Comments on the paper “Best proximity point results with their consequences and applications”

Abstract

Very recently Jain et al. (J. Inequal. Appl. 2022:73, 2022) introduced the concept of multivalued F-contraction with altering distance function and investigated the existence of best proximity points for this class of mappings. In this article we prove that the existence of best proximity points for multivalued F-contraction non-self mappings can be obtained from the corresponding fixed point result for multivalued F-contraction self mappings, and so the main conclusion due to Jain et al. is not a real generalization of fixed point theory.

1 Introduction

In metric fixed point theory, sufficient conditions are derived to ensure the existence of solutions of the equation $U(x)=x$, where U is a self mapping defined on a metric space $(X,d)$. The Banach contraction principle [2] for standard metric spaces is one of the important results in metric fixed point theory, and it has a lot of applications in differential equations and integral equations for the existence of solutions. Let A and B be two nonempty subsets of a metric space $(X,d)$ and $Q:A\rightarrow B$ be a non-self mapping. A necessary condition to guarantee the existence of solutions of the equation $Qx=x$ is $Q(A)\cap A\neq \emptyset$. If $Q(A)\cap A= \emptyset$, then the mapping Q has no fixed points. In this case, one seeks for an element in the domain space whose distance from its image is minimum, i.e., one interesting problem is to minimize $d(x,Qx)$ such that $x\in A$. Since $d(x,Qx)\geq \sigma (A,B):=\inf \{d(x,y):x\in A, y\in B\}$, one searches for an element $x\in A$ for which $d(x,Qx)= \sigma (A,B)$. Best proximity point problems deal with this situation. Authors usually discover best proximity point theorems to generalize the corresponding fixed point theorems.

Just recently, Jain et al. [1] introduced the concept of multivalued F-contraction with altering distance function and investigated the existence of best proximity point for this class of non-self mappings. After proving their main result [1, Theorem 2.1], they stated the corresponding fixed point theorem [1, Corollary 2.2] for self mappings. Then the authors furnished nontrivial examples to validate their claim and also provided applications to fractional calculus and on equation of motion modelling to differential equations.

In this note, we show that the main result of [1], which is related to the existence of a best proximity point for multivalued F-contraction non-self mappings, is a straightforward consequence of the corresponding fixed point result.

2 Preliminaries

We first recall some notations and definitions from [1], which will be needed throughout this paper.

Let W be a nonempty set and $(W, \sigma )$ be a metric space. By $CB(W)$ we denote the collection of all closed and bounded subsets of W.

Suppose that $L~\text{and}~M$ are nonempty subsets of W and $u\in W$. We shall adopt the following notations:

$$\begin{aligned}& D(u,M)=\inf \bigl\{ \sigma (u,v): v\in M\bigr\} ; \\ & \delta (L,M)=\sup \bigl\{ \sigma (u,v): u\in L\text{ and } v\in M\bigr\} ; \\ & \sigma (L,M)=\inf \bigl\{ \sigma (u,v): u\in L \text{ and } v\in M\bigr\} ; \\ & L_{0}=\bigl\{ u\in L : \sigma (u,v)=\sigma (L,M)\text{ for some } v \in M\bigr\} ; \\ & M_{0}=\bigl\{ v \in M : \sigma (u,v)=\sigma (L,M)\text{ for some } u \in L\bigr\} . \end{aligned}$$

Definition 2.1

([3])

Let $(L,M)$ be a nonempty pair of subsets of a metric space $(W, \sigma )$. The pair $(L,M)$ with $L_{0}\neq \emptyset$ is said to have the P-property if for every $u_{1}, u_{2} \in L_{0}$ and $v_{1}, v_{2} \in M_{0}$ we have

$$ \textstyle\begin{cases} \sigma (u_{1}, v_{1})= \sigma (L, M), \\ \sigma (u_{2}, v_{2})= \sigma (L, M), \end{cases}\displaystyle \Longrightarrow\quad \sigma (u_{1},u_{2}) = \sigma (v_{1},v_{2}). $$

Definition 2.2

([4])

Let $(L,M)$ be a nonempty pair of subsets of a metric space $(W, \sigma )$ and $T:L\rightarrow 2^{M}$ be a multivalued mapping. A point $u\in L$ is called a best proximity point of the mapping T if $D(u,Tu)=\sigma (L,M)$.

