# Some new results on fixed and best proximity points in preordered metric spaces

- Alireza Amini-Harandi
^{1, 2}, - Majid Fakhar
^{2, 3}, - Hamid Reza Hajisharifi
^{3}and - Nawab Hussain
^{4}Email author

**2013**:263

https://doi.org/10.1186/1687-1812-2013-263

© Amini-Harandi et al.; licensee Springer. 2013

**Received: **15 July 2013

**Accepted: **9 October 2013

**Published: **7 November 2013

## Abstract

In this paper, we first introduce two new classes of $(\omega ,\delta )$-contractions of the first and second kinds and establish some related new fixed point and best proximity point theorems in preordered metric spaces. Our theorems subsume the corresponding recent results of Samet (J. Optim. Theory Appl. (2013), doi:10.1007/s10957-013-0269-9) and extend and generalize many of the well-known results in the literature. An example is also provided to support our main results.

**MSC:** 47H10, 41A65.

## Keywords

*P*-property$(\omega ,\delta )$-contractionpreordered metric space

## 1 Introduction and preliminaries

Given a metric space $(X,d)$ and a self-mapping *T* on *X*, the theory on the existence of a solution to the equation of the form $Tx=x$ has gained impetus because of its applicability to solve many interesting problems that can be formulated as ordinary differential equations, matrix equations *etc.* For some recent fixed point results, see [1–6] and references therein. Let *A* and *B* be nonempty subsets of *X*, and let $T:A\to B$ be a non-self mapping. The equation $Tx=x$ is unlikely to have a solution, because of the fact that a solution of the preceding equation demands the nonemptiness of $A\cap B$. Eventually, it is quite natural to seek an approximate solution *x* that is optimal in the sense that the distance $d(x,Tx)$ is minimum. The well-known best approximation theorem, due to Fan [7], states that if *A* is a nonempty, compact, and convex subset of a normed linear space *X* and *T* is a continuous function from *A* to *X*, then there exists a point *x* in *A* such that $\parallel x-Tx\parallel =d(Tx,A)=inf\{\parallel Tx-u\parallel :u\in A\}$. Such a point x is called a best approximant point of *T* in *A*. Many generalizations and extensions of this theorem appeared in the literature (see [8–11] and references therein).

Best proximity problem for the pairs $(A,B)$ is to find an element $x\in A$ such that $d(x,Tx)=d(A,B)$, where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$. Since $d(A,B)$ is a lower bound for the function $x\to d(x,Tx)$ on *A*, then the solutions of the best proximity problem are the minimum points of the function $x\to d(x,Tx)$ on *A*. Every solution of the best proximity problem is said to be a best proximity point of *T* in *A*. Moreover, if $A=B$ then every best proximity point of *T* is a fixed point. According to this fact, many authors by motivation of well-known fixed point results obtained sufficient conditions to solving best proximity problems; for more details, see [12–27] and the references therein.

Existence of best proximity and fixed points in partially ordered metric spaces has been considered recently by many authors (see [6, 13, 20, 28]). Recently Samet [29] studied the existence of best proximity points for a class of non-self almost $(\phi ,\theta )$-contractive mappings. In this work we define two new classes of contractions called $(\omega ,\delta )$-contractions of the first and second kind and establish some related new fixed point results in the setting of preordered metric spaces, and then we derive some new best proximity point theorems for these new classes of non-self contractive mappings. The presented theorems extend and generalize many of the well-known fixed point and best proximity point results.

## 2 Fixed point theory

**Definition 2.1**Let $(X,d)$ be a metric space, and let ${R}_{+}=[0,\mathrm{\infty})$.

- (a)Denote by Ω the family of functions $\omega :{R}_{+}\to {R}_{+}$ such that $\omega (0)=0$, $\omega (t)<t$ for each $t>0$ and for each sequence $\{{x}_{n}\}$ in
*X*with,$d({x}_{n},{x}_{n+1})\le \omega (d({x}_{n-1},{x}_{n}))\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{N}\Rightarrow \{{x}_{n}\}\text{is a Cauchy sequence};$ - (b)
Denote by Δ the family of functions $\delta :{R}_{+}^{4}\to R$ such that

*δ*is continuous and if ${t}_{i}=0$ for some $i\in \{1,2,3,4\}$, then $\delta ({t}_{1},{t}_{2},{t}_{3},{t}_{4})=0$; - (c)
Denote by Φ the family of non-decreasing functions $\varphi :{R}_{+}\to {R}_{+}$ such that ${\mathrm{\Sigma}}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}(t)<\mathrm{\infty}$ for each $t>0$;

