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

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 $x,y,z\in X$:

(p₁) $x=y\iff p(x,x)=p(x,y)=p(y,y)$,

(p₂) $p(x,x)\le p(x,y)$,

(p₃) $p(x,y)=p(y,x)$,

(p₄) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

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 (p₁) and (p₂) 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 $(R^{+},p_{\max })$, where the function $p_{\max }:R^{+}\times R^{+}\to R^{+}$ is defined by $p_{\max }(x,y)=max\{x,y\}$ 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:

$$\{B_p(x,\epsilon ):x\in X,\epsilon >0\},$$

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:

(d₁) A sequence $\{x_n\}$ in a partial metric space $(X,p)$ converges to x if and only if $p(x,x)={lim}_{n\to \mathrm{\infty }}p(x,x_n)$.

(d₂) 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.

(d₃) 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 $p(x,x)={lim}_{n,m\to \mathrm{\infty }}p(x_n,x_m)$.

(d₄) 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

$$d_p(x,y)=2p(x,y)-p(x,x)-p(y,y)$$

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

(r₁) 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.

(r₂) 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 $(X,d_p)$.

(r₃) A partial metric space $(X,p)$ is complete if and only if the metric space $(X,d_p)$ is complete.

(r₄) Given a sequence ${\{x_n\}}_{n\in N}$ in a partial metric space $(X,p)$ and $x\in X$, we have that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x,x_n)=0\iff p(x,x)=\underset{n\to \mathrm{\infty }}{lim}p(x,x_n)=\underset{n,m\to \mathrm{\infty }}{lim}p(x_n,x_m).$$

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

$$d(f(x),f(y))\le \overline{\alpha }(x,y)d(x,y)$$

for all $x,y\in X$, where the function $\overline{\alpha }:X\times X\to [0,1)$, holds for every $0<a<b$ that

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le d(x,y)\le b\}<1.$$

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

$$d(f(x),f(y))\le \frac{\alpha }{2}[d(x,f(x))+d(y,f(y))]$$

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

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le d(x,y)\le b\}<1$$

and

$$d(f(x),f(y))\le \frac{\overline{\alpha }(x,y)}{2}[d(x,f(x))+d(y,f(y))].$$

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 $0\le a\le b$,

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le p(x,y)\le b\}<1,$$

and for every $x,y\in X$,

$$p(f(x),f(y))\le \overline{\alpha }(x,y)p(x,y).$$

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

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le p(x,y)\le b\}<1$$

and

$$p(f(x),f(y))\le \frac{\overline{\alpha }(x,y)}{2}[p(x,f(x))+p(y,f(y))].$$

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:

$$\begin{array}{c}{A}_0=\{x\in A:p(x,y)=p(A,B)\text{ for some }y\in B\},\hfill \\ B_0=\{y\in B:p(x,y)=p(A,B)\text{ for some }x\in A\}.\hfill \end{array}$$

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

$$p(f(x),f(y))\le \overline{\alpha }(x,y)p(x,y)$$

for all $x,y\in A$, where the function $\overline{\alpha }:A\times A\to [0,1)$, holds for every $0<a<b$ that

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le p(x,y)\le b\}<1.$$

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

$$p(f(x),f(y))\le \frac{\overline{\alpha }(x,y)}{2}[p(x,f(x))+p(y,f(y))-2p(A,B)]$$

for all $x,y\in A$, where the function $\overline{\alpha }:A\times A\to [0,1)$, holds for every $0<a<b$ that

$$\theta (a,b)=sup\{\overline{\alpha }(x,y):a\le p(x,y)\le b\}<1.$$

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 $y_1,y_2\in B_0$,

$$\{\begin{array}{l}p(x_1,y_1)=p(A,B),\\ p(x_2,y_2)=p(A,B)\end{array}\Rightarrow p(x_1,x_2)=p(y_1,y_2).$$

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

$$\begin{array}{rl}p(x_n,y_n)& \le p(x_n,x)+p(x,y_n)-p(x,x)\\ \le p(x_n,x)+p(x,y)+p(y,y_n)-p(x,x)-p(y,y).\end{array}$$

From the above inequality, we can get that

$$\begin{array}{c}p(x_n,y_n)-p(x,y)\hfill \\ \le [p(x_n,x)-p(x,x)]+[p(y,y_n)-p(y,y)]\to 0\text{as }n\to \mathrm{\infty }.\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{rl}p(x,y)& \le p(x,x_n)+p(x_n,y)-p(x_n,x_n)\\ \le p(x,x_n)+p(x_n,y_n)+p(y_n,y)-p(x_n,x_n)-p(y_n,y_n).\end{array}$$

Then we can obtain

$$\begin{array}{c}p(x,y)-p(x_n,y_n)\hfill \\ \le [p(x,x_n)-p(x_n,x_n)]+[p(y_n,y)-p(y_n,y_n)]\to 0\text{as }n\to \mathrm{\infty }.\hfill \end{array}$$

Above all, we can get that

$$|p(x_n,y_n)-p(x,y)|\to 0\text{as }n\to \mathrm{\infty }.$$

This completes the proof. □

Remark For (r₄) 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

$$x_{2k+1}=T{x}_{2k},p(x_{2k+2},x_{2k+1})=p(A,B),k=0,1,2,\dots$$

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

$$\begin{array}{rl}{d}_p(y_n,y_m)& =2p(y_n,y_m)-p(y_n,y_n)-p(y_m,y_m)\\ =2p(x_n,x_m)-p(x_n,x_n)-p(x_m,x_m)\\ =d_p(x_n,x_m).\end{array}$$

Hence

$$d_p(y_n,y_m)\to 0\Rightarrow d_p(x_n,x_m)\to 0$$

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

$$p(P_{A_0}T{x}_1,P_{A_0}T{x}_2)=p(T{x}_1,T{x}_2)\le \overline{\alpha }(x_1,x_2)p(x_1,x_2)$$

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

$$p(x^{\ast },T{x}^{\ast })=p(A,B).$$

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

$$x_{2k+1}=T{x}_{2k},p(x_{2k+2},x_{2k+1})=p(A,B),k=0,1,2,\dots$$

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

$$x_{2k+1}=T{x}_{2k},p(x_{2k+2},x_{2k+1})=p(A,B),k=0,1,2,\dots$$

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

$$\begin{array}{rcl}p(P_{A_0}T{x}_1,P_{A_0}T{x}_2)& =& p(T{x}_1,T{x}_2)\\ \le & \frac{\overline{\alpha }(x,y)}{2}[p(x_1,T{x}_1)+p(x_2,T{x}_2)-2p(A,B)]\\ \le & \frac{\overline{\alpha }(x,y)}{2}[p(x_1,P_{A_0}T{x}_1)+p(P_{A_0}T{x}_1,T{x}_1)\\ +p(x_2,P_{A_0}T{x}_2)+p(P_{A_0}T{x}_2,T{x}_2)-2p(A,B)]\\ =& \frac{\overline{\alpha }(x,y)}{2}[p(x_1,P_{A_0}T{x}_1)+p(x_2,P_{A_0}T{x}_2)]\end{array}$$

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

$$p(x^{\ast },T{x}^{\ast })=p(A,B).$$

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

$$x_{2k+1}=T{x}_{2k},p(x_{2k+2},x_{2k+1})=p(A,B),k=0,1,2,\dots$$

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. □