Definition 2.3

A mapping $\phi :(0,\infty )\rightarrow (0,\infty )$ is called an altering distance function if ϕ is continuous, monotonically increasing and $\phi (u)>0$ for all $u>0$.

In the year 2012, Wardowski introduced the following class of functions in order to present an interesting extension of the Banach contraction principle (see also [6, 7] for more information).

Definition 2.4

([5])

A function $F:(0,+\infty )\to \mathbb{R}$ is said to be a Wardowski function provided that

$(F_{1})$:

F is strictly increasing;

$(F_{2})$:

For each sequence $\{\gamma _{n}\}$ of positive numbers, $\lim_{n\to \infty}\gamma _{n}=0\Leftrightarrow \lim_{n\to \infty}F( \gamma _{n})=-\infty$;

$(F_{3})$:

There exists $c\in (0,1)$ such that $\lim_{\gamma \to 0}\gamma ^{c}F(\gamma )=0$.

The class of all Wardowski functions will be denoted by $\mathcal{F}$.

Using the class of Wardowski functions, the following family of multivalued contractions was introduced in [1].

Definition 2.5

Let $(W, \sigma , \leq )$ be a partially ordered complete metric space and $(L,M)$ be a nonempty pair of subsets of W such that $L_{0}$ is nonempty. A mapping $T:L\rightarrow CB(M)$ is said to be a multivalued non-self F-contraction depending on an altering distance function ϕ if $Tu_{0}\subseteq M_{0}$ for any $u_{0}\in L_{0}$, and there exist $\tau >0$ and $F\in \mathcal{F}$ such that

$$ \tau +F \bigl(\phi \bigl(\delta (Tu,Tv)\bigr) \bigr)\leq F \bigl(\phi \bigl(N(u,v)\bigr)-\phi \bigl( \sigma (L,M)\bigr) \bigr)$$

for all $u,v \in L$ with $u\leq v$, where $N(u,v)=\max \{\sigma (u,v), D(u,Tu), D(v,Tv), \frac{D(u,Tv)+D(v,Tu)}{2}\}$ and ϕ satisfies $\phi (x+y)\leq \phi (x)+\phi (y)$ for all $x,y>0$.

Notice that if in the above definition $L=M$, then the mapping T is said to be a multivalued F-contraction.

Here we state the main result of [1].

Theorem 2.6

([1, Theorem 2.1])

Let $(W, \sigma , \leq )$ be a partially ordered complete metric space and $(L,M)$ be a nonempty pair of closed subsets of W such that $L_{0}$ is nonempty and $(L,M)$ satisfies the P-property. Let $T:L\rightarrow CB(M)$ be a multivalued non-self F-contraction depending on an altering distance function ϕ such that the following conditions are satisfied:

(i) There exist two elements $u_{0},u_{1} \in L_{0}$ and $v_{0}\in Tu_{0}$ such that $\sigma (u_{1},v_{0})=\sigma (L,M)$ and $u_{0}\leq u_{1}$;

(ii) For all $u,v\in L_{0}$, $u\leq v$ implies $Tu\subseteq Tv$;

(iii) If $\{u_{n}\}$ is a nondecreasing sequence in L such that $u_{n}\rightarrow u$ as $n\rightarrow \infty$, then $u_{n}\leq u$ for all $n\geq 1$.

Then T has a best proximity point.

It is worth noticing that if $L=M$ in Theorem 2.6, then we obtain the following fixed point theorem.

Theorem 2.7

Let $(W, \sigma , \leq )$ be a partially ordered complete metric space and L be a nonempty closed subset of the metric space W, and let $T:L\rightarrow CB(L)$ be a multivalued F-contraction depending on an altering distance function ϕ such that the following conditions are satisfied:

(i) There exist two elements $u_{0},u_{1} \in L$ and $v_{0}\in Tu_{0}$ such that $\sigma (u_{1},v_{0})=0$ and $u_{0}\leq u_{1}=v_{0}$;

(ii) For all $u,v\in L$, $u\leq v$ implies $Tu\subseteq Tv$;

(iii) If $\{u_{n}\}$ is a nondecreasing sequence in L such that $u_{n}\rightarrow u$ as $n\rightarrow \infty$, then $u_{n}\leq u$ for all $n\geq 1$.

