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# Best proximity point results for modified *α*-proximal *C*-contraction mappings

- Poom Kumam
^{1}Email author, - Peyman Salimi
^{2}and - Calogero Vetro
^{3}

**2014**:99

https://doi.org/10.1186/1687-1812-2014-99

© Kumam et al.; licensee Springer. 2014

**Received:**14 October 2013**Accepted:**20 March 2014**Published:**16 April 2014

## Abstract

First we introduce new concepts of contraction mappings, then we establish certain best proximity point theorems for such kind of mappings in metric spaces. Finally, as consequences of these results, we deduce best proximity point theorems in metric spaces endowed with a graph and in partially ordered metric spaces. Moreover, we present an example and some fixed point results to illustrate the usability of the obtained theorems.

**MSC:**46N40, 46T99, 47H10, 54H25.

## Keywords

- best proximity point
- fixed point
- metric space

## 1 Introduction

A wide variety of problems arising in different areas of pure and applied mathematics, such as difference and differential equations, discrete and continuous dynamic systems, and variational analysis, can be modeled as fixed point equations of the form $x=Tx$. Therefore, fixed point theory plays a crucial role for solving equations of above kind, whose solutions are the fixed points of the mapping $T:X\to X$, where *X* is a nonempty set. Areas of potential applications of this theory include physics, economics, and engineering in dealing with the study of equilibrium points (which are fixed points of certain mappings). On the other hand, if *T* is a nonself-mapping, the above fixed point equation could have no solutions and, in this case, it is of a certain interest to determine an approximate solution *x* that is optimal in the sense that the distance between *x* and *Tx* is minimum. In this context, best proximity point theory is an useful tool in studying such kind of element. We recall the following concept.

**Definition 1.1** Let *A*, *B* be two nonempty subsets of a metric space $(X,d)$ and $T:A\to B$ be a nonself-mapping. An element $x\in A$ such that $d(x,Tx)=d(A,B)$ is a best proximity point of the nonself-mapping *T*.

Clearly, if *T* is a self-mapping, a best proximity point is a fixed point, that is, $x=Tx$.

From the beginning, best proximity point theory of nonself-mappings has been studied by many authors; see the pioneering papers of Fan [1] and Kirk *et al.* [2]. The investigation of several variants of conditions for the existence of a best proximity point can be found in [3–12]. In particular, some significant best proximity point results for multivalued mappings are presented in [13]; see also the references therein.

Inspired and motivated by the above facts, in this paper, we introduce new concepts of contraction mappings. Then we establish certain best proximity point theorems for such kind of mappings in metric spaces. As consequences of these results, we deduce best proximity point theorems in metric spaces endowed with a graph and in partially ordered metric spaces. Moreover, we present an example and some fixed point results to illustrate the usability of the obtained theorems.

## 2 Preliminaries

In this section, we collect some useful definitions and results from fixed point theory.

Samet *et al.* [14] defined the notion of *α*-admissible mapping as follows.

**Definition 2.1** ([14])

*α*-admissible if

By using this concept, they proved some fixed point results.

**Theorem 2.1** ([14])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be an*

*α*-

*admissible mapping*.

*Assume that the following conditions hold*:

- (i)
*for all*$x,y\in X$*we have*$\alpha (x,y)d(Tx,Ty)\le \psi (d(x,y)),$(1)*where*$\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$*is a nondecreasing function such that*${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$*for each*$t>0$, - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$, - (iii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

Later on, working on these ideas a wide variety of papers appeared in the literature; see for instance [15–17]. Finally, we recall that Karapinar *et al.* [18] introduced the notion of triangular *α*-admissible mapping as follows.

**Definition 2.2** ([18])

*α*-admissible if

For more details and applications of this line of research, we refer the reader to some related papers of the authors and others [19–25].

## 3 Main results in metric spaces

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. Following the usual notation, we put

If $A\cap B\ne \mathrm{\varnothing}$, then ${A}_{0}$ and ${B}_{0}$ are nonempty. Further, it is interesting to notice that ${A}_{0}$ and ${B}_{0}$ are contained in the boundaries of *A* and *B*, respectively, provided *A* and *B* are closed subsets of a normed linear space such that $d(A,B)>0$ (see [26]). Also, we will use the following definition; see [27] for more details.

**Definition 3.1** Let *A*, *B* be two nonempty subsets of a metric space $(X,d)$. The pair $(A,B)$ is said to have the *V*-property if, for every sequence $\{{y}_{n}\}$ of *B* that satisfies the condition $d(x,{y}_{n})\to d(x,B)$ for some $x\in A$, there is $y\in B$ such that $d(x,y)=d(x,B)$.