- (d)Denote by Σ the family of functions $\sigma :{R}_{+}\to {R}_{+}$ such that $\sigma (t)=\alpha (t)t$ for each $t>0$ and $\alpha :{R}_{+}\to [0,1)$ satisfies$\underset{s\to t}{lim\hspace{0.17em}sup}\alpha (s)<1\phantom{\rule{1em}{0ex}}\text{for each}t0;$(1)
- (e)
Denote by Ψ the family of non-decreasing functions $\psi :{R}_{+}\to {R}_{+}$ such that ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(t)=0$ for each $t>0$;

- (f)
Denote by Λ the family of non-decreasing and upper semicontinuous from the right functions $\lambda :{R}_{+}\to {R}_{+}$ such that $\lambda (t)<t$ for each $t>0$;

- (g)
Let Θ be a collection of the following functions:

$\theta ({t}_{1},{t}_{2},{t}_{3},{t}_{4})=\tau min\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\}$, $\tau >0$;

$\theta ({t}_{1},{t}_{2},{t}_{3},{t}_{4})=\tau ln(1+{t}_{1}{t}_{2}{t}_{3}{t}_{4})$, $\tau >0$;

$\theta ({t}_{1},{t}_{2},{t}_{3},{t}_{4})=\tau {t}_{1}{t}_{2}{t}_{3}{t}_{4}$, $\tau >0$.

**Lemma 2.2**

*Let*$(X,d)$

*be a metric space*.

*Then the following statements hold*:

- (i)
$\mathrm{\Phi}\subseteq \mathrm{\Omega}$,

- (ii)
$\mathrm{\Sigma}\subseteq \mathrm{\Omega}$,

- (iii)
$\mathrm{\Psi}\subseteq \mathrm{\Omega}$,

- (iv)
$\mathrm{\Lambda}\subseteq \mathrm{\Psi}\subseteq \mathrm{\Omega}$,

- (v)
$\mathrm{\Theta}\subseteq \mathrm{\Delta}$.

*Proof*Let $\{{x}_{n}\}$ be a sequence in

*X*. To prove (i), assume that $d({x}_{n},{x}_{n+1})\le \varphi (d({x}_{n-1},{x}_{n}))$ for each $n\in \mathbb{N}$, where $\varphi \in \mathrm{\Phi}$. Since

*ϕ*is non-decreasing, then by induction we get

and so $\{{x}_{n}\}$ is a Cauchy sequence.

*α*satisfying (1). Then

for each $k\in \mathbb{N}$. From the above, we obtain ${lim}_{k\to \mathrm{\infty}}\alpha (d({x}_{{m}_{k}},{x}_{{n}_{k}}))=1$. Then from (1) we get ${lim}_{k\to \mathrm{\infty}}d({x}_{{m}_{k}},{x}_{{n}_{k}})=0$, a contradiction. Therefore, $\{{x}_{n}\}$ is a Cauchy sequence.

(iii) Notice first that $\psi (t)<t$ for each $t>0$. To see this, suppose that there exists ${t}_{0}>0$ with $\psi ({t}_{0})>{t}_{0}$, then since *ψ* is non-decreasing, we see that ${t}_{0}\le {\psi}^{n}({t}_{0})$ for all $n\in \mathbb{N}$ and it is a contradiction with ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(t)=0$ for each $t>0$. Note also that $\psi (0)=0$.

*ψ*is non-decreasing, then by induction we get

This implies that $({x}_{n})$ is Cauchy and the proof of (iii) is complete.

(iv) For each $\lambda \in \mathrm{\Lambda}$, we have ${lim}_{n\to \mathrm{\infty}}{\lambda}^{n}(t)=0$ for each $t>0$ (see Remark 2.2 in [30]). Then the conclusion follows from (iii).

(v) obviously holds. □

Let *X* be a nonempty set. A preorder ⪯ on *X* is a binary relation which is reflexive and transitive. Let $(X,\u2aaf)$ be a preordered set, and let $T:X\to X$ be a mapping. We say that *T* is non-decreasing if for each $x,y\in X$, $x\u2aafy\Rightarrow Tx\u2aafTy$.

**Definition 2.3**Let $(X,\u2aaf)$ be a preordered set and

*d*be a metric on

*X*. We say that $(X,\u2aaf,d)$ is regular if and only if the following condition holds:

**Definition 2.4**Let $(X,\u2aaf,d)$ be a preordered metric space, and let $\omega :{R}_{+}\to {R}_{+}$ and $\delta :{R}_{+}^{4}\to {R}_{+}$ be arbitrary mappings.

