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Best proximity point theorems with Suzuki distances
Journal of Inequalities and Applications volume 2015, Article number: 27 (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.
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.
Preliminaries
Let A, B be two nonempty subsets of a metric space \((X,d)\). The following notations will be used throughout this paper:
We recall that \(x\in A\) is a best proximity point of the mapping \(T:A\longrightarrow B\) if \(d(x,Tx)=d(A,B)\). It can be observed that a best proximity point reduces to a fixed point if the underlying mapping is a selfmapping.
Definition 2.1
([20])
Let \((A, B)\) be a pair of nonempty subsets of a metric space X with \(A\neq\emptyset\). Then the pair \((A,B)\) is said to have the Pproperty if and only if
where \(x_{1}, x_{2}\in A_{0}\) and \(y_{1}, y_{2}\in B_{0}\).
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.
Remark 2.3
([5])
Every set is approximately compact with respect to itself.
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])
For every function \(\psi:[0,\infty)\longrightarrow[0,\infty)\), the following holds:

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])
Let \(T: A\longrightarrow B\) and \(\alpha:A\times A\longrightarrow [0,\infty)\). We say that T is αproximal admissible if
for all \(x_{1},x_{2},u_{1},u_{2}\in A\).
Remark 2.6
Let ‘⪯’ be a partially ordered relation on A and \(\alpha :A\times A\longrightarrow[0,\infty)\) be defined by
If T is αproximal admissible, then T is said to be proximally increasing. In other words, T is proximally increasing if it satisfies the condition that
for all \(x_{1},x_{2},u_{1},u_{2}\in A\).
Definition 2.7
([19])
Let X be a metric space with metric d. A function \(p:X\times X\longrightarrow[0,\infty)\) is called τdistance on X if there exists a function \(\eta:X\times [0,\infty )\longrightarrow[0,\infty)\) such that the following are satisfied:
 (\(\tau_{1}\)):

\(p(x,z)\leq p(x,y)+p(y,z)\) 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
(\(\tau_{2}\)) can be replaced by the following \((\tau_{2})'\).
 \((\tau_{2})'\) :

\(\inf\{\eta(x,t):t>0\}=0\) 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.
Some best proximity point theorems
Now, we define the weak Pproperty with respect to a τdistance as follows.
Definition 3.1
Let \((A, B)\) be a pair of nonempty subsets of a metric space \((X,d)\) with \(A_{0}\neq\emptyset\). Also let p be a τdistance on X. Then the pair \((A, B)\) is said to have the weak Pproperty with respect to p if and only if
where \(x_{1}, x_{2}\in A_{0}\) and \(y_{1}, y_{2}\in B_{0}\).
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
Let \(X=\mathbf{R}^{2}\) with the usual metric and \(p_{1}\), \(p_{2} \) be two τdistances defined in Example 2.11 and Example 2.12, respectively. Consider the following:
Then \((A,B) \) has the weak Pproperty with respect to \(p_{1}\) but has not the weak Pproperty with respect to \(p_{2}\).
By the definition of A and B, we obtain
where \((0,2),(0,3)\in A \) and \((1,1),(1,4)\in B\). We have
Therefore \((A,B) \) has the weak Pproperty with respect to \(p_{1}\). On the other hand, we have
This implies that \((A,B) \) has not the weak Pproperty with respect to \(p_{2}\).
Definition 3.4
Let \((X,d) \) be a metric space and let p be a τdistance on X. A mapping \(T:A\longrightarrow B\) is said to be an αψproximal contraction with respect to p if
where \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(\psi\in \Psi\).
Remark 3.5
([1])
If \(p=d\), then T is said to be an αψproximal contraction.
Example 3.6
Let \((X,d)\) be a metric space and A, B be two subsets of X. Let p be the τdistance defined in Example 2.10. Consider the following:
Then \(T:A\longrightarrow B\) is an \(\alpha_{1}\)ψproximal contraction with respect to p, but it is not an \(\alpha_{2}\)ψproximal contraction with respect to p.
Definition 3.7
\(g:A\longrightarrow A\) is said to be a τdistance preserving with respect to p if
for all \(x_{1}\) and \(x_{2}\) in A.
We first prove the following lemma. Then we state our results.
Lemma 3.8
Let A and B be nonempty, closed subsets of a metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let p be a τdistance on X and \(\alpha:A\times A\longrightarrow[0,\infty) \). Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(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
$$d(gx_{1},Tx_{0})=d(A,B) \quad \textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
Then there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
Proof
By condition (e) there exist \(x_{0},x_{1}\in A \) such that
Since \(Tx_{1}\in T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\), there exists \(x_{2}\in A_{0}\) such that
T is αproximal admissible, therefore by (1) and (2) we have
By condition (c) we obtain
Continuing this process, we can find a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
This completes the proof of the lemma. □
The following result is a special case of Lemma 3.8 obtained by setting α defined in Remark 2.6.
Corollary 3.9
Let A and B be nonempty, closed subsets of a metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let ‘⪯’ be a partially ordered relation on A and p be a τdistance on X. Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(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
$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and}\quad x_{0}\preceq x_{1}. $$
Then there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
The following result is a spacial case of Lemma 3.8 if g is the identity map.
Corollary 3.10
Let A and B be nonempty, closed subsets of a metric space \((X,d)\) such that \(A_{0}\) is nonempty and \(\alpha:A\times A\longrightarrow [0,\infty) \). Suppose that \(T:A\longrightarrow B\) satisfies the following conditions:

