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# Best proximity points for generalized proximal *C*-contraction mappings in metric spaces with partial orders

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 94 (2013)

## Abstract

In this paper we extend the notion of weakly *C*-contraction mappings to the case of non-self mappings and establish the best proximity point theorems for this class. Our results generalize the result due to Harjani *et al.* (Comput. Math. Appl. 61:790-796, 2011) and some other authors.

## 1 Introduction and preliminaries

In 1922, Banach proved that every contractive mapping in a complete metric space has a unique fixed point, which is called Banach’s fixed point theorem or Banach’s contraction principle. Since Banach’s fixed point theorem, many authors have extended, improved and generalized this theorem in several ways and, further, some applications of Banach’s fixed point theorem can be found in [1–6] and many others.

In 1972, Chatterjea [7] introduced the following definition.

**Definition 1.1** Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called a *C*-*contraction* if there exists $\alpha \in (0,\frac{1}{2})$ such that, for all $x,y\in X$,

In 2009, Choudhury [8] introduced a generalization of *C*-contraction given by the following definition.

**Definition 1.2** Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called a *weakly* *C-contraction* if, for all $x,y\in X$,

where $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ is a continuous and nondecreasing function such that $\psi (x,y)=0$ if and only if $x=y=0$.

In 2011, Harjani *et al.* [9] presented some fixed point results for weakly *C*-contraction mappings in a complete metric space endowed with a partial order as follows.

**Theorem 1.3** *Let* $(X,\u2aaf)$ *be a partially ordered set and suppose that there exists a metric* *d* *in* *X* *such that* $(X,d)$ *is a complete metric space*. *Let* $T:X\to X$ *be a continuous and nondecreasing mapping such that*

*for* $x\u2aafy$, *where* $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ *is a continuous and nondecreasing function such that* $\psi (x,y)=0$ *if and only if* $x=y=0$. *If there exists* ${x}_{0}\in X$ *with* ${x}_{0}\u2aafT{x}_{0}$, *then* *T* *has a fixed point*.

On the other hand, most of the results on Banach’s fixed point theorem dilate upon the existence of a fixed point for self-mappings. Nevertheless, if *T* is a non-self mapping, then it is probable that the equation $Tx=x$ has no solution, in which case best approximation theorems explore the existence of an approximate solution, whereas best proximity point theorems analyze the existence of an approximate solution that is optimal.

A classical best approximation theorem was introduced by Fan [10], that is, if *A* is a nonempty compact convex subset of a Hausdorff locally convex topological vector space *B* and $T:A\to B$ is a continuous mapping, then there exists an element $x\in A$ such that $d(x,Tx)=d(Tx,A)$. Afterward, several authors including Prolla [11], Reich [12], Sehgal and Singh [13, 14] have derived the extensions of Fan’s theorem in many directions. Other works on the existence of a best proximity point for some contractions can be seen in [15–19]. In 2005, Eldred, Kirk and Veeramani [20] obtained best proximity point theorems for relatively nonexpansive mappings, and some authors have proved best proximity point theorems for several types of contractions (see, for example, [21–26]).

Let *X* be a nonempty set such that $(X,\u2aaf)$ is a partially ordered set and let $(X,d)$ be a complete metric space. Let *A* and *B* be nonempty subsets of a metric space $(X,d)$. Now, we recall the following notions:

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 [27]).

**Definition 1.4** A mapping $T:A\to B$ is said to be *increasing* if

for all $x,y\in A$.

**Definition 1.5** [28]

A mapping $T:A\to B$ is said to be *proximally order-preserving* if and only if it satisfies the condition that

for all $u,v,x,y\in A$.

It is easy to observe that for a self-mapping, the notion of a proximally order-preserving mapping reduces to that of an increasing mapping.

**Definition 1.6** A point $x\in A$ is called a *best proximity point* of the mapping $T:A\to B$ if

In view of the fact that $d(x,Tx)\ge d(A,B)$ for all *x* in *A*, it can be observed that the global minimum of the mapping $x\mapsto d(x,Tx)$ is attained from a best proximity point. Moreover, it is easy to see that the best proximity point reduces to a fixed point if the underlying mapping *T* is a self-mapping.

In this paper, we introduce a new class of proximal contractions, which extends the class of weakly *C*-contractive mappings to the class of non-self mappings, and also give some examples to illustrate our main results. Our results extend and generalize the corresponding results given by Harjani *et al.* [9] and some authors in the literature.