- (a)A mapping $T:X\to X$ is said to be $(\omega ,\delta ,\u2aaf)$-contraction of the first kind if for all $x,y\in X$ with $x\u2aafy$,$d(Tx,Ty)\le \omega (d(x,y))+\delta (d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx));$
- (b)A mapping $T:X\to X$ is said to be $(\omega ,\delta ,\u2aaf)$-contraction of the second kind if for all $x,y\in X$ with $x\u2aafy$,$d(Tx,Ty)\le \omega (M(x,y))+\delta (d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)),$
where $M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\}$.

**Remark 2.5** If $\u2aaf=X\times X$, that is, $x\u2aafy$ for each $x,y\in X$, then $(\omega ,\delta ,\u2aaf)$-contractions of the first and second kind are called $(\omega ,\delta )$-contractions of the first and second kind in brief. The class of $(\omega ,\delta )$-contraction maps of the first and second kind include the mappings with condition (B) [3] and almost generalized contractions [6], respectively.

**Theorem 2.6**

*Let*$(X,\u2aaf,d)$

*be a complete preordered metric space*,

*and let*$T:X\to X$

*be a mapping*.

*Suppose that the following conditions hold*:

- (i)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*, - (ii)
*T**is non*-*decreasing*, - (iii)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$, - (iv)
*T**is an*$(\omega ,\delta ,\u2aaf)$-*contraction mapping of the first kind*,*where*$\omega \in \mathrm{\Omega}$*and*$\delta \in \mathrm{\Delta}$.

*Then* *T* *has a fixed point*. *Moreover*, *the sequence* $\{{T}^{n}{x}_{0}\}$ *converges to the fixed point of* *T*.

*Proof*Let ${x}_{n}=T{x}_{n-1}$ for any $n\in \mathbb{N}$. Since ${x}_{0}\u2aafT{x}_{0}$ and

*T*is non-decreasing, then we have

*T*is an $(\omega ,\delta ,\u2aaf)$-contraction mapping of the first kind, we get

*T*. If

*T*is continuous, then from the equality ${x}_{n}=T{x}_{n-1}$, we get ${x}^{\ast}=T{x}^{\ast}$. Now assume that $(X,\u2aaf,d)$ is regular. Then, for each $n\in \mathbb{N}$, we have ${x}_{n}\u2aaf{x}^{\ast}$. On the contrary, assume that $d({x}^{\ast},T{x}^{\ast})>0$. For any $n\in \mathbb{N}$,

and so from (5) we get ${x}^{\ast}=T{x}^{\ast}$. □

**Corollary 2.7**

*Let*$(X,d)$

*be a complete metric space*,

*and let*$T:X\to X$

*be an*$(\omega ,\delta )$-

*contraction mapping of the first kind*,

*where*$\omega \in \mathrm{\Omega}$

*and*$\delta \in \mathrm{\Delta}$.

*Then*

- (i)
*T**has a unique fixed point*.*Moreover*,*for all*${x}_{0}\in X$,*the sequence*$\{{T}^{n}{x}_{0}\}$*converges to the fixed point of**T*,*that is*,*T**is the Picard operator*. - (ii)
*T**is continuous at*$Fix(T)=\{{x}^{\ast}\}$.

*Proof*(i) Let $\u2aaf=X\times X$. Then from Theorem 2.6 we deduce that

*T*has a fixed point. To prove the uniqueness, on the contrary, assume that $x,y\in X$ are distinct fixed points of

*T*. So,

a contradiction. By the uniqueness of a fixed point and from Theorem 2.6, we get that the sequence $\{{T}^{n}{x}_{0}\}$ converges to the fixed point of *T* for all ${x}_{0}\in X$.

*X*such that ${y}_{n}\to {x}^{\ast}$. Since

*T*is an $(\omega ,\delta )$-contraction mapping of the first kind, so for all $n\in \mathbb{N}$ we have

Thus $T{y}_{n}\to T{x}^{\ast}$, and so *T* is continuous at ${x}^{\ast}$. □

**Remark 2.8** Theorem 2.6 extends the main result of Babu *et al.* [3], Corollary 1 of Berinde *et al.* [4], Corollary 3.1 of Samet [29] and Theorem 2.1 of Agarwal *et al.* [30].

