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 Open Access
Best proximity point theorems with Suzuki distances
 Mehdi Omidvari^{1},
 Seiyed Mansour Vaezpour^{2},
 Reza Saadati^{3} and
 Sung Jin Lee^{4}Email author
https://doi.org/10.1186/s1366001405387
© Omidvari et al.; licensee Springer 2015
 Received: 26 June 2014
 Accepted: 19 December 2014
 Published: 27 January 2015
Abstract
In this paper, we define the weak Pproperty and the αψproximal contraction by p in which p is a τdistance on a metric space. Then, we prove some best proximity point theorems in a complete metric space X with generalized distance. Also we define two kinds of αpproximal contractions and prove some best proximity point theorems.
Keywords
 weak Pproperty
 best proximity point
 τdistance
 αψproximal contraction
 ordered pproximal contraction
MSC
 90C26
 90C30
 47H09
 47H10
1 Introduction
Let us assume that A and B are two nonempty subsets of a metric space \((X,d)\) and \(T:A\longrightarrow B\). Clearly \(T(A)\cap A\neq \emptyset\) is a necessary condition for the existence of a fixed point of T. The idea of the best proximity point theory is to determine an approximate solution x such that the error of equation \(d(x,Tx)=0 \) is minimum. A solution x for the equation \(d(x,Tx)=d(A,B)\) is called a best proximity point of T. The existence and convergence of best proximity points have been generalized by several authors [1–8] in many directions. Also, Akbar and Gabeleh [9, 10], Sadiq Basha [11] and Pragadeeswarar and Marudai [12] extended the best proximity points theorems in partially ordered metric spaces (see also [13–18]). On the other hand, Suzuki [19] introduced the concept of τdistance on a metric space and proved some fixed point theorems for various contractive mappings by τdistance. In this paper, by using the concept of τdistance, we prove some best proximity point theorems.
2 Preliminaries
Definition 2.1
([20])
It is clear that, for any nonempty subset A of X, the pair \((A,A)\) has the Pproperty.
Definition 2.2
([5])
A is said to be approximately compact with respect to B if every sequence \(\{x_{n}\}\) of A, satisfying the condition that \(d(y,x_{n})\longrightarrow d(y,A)\) for some y in B, has a convergent subsequence.
Samet et al. [21] introduced a class of contractive mappings called αψcontractive mappings. Let Ψ be the family of nondecreasing functions \(\psi:[0,\infty )\longrightarrow[0,\infty)\) such that \(\sum_{n=1}^{\infty}\psi ^{n}(t)<\infty\) for all \(t>0\), where \(\psi^{n}(t) \) is the nth iterate of ψ.
Lemma 2.4
([21])

if ψ is nondecreasing, then, for each \(t > 0\), \(\lim_{n\rightarrow\infty}\psi^{n}(t)=0\) implies \(\psi(t) < t\).
Definition 2.5
([1])
Remark 2.6
Definition 2.7
([19])
 (\(\tau_{1}\)):

\(p(x,z)\leq p(x,y)+p(y,z)\) for all \(x,y,z\in{X}\);
 (\(\tau_{2}\)):

\(\eta(x,0)=0\) and \(\eta(x,t)\geq t\) for all \(x\in{X}\) and \(t\in [0,\infty)\), and η is concave and continuous in its second variable;
 (\(\tau_{3}\)):

\(\lim_{n} x_{n}=x\) and \(\lim_{n}\sup\{\eta(z_{n},(z_{n},x_{m})):m\geq n\}=0\) imply \(p(w,x)\leq\liminf_{n} p(w,x_{n})\) for all \(w\in{X}\);
 (\(\tau_{4}\)):

\(\lim_{n} \sup\{p(x_{n},y_{m}):m\geq n\}=0\) and \(\lim_{n} \eta(x_{n},t_{n})=0\) imply \(\lim_{n} \eta(y_{n},t_{n})=0\)
 (\(\tau_{5}\)):

\(\lim_{n} \eta(z_{n},p(z_{n},x_{n}))=0 \) and \(\lim_{n} \eta(z_{n},p(z_{n},y_{n}))=0 \) imply \(\lim_{n} d(x_{n},y_{n})=0\).
Remark 2.8
 \((\tau_{2})'\) :

