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Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 50 (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.
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].
Following [1], each partial metric 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 $f({B}_{p}({x}_{0},\delta ))\subseteq {B}_{p}(f({x}_{0}),\epsilon )$.
It can be easily verified that the function ${d}_{p}:X\times X\to {R}^{+}$ defined by
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.
(r_{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])
Let $(X,d)$ be a metric space. A mapping $f:X\to X$ is said to be weakly contractive provided that
for all $x,y\in X$, where the function $\overline{\alpha}:X\times X\to [0,1)$, holds for every $0<a<b$ 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])
Let $(X,d)$ be a metric space. A mapping $f:X\to X$ is said to be 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
and
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])
Let $(X,p)$ be a partial metric space. A mapping $f:X\to X$ is said to be weakly contractive provided that there exists $\overline{\alpha}:X\times X\to [0,1)$ such that for every $0\le a\le b$,
and for every $x,y\in X$,
Definition 1.8 ([6])
Let $(X,p)$ be a partial metric space. A mapping $f:X\to X$ is said to be 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
and
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].
Let 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
for all $x,y\in A$, where the function $\overline{\alpha}:A\times A\to [0,1)$, holds for every $0<a<b$ 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
for all $x,y\in A$, where the function $\overline{\alpha}:A\times A\to [0,1)$, holds for every $0<a<b$ 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
From the above inequality, we can get that
On the other hand, we have
Then we can obtain
Above all, we can get that
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
Hence
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, $T({\overline{A}}_{0})\subseteq {B}_{0}$.
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
for any ${x}_{1},{x}_{2}\in {\overline{A}}_{0}$. This shows that ${P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0}$ is a weak contraction from a complete partial metric subspace ${\overline{A}}_{0}$ into itself. Using Theorem 1.9, we can get that ${P}_{{A}_{0}}T$ has a unique fixed point ${x}^{\ast}$; that is, ${P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}$, which implies that
Therefore, ${x}^{\ast}$ is the unique one in ${A}_{0}$ such that $p({x}^{\ast},T{x}^{\ast})=p(A,B)$. And the Picard iteration sequence ${\{{({P}_{{A}_{0}}T)}^{n}{x}_{0}\}}_{n\in N}$ converges, with respect to $\tau ({d}_{p})$, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$. Since the iteration sequence ${\{{x}_{2k}\}}_{n=0}^{\mathrm{\infty}}$ defined by
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
for any ${x}_{1},{x}_{2}\in {\overline{A}}_{0}$. This shows that ${P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0}$ is a weakly Kannan mapping from a complete partial metric subspace ${\overline{A}}_{0}$ into itself. Using Theorem 1.10, we can get that ${P}_{{A}_{0}}T$ has a unique fixed point ${x}^{\ast}$; that is, ${P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}$, which implies that
Therefore, ${x}^{\ast}$ is the unique one in ${A}_{0}$ such that $p({x}^{\ast},T{x}^{\ast})=p(A,B)$. And the Picard iteration sequence ${\{{({P}_{{A}_{0}}T)}^{n}{x}_{0}\}}_{n\in N}$ converges, with respect to $\tau ({d}_{p})$, for every ${x}_{0}\in {A}_{0}$, to ${x}^{\ast}$. Since the iteration sequence ${\{{x}_{2k}\}}_{n=0}^{\mathrm{\infty}}$ defined by
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. □
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Zhang, J., Su, Y. Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl 2014, 50 (2014) doi:10.1186/1687-1812-2014-50
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Keywords
- fixed point
- best proximity point
- weakly contractive mapping
- P-property
- partial metric