**Theorem 2.9**

*Let*$(X,\u2aaf,d)$

*be a complete preordered metric space*,

*and let*$T:X\to X$

*be a mapping*.

*Suppose that the following conditions hold*:

- (i)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*, - (ii)
*T**is non*-*decreasing*, - (iii)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$, - (iv)
*T**is an*$(\omega ,\delta ,\u2aaf)$-*contraction mapping of the second kind*,*where*$\omega \in \mathrm{\Omega}$*and*$\delta \in \mathrm{\Delta}$.

*Then* *T* *has a fixed point*. *Moreover*, *the sequence* $\{{T}^{n}{x}_{0}\}$ *converges to the fixed point of* *T*.

*Proof*Let ${x}_{n}=T{x}_{n-1}$ for any $n\in \mathbb{N}$. If ${x}_{n-1}={x}_{n}$ for some $n\in \mathbb{N}$, then ${x}_{n-1}={x}_{n}=T{x}_{n-1}$, and so ${x}_{n-1}$ is a fixed point of

*T*, and we are finished. So, we may assume that $d({x}_{n-1},{x}_{n})>0$ for all $n\in \mathbb{N}$. Now, since ${x}_{0}\u2aafT{x}_{0}$ and

*T*is non-decreasing, so

*T*is an $(\omega ,\delta ,\u2aaf)$-contraction of the second kind, so for all $n\in \mathbb{N}$ we have

*T*. If

*T*is continuous, then from the equality ${x}_{n}=T{x}_{n-1}$, we get ${x}^{\ast}=T{x}^{\ast}$. Now, assume that $(X,\u2aaf,d)$ is regular. Then, for each $n\in \mathbb{N}$, we have ${x}_{n}\u2aaf{x}^{\ast}$. Now, on the contrary, assume that $d({x}^{\ast},T{x}^{\ast})>0$. So, for any $n\in \mathbb{N}$,

a contradiction. □

**Corollary 2.10** *Let* $(X,d)$ *be a complete metric space*, *and let* $T:X\to X$ *be an* $(\omega ,\delta )$-*contraction mapping of the second kind*, *where* $\omega \in \mathrm{\Omega}$ *and* $\delta \in \mathrm{\Delta}$. *Then* *T* *has a unique fixed point*. *Moreover*, *for all* ${x}_{0}\in X$, *the sequence* $\{{T}^{n}{x}_{0}\}$ *converges to the fixed point of* *T*, *that is*, *T* *is the Picard operator*.

*Proof*By Theorem 2.9 it is sufficient to prove the uniqueness of the fixed point. On the contrary assume that $x,y\in X$ are distinct fixed points of

*T*. Then

a contradiction. □

**Remark 2.11** Theorem 2.9 is a generalization of Theorem 2.2 and Theorem 2.3 of Agarwal *et al.* [30].

**Remark 2.12** When for all $t\in [0,\mathrm{\infty})$ we set $\omega (t)=\alpha t$ where $\alpha \in (0,1)$ and $\delta ({t}_{1},{t}_{2},{t}_{3},{t}_{4})=Lmin\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\}$ where $L\ge 0$, in Corollary 2.10, we obtain Theorem 2.4 of Berinde [5].

## 3 Best proximity point theory

*A*and

*B*be two nonempty subsets of a metric space $(X,d)$. We denote by ${A}_{0}$ and ${B}_{0}$ the following sets:

where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$.

**Definition 3.1**Let $(A,B)$ be a pair of nonempty subsets of the metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Then the pair $(A,B)$ is said to have the P-property [31] if and only if

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

The following lemma is crucial in proving our best proximity point results.

**Lemma 3.2**

*Let*$(A,B)$

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

*such that*${B}_{0}\ne \mathrm{\varnothing}$

*and that*$(A,B)$

*satisfies the*

*P*-

*property*.

*Then there exists a mapping*$Q:{B}_{0}\to {A}_{0}$

*satisfying*

*Furthermore*, ${B}_{0}$ *is closed*.