## 2 Main results

In this section, we first introduce the notion of a generalized proximal *C*-contraction mapping and establish the best proximity point theorems.

**Definition 2.1** A mapping $T:A\to B$ is said to be a *generalized proximal* *C*-*contraction* if, for all $u,v,x,y\in A$, it satisfies

where $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ is a continuous and nondecreasing function such that $\psi (x,y)=0$ if and only if $x=y=0$.

For a self-mapping, it is easy to see that equation (2.1) reduces to (1.1).

**Theorem 2.2** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *such that* ${A}_{0}$ *and* ${B}_{0}$ *are nonempty*. *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is a continuous*, *proximally order*-*preserving and generalized proximal* *C*-*contraction such that* $T({A}_{0})\subseteq {B}_{0}$;

(b) *there exist elements* ${x}_{0}$ *and* ${x}_{1}$ *in* ${A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

*Then there exists a point*
$x\in A$
*such that*

*Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$ *defined by*

*converges to the point* *x*.

*Proof* By the hypothesis (b), there exist ${x}_{0},{x}_{1}\in {A}_{0}$ such that ${x}_{0}\u2aaf{x}_{1}$ and

Since $T({A}_{0})\subseteq {B}_{0}$, there exists a point ${x}_{2}\in {A}_{0}$ such that

By the proximally order-preserving property of *T*, we get ${x}_{1}\u2aaf{x}_{2}$. Continuing this process, we can find a sequence $\{{x}_{n}\}$ in ${A}_{0}$ such that ${x}_{n-1}\u2aaf{x}_{n}$ and

Having found the point ${x}_{n}$, one can choose a point ${x}_{n+1}\in {A}_{0}$ such that ${x}_{n}\u2aaf{x}_{n+1}$ and

Since *T* is a generalized proximal *C*-contraction, for each $n\in \mathbb{N}$, we have

and so it follows that $d({x}_{n},{x}_{n+1})\le d({x}_{n-1},{x}_{n})$, that is, the sequence $\{d({x}_{n+1},{x}_{n})\}$ is non-increasing and bounded below. Then there exists $r\ge 0$ such that

Taking $n\to \mathrm{\infty}$ in (2.3), we have

and so

Again, taking $n\to \mathrm{\infty}$ in (2.3) and using (2.4), (2.5) and the continuity of *ψ*, we get

and hence $\psi (2r,0)=0$. So, by the property of *ψ*, we have $r=0$, which implies that

Next, we prove that $\{{x}_{n}\}$ is a Cauchy sequence. Suppose that $\{{x}_{n}\}$ is not a Cauchy sequence. Then there exist $\epsilon >0$ and subsequences $\{{x}_{{m}_{k}}\}$, $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that ${n}_{k}>{m}_{k}\ge k$ with

for each $k\in \{1,2,3,\dots \}$. For each $n\ge 1$, let ${\alpha}_{n}:=d({x}_{n+1},{x}_{n})$. So, we have

It follows from (2.6) that

Notice also that

Taking $k\to \mathrm{\infty}$ in (2.9), by (2.6) and (2.8), we conclude that

Similarly, we can show that

On the other hand, by the construction of $\{{x}_{n}\}$, we may assume that ${x}_{{m}_{k}}\u2aaf{x}_{{n}_{k}}$ such that

and

By the triangle inequality, (2.12), (2.13) and the generalized proximal *C*-contraction of *T*, we have

Taking $k\to \mathrm{\infty}$ in the above inequality, by (2.6), (2.10), (2.11) and the continuity of *ψ*, we get

Therefore, $\psi (\epsilon ,\epsilon )=0$. By the property of *ψ*, we have that $\epsilon =0$, which is a contradiction. Thus $\{{x}_{n}\}$ is a Cauchy sequence. Since *A* is a closed subset of the complete metric space *X*, there exists $x\in A$ such that

Letting $n\to \mathrm{\infty}$ in (2.2), by (2.14) and the continuity of *T*, it follows that

□

**Corollary 2.3** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *such that* ${A}_{0}$ *and* ${B}_{0}$ *are nonempty*. *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is continuous*, *increasing such that* $T({A}_{0})\subseteq {B}_{0}$ *and*

*where* $\alpha \in (0,\frac{1}{2})$;

(b) *there exist* ${x}_{0},{x}_{1}\in {A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

*Then there exists a point*
$x\in A$
*such that*

*Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$ *defined by*

*converges to the point* *x*.