\(\inf\{\eta(x,t):t>0\}=0\) for all \(x\in{X}\), and η is nondecreasing in its second variable.
Remark 2.9
If \((X,d)\) is a metric space, then the metric d is a τdistance on X.
In the following examples, we define \(\eta:X \times[0,\infty )\longrightarrow[0,\infty)\) by \(\eta(x,t)= t\) for all \(x\in{X}\), \(t\in [0,\infty )\). It is easy to see that p is a τdistance on a metric space X.
Example 2.10
Let \((X,d)\) be a metric space and c be a positive real number. Then \(p:X\times X\longrightarrow[0,\infty)\) by \(p(x,y)=c\) for \(x,y\in X \) is a τdistance on X.
Example 2.11
Let \((X,\\cdot\)\) be a normed space. \(p:X\times X\longrightarrow[0,\infty)\) by \(p(x,y)=\ x\ +\ y\\) for \(x,y\in X \) is a τdistance on X.
Example 2.12
Let \((X,\\cdot\)\) be a normed space. \(p:X\times X\longrightarrow[0,\infty)\) by \(p(x,y)=\ y\\) for \(x,y\in X \) is a τdistance on X.
Definition 2.13
Let \((X,d)\) be a metric space and p be a τdistance on X. A sequence \(\{x_{n}\}\) in X is called pCauchy if there exists a function \(\eta:X \times[0,\infty)\longrightarrow[0,\infty)\) satisfying (\(\tau_{2}\))(\(\tau_{5}\)) and a sequence \(z_{n}\) in X such that \(\lim_{n}\sup \{\eta(z_{n},(z_{n},x_{m})):m\geq n\}=0\).
The following lemmas are essential for the next sections.
Lemma 2.14
([19])
Let \((X,d)\) be a metric space and p be a τdistance on X. If \(\{x_{n}\}\) is a pCauchy sequence, then it is a Cauchy sequence. Moreover, if \(\{y_{n}\}\) is a sequence satisfying \(\lim_{n}\sup\{p(x_{n},y_{m}):m\geq n=0\}\), then \(\{y_{n}\}\) is also a pCauchy sequence and \(\lim_{n} d(x_{n},y_{n})=0\).
Lemma 2.15
([19])
Let \((X,d)\) be a metric space and p be a τdistance on X. If \(\{x_{n}\}\) in X satisfies \(\lim_{n} p(z,x_{n})=0\) for some \(z\in X\), then \(\{x_{n}\}\) is a pCauchy sequence. Moreover, if \(\{y_{n}\}\) in X also satisfies \(\lim_{n}p(z,y_{n})=0\), then \(\lim_{n} d(x_{n},y_{n})=0\). In particular, for \(x,y,z\in X\), \(p(z,x)=0\) and \(p(z,y)=0 \) imply \(x=y\).
Lemma 2.16
([19])
Let \((X,d)\) be a metric space and p be a τdistance on X. If a sequence \(\{x_{n}\}\) in X satisfies \(\lim_{n}\sup\{p(x_{n},x_{m}):m\geq n\} =0\), then \(\{x_{n}\}\) is a pCauchy sequence. Moreover, if \(\{y_{n}\}\) in X satisfies \(\lim_{n} p(x_{n},y_{n})=0\), then \(\{y_{n}\}\) is also a pCauchy sequence and \(\lim_{n} d(x_{n},y_{n})=0\).
The next result is an immediate consequence of Lemma 2.14 and Lemma 2.16.
Corollary 2.17
Let \((X,d)\) be a metric space and p be a τdistance on X. If a sequence \(\{x_{n}\}\) in X satisfies \(\lim_{n}\sup\{p(x_{n},x_{m}):m\geq n\} =0\), then \(\{x_{n}\}\) is a Cauchy sequence.
3 Some best proximity point theorems
Now, we define the weak Pproperty with respect to a τdistance as follows.
Definition 3.1
It is clear that, for any nonempty subset A of X, the pair \((A,A)\) has the weak Pproperty with respect to p.
Remark 3.2
([22])
If \(p=d\), then \((A,B)\) is said to have the weak Pproperty where \(A_{0}\neq\emptyset\).
It is easy to see that if \((A,B)\) has the Pproperty, then \((A,B)\) has the weak Pproperty.
Example 3.3
Definition 3.4
Example 3.6
Definition 3.7
We first prove the following lemma. Then we state our results.
Lemma 3.8
 (a)
T is αproximal admissible.
 (b)
g is a τdistance preserving with respect to p.
 (c)
\(\alpha(gu,gv)\geq1\) implies that \(\alpha(u,v)\geq1\) for all \(u,v\in A \).
 (d)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).
 (e)There exist \(x_{0},x_{1}\in A \) such that$$d(gx_{1},Tx_{0})=d(A,B) \quad \textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
Proof
The following result is a special case of Lemma 3.8 obtained by setting α defined in Remark 2.6.
Corollary 3.9
 (a)
T is proximally increasing.
 (b)
g is a τdistance preserving with respect to p.
 (c)
\(gu\preceq gv\) implies that \(u\preceq v\) for all \(u,v\in A \).
 (d)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).
 (e)There exist \(x_{0},x_{1}\in A \) such that$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and}\quad x_{0}\preceq x_{1}. $$
The following result is a spacial case of Lemma 3.8 if g is the identity map.
Corollary 3.10
 (a)
T is αproximal admissible.
 (b)
\(T(A_{0})\subseteq B_{0}\).
 (c)There exist \(x_{0},x_{1}\in A \) such that$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad \alpha(x_{0},x_{1})\geq1. $$
Theorem 3.11
 (a)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak Pproperty with respect to p.
 (b)
T is αproximal admissible.
 (c)There exist \(x_{0},x_{1}\in A \) such that$$d(x_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
 (d)
T is a continuous αψproximal contraction with respect to p.
Proof
The following result is the special case of Theorem 3.11 obtained by setting \(p=d\).
