Coupled best proximity point theorems for α-ψ-proximal contractive multimaps

Abstract

In this paper, we establish coupled best proximity point theorems for multivalued mappings. Our results extend some recent results by Ali et al. (Abstr. Appl. Anal. 2014:181598, 2014) as well as other results in the literature. We also give examples to support our main results.

1 Introduction and preliminaries

The Banach contraction principle is one of the most well-known and useful tools in analysis. This principle has been generalized by many authors in many different ways (see [1–6]). Recently, Samet et al. [7] introduced the notion of α-ψ-contractive type mappings and proved some fixed point theorems for such mappings within the framework of complete metric spaces. Karapınar and Samet [8] generalized α-ψ-contractive type mappings and obtained some fixed point theorems for generalized α-ψ-contractive type mappings. Some interesting multivalued generalizations of α-ψ-contractive type mappings are available in [9–18]. More recently, Jleli and Samet [19] introduced the notion of α-ψ-proximal contractive type mappings and proved certain best proximity point theorems. Many authors have obtained best proximity point theorems and have done so in a variety of settings; see, for example, [19–41]. Abkar and Gbeleh [22] and Al-Thagafi and Shahzad [24, 26] investigated best proximity points for multivalued mappings. Recently Ali et al. extended the results of Jleli and Samet [19] for nonself multivalued mappings. The concept of coupled best proximity point theorem was introduced by Sintunavarat and Kumam [36], and they proved the coupled best proximity theorem for cyclic contractions.

Inspired and motivated by the recent results of Ali et al. in [42] and by those of Sintunavarat and Kumam in [36], we establish the coupled best proximity points for α-ψ-proximal contractive multimaps. We also give examples to support our main results.

Let $(X,d)$ be a metric space. For $A, B\subset X$, we use the following notations subsequently: $\operatorname{dist}(A,B) = \inf \{ d(a,b) : a\in A, b\in B \}$, $D(x,B) = \inf \{d(x,b) : b\in B \}$, $A_{0} = \{a\in A : d(a,b)=\operatorname{dist}(A,B) \mbox{ for some } b\in B \}$, $B_{0} = \{ b\in B : d(a,b)=\operatorname{dist}(A,B) \mbox{ for some } a\in A\}$, $2^{X}\backslash\emptyset$ is the set of all nonempty subsets of X, $\operatorname{CL}(X)$ is the set of all nonempty closed subsets of X, and $\mathrm{K}(X)$ is the set of all nonempty compact subsets of X. For every $A, B \in\operatorname{CL}(X)$, let

$$ H(A,B) = \begin{cases} \max \{\sup_{x\in A}d(x,B),\sup_{y\in B}d(y,A) \}&\mbox{if the maximum exists};\\ \infty&\mbox{otherwise.} \end{cases} $$

(1)

Such a map H is called the generalized Hausdorff metric induced by d. A point $x^{*}\in X$ is said to be the best proximity point of a mapping $T : A\to B$ if $d(x^{*},Tx^{*})=\operatorname{dist}(A,B)$. When $A=B$, the best proximity point is essentially the fixed point of the mapping T.

Definition 1.1

(see [34])

Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with $A_{0} \ne\emptyset$. Then the pair $(A,B)$ is said to have the weak P-property if and only if, for any $x_{1},x_{2} \in A$ and $y_{1},y_{2} \in B$,

$$ \left . \begin{array}{r@{}} d(x_{1},y_{1})=\operatorname{dist}(A,B),\\ d(x_{2},y_{2})=\operatorname{dist}(A,B) \end{array} \right \} \quad\Rightarrow\quad d(x_{1},x_{2}) \le d(y_{1},y_{2}). $$

(2)

Let Ψ denote the set of all functions $\psi: [0,\infty) \to [0,\infty)$ satisfying the following properties:

• (a) ψ is monotone nondecreasing;

• (b) $\sum_{n=1}^{\infty}\psi^{n}(t)<\infty$ for each $t > 0$.

Definition 1.2

(see [21])

An element $x^{*}\in A$ is said to be the best proximity point of a multivalued nonself mapping T if $D(x^{*},Tx^{*})= \operatorname{dist}(A,B)$.

Definition 1.3

(see [42])

Let A and B be two nonempty subsets of a metric space $(X,d)$. A mapping $T : A\to2^{B}\backslash\emptyset$ is called α-proximal admissible if there exists a mapping $\alpha: A\times A\to [0,\infty)$ such that

$$ \left . \begin{array}{r@{}} \alpha(x_{1},x_{2}) \ge1,\\ d(u_{1},y_{1})=\operatorname{dist}(A,B),\\ d(u_{2},y_{2})=\operatorname{dist}(A,B) \end{array} \right \} \quad\Rightarrow\quad \alpha(u_{1},u_{2}) \ge1, $$

(3)

where $x_{1}, x_{2}, u_{1}, u_{2}\in A$, $y_{1}\in Tx_{1}$ and $y_{2}\in Tx_{2}$.

Definition 1.4

(see [42])

Let A and B be two nonempty subsets of a metric space $(X,d)$. A mapping $T : A \to\operatorname{CL}(B)$ is said to be an α-ψ-proximal contraction if there exist two functions $\psi\in\Psi$ and $\alpha: A\times A\to[0,\infty)$ such that

$$\alpha(x,y)H(Tx,Ty) \le \psi \bigl(d(x,y) \bigr), \quad\forall x,y \in A. $$

(4)

Lemma 1.5

(see [11])

Let $(X,d)$ be a metric space and $B \in\operatorname{CL}(X)$. Then, for each $x\in X$ with $d(x,B)>0$ and $q>1$, there exists an element $b\in B$ such that

$$ d(x,b) < qd(x,B). $$

(5)

• (C) If $\{x_{n} \}$ is a sequence in A such that $\alpha(x_{n},x_{n+1}) \ge1$ for all n and $x_{n}\to x\in A$ as $n\to\infty$, then there exists a subsequence $\{ x_{n_{k}} \}$ of $\{x_{n} \}$ such that $\alpha(x_{n_{k}},x) \ge1$ for all k.

