Skip to content

Advertisement

  • Research
  • Open Access

Best proximity point theorems with Suzuki distances

  • 1,
  • 2,
  • 3 and
  • 4Email author
Journal of Inequalities and Applications20152015:27

https://doi.org/10.1186/s13660-014-0538-7

  • Received: 26 June 2014
  • Accepted: 19 December 2014
  • Published:

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.

Keywords

  • weak P-property
  • best proximity point
  • τ-distance
  • α-ψ-proximal contraction
  • ordered p-proximal 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 [18] 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 [1318]). 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.

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
$$\left.\begin{array}{r@{}} d(x_{1}, y_{1}) = d(A, B),\\ d(x_{2}, y_{2}) = d(A, B) \end{array} \right\} \quad\Longrightarrow\quad d(x_{1},x_{2})=d(y_{1},y_{2}), $$
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
$$\left.\begin{array}{r@{}} \alpha(x_{1},x_{2})\geq1, \\ d(u_{1},Tx_{1}) = d(A, B),\\ d(u_{2},Tx_{2}) = d(A, B) \end{array} \right\} \quad\Longrightarrow\quad \alpha(u_{1},u_{2})\geq1 $$
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
$$\alpha(x,y) = \left \{ \begin{array}{@{}l@{\quad}l} 1, & x\preceq y,\\ 0, & \mbox{otherwise}. \end{array} \right . $$
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
$$\left.\begin{array}{r@{}} x_{1}\preceq x_{2},\\ d(u_{1},Tx_{1}) = d(A, B),\\ d(u_{2},Tx_{2}) = d(A, B) \end{array} \right\} \quad\Longrightarrow\quad u_{1} \preceq u_{2} $$
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
$$\left.\begin{array}{r@{}} d(x_{1}, y_{1}) = d(A, B),\\ d(x_{2}, y_{2}) = d(A, B) \end{array} \right\} \quad\Longrightarrow\quad p(x_{1},x_{2})\leq p(y_{1},y_{2}), $$
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:
$$\begin{aligned}& A= \bigl\{ (a,b)\in\mathbf{R}^{2}\mid a=0,2\leq b\leq3 \bigr\} , \\& B= \bigl\{ (a,b)\in\mathbf{R}^{2}\mid a=1,b\leq1 \bigr\} \cup \bigl\{ (a,b) \in \mathbf{R}^{2}\mid a=1,b\geq4 \bigr\} . \end{aligned}$$
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
$$d \bigl((0,2),(1,1) \bigr)=d \bigl((0,3),(1,4) \bigr)=d(A, B)=\sqrt{2}, $$
where \((0,2),(0,3)\in A \) and \((1,1),(1,4)\in B\). We have
$$\begin{aligned}& p_{1} \bigl((0,2),(0,3) \bigr)=5\quad \mbox{and} \quad p_{1} \bigl((1,1),(1,4) \bigr)=\sqrt {2}+\sqrt{17}, \\& p_{1} \bigl((0,3),(0,2) \bigr)=5 \quad\mbox{and} \quad p_{1} \bigl((1,4),(1,1) \bigr)=\sqrt {17}+\sqrt{2}. \end{aligned}$$
Therefore \((A,B) \) has the weak P-property with respect to \(p_{1}\). On the other hand, we have
$$p_{2} \bigl((0,3),(0,2) \bigr)=2 \quad\mbox{and}\quad p_{2} \bigl((1,4),(1,1) \bigr)=\sqrt{2}. $$
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 \(\psi\in \Psi\).

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:
$$\begin{aligned}& \psi(t)=\frac{t}{2} \quad\mbox{for all } t\geq0 ,\\& \alpha_{1}(x,y)=k_{1},\quad \mbox{where } k_{1}\in\mathbf{R}, 0\leq k_{1}\leq \frac{1}{2},\\& \alpha_{2}(x,y)=k_{2},\quad \mbox{where }k_{2}\in\mathbf{R}, k_{2}> \frac{1}{2}. \end{aligned}$$
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

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:
  1. (a)

    T is α-proximal admissible.

     
  2. (b)

    g is a τ-distance preserving with respect to p.

     
  3. (c)

    \(\alpha(gu,gv)\geq1\) implies that \(\alpha(u,v)\geq1\) for all \(u,v\in A \).

     
  4. (d)

    \(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).

