# Best proximity point results for modified Suzuki *α*-*ψ*-proximal contractions

- Nawab Hussain
^{1}, - Abdul Latif
^{1}Email author and - Peyman Salimi
^{2}

**2014**:10

https://doi.org/10.1186/1687-1812-2014-10

© Hussain et al.; licensee Springer. 2014

**Received: **10 August 2013

**Accepted: **13 December 2013

**Published: **9 January 2014

## Abstract

In this paper, we introduce a modified Suzuki *α*-*ψ*-proximal contraction. Then we establish certain best proximity point theorems for such proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The results presented generalize and improve various known results from best proximity and fixed point theory. Moreover, some examples are given to illustrate the usability of the obtained results.

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

## Keywords

*α*-proximal admissible mapSuzuki

*α*-

*ψ*-proximal contractionbest proximity point

## 1 Introduction and Preliminaries

In the last decade, the answers of the following question has turned into one of the core subjects of applied mathematics and nonlinear functional analysis. Is there a point ${x}_{0}$ in a metric space $(X,d)$ such that $d({x}_{0},T{x}_{0})=d(A,B)$ where *A*, *B* are non-empty subsets of a metric space *X* and $T:A\to B$ is a non-self-mapping where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$? Here, the point ${x}_{0}\in X$ is called the best proximity point. The object of best proximity theory is to determine minimal conditions on the non-self-mapping *T* to guarantee the existence and uniqueness of a best proximal point. The setting of best proximity point theory is richer and more general than the metric fixed point theory in two senses. First, usually the mappings considered in fixed point theory are self-mappings, which is not necessary in the theory of best proximity. Secondly, if one takes $A=B$ in the above setting, the best proximity point becomes a fixed point. It is well known that fixed point theory combines various disciplines of mathematics, such as topology, operator theory, and geometry, to show the existence of solutions of the equation $Tx=x$ under proper conditions. On the other hand, if *T* is not a self-mapping, the equation $Tx=x$ could have no solutions and, in this case, it is of basic interest to determine an element *x* that is in some sense closest to *Tx*. One of the most interesting results in this direction is the following theorem due to Fan [1].

**Theorem F** *Let* *K* *be a non*-*empty compact convex subset of a normed space* *X* *and* $T:K\to X$ *be a continuous non*-*self*-*mapping*. *Then there exists an* *x* *such that* $\parallel x-Tx\parallel =d(K,Tx)=inf\{\parallel Tx-u\parallel :u\in K\}$.

Many generalizations and extensions of this result have appeared in the literature (see [2–6] and references therein).

In fact best proximity point theory has been studied to find necessary conditions such that the minimization problem ${min}_{x\in A}d(x,Tx)$ has at least one solution. For more details on this approach, we refer the reader to [7–13] and [5, 14–25].

One of the interesting generalizations of the Banach contraction principle which characterizes the metric completeness is due to Suzuki [26, 27] (see also [28, 29]). Recently, Abkar and Gabeleh [8] studied best proximity point results for Suzuki contractions. The aim of this paper is to introduce modified Suzuki *α*-*ψ*-proximal contractions and establish certain best proximity point theorems for such proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity and fixed point theory. Moreover, some examples are given to illustrate the usability of the obtained results.

*A*and

*B*are two non-empty subsets of a metric space $(X,d)$. We define

*P*-property [20] if the following condition holds:

for all ${x}_{1},{x}_{2}\in A$ and ${y}_{1},{y}_{2}\in B$.

In 2012, Samet *et al.* [24] introduced the concepts of *α*-*ψ*-contractive and *α*-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces.

Samet *et al.* [24] defined the notion of *α*-admissible mappings as follows.

**Definition 1.1**Let

*T*be a self-mapping on

*X*and $\alpha :X\times X\to [0,+\mathrm{\infty})$ be a function. We say that

*T*is an

*α*-admissible mapping if

Salimi *et al.* [22] modified and generalized the notion of *α*-admissible mappings in the following way.

**Definition 1.2**[22] Let

*T*be a self-mapping on

*X*and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ be two functions. We say that

*T*is an

*α*-admissible mapping with respect to

*η*if

Note that if we take $\eta (x,y)=1$, then this definition reduces to Definition 1.1.

**Definition 1.3** [14]

*T*is called

*α*-proximal admissible if

for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$, where $\alpha :A\times A\to [0,\mathrm{\infty})$.