*Proof*Let $x\in {B}_{0}$, then we show that there exists a unique $y\in {A}_{0}$ such that $d(x,y)=d(A,B)$. To prove the uniqueness, let us assume that there exists $z\in {A}_{0}$ such that $d(x,y)=d(A,B)=d(x,z)$. Since $(A,B)$ has the

*P*-property, we have $d(y,z)=d(x,x)=0$ and so $y=z$. Let $y=Qx$, then $d(x,Qx)=d(x,y)=d(A,B)$. Now, assume that $d(x,Qx)=d(A,B)=d(y,Qy)$, where $x,y\in {B}_{0}$. Then, by the

*P*-property of $(A,B)$, we get $d(x,y)=d(Qx,Qy)$. Therefore, there exists a mapping $Q:{B}_{0}\to {A}_{0}$ such that

*B*is closed). Since

*A*is a closed subset of a complete metric space, $d(Q{x}_{m},Q{x}_{n})=d({x}_{m},{x}_{n})$ for each $m,n\in \mathbb{N}$ and $\{{x}_{n}\}$ is a Cauchy sequence, we deduce that $Q{x}_{n}\to y\in A$. Since $d({x}_{n},Q{x}_{n})=d(A,B)$ for each $n\in \mathbb{N}$, we have

and so $x\in {B}_{0}$. Hence, ${B}_{0}$ is closed. □

**Remark 3.3** It is clear that the mapping *Q* in Lemma 3.2 is a bijection and for any $x\in {A}_{0}$, we have $d(x,{Q}^{-1}x)=d(Q({Q}^{-1}x),{Q}^{-1}x)=d(A,B)$.

**Definition 3.4**Let $(X,\u2aaf)$ be a preordered set. A non-self mapping $T:M\subseteq A\to B$ is said to be proximally non-decreasing if and only if

where ${x}_{1},{x}_{2}\in A$, ${y}_{1},{y}_{2}\in M$.

The following lemma follows from Lemma 14 in [32].

**Lemma 3.5** *Let* $(X,\u2aaf,d)$ *be a preordered metric space*, *and let* $T:A\to B$ *be a non*-*self mapping such that* $T{A}_{0}\subseteq {B}_{0}$. *Let* $(A,B)$ *and* *Q* *be as in the statement of Lemma* 3.2. *Suppose that* $T:{A}_{0}\to {B}_{0}$ *is proximally non*-*decreasing*. *Then the mapping* $S:{A}_{0}\to {A}_{0}$ *defined by* $Sx=QTx$ *for each* $x\in {A}_{0}$ *is non*-*decreasing*.

The following lemma follows from Lemma 15 in [32].

**Lemma 3.6** *Let* $(X,d,\u2aaf)$ *be a preordered metric space* $(A,B)$, *and* *Q* *be as in Lemma* 3.2 *and* $T:A\to B$ *be a non*-*self mapping such that* $T{A}_{0}\subseteq {B}_{0}$. *Suppose that there exist* ${x}_{0},{x}_{1}\in {A}_{0}$ *such that* $d({x}_{1},T{x}_{0})=d(A,B)$ *and* ${x}_{0}\u2aaf{x}_{1}$. *Let the mapping* $S:{A}_{0}\to {A}_{0}$ *be defined by* $Sx=QTx$ *for each* $x\in {A}_{0}$. *Then* ${x}_{0}\u2aafS{x}_{0}$.

Now, we are ready to establish our best proximity point theorems.

**Theorem 3.7**

*Let*$(A,B)$

*be a pair of nonempty closed subsets of a complete preordered metric space*$(X,\u2aaf,d)$

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

*Let*$T:A\to B$

*be a non*-

*self mapping*.

*Suppose that the following conditions hold*:

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

*and*$(A,B)$*satisfy the**P*-*property*, - (ii)
*T**is continuous or*$({A}_{0},\u2aaf,d)$*is regular*, - (iii)
*T**is proximally non*-*decreasing*, - (iv)
*there exist*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$ - (v)
*For all*$x,y\in A$*such that*$x\u2aafy$,*we have*$\begin{array}{rl}d(Tx,Ty)\le & \omega (d(x,y))+\delta (d(x,Tx)-d(A,B),d(y,Ty)-d(A,B),d(x,Ty)\\ -d(A,B),d(y,Tx)-d(A,B)),\end{array}$(18)

*where* $\omega \in \mathrm{\Omega}$, $\delta \in \mathrm{\Delta}$ *and* *δ* *is non*-*decreasing in each of its variables*.