From now on, denote with Ψ the family of all continuous and nondecreasing functions $\psi :[0,+\mathrm{\infty})\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that $\psi (x,y)=0$ if and only if $x=y=0$.

**Definition 3.2**Let

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$ and $\alpha :A\times A\to [0,+\mathrm{\infty})$ be a function. We say that a nonself-mapping $T:A\to B$ is triangular

*α*-proximal admissible if, for all $x,y,z,{x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$,

**Definition 3.3**Let

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$ and $\alpha :A\times A\to [0,+\mathrm{\infty})$ be a function. We say that a nonself-mapping $T:A\to B$ is

- (i)a modified
*α*-proximal*C*-contraction if, for all $u,v,x,y\in A$,$\begin{array}{r}\{\begin{array}{c}\alpha (x,y)\ge 1,\hfill \\ d(u,Tx)=d(A,B),\hfill \\ d(v,Ty)=d(A,B)\hfill \end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d(u,v)\le \frac{1}{2}(d(x,v)+d(y,u))-\psi (d(x,v),d(y,u)),\end{array}$(2) - (ii)an
*α*-proximal*C*-contraction of type (I) if, for all $u,v,x,y\in A$,$\begin{array}{r}\{\begin{array}{c}d(u,Tx)=d(A,B),\hfill \\ d(v,Ty)=d(A,B)\hfill \end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\alpha (x,y)d(u,v)\le \frac{1}{2}(d(x,v)+d(y,u))-\psi (d(x,v),d(y,u)),\end{array}$

- (iii)an
*α*-proximal*C*-contraction of type (II) if, for all $u,v,x,y\in A$,$\begin{array}{r}\{\begin{array}{c}d(u,Tx)=d(A,B),\hfill \\ d(v,Ty)=d(A,B)\hfill \end{array}\\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}{(\alpha (x,y)+\ell )}^{d(u,v)}\le {(\ell +1)}^{\frac{1}{2}(d(x,v)+d(y,u))-\psi (d(x,v),d(y,u))},\end{array}$

where $\ell >0$.

**Remark 3.1** Every *α*-proximal *C*-contraction of type (I) and *α*-proximal *C*-contraction of type (II) mappings are modified *α*-proximal *C*-contraction mappings.

Now we give our main result.

**Theorem 3.1**

*Let*

*A*,

*B*

*be two nonempty subsets of a metric space*$(X,d)$

*such that*

*A*

*is complete and*${A}_{0}$

*is nonempty*.

*Assume that*$T:A\to B$

*is a continuous modified*

*α*-

*proximal*

*C*-

*contraction such that the following conditions hold*:

- (i)
*T**is a triangular**α*-*proximal admissible mapping and*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1.$

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $\alpha (x,y)\ge 1$.

*Proof*By (ii) there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that

*T*is triangular

*α*-proximal admissible, we have $\alpha ({x}_{1},{x}_{2})\ge 1$. Thus

*T*is triangular

*α*-proximal admissible, we obtain $\alpha ({x}_{2},{x}_{3})\ge 1$ and hence

*ψ*, we get

*ψ*, we get $d=0$, that is,

*k*

*T*is a triangular

*α*-proximal admissible mapping and

*T*is a triangular

*α*-proximal admissible mapping and

Thus, by continuing this process, we get (11).

*ψ*, we get

*A*is complete, then there is $z\in A$ such that ${x}_{n}\to z$. Now, from

taking the limit as $n\to +\mathrm{\infty}$, we deduce $d(z,Tz)=d(A,B)$, because of the continuity of *T*.

which implies $d(x,y)=0$, that is, $x=y$. □

**Corollary 3.1**

*Let*

*A*,

*B*

*be two nonempty subsets of a metric space*$(X,d)$

*such that*

*A*

*is complete and*${A}_{0}$

*is nonempty*.

*Assume that*$T:A\to B$

*is a continuous*

*α*-

*proximal*

*C*-

*contraction mapping of type*(

*I*)

*or a continuous*

*α*-

*proximal*

*C*-

*contraction mapping of type*(

*II*)

*such that the following conditions hold*:

- (i)
*T**is a triangular**α*-*proximal admissible mapping and*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1.$

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $\alpha (x,y)\ge 1$.

In analogy to the main result but omitting the continuity hypothesis of *T*, we can state the following theorem.

**Theorem 3.2**

*Let*

*A*,

*B*

*be two nonempty subsets of a metric space*$(X,d)$

*such that*

*A*

*is complete*,

*the pair*$(A,B)$

*has the*

*V*-

*property and*${A}_{0}$

*is nonempty*.