Then T has a fixed point.

3 Main results

The main motivation of the current work is to show that Theorem 2.6 is a special case of Theorem 2.7, and so the main conclusion of [1] is not a real generalization of fixed point theory.

To this end, we first present a brief proof of Theorem 2.7.

Proof of Theorem 2.7

Let $u_{0}\in L$. So, according to condition (i), there exists $u_{1}\in Tu_{0}$ such that $u_{0}\leq u_{1}$. Now, from condition (ii), $Tu_{0}\subseteq Tu_{1}$. So, there exists $u_{2}\in Tu_{1}$ such that $u_{1}\leq u_{2}$. Continuing this process, we obtain a nondecreasing sequence $\{u_{n}\}_{n\geq 0}$ in L such that $u_{n+1}\in Tu_{n}$ for any $n\geq 1$. If for some $n_{0}\in \mathbb{N}$, $u_{n_{0}}=u_{n_{0}+1}$, then

$$\begin{aligned}& D(u_{n_{0}+1}, Tu_{n_{0}})\leq \sigma (u_{n_{0}},u_{n_{0}+1})=0,\\& \quad \Longrightarrow\quad D(u_{n_{0}+1},Tu_{n_{0}})=0,\\& \quad \Longrightarrow\quad D(u_{n_{0}},Tu_{n_{0}})=0 , \end{aligned}$$

that is, $u_{n_{0}}\in L$ is a fixed point of the mapping T, and we are finished. Assume that $u_{n}\neq u_{n+1}$ for all $n\geq 1$. Thus

$$\begin{aligned} F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq & F \bigl(\phi \bigl(\delta (Tu_{n-1}, Tu_{n}) \bigr) \bigr) \\ \leq & F \bigl(\phi \bigl(N(u_{n-1}, u_{n})\bigr) \bigr)- \tau , \end{aligned}$$

where

$$\begin{aligned}& N(u_{n-1}, u_{n}) \\& \quad = \max \biggl\{ \sigma (u_{n-1}, u_{n}), D(u_{n-1},Tu_{n-1}), D(u_{n},Tu_{n}), \frac{D(u_{n-1},Tu_{n})+D(u_{n},Tu_{n-1})}{2}\biggr\} \\& \quad \leq \max \biggl\{ \sigma (u_{n-1}, u_{n}), \sigma (u_{n-1},u_{n}), \sigma (u_{n},u_{n+1}), \frac{D(u_{n-1},u_{n+1})+D(u_{n},u_{n})}{2}\biggr\} \\& \quad \leq \max \bigl\{ \sigma (u_{n-1}, u_{n}),\sigma (u_{n},u_{n+1})\bigr\} , \end{aligned}$$

which means that

$$ F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq F \bigl(\phi \bigl(\max \bigl\{ \sigma (u_{n-1}, u_{n}),\sigma (u_{n},u_{n+1})\bigr\} \bigr) \bigr)- \tau . $$

Note that if $\sigma (u_{n-1},u_{n})<\sigma (u_{n},u_{n+1})$, then

$$ F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr)- \tau ,$$

which is a contradiction. Hence, $\sigma (u_{n-1},u_{n})\geq \sigma (u_{n},u_{n+1})$ for all $n\geq 1$. Let ${\lim_{n\rightarrow \infty}}\sigma (u_{n}, u_{n+1}):=r \geq 0$. Now,

$$\begin{aligned}& F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq F \bigl(\phi \bigl(\sigma (u_{n-1}, u_{n})\bigr) \bigr)-\tau ,\\& \quad \Longrightarrow\quad F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq F \bigl( \phi \bigl(\sigma (u_{n-2}, u_{n-1})\bigr) \bigr)-2\tau . \end{aligned}$$

