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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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:
$$\begin{aligned}& d(y,A):=\inf\bigl\{ d(x,y):x\in A\bigr\} , \\& d(A,B):=\inf\bigl\{ d(x,y):x\in A\mbox{ and }y\in B \bigr\} , \\& A_{0} :=\bigl\{ x \in A : d(x, y)= d(A, B)\mbox{ for some }y \in B \bigr\} , \\& B_{0} :=\bigl\{ y \in B : d(x, y)= d(A, B)\mbox{ for some }x \in A \bigr\} . \end{aligned}$$
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.
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
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.
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.
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
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.
Let\((X,d)\)be a metric space andpbe aτ-distance onX. If\(\{x_{n}\}\)is ap-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 ap-Cauchy sequence and\(\lim_{n} d(x_{n},y_{n})=0\).
Let\((X,d)\)be a metric space andpbe aτ-distance onX. If a sequence\(\{x_{n}\}\)inXsatisfies\(\lim_{n}\sup\{p(x_{n},x_{m}):m\geq n\} =0\), then\(\{x_{n}\}\)is ap-Cauchy sequence. Moreover, if\(\{y_{n}\}\)inXsatisfies\(\lim_{n} p(x_{n},y_{n})=0\), then\(\{y_{n}\}\)is also ap-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 andpbe aτ-distance onX. If a sequence\(\{x_{n}\}\)inXsatisfies\(\lim_{n}\sup\{p(x_{n},x_{m}):m\geq n\} =0\), 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
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:
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
$$\alpha(x,y)p(Tx,Ty)\leq\psi \bigl(p(x,y) \bigr) \quad\mbox{for all }x,y\in A, $$
where \(\alpha:A\times A\longrightarrow[0,\infty)\) and \(\psi\in \Psi\).
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
$$p(gx_{1},gx_{2})=p(x_{1},x_{2}) $$
for all \(x_{1}\) and \(x_{2}\) in A.
We first prove the following lemma. Then we state our results.
Lemma 3.8
LetAandBbe nonempty, closed subsets of a metric space\((X,d)\)such that\(A_{0}\)is nonempty. Letpbe aτ-distance onXand\(\alpha:A\times A\longrightarrow[0,\infty) \). Suppose that\(T:A\longrightarrow B\)and\(g:A\longrightarrow A\)satisfy the following conditions:
(a)
Tisα-proximal admissible.
(b)
gis aτ-distance preserving with respect top.
(c)
\(\alpha(gu,gv)\geq1\)implies that\(\alpha(u,v)\geq1\)for all\(u,v\in A \).
The following result is a special case of Lemma 3.8 obtained by setting α defined in Remark 2.6.
Corollary 3.9
LetAandBbe nonempty, closed subsets of a metric space\((X,d)\)such that\(A_{0}\)is nonempty. Let ‘⪯’ be a partially ordered relation onAandpbe aτ-distance onX. Suppose that\(T:A\longrightarrow B\)and\(g:A\longrightarrow A\)satisfy the following conditions:
(a)
Tis proximally increasing.
(b)
gis aτ-distance preserving with respect top.
(c)
\(gu\preceq gv\)implies that\(u\preceq v\)for all\(u,v\in A \).
The following result is a spacial case of Lemma 3.8 if g is the identity map.
Corollary 3.10
LetAandBbe 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:
Then there exists a sequence\(\{x_{n}\}\)in\(A_{0}\)such that
$$d(x_{n+1},Tx_{n})=d(A,B) \quad\textit{and} \quad \alpha(x_{n},x_{n+1})\geq 1 \quad\textit{for all }n\in\mathbf{N} \cup\{0\}. $$
Theorem 3.11
LetAandBbe 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 thatpis aτ-distance onXand\(T:A\longrightarrow B\)satisfies the following conditions:
(a)
\(T(A_{0})\subseteq B_{0}\)and\((A,B)\)has the weakP-property with respect top.
$$\begin{aligned} p(x_{n},x_{n+1})\leq\psi \bigl(p(x_{n-1},x_{n}) \bigr) \quad \mbox{for all }n\in \mathbf{N}. \end{aligned}$$
(7)
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
$$ p(x_{n},x_{n+1})\leq\psi^{n} \bigl(p(x_{0},x_{1}) \bigr) \quad\mbox{for all }n\in \mathbf{N}. $$
(8)
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
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
$$d(x,Tx)=d(A,B). $$
This completes the proof of the theorem. □
The following result is the special case of Theorem 3.11 obtained by setting \(p=d\).
LetAandBbe 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:
Then there exists an element\(x^{*}\in A_{0}\)such that
$$d\bigl(x^{*},Tx^{*}\bigr)=d(A,B) . $$
Theorem 3.13
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such that\(A_{0}\)is nonempty. Also suppose thatpis aτ-distance onXand\(T:A\longrightarrow B\)satisfies the following conditions:
(a)
\(T(A_{0})\subseteq B_{0}\)and\((A,B)\)has the weakP-property with respect top.
(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)
Tis continuous.