Clearly, if $A=B$, *T* is *α*-proximal admissible implies that *T* is *α*-admissible.

Recently Hussain *et al.* [4] generalized the notion of *α*-proximal admissible as follows.

**Definition 1.4**Let $T:A\to B$ and $\alpha ,\eta :A\times A\to [0,\mathrm{\infty})$ be functions. Then

*T*is called

*α*-proximal admissible with respect to

*η*if

for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$. Note that if we take $\eta (x,y)=1$ for all $x,y\in A$, then this definition reduces to Definition 1.3.

- (i)
*ψ*is non-decreasing; - (ii)there exist ${k}_{0}\in \mathbb{N}$ and $a\in (0,1)$ and a convergent series of nonnegative terms ${\sum}_{k=1}^{\mathrm{\infty}}{v}_{k}$ such that${\psi}^{k+1}(t)\le a{\psi}^{k}(t)+{v}_{k},$

for $k\ge {k}_{0}$ and any $t\in {\mathbb{R}}^{+}$.

In some sources, the Bianchini-Grandolfi gauge function is known as the $(c)$-comparison function (see *e.g.* [31]). We denote by Ψ the family of Bianchini-Grandolfi gauge functions. The following lemma illustrates the properties of these functions.

**Lemma 1.1** (See [31])

*If*$\psi \in \mathrm{\Psi}$,

*then the following hold*:

- (i)
${({\psi}^{n}(t))}_{n\in \mathbb{N}}$

*converges to*0*as*$n\to \mathrm{\infty}$*for all*$t\in {\mathbb{R}}^{+}$; - (ii)
$\psi (t)<t$,

*for any*$t\in (0,\mathrm{\infty})$; - (iii)
*ψ**is continuous at*0; - (iv)
*the series*${\sum}_{k=1}^{\mathrm{\infty}}{\psi}^{k}(t)$*converges for any*$t\in {\mathbb{R}}^{+}$.

## 2 Best proximity point results in metric spaces

We start this section with the following definition.

**Definition 2.1**Suppose that

*A*and

*B*are two non-empty subsets of a metric space $(X,d)$. A non-self-mapping $T:A\to B$ is said to be modified Suzuki

*α*-

*ψ*-proximal contraction, if

for all $x,y\in A$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$, $\alpha :A\times A\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$.

The following is our first main result of this section.

**Theorem 2.1**

*Suppose that*

*A*

*and*

*B*

*are two non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*Let*$T:A\to B$

*be a modified Suzuki*

*α*-

*ψ*-

*proximal contraction satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is**α*-*proximal admissible with respect to*$\eta (x,y)=2$; - (iii)
*the elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfy}}\alpha ({x}_{0},{x}_{1})\ge 2;$ - (iv)
*T**is continuous*.

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

*Proof*As ${A}_{0}$ is non-empty and $T({A}_{0})\subseteq {B}_{0}$, there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ such that $d({x}_{1},T{x}_{0})=d(A,B)$ and by (iii) $\alpha ({x}_{0},{x}_{1})\ge 2$. Owing to the fact that $T({A}_{0})\subseteq {B}_{0}$, there exists ${x}_{2}\in {A}_{0}$ such that

*T*is

*α*-proximal admissible, we have $\alpha ({x}_{1},{x}_{2})\ge 2$. Again, by using the fact that $T({A}_{0})\subseteq {B}_{0}$, there exists ${x}_{3}\in {A}_{0}$ such that

*T*is

*α*-proximal admissible, we derive that $\alpha ({x}_{2},{x}_{3})\ge 2$, that is,

*P*-property together with (2.2), we conclude that

*T*. Hence, we assume that

*ψ*is non-decreasing together with the assumption (2.1), inductively, we conclude that

*X*is complete, there is $z\in X$ such that ${x}_{n}\to z$. By the continuity of

*T*, we derive that $T{x}_{n}\to Tz$ as $n\to \mathrm{\infty}$. Hence, we get the desired result:

*T*has a unique best proximity point. Suppose, on the contrary, that $y,z\in {A}_{0}$ are two best proximity points of

*T*with $y\ne z$, that is,

*P*-property and (2.6) we get

which is a contradiction. Hence, $y=z$. This completes the proof of the theorem. □

In the following theorem, we replace the continuity condition on Suzuki *α*-*ψ*-proximal contraction *T* by regularity of the space $(X,d)$.

