Abstract
In this paper, we define the weak P-property and the α-ψ-proximal contraction by p in which p is a τ-distance 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 α-p-proximal contractions and prove some best proximity point theorems.
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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 τ-distance on a metric space and proved some fixed point theorems for various contractive mappings by τ-distance. In this paper, by using the concept of τ-distance, we prove some best proximity point theorems.
2 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 self-mapping.
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 P-property 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 P-property.
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 α-ψ-contractive 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 α-proximal 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 α-proximal 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 τ-distance 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 τ-distance 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 τ-distance 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 τ-distance 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 τ-distance 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 τ-distance on X.
Definition 2.13
Let \((X,d)\) be a metric space and p be a τ-distance on X. A sequence \(\{x_{n}\}\) in X is called p-Cauchy 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 τ-distance on X. If \(\{x_{n}\}\) is a p-Cauchy 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 p-Cauchy 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 τ-distance 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 p-Cauchy 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 τ-distance 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 p-Cauchy sequence. Moreover, if \(\{y_{n}\}\) in X satisfies \(\lim_{n} p(x_{n},y_{n})=0\), then \(\{y_{n}\}\) is also a p-Cauchy 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 τ-distance 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 P-property with respect to a τ-distance 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 τ-distance on X. Then the pair \((A, B)\) is said to have the weak P-property 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 P-property with respect to p.
Remark 3.2
([22])
If \(p=d\), then \((A,B)\) is said to have the weak P-property where \(A_{0}\neq\emptyset\).
It is easy to see that if \((A,B)\) has the P-property, then \((A,B)\) has the weak P-property.
Example 3.3
Let \(X=\mathbf{R}^{2}\) with the usual metric and \(p_{1}\), \(p_{2} \) be two τ-distances defined in Example 2.11 and Example 2.12, respectively. Consider the following:
Then \((A,B) \) has the weak P-property with respect to \(p_{1}\) but has not the weak P-property 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 P-property with respect to \(p_{1}\). On the other hand, we have
This implies that \((A,B) \) has not the weak P-property with respect to \(p_{2}\).
Definition 3.4
Let \((X,d) \) be a metric space and let p be a τ-distance on X. A mapping \(T:A\longrightarrow B\) is said to be an α-ψ-proximal 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 α-ψ-proximal contraction.
Example 3.6
Let \((X,d)\) be a metric space and A, B be two subsets of X. Let p be the τ-distance defined in Example 2.10. Consider the following:
Then \(T:A\longrightarrow B\) is an \(\alpha_{1}\)-ψ-proximal contraction with respect to p, but it is not an \(\alpha_{2}\)-ψ-proximal contraction with respect to p.
Definition 3.7
\(g:A\longrightarrow A\) is said to be a τ-distance 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 τ-distance 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 α-proximal admissible.
-
(b)
g is a τ-distance 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 α-proximal 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 τ-distance 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 τ-distance 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 α-proximal 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 τ-distance on X and \(T:A\longrightarrow B\) satisfies the following conditions:
-
(a)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak P-property with respect to p.
-
(b)
T is α-proximal 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 α-ψ-proximal 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 P-property with respect to p, therefore by (4) we obtain that
Also, by the definition of T, we have
On the other hand, we have \(\alpha(x_{n-1},x_{n})\geq1\) 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 \(p(x_{n-1},x_{n})\neq0\) 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 P-property.
-
(b)
T is α-proximal 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 α-ψ-proximal 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 τ-distance on X and \(T:A\longrightarrow B\) satisfies the following conditions:
-
(a)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak P-property 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 P-property 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 self-map. Then T has a unique fixed point \(x^{*}\) in A.
4 α-p-Proximal contractions
Definition 4.1
Let A, B be subsets of a metric space \((X,d)\) and p be a τ-distance on X. A mapping \(T:A\longrightarrow B\) is said to be an α-p-proximal 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 α-p-proximal contraction of the first kind, then
-
(i)
T is said to be an ordered p-proximal 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 p-proximal 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 p-proximal 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 p-proximal 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 τ-distance on X. A mapping \(T:A\longrightarrow B\) is said to be an α-p-proximal 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 α-p-proximal contraction of the second kind, then
-
(i)
T is said to be an ordered p-proximal 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 p-proximal contraction of the second kind if \(\alpha(x,y)=1 \) for all \(x,y\in A \).
Remark 4.5
If T is an ordered p-proximal 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 p-proximal 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 τ-distance 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 p-proximal 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 p-proximal contraction of the first kind. Also,
for all \(r\in[0,1) \). This implies that T is a p-proximal contraction of the second kind.
Example 4.8
Let \(X=\mathbf{R}\) with the usual metric and p be the τ-distance 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 p-proximal contraction of the first and second kind, but it is not a p-proximal 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 p-proximal contraction of the first kind. But T is not a p-proximal 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 \(3\npreceq-2 \). This implies that T is an ordered p-proximal contraction of the second kind. But T is not a p-proximal 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 w-distance 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 α-proximal admissible and continuous α-p-proximal contraction of the first kind.
-
(b)
g is a continuous τ-distance 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 α-p-proximal contraction of the first kind and (3) holds, hence, for any positive integer n, we have
Also g is a τ-distance 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 τ-distance 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 p-proximal contraction of the first kind.
-
(b)
g is a continuous τ-distance 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 τ-distance on X. Suppose that \(T:A\longrightarrow B\) and \(g:A\longrightarrow A\) satisfy the following conditions:
-
(a)
T is a continuous p-proximal contraction of the first kind.
-
(b)
g is a continuous τ-distance 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 p-proximal contraction of the first kind and g is a τ-distance 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 α-p-proximal 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 τ-distance on X and \(\alpha :A\times A\longrightarrow[0,\infty) \). Suppose that \(T:A\longrightarrow B\) satisfies the following conditions:
-
(a)
T is an α-proximal admissible and continuous α-p-proximal 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 α-p-proximal 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 τ-distance on X. Suppose that \(T:A\longrightarrow B\) satisfies the following conditions:
-
(a)
T is a proximally increasing and continuous ordered p-proximal 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 τ-distance on X. Further, suppose that \(A_{0}\) is nonempty. Let \(T:A\longrightarrow B\) satisfy the following conditions:
-
(a)
T is a continuous p-proximal 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 p-proximal 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/s13660-014-0538-7
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DOI: https://doi.org/10.1186/s13660-014-0538-7