Corollary 3.12
([1])
 (a)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the Pproperty.
 (b)
T is αproximal admissible.
 (c)There exist \(x_{0},x_{1}\in A \) such that$$d(x_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
 (d)
T is a continuous αψproximal contraction.
Theorem 3.13
 (a)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak Pproperty with respect to p.
 (b)There exists \(r\in[0,1)\) such that$$\begin{aligned} p(Tx,Ty)\leq rp(x,y), \quad \forall x,y\in A. \end{aligned}$$
 (c)
T is continuous.
Proof
The next result is an immediate consequence of Theorem 3.13 by taking \(A=B\) and \(p=d\).
Corollary 3.14
(Banach’s contraction principle)
Let \((X,d)\) be a complete metric space and A be a nonempty closed subset of X. Let \(T:A\longrightarrow A\) be a contractive selfmap. Then T has a unique fixed point \(x^{*}\) in A.
4 αpProximal contractions
Definition 4.1
 (i)
T is said to be an ordered pproximal contraction of the first kind if ‘⪯’ is a partially ordered relation on A and α is defined in Remark 2.6.
 (ii)
T is said to be pproximal contraction of the first kind if \(\alpha(x,y)=1 \) for all \(x,y\in A \).
Remark 4.2
([11])
If T is an ordered pproximal contraction of the first kind and \(p=d\), then T is said to be an ordered proximal contraction of the first kind.
Remark 4.3
If T is a pproximal contraction of the first kind and \(p=d\), then T is said to be a proximal contraction of the first kind (see [5]).
Definition 4.4
 (i)
T is said to be an ordered pproximal contraction of the second kind if ‘⪯’ is a partially ordered relation on A and α is defined in Remark 2.6.
 (ii)
T is said to be a pproximal contraction of the second kind if \(\alpha(x,y)=1 \) for all \(x,y\in A \).
Remark 4.5
If T is an ordered pproximal contraction of the second kind and \(p=d\), then T is said to be an ordered proximal contraction of the second kind.
Remark 4.6
If T is a pproximal contraction of the second kind and \(p=d\), then T is said to be a proximal contraction of the second kind.
Example 4.7
Example 4.8
Theorem 4.9
 (a)
T is an αproximal admissible and continuous αpproximal contraction of the first kind.
 (b)
g is a continuous τdistance preserving with respect to p.
 (c)
\(\alpha(gu,gv)\geq1\) implies that \(\alpha(u,v)\geq1\) for all \(u,v\in A \).
 (d)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).
 (e)There exist \(x_{0},x_{1}\in A \) such that$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1. $$
Proof
The next result is an immediate consequence of Theorem 4.9 by setting α defined in Remark 2.6.
Corollary 4.10
 (a)
T is a proximally increasing and continuous ordered pproximal contraction of the first kind.
 (b)
g is a continuous τdistance preserving with respect to p.
 (c)
\(gu\preceq gv\) implies that \(u\preceq v\) for all \(u,v\in A \).
 (d)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).
 (e)There exist \(x_{0},x_{1}\in A \) such that$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad x_{0}\preceq x_{1}. $$
Theorem 4.11
 (a)
T is a continuous pproximal contraction of the first kind.
 (b)
g is a continuous τdistance preserving with respect to p.
 (c)
\(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).
Proof
The next result is obtained by taking \(p=d\) in Theorem 4.11.
Corollary 4.12
([5])
 (a)
T is a continuous proximal contraction of the first kind.
 (b)
g is an isometry.
 (c)
\(T(A_{0})\subseteq B_{0}\).
 (d)
\(A_{0}\subseteq g(A_{0})\).
The following result is a best proximity point theorem for nonself αpproximal contraction of the second kind.
Theorem 4.13
 (a)
T is an αproximal admissible and continuous αpproximal contraction of the second kind.
 (b)
\(T(A_{0})\subseteq B_{0}\).
 (c)There exist \(x_{0},x_{1}\in A \) such that$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad \alpha(x_{0},x_{1})\geq1. $$
Proof
The next result is an immediate consequence of Theorem 4.13 by setting α defined in Remark 2.6.
Corollary 4.14
 (a)
T is a proximally increasing and continuous ordered pproximal contraction of the second kind.
 (b)
\(T(A_{0})\subseteq B_{0}\).
 (c)There exist \(x_{0},x_{1}\in A \) such that$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad x_{0}\preceq x_{1} . $$
Theorem 4.15
 (a)
T is a continuous pproximal contraction of the second kind.
 (b)
\(T(A_{0})\subseteq B_{0}\).
Proof
The following result is obtained by taking \(p=d\) in Theorem 4.15.
Corollary 4.16
([5])
 (a)
T is a continuous proximal contraction of the second kind.
 (b)
\(T(A_{0})\) is contained in \(B_{0}\).
Declarations
Acknowledgements
The authors are grateful to reviewers for their valuable comments and suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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