# Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces

- Jingling Zhang
^{1}and - Yongfu Su
^{1}Email author

**2014**:50

https://doi.org/10.1186/1687-1812-2014-50

© Zhang and Su; licensee Springer. 2014

**Received: **24 October 2013

**Accepted: **10 February 2014

**Published: **25 February 2014

## Abstract

The purpose of this paper is to obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. In this paper, the *P*-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.

**MSC:**47H05, 47H09, 47H10.

## Keywords

*P*-propertypartial metric

## 1 Introduction and preliminaries

Let us recall some basic definitions of a partial metric space and its properties which can be found in [1].

**Definition 1.1** A *partial metric* on a nonempty set *X* is a function $p:X\times X\to {R}^{+}$ such that for all $x,y,z\in X$:

(p_{1}) $x=y\iff p(x,x)=p(x,y)=p(y,y)$,

(p_{2}) $p(x,x)\le p(x,y)$,

(p_{3}) $p(x,y)=p(y,x)$,

(p_{4}) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

A partial metric space is a pair $(X,p)$ such that *X* is a nonempty set and *p* is a partial metric on *X*.

We can see from (p_{1}) and (p_{2}) that $p(x,y)=0$ implies $x=y$. However, the converse is not necessarily true. A typical example of this situation is provided by the partial metric space $({R}^{+},{p}_{\mathrm{max}})$, where the function ${p}_{\mathrm{max}}:{R}^{+}\times {R}^{+}\to {R}^{+}$ is defined by ${p}_{\mathrm{max}}(x,y)=max\{x,y\}$ for all $x,y\in {R}^{+}$. Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1] and [2].

*p*on

*X*generates a ${T}_{0}$ topology $\tau (p)$ on

*X*, whose base is a family of open

*p*-balls:

where ${B}_{p}(x,\epsilon )=\{y\in X:p(x,y)\le p(x,x)+\epsilon \}$ for all $x\in X$ and $\epsilon >0$. Definitions of convergence, Cauchy sequence, completeness and continuity on partial metric spaces are as follows:

(d_{1}) A sequence $\{{x}_{n}\}$ in a partial metric space $(X,p)$ converges to *x* if and only if $p(x,x)={lim}_{n\to \mathrm{\infty}}p(x,{x}_{n})$.

(d_{2}) A sequence $\{{x}_{n}\}$ in a partial metric space $(X,p)$ is called a Cauchy sequence if ${lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})$ exists and is finite.

(d_{3}) A partial metric space $(X,p)$ is called complete if every Cauchy sequence $\{{x}_{n}\}$ in *X* converges, with respect to $\tau (p)$, to a point $x\in X$ such that $p(x,x)={lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})$.

(d_{4}) A mapping $f:X\to X$ is said to be continuous at ${x}_{0}\in X$ if for every $\epsilon >0$, there exists $\delta >0$ such that $f({B}_{p}({x}_{0},\delta ))\subseteq {B}_{p}(f({x}_{0}),\epsilon )$.

is a metric on *X*. The following useful remarks were introduced in [1]:

(r_{1}) If a sequence converges in a partial metric space $(X,p)$ with respect to $\tau ({d}_{p})$, then it converges with respect to $\tau (p)$. Of course, the converse is not true.

(r_{2}) A sequence ${\{{x}_{n}\}}_{n\in N}$ in a partial metric space $(X,p)$ is a Cauchy sequence if and only if it is a Cauchy sequence in the metric space $(X,{d}_{p})$.

(r_{3}) A partial metric space $(X,p)$ is complete if and only if the metric space $(X,{d}_{p})$ is complete.

_{4}) Given a sequence ${\{{x}_{n}\}}_{n\in N}$ in a partial metric space $(X,p)$ and $x\in X$, we have that

Let *A* and *B* be nonempty subsets of a metric space $(X,d)$. An operator $T:A\to B$ is said to be contractive if there exists $k\in [0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for any $x,y\in A$. The well-known Banach contraction principle says: Let $(X,d)$ be a complete metric space, and let $T:X\to X$ be a contraction of *X* into itself; then *T* has a unique fixed point in *X*.