(a)
T is αproximal admissible.

(b)
\(T(A_{0})\subseteq B_{0}\).

(c)
There exist \(x_{0},x_{1}\in A \) such that
$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad \alpha(x_{0},x_{1})\geq1. $$
Then there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
Theorem 3.11
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(\psi\in\Psi\). Also suppose that p is a τdistance on X and \(T:A\longrightarrow B\) satisfies the following conditions:

(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.
Then T has a best proximity point in A.
Proof
By Corollary 3.10 there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
\((A,B)\) satisfies the weak Pproperty with respect to p, therefore by (4) we obtain that
Also, by the definition of T, we have
On the other hand, we have \(\alpha(x_{n1},x_{n})\geq1\) for all \(n\in \mathbf{N}\), which implies that
If there exists \(n_{0}\in\mathbf{N}\) such that \(p(x_{n_{0}},x_{n_{0}1})=0 \), then, by the definition of ψ, we obtain that \(\psi(p(x_{n_{0}1},x_{n_{0}}))=0 \). Therefore by (7) we have \(p(x_{n},x_{n+1})=0\) for all \(n>n_{0} \). Thus by Lemma 3.8 the sequence \(\{x_{n}\} \) is Cauchy.
Now, let \(p(x_{n1},x_{n})\neq0\) for all \(n\in\mathbf{N}\). By the monotony of ψ and using induction in (7), we obtain
By the definition of ψ, we have \(\sum_{k=1}^{\infty}\psi ^{k}(p(x_{0},x_{1}))<\infty\). So, for all \(\varepsilon>0\), there exists some positive integer \(h=h(\varepsilon)\) such that
Now let \(m>n>h\). By the triangle inequality and (8), we have
This implies that
By Corollary 2.17 \(\{x_{n}\}\) is a Cauchy sequence in A. Since X is a complete metric space and A is a closed subset of X, there exists \(x\in A\) such that \(\lim_{n\rightarrow\infty}x_{n}=x\).
T is continuous, therefore, by letting \(n\longrightarrow\infty\) in (4), we obtain
This completes the proof of the theorem. □
The following result is the special case of Theorem 3.11 obtained by setting \(p=d\).
Corollary 3.12
([1])
Let A and B be nonempty closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(\psi\in\Psi\). Suppose that \(T:A\longrightarrow B\) is a nonself mapping satisfying the following conditions:

(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.
Then there exists an element \(x^{*}\in A_{0}\) such that
Theorem 3.13
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Also suppose that p is a τdistance on X and \(T:A\longrightarrow B\) satisfies the following conditions:

(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.
Then T has a best proximity point in A. Moreover, if \(d(x,Tx)=d(x^{*},Tx^{*})=d(A,B)\) for some \(x,x^{*}\in A\), then \(p(x,x^{*})=0\).
Proof
Define \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(\psi :[0,\infty)\longrightarrow[0,\infty)\) by \(\alpha(x,y) = 1\) for all \(x,y\in A\) and \(\psi(t)=t\) for all \(t\geq0\). Therefore by Theorem 3.11, T has a best proximity point in A. Now let x, \(x^{*}\) be best proximity points in A. Therefore we have
The pair \((A,B)\) has the weak Pproperty with respect to p, hence by the definition of T we obtain that
Hence \(p(x,x^{*})=0\) and this completes the proof of the theorem. □
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.
αpProximal contractions
Definition 4.1
Let A, B be subsets of a metric space \((X,d)\) and p be a τdistance on X. A mapping \(T:A\longrightarrow B\) is said to be an αpproximal contraction of the first kind if there exists \(r\in [0,1)\) such that
where \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(u_{1}, u_{2},x_{1}, x_{2}\in A\).
Also if T is an αpproximal contraction of the first kind, then