     
  5. (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
$$d(gx_{n+1},Tx_{n})=d(A,B) \quad\textit{and} \quad \alpha(x_{n},x_{n+1})\geq 1 \quad\textit{for all }n\in\mathbf{N}\cup \{0\}. $$

Proof

By condition (e) there exist \(x_{0},x_{1}\in A \) such that
$$\begin{aligned} d(gx_{1},Tx_{0})=d(A,B) \quad\mbox{and} \quad \alpha(x_{0},x_{1})\geq1. \end{aligned}$$
(1)
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
$$\begin{aligned} d(gx_{2}, Tx_{1})=d(A,B). \end{aligned}$$
(2)
T is α-proximal admissible, therefore by (1) and (2) we have
$$\alpha(gx_{1},gx_{2})\geq1 . $$
By condition (c) we obtain
$$\alpha(x_{1},x_{2})\geq1 . $$
Continuing this process, we can find a sequence \(\{x_{n}\}\) in \(A_{0}\) such that
$$\begin{aligned} d(gx_{n+1},Tx_{n})=d(A,B) \quad\mbox{and}\quad \alpha(x_{n},x_{n+1})\geq 1 \quad \mbox{for all }n\in\mathbf{N} \cup\{0\}. \end{aligned}$$
(3)
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:
  1. (a)

    T is proximally increasing.

     
  2. (b)

    g is a τ-distance preserving with respect to p.

     
  3. (c)

    \(gu\preceq gv\) implies that \(u\preceq v\) for all \(u,v\in A \).

     
  4. (d)

    \(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).

     
  5. (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
$$d(gx_{n+1},Tx_{n})=d(A,B) \quad \textit{and}\quad x_{n}\preceq x_{n+1} \quad \textit{for all }n\in \mathbf{N}\cup \{0\}. $$

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:
  1. (a)

    T is α-proximal admissible.

     
  2. (b)

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

     
  3. (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
$$d(x_{n+1},Tx_{n})=d(A,B) \quad\textit{and} \quad \alpha(x_{n},x_{n+1})\geq 1 \quad\textit{for all }n\in\mathbf{N} \cup\{0\}. $$

Theorem 3.11

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:
  1. (a)

    \(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak P-property with respect to p.

     
  2. (b)

    T is α-proximal admissible.

     
  3. (c)
    There exist \(x_{0},x_{1}\in A \) such that
    $$d(x_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
     
  4. (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
$$\begin{aligned} d(x_{n+1},Tx_{n})=d(A,B) \quad\mbox{and}\quad \alpha(x_{n},x_{n+1})\geq 1 \quad \mbox{for all }n\in\mathbf{N} \cup\{0\}. \end{aligned}$$
(4)
\((A,B)\) satisfies the weak P-property with respect to p, therefore by (4) we obtain that
$$\begin{aligned} p(x_{n},x_{n+1})\leq p(Tx_{n-1},Tx_{n}) \quad \mbox{for all }n\in\mathbf{N}. \end{aligned}$$
(5)
Also, by the definition of T, we have
$$\alpha(x_{n-1},x_{n})p(Tx_{n-1},Tx_{n}) \leq\psi \bigl(p(x_{n-1},x_{n}) \bigr) \quad\mbox{for all }n \in\mathbf{N}. $$
On the other hand, we have \(\alpha(x_{n-1},x_{n})\geq1\) for all \(n\in \mathbf{N}\), which implies that
$$\begin{aligned} p(Tx_{n-1},Tx_{n})\leq\psi \bigl(p(x_{n-1},x_{n}) \bigr) \quad \mbox{for all }n\in \mathbf{N}. \end{aligned}$$
(6)
From (5) and (6), we get that
$$\begin{aligned} p(x_{n},x_{n+1})\leq\psi \bigl(p(x_{n-1},x_{n}) \bigr) \quad \mbox{for all }n\in \mathbf{N}. \end{aligned}$$
(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
$$\sum_{k\geq h}^{\infty}\psi^{k} \bigl(p(x_{0},x_{1}) \bigr)<\varepsilon. $$
Now let \(m>n>h\). By the triangle inequality and (8), we have
$$\begin{aligned} p(x_{n},x_{m})\leq\sum_{k=n}^{m-1}p(x_{k},x_{k+1}) \leq\sum_{k=n}^{m-1}\psi^{k} \bigl(p(x_{0},x_{1}) \bigr)\leq\sum _{k\geq h} \psi ^{k} \bigl(p(x_{0},x_{1}) \bigr)<\varepsilon. \end{aligned}$$
This implies that
$$\begin{aligned} \lim_{n}\sup\bigl\{ p(x_{n},x_{m}):m \geq n\bigr\} =0. \end{aligned}$$
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 \(p=d\).

