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Best proximity point results for modified Suzuki αψproximal contractions
Fixed Point Theory and Applications volume 2014, Article number: 10 (2014)
Abstract
In this paper, we introduce a modified Suzuki αψproximal 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.
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 nonempty subsets of a metric space X and $T:A\to B$ is a nonselfmapping 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 nonselfmapping 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 selfmappings, 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 selfmapping, 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 nonempty compact convex subset of a normed space X and $T:K\to X$ be a continuous nonselfmapping. Then there exists an x such that $\parallel xTx\parallel =d(K,Tx)=inf\{\parallel Txu\parallel :u\in K\}$.
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 αψproximal 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.
We recollect some essential notations, required definitions and primary results to coherence with the literature. Suppose that A and B are two nonempty subsets of a metric space $(X,d)$. We define
Under the assumption of ${A}_{0}\ne \mathrm{\varnothing}$, we say that the pair $(A,B)$ has the Pproperty [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 αψcontractive and αadmissible mappings and established various fixed point theorems for such mappings in complete metric spaces.
Samet et al. [24] defined the notion of αadmissible mappings as follows.
Definition 1.1 Let T be a selfmapping on X and $\alpha :X\times X\to [0,+\mathrm{\infty})$ be a function. We say that T is an αadmissible mapping if
Salimi et al. [22] modified and generalized the notion of αadmissible mappings in the following way.
Definition 1.2 [22] Let T be a selfmapping on X and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ be two functions. We say that T is an αadmissible 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]
A nonselfmapping T is called αproximal 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 αproximal admissible implies that T is αadmissible.
Recently Hussain et al. [4] generalized the notion of αproximal 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 αproximal 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.
A function $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is called BianchiniGrandolfi gauge function [17, 18, 30] if the following conditions hold:

(i)
ψ is nondecreasing;

(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 BianchiniGrandolfi gauge function is known as the $(c)$comparison function (see e.g. [31]). We denote by Ψ the family of BianchiniGrandolfi 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 nonempty subsets of a metric space $(X,d)$. A nonselfmapping $T:A\to B$ is said to be modified Suzuki αψproximal 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 nonempty closed subsets of a complete metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Let $T:A\to B$ be a modified Suzuki αψproximal contraction satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is αproximal 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 nonempty 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
Since T is αproximal 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
So we conclude that
As T is αproximal admissible, we derive that $\alpha ({x}_{2},{x}_{3})\ge 2$, that is,
By repeating this process, we observe that
By the triangle inequality, we have
which implies
From (2.1), we derive that
Due the fact that the pair $(A,B)$ has the Pproperty together with (2.2), we conclude that
Consequently, from (2.3), we obtain
If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ for some ${n}_{0}\in \mathbb{N}$, then (2.2) implies that
that is, ${x}_{{n}_{0}}$ is a best proximity point of T. Hence, we assume that
By using the fact that ψ is nondecreasing together with the assumption (2.1), inductively, we conclude that
Fix $\u03f5>0$; there exists $N\in \mathbb{N}$ such that
Let $m,n\in \mathbb{N}$ with $m>n\ge N$. By the triangle inequality, we have
which yields ${lim}_{m,n,\to +\mathrm{\infty}}d({x}_{n},{x}_{m})=0$. Hence, $\{{x}_{n}\}$ is a Cauchy sequence. Since 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:
We now show that 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,
By applying the Pproperty and (2.6) we get
Also from (2.6) we get
which implies that ${d}^{\ast}(y,Ty)=0\le \alpha (y,z)d(y,z)$. Applying (2.1), we have
From (2.7) we deduce
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 αψproximal contraction T by regularity of the space $(X,d)$.
Theorem 2.2 Suppose that A and B are two nonempty closed subsets of a complete metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Let $T:A\to B$ be a modified Suzuki αψproximal contraction satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is αproximal 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
for all $n\in \mathbb{N}$. By using (2.2), we have
and
Hence, (2.8) and (2.9) imply that
We suppose that the inequalities
and
hold for some $n\in \mathbb{N}$. Then, by using (2.10) we can write
a contradiction. Hence, for all $n\in \mathbb{N}$, we have either
or
Using (2.1), we obtain either
or
If we take the limit as $n\to +\mathrm{\infty}$ in each of these inequalities, we have
Consequently, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that $T{x}_{{n}_{k}}\to Tz$ as ${x}_{{n}_{k}}\to z$. Therefore,
The uniqueness of best proximity point follows as in the proof of Theorem 2.1. □
Example 2.1 Let $X=\mathbb{R}$ and $d(x,y)=xy$ be a usual metric on X. Suppose $A=(\mathrm{\infty},1]$ and $B=[5/4,+\mathrm{\infty})$. Define $T:A\to B$ by
Also, define $\alpha :{X}^{2}\to [0,\mathrm{\infty})$ by
and $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\psi (t)=\frac{1}{2}t$. Clearly, $d(A,B)=9/4$. Now we have:
Also, $T({A}_{0})\subseteq {B}_{0}$ and clearly, the pair $(A,B)$ has the Pproperty. Suppose
then
Note that $Tw\in [5/4,3/2]$ for all $w\in [2,1]$. Hence, ${u}_{1}={u}_{2}=1$, i.e., $\alpha ({u}_{1},{u}_{2})\ge 2$. That is, T is a αproximal 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
Otherwise, $\alpha (x,y)=0$. That is, $\frac{1}{2}{d}^{\ast}(x,Tx)>\alpha (x,y)d(x,y)=0$. Hence,
All conditions of Theorem 2.2 hold for this example and there is a unique best proximity point $z=1$ such that $d(1,T(1))=d(A,B)$. Note that in this example the contractive condition of Theorems 3.1 and 3.2 of Jleli and Samet [14] is not satisfied and so these are not applicable here. Indeed, if, $x=2$ and $y=1$, then we have
The following results are nice consequences of Theorem 2.2.
Theorem 2.3 Let A and B be nonempty closed subsets of a complete metric space $(X,d)$ such that ${A}_{0}$ is nonempty. Assume $T:A\to B$ is a nonselfmapping satisfying the following assertions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(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
then
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 nonempty closed subsets of a complete metric space $(X,d)$ such that ${A}_{0}$ is nonempty. Assume $T:A\to B$ is a nonselfmapping satisfying the following assertions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(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 nonempty closed subsets of a complete metric space $(X,d)$ such that ${A}_{0}$ is nonempty. Assume $T:A\to B$ is a nonselfmapping satisfying the following assertions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
define a nonincreasing function $\theta :[0,1)\to (1/2,1]$ by
$$\theta (r)=\{\begin{array}{cc}1\hfill & \mathit{\text{if}}0\le r\le (\sqrt{5}1)/2,\hfill \\ (1r){r}^{2}\hfill & \mathit{\text{if}}(\sqrt{5}1)/2r{2}^{1/2},\hfill \\ {(1+r)}^{1}\hfill & \mathit{\text{if}}{2}^{1/2}\le r1.\hfill \end{array}$$(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 nonempty closed subsets of a complete metric space $(X,d)$ such that ${A}_{0}$ is nonempty. Assume $T:A\to B$ is a nonselfmapping satisfying the following assertions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
define a nonincreasing 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 nonempty closed subsets of a complete metric space $(X,d)$ such that ${A}_{0}$ is nonempty. Assume $T:A\to B$ is a nonself mapping satisfying the following assertions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(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 orderpreserving 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 orderpreserving map $T:A\to A$ reduces to a nondecreasing map.
Theorem 3.1 Let A and B be two nonempty 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 nonselfmapping satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is proximally orderpreserving;