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

*Proof*Since ${A}_{0}\ne \mathrm{\varnothing}$, so ${B}_{0}\ne \mathrm{\varnothing}$. By Lemma 3.2, ${B}_{0}$ is closed and there exists an isometry $Q:{B}_{0}\to {A}_{0}$ which satisfies (17). Let $S:{A}_{0}\to {A}_{0}$ be defined by $Sx=QTx$ for each $x\in {A}_{0}$. Let $x,y\in {A}_{0}$ and $x\u2aafy$, then from (18) we have

Thus *S* is an ordered $(\omega ,\delta ,\u2aaf)$-contraction mapping of the first kind. Now conditions (ii), (iii) and (iv) with Lemma 3.5 and Lemma 3.6 imply that *S* satisfies conditions (i), (ii) and (iii) of Theorem 2.6. Consequently, *S* has a fixed point ${x}^{\ast}\in {A}_{0}$ such that ${x}^{\ast}=S{x}^{\ast}=QT{x}^{\ast}$ and ${Q}^{-1}{x}^{\ast}=T{x}^{\ast}$. That is, $d({x}^{\ast},T{x}^{\ast})=d({x}^{\ast},{Q}^{-1}{x}^{\ast})=d(Q({Q}^{-1}{x}^{\ast}),{Q}^{-1}{x}^{\ast})=d(A,B)$. Thus ${x}^{\ast}\in {A}_{0}$ is the required best proximity point for *T*. □

**Corollary 3.8**

*Let*$(A,B)$

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

*such that*${A}_{0}\ne \mathrm{\varnothing}$

*and*$(A,B)$

*satisfies the*

*P*-

*property*.

*Let*$T:A\to B$

*such that for all*$x,y\in A$,

*where* $\omega \in \mathrm{\Omega}$, $\delta \in \mathrm{\Delta}$ *and* *δ* *is non*-*decreasing in each of its variables*. *Moreover*, *assume that* $T{A}_{0}\subseteq {B}_{0}$. *Then* *T* *has a best proximity point in* *A*.

**Theorem 3.9**

*Let*$(A,B)$

*be a pair of nonempty closed subsets of a complete preordered metric space*$(X,\u2aaf,d)$

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

*Let*$T:A\to B$

*be a non*-

*self mapping*.

*Suppose that the following conditions hold*:

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

*and*$(A,B)$*satisfy the**P*-*property*, - (ii)
*T**is continuous or*$({A}_{0},\u2aaf,d)$*is regular*, - (iii)
*T**is proximally non*-*decreasing*, - (iv)
*there exist*${x}_{0},{x}_{1}\in {A}_{0}$*such that*$d({x}_{1},T{x}_{0})=d(A,B),\phantom{\rule{1em}{0ex}}{x}_{0}\u2aaf{x}_{1},$ - (v)
*For all*$x,y\in A$*such that*$y\u2aafx$,*we have*$\begin{array}{rl}d(Tx,Ty)\le & \omega (max\{d(x,y),d(x,Tx)-d(A,B),d(y,Ty)-d(A,B),\\ \frac{d(x,Ty)+d(y,Tx)}{2}-d(A,B)\left\}\right)\\ +\delta (d(x,Tx)-d(A,B),d(y,Ty)-d(A,B),d(x,Ty)\\ -d(A,B),d(y,Tx)-d(A,B)),\end{array}$(21)

*where* $\omega \in \mathrm{\Omega}$ *is non*-*decreasing*, $\delta \in \mathrm{\Delta}$ *and* *δ* *is non*-*decreasing in each of its variables*.

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

*Proof*Since ${A}_{0}\ne \mathrm{\varnothing}$, so ${B}_{0}\ne \mathrm{\varnothing}$. By Lemma 3.2, ${B}_{0}$ is closed and there exists an isometry $Q:{B}_{0}\to {A}_{0}$ which satisfies (17). Let $S:{A}_{0}\to {A}_{0}$ be defined by $Sx=QTx$ for each $x\in {A}_{0}$. Let $x,y\in {A}_{0}$ and $y\u2aafx$, then from (21) we have

*ω*is non-decreasing and

*δ*is non-decreasing in each of its variables, in view of the proof of Theorem 3.7, we get

for each $x,y\in {A}_{0}$, where $M(x,y)=max\{d(x,y),d(x,Sx),d(y,Sy),\frac{d(x,Sy)+d(y,Sx)}{2}\}$. Thus *S* is an ordered $(\omega ,\delta ,\u2aaf)$-contraction mapping of the second kind. Now conditions (ii), (iii) and (iv) with Lemma 3.5 and Lemma 3.6 imply that *S* satisfies conditions (i), (ii) and (iii) of Theorem 2.9, so by Theorem 2.9 *S* has a fixed point ${x}^{\ast}\in {A}_{0}$ such that ${x}^{\ast}=S{x}^{\ast}=QT{x}^{\ast}$ and ${Q}^{-1}{x}^{\ast}=T{x}^{\ast}$. Thus $d({x}^{\ast},T{x}^{\ast})=d({x}^{\ast},{Q}^{-1}{x}^{\ast})=d(Q({Q}^{-1}{x}^{\ast}),{Q}^{-1}{x}^{\ast})=d(A,B)$, as required. □

**Corollary 3.10**

*Let*$(A,B)$

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

*such that*${A}_{0}\ne \mathrm{\varnothing}$

*and*$(A,B)$

*satisfies the*

*P*-

*property*.