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:
  1. (a)

    \(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the P-property.

     
  2. (b)

    T is α-proximal admissible.

     
  3. (c)
    There exist \(x_{0},x_{1}\in A \) such that
    $$d(x_{1},Tx_{0})=d(A,B) \quad\textit{and} \quad \alpha(x_{0},x_{1})\geq1 . $$
     
  4. (d)

    T is a continuous α-ψ-proximal contraction.

     
Then there exists an element \(x^{*}\in A_{0}\) such that
$$d\bigl(x^{*},Tx^{*}\bigr)=d(A,B) . $$

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:
  1. (a)

    \(T(A_{0})\subseteq B_{0}\) and \((A,B)\) has the weak P-property with respect to p.

     
  2. (b)
    There exists \(r\in[0,1)\) such that
    $$\begin{aligned} p(Tx,Ty)\leq rp(x,y), \quad \forall x,y\in A. \end{aligned}$$
     
  3. (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
$$d(x,Tx)=d\bigl(x^{*},Tx^{*}\bigr)=d(A,B) . $$
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 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
$$\left.\begin{array}{r@{}} \alpha(x_{1},x_{2})\geq1,\\ d(u_{1},Tx_{1}) = d(A, B),\\ d(u_{2},Tx_{2}) = d(A, B) \end{array} \right\} \quad\Longrightarrow\quad p(u_{1},u_{2})\leq rp(x_{1},x_{2}), $$
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
  1. (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.

     
  2. (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
$$\left.\begin{array}{r@{}} \alpha(x_{1},x_{2})\geq1,\\ d(u_{1},Tx_{1}) = d(A, B),\\ d(u_{2},Tx_{2}) = d(A, B) \end{array} \right\}\quad \Longrightarrow\quad p(Tu_{1},Tu_{2})\leq rp(Tx_{1},Tx_{2}), $$
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
  1. (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.

     
  2. (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
$$T(x) = \left \{ \begin{array}{@{}l@{\quad}l} x+2, & -3\leq x \leq-2,\\ x-2, & 2\leq x\leq3, \end{array} \right . $$
then T is a p-proximal contraction of the first and second kind.
It is easy to see that
$$d \bigl(-2,T(-3) \bigr)=d \bigl(2,T(3) \bigr)=d(A,B)=1. $$
If \(r\in[\frac{2}{3},1) \), then we have
$$\begin{aligned}& p(-2,2)\leq rp(-3,3), \\& p(2,-2)\leq rp(3,-3). \end{aligned}$$
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
$$T(x) = \left \{ \begin{array}{@{}l@{\quad}l} -1, & x =-2,\\ x-2, & 2\leq x\leq3, \end{array} \right . $$
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
$$d \bigl(-2,T(-2) \bigr)=d \bigl(2,T(3) \bigr)=d(A,B)=1 \quad\mbox{and}\quad {-}2 \preceq3 . $$
If \(r\in[\frac{2}{3},1) \), then we have
$$p(-2,2)\leq rp(-2,3) . $$
\(p(2,-2)\nleq rp(3,-2)\), 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

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:
  1. (a)

    T is an α-proximal admissible and continuous α-p-proximal contraction of the first kind.

     
  2. (b)

    g is a continuous τ-distance preserving with respect to p.

     
  3. (c)

    \(\alpha(gu,gv)\geq1\) implies that \(\alpha(u,v)\geq1\) for all \(u,v\in A \).

     
  4. (d)

    \(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).