(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
Let A, B be subsets of a metric space \((X,d)\) and p be a τdistance on X. A mapping \(T:A\longrightarrow B\) is said to be an αpproximal contraction of the second kind if there exists \(r\in [0,1)\) such that
where \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(u_{1}, u_{2},x_{1}, x_{2}\in A\).
Also if T is an αpproximal contraction of the second kind, then

(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
Let \(X=\mathbf{R}\) with the usual metric and p be the τdistance defined in Example 2.11. Given \(A=[3,2]\cup[2,3]\), \(B=[1,1]\) and \(T:A\longrightarrow B \) by
then T is a pproximal contraction of the first and second kind.
It is easy to see that
If \(r\in[\frac{2}{3},1) \), then we have
Hence T is a pproximal contraction of the first kind. Also,
for all \(r\in[0,1) \). This implies that T is a pproximal contraction of the second kind.
Example 4.8
Let \(X=\mathbf{R}\) with the usual metric and p be the τdistance defined in Example 2.12. Let ‘⪯’ be the usual partially ordered relation in R. Given \(A=\{2\}\cup[2,3]\), \(B=[1,1]\) and \(T:A\longrightarrow B \) by
then T is an ordered pproximal contraction of the first and second kind, but it is not a pproximal contraction of the first and second kind.
It is easy to see that
If \(r\in[\frac{2}{3},1) \), then we have
\(p(2,2)\nleq rp(3,2)\), but it is not necessary because \(3\npreceq 2 \). Hence T is an ordered pproximal contraction of the first kind. But T is not a pproximal contraction of the first kind because \(p(2,2)\nleq rp(3,2)\) for all \(r\in[0,1)\). Also,
for all \(r\in[0,1) \). Notice that \(p (T(2),T(2) )\nleq rp (T(3),T(2) ) \), but it is not necessary because \(3\npreceq2 \). This implies that T is an ordered pproximal contraction of the second kind. But T is not a pproximal contraction of the second kind because \(p (T(2),T(2) )\nleq rp (T(3),T(2) )\) for all \(r\in[0,1)\).
Theorem 4.9
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let p be a wdistance on X and \(\alpha:A\times A\longrightarrow[0,\infty) \). Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(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
$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1. $$
Then there exists an element \(x\in A\) such that
Proof
By Lemma 3.8 there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
We will prove the convergence of a sequence \(\{x_{n}\}\) in A. T is an αpproximal contraction of the first kind and (3) holds, hence, for any positive integer n, we have
Also g is a τdistance preserving with respect to p, so we get that
for every \(n\in\mathbf{N}\). Hence, if \(m>n\),
This implies that
By Corollary 2.17, \(\{x_{n}\}\) is a Cauchy sequence in A. Since X is a complete metric space and A is a closed subset of X, there exists \(x\in A\) such that \(\lim_{n\rightarrow\infty}x_{n}=x\).
T and g are continuous, therefore by letting \(n\longrightarrow \infty\) in (3), we obtain
This completes the proof of the theorem. □
The next result is an immediate consequence of Theorem 4.9 by setting α defined in Remark 2.6.
Corollary 4.10
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let ‘⪯’ be a partially ordered relation on A and p be a τdistance on X. Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(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
$$d(gx_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad x_{0}\preceq x_{1}. $$
Then there exists an element \(x\in A\) such that
Theorem 4.11
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let p be a τdistance on X. Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(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 \(A_{0}\subseteq g(A_{0})\).
Then there exists an element \(x\in A\) such that
Moreover, if \(d(gx,Tx)=d(gx^{*},Tx^{*})=d(A,B)\) for some \(x,x^{*}\in A\), then \(p(x,x^{*})=0\).
Proof
By Theorem 4.9 there exists an element \(x\in A\) such that
Now let \(x^{*}\) be in A such that
T is a pproximal contraction of the first kind and g is a τdistance preserving with respect to p, therefore
Hence \(p(x,x^{*})=0\) and this completes the proof of the theorem. □
The next result is obtained by taking \(p=d\) in Theorem 4.11.
Corollary 4.12
([5])
Let X be a complete metric space. Let A and B be nonempty, closed subsets of X. Further, suppose that \(A_{0}\) and \(B_{0}\) are nonempty. Let \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:

(a)
T is a continuous proximal contraction of the first kind.