Continuing in this way, we obtain

$$\begin{aligned}& F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr) \leq F \bigl(\phi \bigl(\sigma (u_{0}, u_{1})\bigr) \bigr) -n\tau ,\\& \quad \Longrightarrow\quad {\lim_{n\rightarrow \infty}} F \bigl( \phi \bigl( \sigma (u_{n}, u_{n+1})\bigr) \bigr)=- \infty ,\\& \quad \Longrightarrow\quad {\lim_{n\rightarrow \infty}} \phi \bigl( \sigma (u_{n}, u_{n+1})\bigr)=0,\\& \quad \Longrightarrow\quad {\lim_{n\rightarrow \infty}} \sigma (u_{n}, u_{n+1})=0. \end{aligned}$$

This shows that $r=0$. Corresponding to the function F, there exists $k\in (0,1)$ such that

$$\begin{aligned}& {\lim_{\phi (\sigma (u_{n}, u_{n+1}))\rightarrow 0}} \phi \bigl(\sigma (u_{n}, u_{n+1})\bigr)^{k} F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr)=0,\\& \quad \Longrightarrow\quad {\lim_{n \rightarrow \infty}}\phi \bigl( \sigma (u_{n}, u_{n+1})\bigr)^{k} F \bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr)=0. \end{aligned}$$

Therefore,

$$\begin{aligned}& \phi \bigl(\sigma (u_{n}, u_{n+1})\bigr)^{k} \bigl(F\bigl(\phi \bigl(\sigma (u_{n}, u_{n+1})\bigr) \bigr)-F\bigl( \phi \bigl(\sigma (u_{0}, u_{1})\bigr) \bigr) \bigr)\leq -\phi \bigl(\sigma (u_{n}, u_{n+1}) \bigr)^{k} n \tau,\\& \quad \Longrightarrow \quad {\lim_{n \rightarrow \infty}}\phi \bigl( \sigma (u_{n}, u_{n+1})\bigr)^{k}n \tau =0,\\& \quad \Longrightarrow \quad {\lim_{n \rightarrow \infty}}nB_{n}^{k}=0, \end{aligned}$$

where $B_{n}:=\phi (\sigma (u_{n}, u_{n+1}))$. Thus, for given $\varepsilon >0$, there exists $n_{1}\in \mathbb{N}$ such that

$$\begin{aligned}& \bigl\vert nB_{n}^{k} \bigr\vert < \varepsilon ^{k},\quad \forall n\geq n_{1},\\& \qquad \Longrightarrow\quad B_{n} < \frac{\varepsilon}{n^{\frac{1}{k}}},\quad \forall n \geq n_{1}. \end{aligned}$$

Let $m>n\geq n_{1}$. Then

$$\begin{aligned}& \phi \bigl(\sigma (u_{m}, u_{n})\bigr)\leq \phi \bigl( \sigma (u_{m}, u_{m-1})\bigr)+\phi \bigl( \sigma (u_{m-1}, u_{m-2})\bigr)+\cdots +\phi \bigl(\sigma (u_{n+1}, u_{n})\bigr),\\& \quad \Longrightarrow \quad \phi \bigl(\sigma (u_{m}, u_{n}) \bigr)< \frac{\varepsilon}{(m-1)^{\frac{1}{k}}}+ \frac{\varepsilon}{(m-2)^{\frac{1}{k}}}+\cdots + \frac{\varepsilon}{{n}^{\frac{1}{k}}},\\& \quad \Longrightarrow\quad \phi \bigl(\sigma (u_{m}, u_{n}) \bigr)< \varepsilon \sum_{i=n}^{ \infty} \frac{1}{i^{\frac{1}{k}}}\rightarrow 0, \quad \text{as } m,n \rightarrow \infty ,\\& \quad \Longrightarrow\quad \sigma (u_{m}, u_{n})\rightarrow 0\quad \text{as }m,n \rightarrow \infty . \end{aligned}$$