     
  5. (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
$$d(gx,Tx)=d(A,B). $$

Proof

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
$$p(x_{n},x_{n+1})\leq rp(x_{n},x_{n-1})\leq \cdots\leq r^{n}p(x_{0},x_{1}) $$
for every \(n\in\mathbf{N}\). Hence, if \(m>n\),
$$\begin{aligned} p(x_{n},x_{m}) \leq& p(x_{n},x_{n+1})+ \cdots+p(x_{m-1},x_{m}) \\ \leq& r^{n}p(x_{0},x_{1})+\cdots+r^{m-1}p(x_{0},x_{1}) \\ \leq& \frac{r^{n}}{1-r}p(x_{0},x_{1}). \end{aligned}$$
This implies that
$$\begin{aligned} \lim_{n}\sup\bigl\{ p(x_{n},x_{m}):m \geq n\bigr\} =0. \end{aligned}$$
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

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:
  1. (a)

    T is a proximally increasing and continuous ordered p-proximal contraction of the first kind.

     
  2. (b)

    g is a continuous τ-distance preserving with respect to p.

     
  3. (c)

    \(gu\preceq gv\) implies that \(u\preceq v\) for all \(u,v\in A \).

     
  4. (d)

    \(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).

     
  5. (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
$$d(gx,Tx)=d(A,B). $$

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:
  1. (a)

    T is a continuous p-proximal contraction of the first kind.

     
  2. (b)

    g is a continuous τ-distance preserving with respect to p.

     
  3. (c)

    \(T(A_{0})\subseteq B_{0}\) and \(A_{0}\subseteq g(A_{0})\).

     
Then there exists an element \(x\in A\) such that
$$d(gx,Tx)=d(A,B). $$
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
$$d(gx,Tx)=d(A,B). $$
Now let \(x^{*}\) be in A such that
$$d\bigl(gx^{*},Tx^{*}\bigr)=d(A,B). $$
T is a p-proximal contraction of the first kind and g is a τ-distance preserving with respect to p, therefore
$$p\bigl(x,x^{*}\bigr)\leq rp\bigl(x,x^{*}\bigr). $$
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:
  1. (a)

    T is a continuous proximal contraction of the first kind.

     
  2. (b)

    g is an isometry.

     
  3. (c)

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

     
  4. (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

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:
  1. (a)

    T is an α-proximal admissible and continuous α-p-proximal contraction of the second kind.

     
  2. (b)

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

     
  3. (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
$$d(x,Tx)=d(A,B). $$

Proof

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
$$p(Tx_{n},Tx_{n+1})\leq rp(Tx_{n-1},Tx_{n}) \leq\cdots\leq r^{n}p(Tx_{0},Tx_{1}) $$
for every \(n\in\mathbf{N}\). Hence, if \(m>n\),
$$\begin{aligned} p(Tx_{n},Tx_{m}) \leq& p(Tx_{n},Tx_{n+1})+ \cdots+p(Tx_{m-1},Tx_{m}) \\ \leq& r^{n}p(Tx_{0},Tx_{1})+ \cdots+r^{m-1}p(Tx_{0},Tx_{1}) \\ \leq& \frac{r^{n}}{1-r}p(Tx_{0},Tx_{1}). \end{aligned}$$
This implies that
$$\begin{aligned} \lim_{n}\sup\bigl\{ p(Tx_{n},Tx_{m}):m \geq n\bigr\} =0. \end{aligned}$$
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
$$\begin{aligned} d(y,A) \leq& d(y,x_{n}) \\ \leq& d(y,Tx_{n-1})+d(Tx_{n-1},x_{n}) \\ =&d(y,Tx_{n-1})+d(A,B) \\ \leq&(y,Tx_{n-1})+d(y,A). \end{aligned}$$
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
$$d(x,y)=\lim_{k\rightarrow\infty}d(x_{n_{k}},Tx_{n_{k}-1})=d(A,B). $$
This implies that \(x\in A_{0}\). T is continuous and \(\{Tx_{n}\}\) is convergent to y, therefore
$$\lim_{n_{k}\rightarrow\infty}Tx_{n_{k}}=Tx=y. $$
Thus, it follows that
$$d(x,Tx)=\lim_{n_{k}\rightarrow\infty}d(x_{n_{k}},Tx_{n_{k}-1})=d(A,B). $$
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:
  1. (a)

    T is a proximally increasing and continuous ordered p-proximal contraction of the second kind.

     
  2. (b)

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

     
  3. (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
$$d(x,Tx)=d(A,B). $$

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:
  1. (a)

    T is a continuous p-proximal contraction of the second kind.