In the last fifty years, the Banach contraction principle has been extensively studied and generalized on many settings. One of the generalizations is a weakly contractive mapping.

**Definition 1.2** ([3])

*weakly contractive*provided that

The fixed point theorem for a weakly contractive mapping was presented in [3].

**Theorem 1.3** *Let* $(X,d)$ *be a complete metric space*. *If* $f:X\to X$ *is a weakly contractive mapping*, *then* *f* *has a unique fixed point* ${x}^{\ast}$ *and the Picard sequence of iterates* ${\{{f}^{n}(x)\}}_{n\in N}$ *converges*, *for every* $x\in X$, *to* ${x}^{\ast}$.

One type of contraction which is different from the Banach contraction is Kannan mappings. In [4], Kannan obtained the following fixed point theorem.

**Theorem 1.4** ([4])

*Let*$(X,d)$

*be a complete metric space*,

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

*be a mapping such that*

*for all* $x,y\in X$ *and some* $\alpha \in [0,1]$, *then* *f* *has a unique fixed point* ${x}^{\ast}\in X$. *Moreover*, *the Picard sequence of iterates* ${\{{f}^{n}(x)\}}_{n\in N}$ *converges*, *for every* $x\in X$, *to* ${x}^{\ast}$.

In [5], the authors introduced a more general weakly Kannan mapping and obtained its fixed point theorem.

**Definition 1.5** ([5])

*weakly Kannan*if there exists $\overline{\alpha}:X\times X\to [0,1)$ which satisfies, for every $0<a\le b$ and for all $x,y\in X$, that

**Theorem 1.6** ([5])

*Let* $(X,d)$ *be a complete metric space*. *If* $f:X\to X$ *is a weakly Kannan mapping*, *then* *f* *has a unique fixed point* ${x}^{\ast}$ *and the Picard sequence of iterates* ${\{{f}^{n}(x)\}}_{n\in N}$ *converges*, *for every* $x\in X$, *to* ${x}^{\ast}$.

Recently, Alghamdi *et al.* [6] generalized the weakly contractive and weakly Kannan mappings to partial metric spaces and obtained the following fixed point theorems.

**Definition 1.7** ([6])

*weakly contractive*provided that there exists $\overline{\alpha}:X\times X\to [0,1)$ such that for every $0\le a\le b$,

**Definition 1.8** ([6])

*weakly Kannan*if there exists $\overline{\alpha}:X\times X\to [0,1)$ which satisfies for every $0<a\le b$ and for all $x,y\in X$ that

**Theorem 1.9** ([6])

*Let* $(X,p)$ *be a complete partial metric space*, *and let* $f:X\to X$ *be a weakly contractive mapping*. *Then* *f* *has a unique fixed point* ${x}^{\ast}\in X$ *and the Picard sequence of iterates* ${\{{f}^{n}(x)\}}_{n\in N}$ *converges*, *with respect to* $\tau ({d}_{p})$, *for every* $x\in X$, *to* ${x}^{\ast}$. *Moreover*, $p({x}^{\ast},{x}^{\ast})=0$.

**Theorem 1.10** ([6])

*Let* $(X,p)$ *be a complete partial metric space*, *and let* $f:X\to X$ *be a weakly Kannan mapping*. *Then* *f* *has a unique fixed point* ${x}^{\ast}\in X$ *and the Picard sequence of iterates* ${\{{f}^{n}(x)\}}_{n\in N}$ *converges*, *with respect to* $\tau ({d}_{p})$, *for every* $x\in X$, *to* ${x}^{\ast}$. *Moreover*, $p({x}^{\ast},{x}^{\ast})=0$.

In this paper, we first obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. The *P*-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.