This implies that the sequence $\{u_{n}\}$ is a Cauchy sequence. Since L is complete, there exists $u\in L$ such that $u_{n}\rightarrow u$ as $n\rightarrow \infty$. It now follows from condition (iii) that $u_{n}\leq u$ for all $n\geq 1$. Suppose that $u\notin Tu$. Then

$$\begin{aligned}& F \bigl(\phi \bigl(D(u_{n+1}, Tu)\bigr) \bigr) \\& \quad \leq F \bigl(\phi \bigl(\delta (Tu_{n}, Tu)\bigr) \bigr) \\& \quad \leq F \bigl(\phi \bigl(N(u_{n}, u)\bigr) \bigr)- \tau \\& \quad \leq F \biggl(\phi \biggl(\max \biggl\{ \sigma (u_{n}, u), \sigma (u_{n},u_{n+1}), D(u,Tu), \frac{D(u_{n},Tu)+D(u_{n+1},u)}{2}\biggr\} \biggr) \biggr)- \tau . \end{aligned}$$

Letting $n\rightarrow \infty$ in the above relation, we obtain

$$ F \bigl(\phi \bigl(D(u, Tu)\bigr) \bigr) \leq F \bigl(\phi \bigl(D(u, Tu)\bigr) \bigr)-\tau ,$$

which is a contradiction. Therefore, $u\in Tu$, and this completes the proof. □

We are now ready to state the main result of this article.

Theorem 3.1

Theorem 2.6is a straightforward consequence of Theorem 2.7.

Proof

Let $x_{0} \in L_{0}$. Then $Tx_{0}\subseteq M_{0}$. As the pair $(L,M)$ has the P-property, for every $y_{0}\in Tx_{0}$, there exists unique $z_{0}\in L_{0}$ such that $\sigma (z_{0},y_{0})=\sigma (L,M)$. Let

$$ D_{x_{0}}= \bigl\{ z\in L_{0} : \text{there exists}~y\in Tx_{0}~ \text{such that}~\sigma (z,y)=\sigma (L,M) \bigr\} . $$

We show that $D_{x_{0}}\in CB(L_{0})$. Let $p,q \in D_{x_{0}}$. Then there exist $p',q'\in Tx_{0}$ such that

$$ \sigma \bigl(p,p'\bigr)=\sigma (L,M)=\sigma \bigl(q,q' \bigr).$$

Since the pair $(L,M)$ has the P-property, $\sigma (p,q)=\sigma (p',q')\leq {\mathrm{diam}}(Tx_{0})$. This shows that $D_{x_{0}}$ is bounded. Now, let $x\in \overline{D_{x_{0}}}$. Then there exists a sequence $\{x_{n}\}\subset D_{x_{0}}$ such that $x_{n} \rightarrow x$ as $n\rightarrow \infty$. By definition of the set $D_{x_{0}}$, for all $n\geq 1$, there exists $y_{n}\in Tx_{0}$ such that $\sigma (x_{n},y_{n})=\sigma (L,M)$. Thus, for any $m,n\geq 1$, we have $\sigma (x_{n},y_{n})=\sigma (L,M)=\sigma (x_{m},y_{m})$. Again using the fact that $(L,M)$ has the P-property, we must have $\sigma (y_{m},y_{n})=\sigma (x_{m},x_{n})$, which ensures that the sequence $\{y_{n}\}$ is Cauchy. Since $Tx_{0}$ is closed, there exists $y\in Tx_{0}$ such that $y_{n} \rightarrow y$ as $n\rightarrow \infty$. Thereby, $\sigma (x,y)=\sigma (L,M)$, which concludes that $x\in D_{x_{0}}$, i.e., $D_{x_{0}}$ is a closed subset of $L_{0}$. Now, let us define a multivalued mapping $S:L_{0}\rightarrow CB(L_{0})$ by $Sx_{0}=D_{x_{0}}$. Since for every $y_{0}\in Tx_{0}$ there exists unique $x\in L_{0}$ with $\sigma (x,y_{0})=\sigma (L,M)$, so this mapping is well defined.