(b)
g is an isometry.

(c)
\(T(A_{0})\subseteq B_{0}\).

(d)
\(A_{0}\subseteq g(A_{0})\).
Then there exists a unique element \(x\in A\) such that
The following result is a best proximity point theorem for nonself αpproximal contraction of the second kind.
Theorem 4.13
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that A is approximately compact with respect to B and \(A_{0}\) is nonempty. Let p be a τdistance on X and \(\alpha :A\times A\longrightarrow[0,\infty) \). Suppose that \(T:A\longrightarrow B\) satisfies the following conditions:

(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
$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad \alpha(x_{0},x_{1})\geq1. $$
Then there exists an element \(x\in A\) such that
Proof
By Corollary 3.10 there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
We will prove the convergence of a sequence \(\{x_{n}\}\) in A. T is an αpproximal contraction of the second kind and (10) holds, hence, for any positive integer n, we have
for every \(n\in\mathbf{N}\). Hence, if \(m>n\),
This implies that
By Corollary 2.17, \(\{Tx_{n}\}\) is a Cauchy sequence in B. Since X is a complete metric space and B is a closed subset of X, there exists \(y\in B\) such that \(\lim_{n\rightarrow\infty}Tx_{n}=y \). By the triangle inequality, we have
Letting \(n\longrightarrow\infty\) in the above inequality, we obtain
Since A is approximately compact with respect to B, there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n}\}\) converging to some \(x\in A\). Therefore
This implies that \(x\in A_{0}\). T is continuous and \(\{Tx_{n}\}\) is convergent to y, therefore
Thus, it follows that
This completes the proof of the theorem. □
The next result is an immediate consequence of Theorem 4.13 by setting α defined in Remark 2.6.
Corollary 4.14
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that A is approximately compact with respect to B and \(A_{0}\) is nonempty. Let ‘⪯’ be a partially ordered relation on A and p be a τdistance on X. Suppose that \(T:A\longrightarrow B\) satisfies the following conditions:

(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
$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and}\quad x_{0}\preceq x_{1} . $$
Then there exists an element \(x\in A\) such that
Theorem 4.15
Let A and B be nonempty, closed subsets of a complete metric space \((X,d)\) such that A is approximately compact with respect to B, and let p be a τdistance on X. Further, suppose that \(A_{0}\) is nonempty. Let \(T:A\longrightarrow B\) satisfy the following conditions:

(a)
T is a continuous pproximal contraction of the second kind.

(b)
\(T(A_{0})\subseteq B_{0}\).
Then there exists an element \(x\in A\) such that
Moreover, if \(d(x,Tx)=d(x^{*},Tx^{*})=d(A,B)\) for some \(x,x^{*}\in A\), then \(p(Tx,Tx^{*})=0\).
Proof
By Theorem 4.13 there exists an element \(x\in A\) such that
Now let \(x^{*}\) be an element in A such that
We will show that \(p(Tx,Tx^{*})=0\). T is a pproximal contraction of the second kind, therefore
Hence \(p(Tx,Tx^{*})=0\) and this completes the proof of the theorem. □
The following result is obtained by taking \(p=d\) in Theorem 4.15.
Corollary 4.16
([5])
Let A and B be nonempty, closed subsets of a complete metric space such that A is approximately compact with respect to B. Further, suppose that \(A_{0}\) and \(B_{0}\) are nonempty. Let \(T:A\longrightarrow B\) satisfy the following conditions:

(a)
T is a continuous proximal contraction of the second kind.

(b)
\(T(A_{0})\) is contained in \(B_{0}\).
Then there exists an element \(x\in A\) such that
Moreover, if \(x^{*}\) is another best proximity point of T, then Tx and \(Tx^{*}\) are identical.
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Omidvari, M., Vaezpour, S.M., Saadati, R. et al. Best proximity point theorems with Suzuki distances. J Inequal Appl 2015, 27 (2015). https://doi.org/10.1186/s1366001405387
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DOI: https://doi.org/10.1186/s1366001405387
MSC
 90C26
 90C30
 47H09
 47H10
Keywords
 weak Pproperty
 best proximity point
 τdistance
 αψproximal contraction
 ordered pproximal contraction