**Theorem 2.2**

*Suppose that*

*A*

*and*

*B*

*are two non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*Let*$T:A\to B$

*be a modified Suzuki*

*α*-

*ψ*-

*proximal contraction satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is**α*-*proximal admissible with respect to*$\eta (x,y)=2$; - (iii)
*there exist elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}\alpha ({x}_{0},{x}_{1})\ge 2;$ - (iv)
*if*$\{{x}_{n}\}$*is a sequence in**A**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 2$*and*${x}_{n}\to x\in A$*as*$n\to \mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 2$*for all*$n\in \mathbb{N}$.

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

*Proof*Following the lines of proof of Theorem 2.1, we obtain a Cauchy sequence $\{{x}_{n}\}$ which converges to $z\in X$. Suppose that the condition (iv) holds, that is, $\alpha ({x}_{n},z)\ge 2$ for all $n\in \mathbb{N}$. From (2.4) and (2.5) we obtain

The uniqueness of best proximity point follows as in the proof of Theorem 2.1. □

**Example 2.1**Let $X=\mathbb{R}$ and $d(x,y)=|x-y|$ be a usual metric on

*X*. Suppose $A=(-\mathrm{\infty},-1]$ and $B=[5/4,+\mathrm{\infty})$. Define $T:A\to B$ by

*P*-property. Suppose

*i.e.*, $\alpha ({u}_{1},{u}_{2})\ge 2$. That is,

*T*is a

*α*-proximal admissible mapping with respect to $\eta (x,y)=2$. Also, assume that $\alpha ({x}_{n},{x}_{n+1})\ge 2$ 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 $x\in [-2,-1]$. That is, $\alpha ({x}_{n},x)\ge 2$ for all $n\in \mathbb{N}\cup \{0\}$. If $x,y\in [-1,-2]$, then

The following results are nice consequences of Theorem 2.2.

**Theorem 2.3**

*Let*

*A*

*and*

*B*

*be non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*is non*-

*empty*.

*Assume*$T:A\to B$

*is a non*-

*self*-

*mapping satisfying the following assertions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*for a function*$\delta :[0,1)\to (0,1/2]$,*there exists*$r\in [0,1)$*such that*$\delta (r){d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(2.11)

*for* $x,y\in A$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ *and* $\psi \in \mathrm{\Psi}$.

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

*Proof*First, we fix

*r*and define ${\alpha}_{r}:A\times A\to [0,\mathrm{\infty})$ by ${\alpha}_{r}(x,y)=\frac{1}{\delta (r)}$ for all $x,y\in A$. Since $\frac{1}{\delta (r)}\ge 2$ for all $r\in [0,1)$, ${\alpha}_{r}(w,v)\ge 2$ for all $w,v\in A$. Now, since ${\alpha}_{r}(w,v)$ is constant and ${\alpha}_{r}(w,v)\ge 2$ for all $w,v\in A$,

*T*is an ${\alpha}_{r}$-proximal admissible mapping with respect to $\eta (x,y)=2$ and hence conditions (ii)-(iv) of Theorem 2.2 hold. Furthermore, if

and so by (2.11) we deduce $d(Tx,Ty)\le \psi (d(x,y))$. Hence all conditions of Theorem 2.2 hold and *T* has a unique best proximity point. □

If we take $\psi (t)=rt$ in Theorem 2.3, where $0\le r<1$, then we obtain the following result.

**Corollary 2.1**

*Let*

*A*

*and*

*B*

*be non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*is non*-

*empty*.

*Assume*$T:A\to B$

*is a non*-

*self*-

*mapping satisfying the following assertions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*for a function*$\delta :[0,1)\to (0,1/2]$,*there exists*$r\in [0,1)$*such that*$\delta (r){d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le rd(x,y)$(2.12)

*for* $x,y\in A$.

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

**Corollary 2.2**

*Let*

*A*

*and*

*B*

*be non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*is non*-

*empty*.

