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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
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
- fixed point
- best proximity point
- weakly contractive mapping
- P-property
- partial 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].
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.
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])
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])
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])
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])
Definition 1.8 ([6])
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].
2 Best proximity point theorems in partial metric spaces
We rewrite the P-property in the setting of partial metric spaces as follows.
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 $n\to \mathrm{\infty}$.
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 $n\to \mathrm{\infty}$.
converges, with respect to $\tau ({d}_{p})$, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$.
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, $T({\overline{A}}_{0})\subseteq {B}_{0}$.
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. □
converges, with respect to $\tau ({d}_{p})$, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$.
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
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