(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
Now we prove that T is a αproximal admissible mapping with respect to $\eta (x,y)=2$. For this, assume
So
Now, since T is proximally orderpreserving, $u\u2aafv$. Thus, $\alpha (u,v)\ge 2$. Furthermore, by (iii) the elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with
Let ${d}^{\ast}(x,Tx)\le \alpha (x,y)d(x,y)$. Then for all $x,y\in A$ with $x\u2aafy$, we have $\alpha (x,y)\ge 2$, and hence
From (v) we get $d(Tx,Ty)\le \psi (d(x,y))$. That is, T is a modified Suzuki αψproximal 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 nonempty 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 nonselfmapping satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is proximally orderedpreserving;

(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 nonempty closed subsets of partially ordered complete metric space $(X,d,\u2aaf)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Let $T:A\to B$ be a nonself mapping satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is proximally orderpreserving;

(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 nondecreasing 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 αproximal admissible mapping with respect to $\eta (x,y)=2$ and is modified Suzuki αψproximal 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 nonempty closed subsets of partially ordered complete metric space $(X,d,\u2aaf)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Let $T:A\to B$ be a nonselfmapping satisfying the following conditions:

(i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the Pproperty;

(ii)
T is proximally orderedpreserving;

(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 nondecreasing 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 Suzukitype 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 αadmissible 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 αadmissible 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 selfmapping on X. Define a nonincreasing 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 nondecreasing;

(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 nondecreasing;

(ii)
there exists ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$;

(iii)
if $\{{x}_{n}\}$ is a nonincreasing 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.
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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.
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Hussain, N., Latif, A. & Salimi, P. Best proximity point results for modified Suzuki αψproximal contractions. Fixed Point Theory Appl 2014, 10 (2014). https://doi.org/10.1186/16871812201410
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Keywords
 αproximal admissible map
 Suzuki αψproximal contraction
 best proximity point