*Assume*$T:A\to B$

*is a non*-

*self*-

*mapping satisfying the following assertions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*define a non*-*increasing function*$\theta :[0,1)\to (1/2,1]$*by*$\theta (r)=\{\begin{array}{cc}1\hfill & \mathit{\text{if}}0\le r\le (\sqrt{5}-1)/2,\hfill \\ (1-r){r}^{-2}\hfill & \mathit{\text{if}}(\sqrt{5}-1)/2r{2}^{-1/2},\hfill \\ {(1+r)}^{-1}\hfill & \mathit{\text{if}}{2}^{-1/2}\le r1.\hfill \end{array}$(2.13)

*Assume that there exists*$r\in [0,1)$

*such that*

*for* $x,y\in A$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$.

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

*Proof* If we take $\delta (r)=\frac{1}{2}\theta (r)$ in Corollary 2.1, we obtain the required result. □

If we take $\delta (r)=\frac{1}{2(1+r)}$ in Corollary 2.1, we obtain the main result of [8] in the following form.

**Corollary 2.3**

*Let*

*A*

*and*

*B*

*be non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*is non*-

*empty*.

*Assume*$T:A\to B$

*is a non*-

*self*-

*mapping satisfying the following assertions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*define a non*-*increasing function*$\beta :[0,1)\to (1/2,1]$*by*$\beta (r)=\frac{1}{2(1+r)}.$(2.14)

*Assume that there exists*$r\in [0,1)$

*such that*

*for* $x,y\in A$.

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

If we take $\delta (r)=\frac{1}{2}$ in Corollary 2.1 we have following result.

**Corollary 2.4**

*Let*

*A*

*and*

*B*

*be non*-

*empty closed subsets of a complete metric space*$(X,d)$

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

*is non*-

*empty*.

*Assume*$T:A\to B$

*is a non*-

*self mapping satisfying the following assertions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le rd(x,y)$

*for all* $x,y\in A$.

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

## 3 Best proximity point results in partially ordered metric spaces

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [2, 9, 32] and references therein). The existence of best proximity and fixed point results in partially ordered metric spaces has been considered recently by many authors [4, 7, 21, 33, 34]. The aim of this section is to deduce some best proximity and fixed point results in the context of partially ordered metric spaces. Moreover, we obtain certain recent fixed point results as corollaries in partially ordered metric spaces.

**Definition 3.1**[21] A mapping $T:A\to B$ is said to be proximally order-preserving if and only if it satisfies the condition

for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$.

Clearly, if $B=A$, then the proximally order-preserving map $T:A\to A$ reduces to a non-decreasing map.

**Theorem 3.1**

*Let*

*A*

*and*

*B*

*be two non*-

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

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

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

*is a non*-

*self*-

*mapping satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is proximally order*-*preserving*; - (iii)
*there exist elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$ - (iv)
*T**is continuous*; - (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$

*for all* $x,y\in A$ *with* $x\u2aafy$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ *and* $\psi \in \mathrm{\Psi}$.

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

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

*T*is a

*α*-proximal admissible mapping with respect to $\eta (x,y)=2$. For this, assume

*T*is proximally order-preserving, $u\u2aafv$. Thus, $\alpha (u,v)\ge 2$. Furthermore, by (iii) the elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with

From (v) we get $d(Tx,Ty)\le \psi (d(x,y))$. That is, *T* is a modified Suzuki *α*-*ψ*-proximal contraction. Thus all conditions of Theorem 2.1 hold and *T* has a unique best proximity point. □

**Corollary 3.1**

*Let*

*A*

*and*

*B*

*be two non*-

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

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

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

*be a non*-

*self*-

*mapping satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is proximally ordered*-*preserving*; - (iii)
*there exist elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$ - (iv)
*T**is continuous*; - (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le r(d(x,y))$

*for all* $x,y\in A$ *with* $x\u2aafy$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ *and* $0\le r<1$.

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

**Theorem 3.2**

*Suppose that*

*A*

*and*

*B*

*are two non*-

*empty closed subsets of partially ordered complete metric space*$(X,d,\u2aaf)$

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

*Let*$T:A\to B$

*be a non*-

*self mapping satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is proximally order*-*preserving*; - (iii)
*the elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfy}}{x}_{0}\u2aaf{x}_{1};$ - (iv)
*if*$\{{x}_{n}\}$*is a non*-*decreasing sequence in**A**such that*${x}_{n}\to x\in A$*as*$n\to \mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$; - (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(3.1)

*for all* $x,y\in A$ *with* $x\u2aafy$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ *and* $\psi \in \mathrm{\Psi}$.