     
  2. (b)

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

     
Then there exists an element \(x\in A\) such that
$$d(x,Tx)=d(A,B). $$
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
$$d(x,Tx)=d(A,B). $$
Now let \(x^{*}\) be an element in A such that
$$d\bigl(x^{*},Tx^{*}\bigr)=d(A,B). $$
We will show that \(p(Tx,Tx^{*})=0\). T is a p-proximal contraction of the second kind, therefore
$$p\bigl(Tx,Tx^{*}\bigr)\leq rp\bigl(Tx,Tx^{*}\bigr). $$
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:
  1. (a)

    T is a continuous proximal contraction of the second kind.

     
  2. (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 of T, then Tx and \(Tx^{*}\) are identical.

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

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University, Ashrafi Esfahani Ave., Tehran, 14778, Iran
(2)
Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
(3)
Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
(4)
Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

References

  1. Jleli, M, Samet, B: Best proximity points for α-ψ-proximal contractive type mappings and applications. Bull. Sci. Math. 137(8), 977-995 (2013) View ArticleMATHMathSciNetGoogle Scholar
  2. Prolla, JB: Fixed-point theorems for set-valued mappings and existence of best approximants. Numer. Funct. Anal. Optim. 5(4), 449-455 (1982/1983) Google Scholar
  3. Reich, S: Approximate selections, best approximations, fixed points, and invariant sets. J. Math. Anal. Appl. 62(1), 104-113 (1978) View ArticleMATHMathSciNetGoogle Scholar
  4. Sadiq Basha, S: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 74(17), 5844-5850 (2011) View ArticleMATHMathSciNetGoogle Scholar
  5. Sadiq Basha, S: Best proximity points: global optimal approximate solutions. J. Glob. Optim. 49(1), 15-21 (2011) View ArticleMATHMathSciNetGoogle Scholar
  6. Sehgal, VM, Singh, SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 102(3), 534-537 (1988) MATHMathSciNetGoogle Scholar
  7. Sehgal, VM, Singh, SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 10(1-2), 181-184 (1989) View ArticleMATHMathSciNetGoogle Scholar
  8. Vetrivel, V, Veeramani, P, Bhattacharyya, P: Some extensions of Fan’s best approximation theorem. Numer. Funct. Anal. Optim. 13(3-4), 397-402 (1992) View ArticleMATHMathSciNetGoogle Scholar
  9. Abkar, A, Gabeleh, M: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 150(1), 188-193 (2011) View ArticleMATHMathSciNetGoogle Scholar
  10. Abkar, A, Gabeleh, M: Generalized cyclic contractions in partially ordered metric spaces. Optim. Lett. 6(8), 1819-1830 (2012) View ArticleMATHMathSciNetGoogle Scholar
  11. Sadiq Basha, S: Global optimal approximate solutions. Optim. Lett. 5(4), 639-645 (2011) View ArticleMATHMathSciNetGoogle Scholar
  12. Pragadeeswarar, V, Marudai, M: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. 7(8), 1883-1892 (2013) View ArticleMATHMathSciNetGoogle Scholar
  13. Haddadi, MZ: Best proximity point iteration for nonexpansive mapping in Banach spaces. J. Nonlinear Sci. Appl. 7(2), 126-130 (2014) MATHMathSciNetGoogle Scholar
  14. Mongkolkeha, C, Cho, YJ, Kumam, P: Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013, 94 (2013) View ArticleMathSciNetGoogle Scholar
  15. Mongkolkeha, C, Cho, YJ, Kumam, P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl. 2013, 180 (2013) View ArticleMathSciNetGoogle Scholar
  16. Mongkolkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 155(1), 215-226 (2012) View ArticleMATHMathSciNetGoogle Scholar
  17. Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, 93 (2012) View ArticleGoogle Scholar
  18. Mongkolkeha, C, Kumam, P: Some common best proximity points for proximity commuting mappings. Optim. Lett. 7(8), 1825-1836 (2013) View ArticleMATHMathSciNetGoogle Scholar
  19. Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253(2), 440-458 (2001) View ArticleMATHMathSciNetGoogle Scholar
  20. Raj, VS: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74(14), 4804-4808 (2011) View ArticleMATHMathSciNetGoogle Scholar
  21. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 75(4), 2154-2165 (2012) View ArticleMATHMathSciNetGoogle Scholar
  22. Zhang, J, Su, Y, Cheng, Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013, 99 (2013) View ArticleGoogle Scholar

Copyright

© Omidvari et al.; licensee Springer 2015

Advertisement