Before giving the main results, we need the following notations and basic facts.

Let *A*, *B* be two nonempty subsets of a complete partial metric space $(X,p)$ and consider a mapping $T:A\to B$. The best proximity point problem is whether we can find an element ${x}_{0}\in A$ such that $p({x}_{0},T{x}_{0})=p(A,B)$, where $p(A,B)=inf\{p(x,y):x\in A\text{and}y\in B\}$. Since $p(x,Tx)\ge p(A,B)$ for any $x\in A$, in fact, the optimal solution to this problem is the one for which the value $p(A,B)$ is attained. Some works on the best proximity point problem can be found in [7–11].

*A*and

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

## 2 Best proximity point theorems in partial metric spaces

**Definition 2.1**Let $(A,B)$ be a pair of nonempty subsets of a partial metric space $(X,p)$. A mapping $f:A\to B$ is said to be

*weakly contractive*provided that

**Definition 2.2**Let $(A,B)$ be a pair of nonempty subsets of a partial metric space $(X,p)$. A mapping $f:A\to B$ is said to be

*weakly Kannan*provided that

We rewrite the *P*-property in the setting of partial metric spaces as follows.

**Definition 2.3**Let $(A,B)$ be a pair of nonempty subsets of a partial metric space $(X,p)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Then the pair $(A,B)$ is said to have the

*P*-

*property*if and only if, for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

**Lemma 2.4** *Let* $(X,p)$ *be a partial metric space*, *then* *p* *is a continuous function*, *that is*, *for any* ${x}_{n},{y}_{n},x,y\subseteq X$, *if* ${x}_{n}\to x$, ${y}_{n}\to y$, *then* $p({x}_{n},{y}_{n})\to p(x,y)$ *as* $n\to \mathrm{\infty}$.

*Proof*Since

This completes the proof. □

**Remark** For (r_{4}) we know that, for any ${x}_{n},{y}_{n},x,y\subseteq X$, if ${d}_{p}({x}_{n},x)\to 0$, ${d}_{p}({y}_{n},y)\to 0$, then $p({x}_{n},{y}_{n})\to p(x,y)$ as $n\to \mathrm{\infty}$.

**Theorem 2.5**

*Let*$(A,B)$

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

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

*Let*$T:A\to B$

*be a continuous weakly contractive mapping*.

*Suppose that*$T({A}_{0})\subseteq {B}_{0}$

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

*has the*

*P*-

*property*.

*Then*

*T*

*has a unique best proximity point*${x}^{\ast}\in {A}_{0}$

*and the iteration sequence*${\{{x}_{2k}\}}_{n=0}^{\mathrm{\infty}}$

*defined by*

*converges*, *with respect to* $\tau ({d}_{p})$, *for every* ${x}_{0}\in {A}_{0}$, *to* ${x}^{\ast}$.

*Proof*We first prove that ${B}_{0}$ is closed with respect to $(X,{d}_{p})$. Let $\{{y}_{n}\}\subseteq {B}_{0}$ be a sequence such that ${y}_{n}\to q\in B$. It follows from the

*P*-property that

as $n,m\to \mathrm{\infty}$, where ${x}_{n},{x}_{m}\in {A}_{0}$ and $p({x}_{n},{y}_{n})=p(A,B)$, $p({x}_{m},{y}_{m})=p(A,B)$. Then $\{{x}_{n}\}$ is a Cauchy sequence in $(X,{d}_{p})$, so that $\{{x}_{n}\}$ converges to a point $p\in A$. By the continuity of a partial metric *p*, we have $p(p,q)=p(A,B)$, that is, $q\in {B}_{0}$, and hence ${B}_{0}$ is closed with respect to $(X,{d}_{p})$.