*Proof* Let $\alpha \in (0,\frac{1}{2})$ and the function *ψ* in Theorem 2.2 be defined by

Obviously, it follows that $\psi (a,b)=0$ if and only if $a=b=0$ and (2.1) become (2.15). Hence we obtain Corollary 2.3. □

For a self-mapping, the condition (b) implies that ${x}_{0}\u2aafT{x}_{0}$ and so Theorem 2.2 includes the results of Harjani *et al.* [9] as follows.

**Corollary 2.4** [9]

*Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* $T:X\to X$ *be a continuous and nondecreasing mapping such that*, *for all* $x,y\in X$,

*for* $x\u2aafy$, *where* $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ *is a continuous and nondecreasing function such that* $\psi (x,y)=0$ *if and only if* $x=y=0$. *If there exists* ${x}_{0}\in X$ *with* ${x}_{0}\u2aafT{x}_{0}$, *then* *T* *has a fixed point*.

Now, we give an example to illustrate Theorem 2.2.

**Example 2.5** Consider the complete metric space ${\mathbb{R}}^{2}$ with an Euclidean metric. Let

Then $d(A,B)=1$, ${A}_{0}=\{(0,0)\}$ and ${B}_{0}=\{(0,1)\}$. Define a mapping $T:A\to B$ as follows:

for all $(x,0)\in A$. Clearly, *T* is continuous and $T({A}_{0})\subseteq {B}_{0}$. If ${x}_{1}\u2aaf{x}_{2}$ and

for some ${u}_{1},{u}_{2},{x}_{1},{x}_{2}\in A$, then we have

Therefore, *T* is a generalized proximal *C*-contraction with $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ defined by

Further, observe that $(0,0)\in A$ such that

In Theorem 2.6, we do not need the condition that *T* is continuous. Now, we improve the condition in Theorem 2.2 to prove the new best proximity point theorem as follows.

**Theorem 2.6** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *such that* ${A}_{0}$ *and* ${B}_{0}$ *are nonempty*. *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is a proximally order*-*preserving and generalized proximal* *C*-*contraction such that* $T({A}_{0})\subseteq {B}_{0}$;

(b) *there exist elements* ${x}_{0},{x}_{1}\in {A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

(c) *if* $\{{x}_{n}\}$ *is an increasing sequence in* *A* *converging to* *x*, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$.

*Then there exists a point*
$x\in A$
*such that*

*Proof*

As in the proof of Theorem 2.2, we have

for all $n\ge 0$. Moreover, $\{{x}_{n}\}$ is a Cauchy sequence and converges to some point $x\in A$. Observe that for each $n\in \mathbb{N}$,

Taking $n\to \mathrm{\infty}$ in the above inequality, we obtain ${lim}_{n\to \mathrm{\infty}}d(x,T{x}_{n})=d(A,B)$ and hence $x\in {A}_{0}$. Since $T({A}_{0})\subseteq {B}_{0}$, there exists $v\in A$ such that

Next, we prove that $x=v$. By the condition (c), we have ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$. Using (2.16), (2.17) and the generalized proximal *C*-contraction of *T*, we have

Letting $n\to \mathrm{\infty}$ in (2.18), we get

which implies that $d(x,v)=0$, that is, $x=v$. If we replace *v* by *x* in (2.17), we have

□

**Corollary 2.7** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *such that* ${A}_{0}$ *and* ${B}_{0}$ *are nonempty*. *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is an increasing mapping such that* $T({A}_{0})\subseteq {B}_{0}$ *and*

*where* $\alpha \in (0,\frac{1}{2})$;

(b) *there exist* ${x}_{0},{x}_{1}\in {A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

(c) *if* $\{{x}_{n}\}$ *is an increasing sequence in* *A* *converging to a point* $x\in X$, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$.

*Then there exists a point*
$x\in A$
*such that*

**Corollary 2.8** [9]

*Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Assume that if* $\{{x}_{n}\}\subseteq X$ *is a nondecreasing sequence such that* ${x}_{n}\to x$ *in* *X*, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$. *Let* $T:X\to X$ *be a nondecreasing mapping such that*

*for* $x\u2aafy$, *where* $\psi :{[0,\mathrm{\infty})}^{2}\to [0,\mathrm{\infty})$ *is a continuous and nondecreasing function such that* $\psi (x,y)=0$ *if and only if* $x=y=0$. *If there exists* ${x}_{0}\in X$ *with* ${x}_{0}\u2aafT{x}_{0}$, *then* *T* *has a fixed point*.

Now, we recall the condition defined by Nieto and Rodríguez-López [3] for the uniqueness of the best proximity point in Theorems 2.2 and 2.6.