By the given condition there exist $u_{0}, u_{1}\in L_{0}$ and $v_{0}\in Tu_{0}$ such that $\sigma (u_{1},v_{0})=\sigma (L,M)$ and $u_{0}\leq u_{1}$. By the definition of the mapping S, $u_{1}\in Su_{0}$ and $u_{0}\leq u_{1}$. Assume that $u,v \in L_{0}$ with $u\leq v$. We assert that $Su\subseteq Sv$. Let $y\in Su=D_{u}$. Then there exists an element $w\in Tu$ for which $\sigma (y,w)=\sigma (L,M)$. By the fact that $Tu\subseteq Tv$, we conclude that $w\in Tv$, and so

$$ y\in \bigl\{ z\in L_{0} : \text{there exists}~q\in Tv~\text{such that}~ \sigma (z,q)=\sigma (L,M) \bigr\} =D_{v}=Sv. $$

So, the first and second conditions of Theorem 2.7 are satisfied. Now let $u,v\in L_{0}$ with $u\leq v$. Since $T:L\rightarrow CB(M)$ is a multivalued non-self F-contraction with altering distance function ϕ,

$$ \tau +F \bigl(\phi \bigl(\delta (Tu,Tv)\bigr) \bigr)\leq F \bigl(\phi \bigl(N(u,v)\bigr)-\phi \bigl( \sigma (L,M)\bigr) \bigr),$$

where $N(u,v)=\max \{\sigma (u,v), D(u,Tu), D(v,Tv), \frac{D(u,Tv)+D(v,Tu)}{2}\}$.

Let $p_{0}\in Su~\text{and}~q_{0} \in Sv$. Then there exists $p_{1}\in Tu$ such that $\sigma (p_{0},p_{1})=\sigma (L,M)$. Similarly, there exists $q_{1}\in Tv$ such that $\sigma (q_{0},q_{1})=\sigma (L,M)$. Since the pair $(L,M)$ has the P-property, we obtain

$$\begin{aligned}& \sigma (p_{0},q_{0})=\sigma (p_{1},q_{1}) \leq \delta (Tu,Tv),\\& \quad \Longrightarrow\quad \delta (Su,Sv)\leq \delta (Tu,Tv). \end{aligned}$$

Besides, if $p\in Tu$, $q\in Tv$, then there exists unique $(p',q') \in L_{0}\times L_{0}$ such that $\sigma (p',p)=\sigma (L,M)$ and $\sigma (q',q)=\sigma (L,M)$. Thus $(p',q')\in Su\times Sv$. Since $(L,M)$ has the P-property,

$$\begin{aligned}& \sigma (p,q)=\sigma \bigl(p',q'\bigr)\leq \delta (Su,Sv),\\& \quad \Longrightarrow\quad \delta (Tu,Tv)\leq \delta (Su,Sv). \end{aligned}$$

Thereby, $\delta (Tu,Tv)= \delta (Su,Sv)$. Let $p'\in Su$. Then there exists $p\in Tu$ such that $\sigma (p',p)=\sigma (L,M)$, which deduces that

$$\begin{aligned}& D(u,Tu)\leq \sigma (u,p)\leq \sigma \bigl(u,p'\bigr)+\sigma (L,M),\\& \quad \Longrightarrow\quad D(u,Tu)\leq D(u,Su)+\sigma (L,M). \end{aligned}$$

Moreover, if $h'\in Sv$, then there exists $h\in Tv$ such that $\sigma (h',h)=\sigma (L,M)$, which implies that

$$\begin{aligned}& D(v,Tv)\leq \sigma (v,h)\leq \sigma \bigl(v,h'\bigr)+\sigma (L,M),\\& \quad \Longrightarrow\quad D(v,Tv)\leq D(v,Sv)+\sigma (L,M). \end{aligned}$$

Similarly, if $q'\in Sv$, then there exists $q\in Tv$ such that $\sigma (q',q)=\sigma (L,M)$, and so

$$\begin{aligned}& D(u,Tv)\leq \sigma (u,q)\leq \sigma \bigl(u,q'\bigr)+\sigma (L,M),\\& \quad \Longrightarrow\quad D(u,Tv)\leq D(u,Sv)+\sigma (L,M). \end{aligned}$$