Let ${\overline{A}}_{0}$ be the closure of ${A}_{0}$ in a metric space $(X,{d}_{p})$, we claim that $T({\overline{A}}_{0})\subseteq {B}_{0}$. In fact, if $x\in {\overline{A}}_{0}\setminus {A}_{0}$, then there exists a sequence $\{{x}_{n}\}\subseteq {A}_{0}$ such that ${x}_{n}\to x$. By the continuity of *T* and the closedness of ${B}_{0}$, we have $Tx={lim}_{n\to \mathrm{\infty}}T{x}_{n}\in {B}_{0}$; that is, $T({\overline{A}}_{0})\subseteq {B}_{0}$.

*P*-property, we have

is exactly the subsequence of $\{{x}_{n}\}$, so that it converges, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$. This completes the proof. □

**Theorem 2.6**

*Let*$(A,B)$

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

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

*Let*$T:A\to B$

*be a continuous weakly Kannan mapping*.

*Suppose that*$T({A}_{0})\subseteq {B}_{0}$

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

*has the*

*P*-

*property*.

*Then*

*T*

*has a unique best proximity point*${x}^{\ast}\in {A}_{0}$

*and the iteration sequence*${\{{x}_{2k}\}}_{n=0}^{\mathrm{\infty}}$

*defined by*

*converges*, *with respect to* $\tau ({d}_{p})$, *for every* ${x}_{0}\in {A}_{0}$, *to* ${x}^{\ast}$.

*Proof*We can prove that ${B}_{0}$ is closed and $T({\overline{A}}_{0})\subseteq {B}_{0}$ in the same way as in Theorem 2.5. Now define an operator ${P}_{{A}_{0}}:T({\overline{A}}_{0})\to {A}_{0}$ by ${P}_{{A}_{0}}y=\{x\in {A}_{0}:p(x,y)=p(A,B)\}$. Since the pair $(A,B)$ has the

*P*-property, we have

is exactly the subsequence of $\{{x}_{n}\}$, so that it converges, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$. This completes the proof. □

## Declarations

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

## Authors’ Affiliations

## References

- Matthews SG: Partial metric topology.
*Ann. N.Y. Acad. Sci.*1994, 728: 183–197. 10.1111/j.1749-6632.1994.tb44144.xView ArticleMathSciNetGoogle Scholar - Escardo MH: PCF extended with real numbers.
*Theor. Comput. Sci.*1996, 162: 79–115. 10.1016/0304-3975(95)00250-2View ArticleMathSciNetGoogle Scholar - Dugundji J, Granas A: Weakly contractive mappings and elementary domain invariance theorem.
*Bull. Greek Math. Soc.*1978, 19: 141–151.MathSciNetGoogle Scholar - Kannan R: Some results on fixed points.
*Bull. Calcutta Math. Soc.*1968, 60: 71–76.MathSciNetGoogle Scholar - Ariza-Ruiz D, Jimenez-Melando A: A continuation method for weakly Kannan mappings.
*Fixed Point Theory Appl.*2010., 2010: Article ID 321594Google Scholar - Alghamdi MA, Shahzad N, Valero O: On fixed point theory in partial metric spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 175Google Scholar - Gabeleh M: Global optimal solutions of non-self mappings.
*U.P.B. Sci. Bull., Ser. A*2013, 75: 67–74.MathSciNetGoogle Scholar - Zhang J, Su Y, Cheng Q: A note on ‘A best proximity point theorem for Geraghty-contractions’.
*Fixed Point Theory Appl.*2013., 2013: Article ID 99Google Scholar - Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points.
*Nonlinear Anal.*2009, 70: 3665–3671. 10.1016/j.na.2008.07.022View ArticleMathSciNetGoogle Scholar - Eldred AA, Veeramani P: Existence and convergence of best proximity points.
*J. Math. Anal. Appl.*2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081View ArticleMathSciNetGoogle Scholar - Alghamdi MA, Shahzad N, Vetro F: Best proximity points for some classes of proximal contractions.
*Abstr. Appl. Anal.*2013., 2013: Article ID 713252Google Scholar

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