.$$

(4.7)

Now, from (4.7) and by continuity of 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 $H^J(T(x),T(y))=H(T(x),T(y))⩽1$, 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 $H^J(T(x),T(y))⩽1<10/4=(1/4)\cdot 10=(\lambda /s)J(x,y)$. In consequence, $s{H}^J(T(x),T(y))⩽\lambda J(x,y)$.

Case 2. If $\{x,y\}\cap E=\{x,y\}$, then $x,y\in E\cap [0,1]=[0,/1/4]\cup \{1\}$. From the obvious property

$${\mathrm{\forall }}_{x,y\in [0,/1/4]}\{T(x)=T(y)\wedge T(1)\subset T(x)\}$$

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

In consequence, 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 $x\in A$.

Indeed, it is an easy consequence of (4.5).

• (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. ${\mathrm{\exists }}_{0⩽\lambda <1}{\mathrm{\forall }}_{x,y\in X}\{sH(T(x),T(y))⩽\lambda d(x,y)\}$. 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 $2=2H(T(x_0),T(y_0))=sH(T(x_0),T(y_0))⩽\lambda d(x_0,y_0)=\lambda {|1/2-1|}^2=\lambda \cdot 1/4<1/4$. 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. ${\mathrm{\exists }}_{0⩽\lambda <1}{\mathrm{\forall }}_{x,y\in X}\{H(T(x),T(y))⩽\lambda d(x,y)\}$. In particular, for $x_0=\frac{1}{2}$ and $y_0=1$ we have $2=2H(T(x_0),T(y_0))=sH(T(x_0),T(y_0))⩽\lambda d(x_0,y_0)=\lambda |1/2-1|=\lambda \cdot 1/2<1/2$. This is absurd. Hence, we find that our theorem is more general than Theorem 1.1 (Abkar and Gabeleh [13]).