**Theorem 2.9** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *and let* ${A}_{0}$ *and* ${B}_{0}$ *be nonempty such that* ${A}_{0}$ *satisfies the condition* (2.20). *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is a continuous*, *proximally order*-*preserving and generalized proximal* *C*-*contraction such that* $T({A}_{0})\subseteq {B}_{0}$;

(b) *there exist elements* ${x}_{0}$ *and* ${x}_{1}$ *in* ${A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

*Then there exists a unique point*
$x\in A$
*such that*

*Proof* We will only prove the uniqueness of the point $x\in A$ such that $d(x,Tx)=d(A,B)$. Suppose that there exist *x* and ${x}^{\ast}$ in *A* which are best proximity points, that is,

Case I: *x* is comparable to ${x}^{\ast}$, that is, $x\u2aaf{x}^{\ast}$ (or ${x}^{\ast}\u2aafx$). By the generalized proximal *C*-contraction of *T*, we have

which implies that $\psi (d(x,{x}^{\ast}),d({x}^{\ast},x))=0$. Using the property of *ψ*, we get $d({x}^{\ast},x)=0$ and hence $x={x}^{\ast}$.

Case II: *x* is not comparable to ${x}^{\ast}$. Since ${A}_{0}$ satisfies the condition (2.20), then there exists $z\in {A}_{0}$ such that *z* is comparable to *x* and ${x}^{\ast}$, that is, $x\u2aafz$ (or $z\u2aafx$) and ${x}^{\ast}\u2aafz$ (or $z\u2aaf{x}^{\ast}$). Suppose that $x\u2aafz$ and ${x}^{\ast}\u2aafz$. Since $T({A}_{0})\subseteq {B}_{0}$, there exists a point ${v}_{0}\in {A}_{0}$ such that

By the proximally order-preserving property of *T*, we get $x\u2aaf{v}_{0}$ and ${x}^{\ast}\u2aaf{v}_{0}$. Since $T({A}_{0})\subseteq {B}_{0}$, there exists a point ${v}_{1}\in {A}_{0}$ such that

Again, by the proximally order-preserving property of *T*, we get $x\u2aaf{v}_{1}$ and ${x}^{\ast}\u2aaf{v}_{1}$. One can proceed further in a similar fashion to find ${v}_{n}$ in ${A}_{0}$ with ${v}_{n+1}\in {A}_{0}$ such that

Hence $x\u2aaf{v}_{n}$ and ${x}^{\ast}\u2aaf{v}_{n}$ for all $n\in \mathbb{N}$. By the generalized proximal *C*-contraction of *T*, we have

It follows from (2.21), (2.22) and the property of *ψ* that

By the uniqueness of limit, we conclude that $x={x}^{\ast}$. Other cases can we proved similarly and this completes the proof. □

**Theorem 2.10** *Let* *X* *be a nonempty set such that* $(X,\u2aaf)$ *is a partially ordered set and let* $(X,d)$ *be a complete metric space*. *Let* *A* *and* *B* *be nonempty closed subsets of* *X* *and let* ${A}_{0}$ *and* ${B}_{0}$ *be nonempty such that* ${A}_{0}$ *satisfies the condition* (2.20). *Let* $T:A\to B$ *satisfy the following conditions*:

(a) *T* *is a proximally order*-*preserving and generalized proximal* *C*-*contraction such that* $T({A}_{0})\subseteq {B}_{0}$;

(b) *there exist elements* ${x}_{0},{x}_{1}\in {A}_{0}$ *such that* ${x}_{0}\u2aaf{x}_{1}$ *and*

(c) *if* $\{{x}_{n}\}$ *is an increasing sequence in* *A* *converging to* *x*, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$.

*Then there exists a unique point*
$x\in A$
*such that*

*Proof* For the proof, combine the proofs of Theorems 2.6 and 2.9. □

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## Acknowledgements

This research was partially finished at the Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Mr. Chirasak Mongkolkeha was supported by the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for the Ph.D. program at KMUTT, Thailand. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2012-0008170). The third author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).

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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Mongkolkeha, C., Cho, Y.J. & Kumam, P. Best proximity points for generalized proximal *C*-contraction mappings in metric spaces with partial orders.
*J Inequal Appl* **2013, **94 (2013). https://doi.org/10.1186/1029-242X-2013-94

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### Keywords

- Point Theorem
- Nonexpansive Mapping
- Cauchy Sequence
- Nondecreasing Function
- Unique Fixed Point