*Assume that*$T:A\to B$

*is a modified*

*α*-

*proximal*

*C*-

*contraction such that the following conditions hold*:

- (i)
*T**is a triangular**α*-*proximal admissible mapping and*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,$ - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**A**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*and*${x}_{n}\to x\in A$*as*$n\to +\mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $\alpha (x,y)\ge 1$.

*Proof*Following the proof of Theorem 3.1, there exist a Cauchy sequence $\{{x}_{n}\}\subseteq A$ and $z\in A$ such that (3) holds and ${x}_{n}\to z$ as $n\to +\mathrm{\infty}$. On the other hand, for all $n\in \mathbb{N}$, we can write

*V*-property, then there exists $w\in B$ such that $d(z,w)=d(A,B)$ and hence $z\in {A}_{0}$. Moreover, since $T({A}_{0})\subseteq {B}_{0}$, then there exists $v\in A$ such that

*T*is a modified

*α*-proximal

*C*-contraction, we get

which implies, $d(z,v)=0$, that is, $v=z$. Hence *z* is a best proximity point of *T*. The uniqueness of the best proximity point follows easily proceeding as in Theorem 3.1. □

Next, we use an example to illustrate the efficiency of the new theorem.

**Example 3.1**Let $X=\mathbb{R}$ be endowed with the usual metric $d(x,y)=|x-y|$, for all $x,y\in X$. Consider $A=(-\mathrm{\infty},-1]$, $B=[1,+\mathrm{\infty})$ and define $T:A\to B$ by

*V*-property and $d(A,B)=2$. Now, we have

It is immediate to see that $T({A}_{0})\subseteq {B}_{0}$, $d(-1,T(-1))=d(A,B)=2$ and $\alpha (-1,-1)\ge 1$.

that is, *T* is a triangular *α*-proximal admissible and modified *α*-proximal *C*-contraction mapping. Moreover, if $\{{x}_{n}\}$ is a sequence such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subseteq [-2,-1]$ and hence $x\in [-2,-1]$. Consequently, $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. Therefore all the conditions of Theorem 3.2 hold for this example and *T* has a best proximity point. Here $z=-1$ is the best proximity point of *T*.

We conclude this section with another corollary.

**Corollary 3.2**

*Let*

*A*,

*B*

*be two nonempty subsets of a metric space*$(X,d)$

*such that*

*A*

*is complete*,

*the pair*$(A,B)$

*has the*

*V*-

*property and*${A}_{0}$

*is nonempty*.

*Assume that*$T:A\to B$

*is a continuous*

*α*-

*proximal*

*C*-

*contraction mapping of type*(

*I*)

*or a continuous*

*α*-

*proximal*

*C*-

*contraction mapping of type*(

*II*)

*such that the following conditions hold*:

- (i)
*T**is a triangular**α*-*proximal admissible mapping and*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist elements*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1})\ge 1,$ - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**A**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*and*${x}_{n}\to x\in A$*as*$n\to +\mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $\alpha (x,y)\ge 1$.

## 4 Some results in metric spaces endowed with a graph

Consistent with Jachymski [28], let $(X,d)$ be a metric space and Δ denotes the diagonal of the Cartesian product $X\times X$. Consider a directed graph *G* such that the set $V(G)$ of its vertices coincides with *X*, and the set $E(G)$ of its edges contains all loops, that is, $E(G)\supseteq \mathrm{\Delta}$. We assume that *G* has no parallel edges, so we can identify *G* with the pair $(V(G),E(G))$. Moreover, we may treat *G* as a weighted graph (see [29], p.309) by assigning to each edge the distance between its vertices. If *x* and *y* are vertices in a graph *G*, then a path in *G* from *x* to *y* of length *N* ($N\in \mathbb{N}$) is a sequence ${\{{x}_{i}\}}_{i=0}^{N}$ of $N+1$ vertices such that ${x}_{0}=x$, ${x}_{N}=y$ and $({x}_{i-1},{x}_{i})\in E(G)$ for $i=1,\dots ,N$. A graph *G* is connected if there is a path between any two vertices. *G* is weakly connected if $\tilde{G}$ is connected (see for details [28, 30]).

Recently, some results have appeared providing sufficient conditions for a mapping to be a Picard operator if $(X,d)$ is endowed with a graph. The first result in this direction was given by Jachymski [28].