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

*Proof* Defining $\alpha :X\times X\to [0,\mathrm{\infty})$ as in the proof of Theorem 3.1, we find that *T* is an *α*-proximal admissible mapping with respect to $\eta (x,y)=2$ and is modified Suzuki *α*-*ψ*-proximal contraction. Assume $\alpha ({x}_{n},{x}_{n+1})\ge 2$ for all $n\in \mathbb{N}$ such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. Then ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}$. Hence, by (iv) we get ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$ and so $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$. That is, all conditions of Theorem 2.2 hold and *T* has a unique best proximity point. □

**Corollary 3.2**

*Suppose that*

*A*

*and*

*B*

*are two non*-

*empty closed subsets of partially ordered complete metric space*$(X,d,\u2aaf)$

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

*Let*$T:A\to B$

*be a non*-

*self*-

*mapping satisfying the following conditions*:

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

*and*$(A,B)$*satisfies the**P*-*property*; - (ii)
*T**is proximally ordered*-*preserving*; - (iii)
*there exist elements*${x}_{0}$*and*${x}_{1}$*in*${A}_{0}$*with*$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$ - (iv)
*if*$\{{x}_{n}\}$*is a non*-*decreasing sequence in**A**such that*${x}_{n}\to x\in A$*as*$n\to \mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$; - (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le r(d(x,y))$(3.2)

*for all* $x,y\in A$ *with* $x\u2aafy$ *where* ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ *and* $0\le r<1$.

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

## 4 Applications

As an application of our results, we deduce new fixed point results for Suzuki-type contractions in the set up of metric and partially ordered metric spaces.

If we take $A=B=X$ in Theorems 2.1 and 2.2, then we deduce the following result.

**Theorem 4.1**

*Let*$(X,d)$

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

*be an*

*α*-

*admissible mapping with respect to*$\eta (x,y)=2$

*such that*

*for all*$x,y\in X$

*where*$\psi \in \mathrm{\Psi}$.

*Also suppose that the following assertions holds*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 2$; - (ii)
*either**T**is continuous or if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 2$*and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 2$*for all*$n\in \mathbb{N}$.

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

If we take $\psi (t)=kt$ in Theorem 4.1, where $0\le k<1$, then we conclude to the following theorem.

**Theorem 4.2**

*Let*$(X,d)$

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

*be an*

*α*-

*admissible mapping with respect to*$\eta (x,y)=2$

*such that*

*for all*$x,y\in X$

*where*$k\in [0,1)$.

*Also suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 2$; - (ii)
*either**T**is continuous or if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 2$*and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then*$\alpha ({x}_{n},x)\ge 2$*for all*$n\in \mathbb{N}$.

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

As a consequence of Theorem 4.2, by taking $\alpha (x,y)=2/\theta (r)$, we derive the following theorem.

**Theorem 4.3**

*Let*$(X,d)$

*be a complete metric space and*

*T*

*be a self*-

*mapping on*

*X*.

*Define a non*-

*increasing function*$\theta :[0,1)\to (1/2,1]$

*by*

*Assume that there exists*$r\in [0,1)$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

Furthermore, if we take $A=B=X$ in Theorems 3.1 and 3.2, then we deduce the following results.

**Theorem 4.4**

*Suppose that*$(X,d,\u2aaf)$

*is a partially ordered complete metric space and*$T:X\to X$

*is a mapping satisfying the following conditions*:

- (i)
*T**is non*-*decreasing*; - (ii)
*there exists*${x}_{0}$*in**X**such that*${x}_{0}\u2aafT{x}_{0}$; - (iii)
*T**is continuous*; - (iv)$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(4.3)

*for all* $x,y\in X$ *with* $x\u2aafy$ *where* $\psi \in \mathrm{\Psi}$.

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

**Theorem 4.5**

*Suppose that*$(X,d,\u2aaf)$

*is a partially ordered complete metric space and let*$T:X\to X$

*be a mapping satisfying the following conditions*:

- (i)
*T**is non*-*decreasing*; - (ii)
*there exists*${x}_{0}$*in**X**such that*${x}_{0}\u2aafT{x}_{0}$; - (iii)
*if*$\{{x}_{n}\}$*is a non*-*increasing sequence in**X**such that*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$; - (iv)$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(4.4)

*for all* $x,y\in X$ *with* $x\u2aafy$ *where* $\psi \in \mathrm{\Psi}$.

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

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.

## Authors’ Affiliations

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