*Let*$T:A\to B$

*be such that for all*$x,y\in A$,

*where* $\omega \in \mathrm{\Omega}$ *is non*-*decreasing*, $\delta \in \mathrm{\Delta}$ *and* *δ* *is non*-*decreasing in each of its variables*. *Moreover*, *assume that* $T{A}_{0}\subseteq {B}_{0}$. *Then* *T* *has a best proximity point in* *A*.

**Remark 3.11**

From Lemma 2.2 and Corollary 3.8, we deduce the following result due to Samet [29].

**Theorem 3.12**

*Let*$(A,B)$

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

*such that*${A}_{0}\ne \mathrm{\varnothing}$, $(A,B)$

*satisfies the*

*P*-

*property*.

*Let*$T:A\to B$

*such that for all*$x,y\in A$,

*where* $\phi \in \mathrm{\Phi}$, $\theta \in \mathrm{\Theta}$. *Moreover*, *assume that* $T{A}_{0}\subseteq {B}_{0}$. *Then* *T* *has a best proximity point in* *A*.

Now we provide the following example to show that Corollary 3.8 is an essential extension of the above mentioned theorem of Samet.

**Example 3.13** Consider the complete metric space $X=[0,2]\times [0,\mathrm{\infty})$ with the Euclidean metric. Let $A=\{(0,x):0\le x\}$ and $B=\{(2,y):0\le y\}$. Then $d(A,B)=2$, ${A}_{0}=A$, ${B}_{0}=B$ and $(A,B)$ has the *P*-property.

*T*has a best proximity point (indeed, $P=(0,0)$ is a best proximity point of

*T*). But we cannot invoke the above mentioned theorem of Samet to show that the mapping

*T*has a best proximity point in

*A*because

*T*is not an almost $(\phi ,\theta )$ contraction. On the contrary, assume that there exist $\phi \in \mathrm{\Phi}$ and $\theta \in \mathrm{\Theta}$ such that for all $x,y\in A$,

*f*on $(0,\mathrm{\infty})$ is an increasing positive function. So, we have

*φ*is non-decreasing, then from the above, we get

and so (24) holds for each $n\in \mathbb{N}$. Since ${x}_{n}>\frac{c}{n}$ for each $n\in \mathbb{N}$ and $\sum \frac{c}{n}=\mathrm{\infty}$, then we get $\sum {x}_{n}=\mathrm{\infty}$, a contradiction.

## Declarations

### Acknowledgements

The first author was partially supported by a grant from IPM (No. 92470412). The second author was partially supported by a grant from IPM (No. 92550414). The first and the second author were also partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the fourth author acknowledges with thanks DSR, KAU for financial support.