**Definition 4.1** ([28])

*G*. We say that a self-mapping $T:X\to X$ is a Banach

*G*-contraction or simply a

*G*-contraction if

*T*preserves the edges of

*G*, that is,

*T*decreases weights of the edges of

*G*in the following way:

**Definition 4.2**Let $A,B$ be two nonempty closed subsets of a metric space $(X,d)$ endowed with a graph

*G*. We say that a nonself-mapping $T:A\to B$ is a

*G*-proximal

*C*-contraction if, for all $u,v,x,y\in A$,

**Theorem 4.1**

*Let*

*A*,

*B*

*be two nonempty closed subsets of a metric space*$(X,d)$

*endowed with a graph*

*G*.

*Assume that*

*A*

*is complete*, ${A}_{0}$

*is nonempty and*$T:A\to B$

*is a continuous*

*G*-

*proximal*

*C*-

*contraction mapping such that the following conditions hold*:

- (i)
$T({A}_{0})\subseteq {B}_{0}$,

- (ii)
*there exist elements*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G),$ - (iii)
*for all*$(x,y)\in E(G)$*and*$(y,z)\in E(G)$,*we have*$(x,z)\in E(G)$.

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $(x,y)\in E(G)$.

*Proof*Define $\alpha :X\times X\to [0,+\mathrm{\infty})$ by

*T*is a triangular

*α*-proximal admissible mapping. To this aim, assume

*T*is a

*G*-proximal

*C*-contraction mapping, we get $(u,v)\in E(G)$, that is, $\alpha (u,v)\ge 1$ and

Also, let $\alpha (x,z)\ge 1$ and $\alpha (z,y)\ge 1$, then $(x,z)\in E(G)$ and $(z,y)\in E(G)$. Consequently, from (iii), we deduce that $(x,y)\in E(G)$, that is, $\alpha (x,y)\ge 1$. Thus *T* is a triangular *α*-proximal admissible mapping with $T({A}_{0})\subseteq {B}_{0}$. Moreover, *T* is a continuous modified *α*-proximal *C*-contraction. From (ii) there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that $d({x}_{1},T{x}_{0})=d(A,B)$ and $({x}_{0},{x}_{1})\in E(G)$, that is, $d({x}_{1},T{x}_{0})=d(A,B)$ and $\alpha ({x}_{0},{x}_{1})\ge 1$. Hence, all the conditions of Theorem 3.1 are satisfied and *T* has a unique fixed point. □

Similarly, by using Theorem 3.2, we can prove the following theorem.

**Theorem 4.2**

*Let*

*A*,

*B*

*be two nonempty closed subsets of a metric space*$(X,d)$

*endowed with a graph*

*G*.

*Assume that*

*A*

*is complete*,

*the pair*$(A,B)$

*has the*

*V*-

*property and*${A}_{0}$

*is nonempty*.

*Also suppose that*$T:A\to B$

*is a*

*G*-

*proximal*

*C*-

*contraction mapping such that the following conditions hold*:

- (i)
$T({A}_{0})\subseteq {B}_{0}$,

- (ii)
*there exist elements*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}({x}_{0},{x}_{1})\in E(G),$ - (iii)
*for all*$(x,y)\in E(G)$*and*$(y,z)\in E(G)$,*we have*$(x,z)\in E(G)$, - (iv)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in E(G)$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$({x}_{n},x)\in E(G)$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a best proximity point*. *Further*, *the best proximity point is unique if*, *for every* $x,y\in A$ *such that* $d(x,Tx)=d(A,B)=d(y,Ty)$, *we have* $(x,y)\in E(G)$.

## 5 Some results in partially ordered metric spaces

In recent years, Ran and Reurings [31] initiated the study of weaker contraction conditions by considering self-mappings in partially ordered metric space. Further these results were generalized by many authors; see for instance [32, 33]. Here we consider some recent results of Mongkolkeha *et al.* [34] and Sadiq Basha *et al.* [35].

**Definition 5.1** ([35])

**Theorem 5.1** (Theorem 2.2 of [34])

*Let*

*A*,

*B*

*be two nonempty closed subsets of a partially ordered complete metric space*$(X,d,\u2aaf)$

*such that*${A}_{0}$

*is nonempty*.

*Assume that*$T:A\to B$

*satisfies the following conditions*:

- (i)
*T**is continuous and proximally ordered*-*preserving such that*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist elements*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$ - (iii)
*for all*$x,y,u,v\in A$,$\{\begin{array}{c}x\u2aafy,\hfill \\ d(u,Tx)=d(A,B),\hfill \\ d(y,Ty)=d(A,B)\hfill \end{array}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d(u,v)\le \frac{1}{2}(d(x,v)+d(y,u))-\psi (d(x,v),d(y,u)).$

*Then* *T* *has a best proximity point*.