Equivalently,

$$ D(v,Tu)\leq D(v,Su)+\sigma (L,M).$$

This shows that $N(u,v)\leq N'(u,v)+\sigma (L,M)$, where

$$ N'(u,v)=\max \biggl\{ \sigma (u,v), D(u,Su), D(v,Sv), \frac{D(u,Sv)+D(v,Su)}{2} \biggr\} .$$

So, we have

$$\begin{aligned}& \tau +F \bigl(\phi \bigl(\delta \bigl(S(u),S(v)\bigr)\bigr) \bigr)\leq F \bigl( \phi \bigl(N'(u,v)+ \sigma (L,M)\bigr)-\phi \bigl(\sigma (L,M) \bigr) \bigr),\\& \quad \Longrightarrow\quad \tau +F \bigl(\phi \bigl(\delta \bigl(S(u),S(v)\bigr) \bigr) \bigr)\leq F \bigl( \phi \bigl(N'(u,v)\bigr)+\phi \bigl( \sigma (L,M)\bigr)-\phi \bigl(\sigma (L,M)\bigr) \bigr),\\& \quad \Longrightarrow \quad \tau +F \bigl(\phi \bigl(\delta \bigl(S(u),S(v) \bigr)\bigr) \bigr)\leq F \bigl( \phi \bigl(N'(u,v)\bigr) \bigr). \end{aligned}$$

This shows that the mapping $S:L_{0}\rightarrow CB(L_{0})$ is a multivalued F-contraction with altering distance function ϕ. Also, the third condition of Theorem 2.7 is satisfied. It now follows from Theorem 2.7 that there exists $x^{*}\in L_{0}$ such that $x^{*}\in Sx^{*}$, which deduces that there exists $y^{*}\in Tx^{*}$ such that $\sigma (x^{*},y^{*})=\sigma (L,M)$. Since

$$ D\bigl(x^{*},Tx^{*}\bigr)\leq \sigma \bigl(x^{*},y^{*}\bigr)=\sigma (L,M),$$

we conclude that $D(x^{*},Tx^{*})=\sigma (L,M)$, that is, $x^{*}\in L_{0}$ is a best proximity point of the mapping $T:L\rightarrow CB(M)$, and we are finished. □

We now apply our technique to illustrate Example 2.1 of [1].

Example 3.2

(Example 2.1 of [1])

Let $W=\mathbb{R}^{2}$ with the order ≤ defined with $(x,y)\leq (z,t)\Leftrightarrow x\leq z$, $y\leq t$. Consider the metric σ on W as follows:

$$ \sigma \bigl((u_{1},v_{1}), (u_{2},v_{2}) \bigr)= \vert u_{1}-u_{2} \vert + \vert v_{1}-v_{2} \vert . $$

Then $(W,\sigma )$ is a complete partially ordered metric space. Assume that

$$ L=\bigl\{ (-7,0), (0,-7), (0,5)\bigr\} , \qquad M=\bigl\{ (-2,0), (0,-2), (0,0), (-2,2), (2,2) \bigr\} . $$

Clearly, L and M are closed with $\sigma (L,M)=5$ and $(L_{0},M_{0})=(L,M)$. Define the multivalued non-self-mapping $T:L\rightarrow CB(M)$ as

$$ T(u,v)= \textstyle\begin{cases} \{(0,-2), (0,0)\}, & \text{if } (u,v)=(-7,0); \\ \{(2,2), (-2,2)\}, & \text{if } (u,v)=(0,-7); \\ \{(-2,2), (0,0), (0,-2), (2,2)\}, & \text{if } (u,v)=(0,5). \end{cases} $$

By the used technique as in the proof of Theorem 3.1, the mapping $S:L_{0}\rightarrow CB(L_{0})$ is defined by

$$ S(u,v)= \textstyle\begin{cases} \{(0,-7),(0,5)\}, & \text{if } (u,v)=(-7,0); \\ \{(0,5)\}, & \text{if } (u,v)=(0,-7); \\ \{(0,5),(0,-7)\}, & \text{if } (u,v)=(0,5). \end{cases} $$

Obviously, $(0,5)\in L_{0}$ is a fixed point of the mapping S which implies that it is a best proximity point of the mapping T.