## Authors’ Affiliations

## References

- Amini-Harandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem.
*Math. Comput. Model.*2012. 10.1016/j.mcm.2011.12.006Google Scholar - Amini-Harandi A: Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 215 10.1186/1687-1812-2012-215Google Scholar - Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions.
*Carpath. J. Math.*2008, 24(1):8–12.MathSciNetMATHGoogle Scholar - Berinde V, Pacurar M: Fixed points and continuity of almost contractions.
*Fixed Point Theory*2008, 9: 23–34.MathSciNetMATHGoogle Scholar - Berinde V: Some remarks on fixed points theorem for Ćirić-type almost contractions.
*Carpath. J. Math.*2009, 25: 157–162.MathSciNetMATHGoogle Scholar - Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mapping in ordered metric spaces.
*Appl. Math. Comput.*2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060MathSciNetView ArticleMATHGoogle Scholar - Fan K: Extensions of two fixed point theorems of F. E. Browder.
*Math. Z.*1969, 112: 234–240. 10.1007/BF01110225MathSciNetView ArticleMATHGoogle Scholar - Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations.
*Numer. Funct. Anal. Optim.*1982/1983, 5: 449–455.MathSciNetView ArticleMATHGoogle Scholar - Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces.
*J. Nonlinear Convex Anal.*2010, 11(3):475–489.MathSciNetMATHGoogle Scholar - Hussain N, Khan AR: Applications of the best approximation operator to ∗-nonexpansive maps in Hilbert spaces.
*Numer. Funct. Anal. Optim.*2003, 24(3–4):327–338. 10.1081/NFA-120022926MathSciNetView ArticleMATHGoogle Scholar - Takahashi W: Fan’s existence theorem for inequalities concerning convex functions and its applications. In
*Minimax Theory and Applications*. Edited by: Ricceri B, Simons S. Kluwer Academic, Dordrecht; 1998:597–602.Google Scholar - Abkar A, Gabeleh M: The existence of best proximity points for multivalued mappings.
*Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.*2013. 10.1007/s13398-012-0074-6Google Scholar - Abkar A, Gabeleh M: Best proximity points for cyclic mappings in ordered metric spaces.
*J. Optim. Theory Appl.*2011, 150(1):188–193. 10.1007/s10957-011-9818-2MathSciNetView ArticleMATHGoogle Scholar - Amini-Harandi A: Best proximity points theorems for cyclic strongly quasi-contraction mappings.
*J. Glob. Optim.*2013. 10.1007/s10898-012-9953-9Google Scholar - Amini-Harandi A: Best proximity points for proximal generalized contractions in metric spaces.
*Optim. Lett.*2013. 10.1007/s11590-012-0470-zGoogle Scholar - Amini-Harandi A: Common best proximity points theorems in metric spaces.
*Optim. Lett.*2012. 10.1007/s11590-012-0600-7Google Scholar - Amini-Harandi A, Hussain N, Akbar F: Best proximity point results for generalized contractions in metric spaces.
*Fixed Point Theory Appl.*2013., 2013: Article ID 164Google Scholar - Caballero C, Harjani J, Sadarangani K: A best proximity point theorem for Geraghty-contraction.
*Fixed Point Theory Appl.*2012., 2012: Article ID 231 10.1186/1687-1812-2012-231Google Scholar - Eldred AA, Veeramani P: Existence and convergence of best proximity points.
*J. Math. Anal. Appl.*2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetView ArticleMATHGoogle Scholar - Hussain N, Kutbi MA, Salimi P: Best proximity point results for modified
*α*-*ψ*-proximal rational contractions.*Abstr. Appl. Anal.*2013., 2013: Article ID 927457Google Scholar - Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC.
*Nonlinear Anal.*2009, 71: 2918–2926. 10.1016/j.na.2009.01.173MathSciNetView ArticleMATHGoogle Scholar - Jleli M, Samet B: Best proximity point for
*α*-*ψ*-proximal contractive type mappings and applications.*Bull. Sci. Math.*2013. 10.1016/j.bulsci.2013.02.003Google Scholar - Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems.
*Numer. Funct. Anal. Optim.*2003, 24: 851–862. 10.1081/NFA-120026380MathSciNetView ArticleMATHGoogle Scholar - Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces.
*J. Optim. Theory Appl.*2012. 10.1007/s10957-012-9991-yGoogle Scholar - Sadiq Basha S: Best proximity points: global optimal approximate solution.
*J. Glob. Optim.*2011, 49: 15–21. 10.1007/s10898-009-9521-0MathSciNetView ArticleMATHGoogle Scholar - Sadiq Basha S: Best proximity points: optimal solutions.
*J. Optim. Theory Appl.*2011, 151: 210–216. 10.1007/s10957-011-9869-4MathSciNetView ArticleMATHGoogle Scholar - Sadiq Basha S: Best proximity point theorems: unriddling a special nonlinear programming problem.
*Top*2012. 10.1007/s11750-012-0269-1Google Scholar - Agarwal RP, Hussain N, Taoudi MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations.
*Abstr. Appl. Anal.*2012., 2012: Article ID 245872Google Scholar - Samet B: Some results on best proximity points.
*J. Optim. Theory Appl.*2013. 10.1007/s10957-013-0269-9Google Scholar - Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces.
*Appl. Anal.*2008, 87(1):109–116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar - Sankar Raj V: A best proximity point theorems for weakly contractive non-self mappings.
*Nonlinear Anal.*2011, 74: 4804–4808. 10.1016/j.na.2011.04.052MathSciNetView ArticleMATHGoogle Scholar - Jleli M, Karapinar E, Samet B: On best proximity points under the
*P*-property on partially ordered metric spaces.*Abstr. Appl. Anal.*2013., 2013: Article ID 150970Google Scholar

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