# Best proximity point theorems with Suzuki distances

## 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.

## 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.

## 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.

## 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)

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)

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)

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)

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.

## 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.

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## Acknowledgements

The authors are grateful to reviewers for their valuable comments and suggestions.

## Author information

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### Corresponding author

Correspondence to Sung Jin Lee.

### 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.

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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

• 90C26
• 90C30
• 47H09
• 47H10

### Keywords

• weak P-property
• best proximity point
• τ-distance
• α-ψ-proximal contraction
• ordered p-proximal contraction