*Proof*Define $\alpha :A\times A\to [0,+\mathrm{\infty})$ by

*T*is a triangular

*α*-proximal admissible mapping. To this aim, assume

*T*is proximally ordered-preserving, then $u\u2aafv$, that is, $\alpha (u,v)\ge 1$. Consequently, condition (T1) of Definition 3.2 holds. Also, assume

Thus all the conditions of Theorem 3.1 hold and *T* has a best proximity point. □

Similarly, omitting the continuity hypothesis of *T*, we can give the following result.

**Theorem 5.2** (see Theorem 2.6 of [34])

*Let*

*A*,

*B*

*be two nonempty closed subsets of a partially ordered complete metric space*$(X,d,\u2aaf)$

*such that*${A}_{0}$

*is nonempty and the pair*$(A,B)$

*has the*

*V*-

*property*.

*Assume that*$T:A\to B$

*satisfies the following conditions*:

- (i)
*T**is proximally ordered*-*preserving such that*$T({A}_{0})\subseteq {B}_{0}$, - (ii)
*there exist elements*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$ - (iii)
*for all*$x,y,u,v\in A$,$\{\begin{array}{c}x\u2aafy,\hfill \\ d(u,Tx)=d(A,B),\hfill \\ d(y,Ty)=d(A,B)\hfill \end{array}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d(u,v)\le \frac{1}{2}(d(x,v)+d(y,u))-\psi (d(x,v),d(y,u)),$ - (iv)
*if*$\{{x}_{n}\}$*is an increasing sequence in**A**converging to*$x\in A$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *T* *has a best proximity point*.

## 6 Application to fixed point theorems

In this section we briefly collect some fixed point results which are consequences of the results presented in the main section. Stated precisely, from Theorem 3.1, we obtain the following theorems.

**Theorem 6.1**

*Let*$(X,d)$

*be a complete metric space*.

*Assume that*$T:X\to X$

*is a continuous self*-

*mapping satisfying the following conditions*:

- (i)
*T**is triangular**α*-*admissible*, - (ii)
*there exists*${x}_{0}$*in**X**such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$, - (iii)
*for all*$x,y\in X$,$\alpha (x,y)d(Tx,Ty)\le \frac{1}{2}(d(x,Ty)+d(y,Tx))-\psi (d(x,Ty),d(y,Tx)).$

*Then* *T* *has a fixed point*.

**Theorem 6.2**

*Let*$(X,d)$

*be a complete metric space*.

*Assume that*$T:X\to X$

*is a continuous self*-

*mapping satisfying the following conditions*:

- (i)
*T**is triangular**α*-*admissible*, - (ii)
*there exists*${x}_{0}$*in**X**such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$, - (iii)
*for all*$x,y\in X$,${(\alpha (x,y)+\ell )}^{d(Tx,Ty)}\le {(\ell +1)}^{\frac{1}{2}(d(x,Ty)+d(y,Tx))-\psi (d(x,Ty),d(y,Tx))},$

*where* $\ell >0$.

*Then* *T* *has a fixed point*.

Analogously, from Theorem 3.2, we obtain the following theorems, which do not require the continuity of *T*.

**Theorem 6.3**

*Let*$(X,d)$

*be a complete metric space*.

*Assume that*$T:X\to X$

*is a self*-

*mapping satisfying the following conditions*:

- (i)
*T**is triangular**α*-*admissible*, - (ii)
*there exists*${x}_{0}$*in**X**such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$, - (iii)
*for all*$x,y\in X$,$\alpha (x,y)d(Tx,Ty)\le \frac{1}{2}(d(x,Ty)+d(y,Tx))-\psi (d(x,Ty),d(y,Tx)),$ - (iv)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

*Then* *T* *has a fixed point*.

**Theorem 6.4**

*Let*$(X,d)$

*be a complete metric space*.

*Assume that*$T:X\to X$

*is a self*-

*mapping satisfying the following conditions*:

- (i)
*T**is triangular**α*-*admissible*, - (ii)
*there exists*${x}_{0}$*in**X**such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$, - (iii)
*for all*$x,y\in X$,${(\alpha (x,y)+1)}^{d(Tx,Ty)}\le {2}^{[\frac{1}{2}(d(x,Ty)+d(y,Tx))-\psi (d(x,Ty),d(y,Tx))]},$ - (iv)
*if*$\{{x}_{n}\}$*is a sequence in**A**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*and*${x}_{n}\to x\in A$*as*$n\to +\mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

*Then* *T* *has a fixed point*.

## Declarations

### Acknowledgements

First author was supported by the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant no. MRG5580213). Third author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

## Authors’ Affiliations

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