ThenThas a best proximity point inA. 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
$$d(x,Tx)=d\bigl(x^{*},Tx^{*}\bigr)=d(A,B) . $$
The pair \((A,B)\) has the weak P-property with respect to p, hence by the definition of T we obtain that
$$p\bigl(x,x^{*}\bigr)\leq p\bigl(Tx,Tx^{*}\bigr)\leq rp \bigl(x,x^{*}\bigr). $$
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 andAbe a nonempty closed subset ofX. Let\(T:A\longrightarrow A\)be a contractive self-map. ThenThas a unique fixed point\(x^{*}\)inA.
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
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
Hence T is a p-proximal contraction of the first kind. Also,
$$\begin{aligned}& p \bigl(T(-2),T(2) \bigr)\leq rp \bigl(T(-3),T(3) \bigr), \\& p \bigl(T(2),T(-2) \bigr)\leq rp \bigl(T(3),T(-3) \bigr) \end{aligned}$$
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
\(p(2,-2)\nleq rp(3,-2)\), 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,
$$p \bigl(T(-2),T(2) \bigr)\leq rp \bigl(T(-2),T(3) \bigr) $$
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
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such that\(A_{0}\)is nonempty. Letpbe aw-distance onXand\(\alpha:A\times A\longrightarrow[0,\infty) \). Suppose that\(T:A\longrightarrow B\)and\(g:A\longrightarrow A\)satisfy the following conditions:
(a)
Tis anα-proximal admissible and continuousα-p-proximal contraction of the first kind.
(b)
gis a continuousτ-distance preserving with respect top.
(c)
\(\alpha(gu,gv)\geq1\)implies that\(\alpha(u,v)\geq1\)for all\(u,v\in A \).
By Lemma 3.8 there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
$$\begin{aligned} d(gx_{n+1},Tx_{n})=d(A,B) \quad\mbox{and}\quad \alpha(x_{n},x_{n+1})\geq 1 \quad\mbox{for all }n\in\mathbf{N}\cup \{0\}. \end{aligned}$$
(9)
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
$$p(gx_{n},gx_{n+1})\leq rp(x_{n},x_{n-1}). $$
Also g is a τ-distance preserving with respect to p, so we get 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
$$d(gx,Tx)=d(A,B). $$
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
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such that\(A_{0}\)is nonempty. Let ‘⪯’ be a partially ordered relation onAandpbe aτ-distance onX. Suppose that\(T:A\longrightarrow B\)and\(g:A\longrightarrow A\)satisfy the following conditions:
(a)
Tis a proximally increasing and continuous orderedp-proximal contraction of the first kind.
(b)
gis a continuousτ-distance preserving with respect top.
(c)
\(gu\preceq gv\)implies that\(u\preceq v\)for all\(u,v\in A \).
LetXbe a complete metric space. LetAandBbe nonempty, closed subsets ofX. 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)
Tis a continuous proximal contraction of the first kind.
(b)
gis 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
$$d(gx,Tx)=d(A,B). $$
The following result is a best proximity point theorem for nonself α-p-proximal contraction of the second kind.
Theorem 4.13
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such thatAis approximately compact with respect toBand\(A_{0}\)is nonempty. Letpbe aτ-distance onXand\(\alpha :A\times A\longrightarrow[0,\infty) \). Suppose that\(T:A\longrightarrow B\)satisfies the following conditions:
(a)
Tis anα-proximal admissible and continuousα-p-proximal contraction of the second kind.
By Corollary 3.10 there exists a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
$$\begin{aligned} d(x_{n+1},Tx_{n})=d(A,B) \quad\mbox{and}\quad \alpha(x_{n},x_{n+1})\geq 1 \quad\mbox{for all }n\in\mathbf{N}\cup \{0\}. \end{aligned}$$
(10)
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
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
$$\lim_{n\rightarrow\infty}d(y,x_{n})=d(y,A). $$
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
The next result is an immediate consequence of Theorem 4.13 by setting α defined in Remark 2.6.
Corollary 4.14
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such thatAis approximately compact with respect toBand\(A_{0}\)is nonempty. Let ‘⪯’ be a partially ordered relation onAandpbe aτ-distance onX. Suppose that\(T:A\longrightarrow B\)satisfies the following conditions:
(a)
Tis a proximally increasing and continuous orderedp-proximal contraction of the second kind.
LetAandBbe nonempty, closed subsets of a complete metric space\((X,d)\)such thatAis approximately compact with respect toB, and letpbe aτ-distance onX. Further, suppose that\(A_{0}\)is nonempty. Let\(T:A\longrightarrow B\)satisfy the following conditions:
(a)
Tis a continuousp-proximal contraction of the second kind.
LetAandBbe nonempty, closed subsets of a complete metric space such thatAis approximately compact with respect toB. Further, suppose that\(A_{0}\)and\(B_{0}\)are nonempty. Let\(T:A\longrightarrow B\)satisfy the following conditions:
(a)
Tis 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
$$d(x,Tx)=d(A,B). $$
Moreover, if\(x^{*}\)is another best proximity point ofT, thenTxand\(Tx^{*}\)are identical.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.