The main results of Ali et al. in [42] are the following.

Theorem 1.6

(see [42])

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and $\psi\in\Psi$ be a strictly increasing map. Suppose that $T : A \to\operatorname{CL}(B)$ is a mapping satisfying the following conditions:

• (i) $Tx \subseteq B_{0}$ for each $x\in A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $x_{0}$, $x_{1}$ in $A_{0}$ and $y_{1}\in Tx_{0}$ such that

  $$ d(x_{1},y_{1}) = d(A,B),\qquad \alpha(x_{0},x_{1})\ge1; $$

  (6)

• (iv) T is a continuous α-ψ-proximal contraction.

Then there exists an element $x^{*}\in A_{0}$ such that

$$ D \bigl(x^{*},Tx^{*} \bigr) = \operatorname{dist}(A,B). $$

Theorem 1.7

(see [42])

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and let $\psi\in\Psi$ be a strictly increasing map. Suppose that $T : A \to\operatorname{CL}(B)$ is a mapping satisfying the following conditions:

• (i) $Tx \subseteq B_{0}$ for each $x\in A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $x_{0}$, $x_{1}$ in $A_{0}$ and $y_{1}\in Tx_{0}$ such that

  $$ d(x_{1},y_{1}) = d(A,B),\qquad \alpha(x_{0},x_{1})\ge1; $$

  (7)

• (iv) property (C) holds and T is an α-ψ-proximal contraction.

Then there exists an element $x^{*}\in A_{0}$ such that

$$ D \bigl(x^{*},Tx^{*} \bigr) = \operatorname{dist}(A,B). $$

The purpose of this paper is to extend the recent results of Ali et al. [42] to a coupled best proximity point of nonself multivalued mappings.

2 Main results

We begin this section by introducing the following definitions.

Definition 2.1

Let A and B be two nonempty subsets of a metric space $(X,d)$. A mapping $T : A\times A \to2^{B}\backslash\emptyset$ is called α-proximal admissible if there exists a mapping $\alpha: A\times A\to[0,\infty)$ such that

$$ \left . \begin{array}{r@{}} \alpha(x_{1},x_{2}) \ge1,\\ d(w_{1},u_{1})=\operatorname{dist}(A,B),\\ d(w_{2},u_{2})=\operatorname{dist}(A,B) \end{array} \right \} \quad\Rightarrow\quad \alpha(w_{1},w_{2}) \ge1, $$

(8)

where $x_{1}, x_{2}, w_{1}, w_{2}, y_{1}, y_{2}\in A$, $u_{1}\in T(x_{1},y_{1})$ and $u_{2}\in T(x_{2},y_{2})$, and

$$ \left . \begin{array}{r@{}} \alpha(y_{1},y_{2}) \ge1,\\ d(w'_{1},v_{1})=\operatorname{dist}(A,B),\\ d(w'_{2},v_{2})=\operatorname{dist}(A,B) \end{array} \right \} \quad\Rightarrow\quad \alpha\bigl(w'_{1},w'_{2}\bigr) \ge1, $$

(9)

where $y_{1}, y_{2}, w'_{1}, w'_{2}, x_{1}, x_{2}\in A$, $v_{1}\in T(y_{1},x_{1})$ and $v_{2}\in T(y_{2},x_{2})$.

Definition 2.2

Let A and B be two nonempty subsets of a metric space $(X,d)$. A mapping $T : A\times A \to\operatorname{CL}(B)$ is said to be an α-ψ-proximal contraction if there exist two functions $\psi\in \Psi$ and $\alpha: A\times A\to[0,\infty)$ such that

$$ \alpha(x,y)H\bigl(T\bigl(x,x'\bigr),T \bigl(y,y'\bigr)\bigr) \le \psi \bigl(d(x,y) \bigr),\quad \forall x,x',y,y' \in A. $$

(10)

Definition 2.3

An element $(x^{*},y^{*})\in A \times A$ is said to be the coupled best proximity point of a multivalued nonself mapping T if $D(x^{*},T(x^{*},y^{*}))= \operatorname{dist}(A,B)$ and $D(y^{*},T(y^{*},x^{*}))= \operatorname{dist}(A,B)$.

The following are our main results.

Theorem 2.4

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and let $\psi\in\Psi$ be a strictly increasing map. Suppose that $T : A\times A \to\operatorname{CL}(B)$ is a mapping satisfying the following conditions:

• (i) $T(x,y) \subseteq B_{0}$ for each $x,y\in A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $(x_{0},y_{0})$, $(x_{1},y_{1})$ in $A_{0}\times A_{0}$ and $u_{1}\in T(x_{0},y_{0})$, $v_{1}\in T(y_{0},x_{0})$ such that

  $$ \begin{aligned} &d(x_{1},u_{1}) = d(A,B), \qquad\alpha(x_{0},x_{1})\ge1 \quad\textit{and}\\ &d(y_{1},v_{1}) = d(A,B),\qquad \alpha(y_{0},y_{1}) \ge1; \end{aligned} $$

  (11)

• (iv) T is a continuous α-ψ-proximal contraction.

Then there exists an element $(x^{*},y^{*})\in A_{0}\times A_{0}$ such that

$$ \begin{aligned} &D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) = \operatorname{dist}(A,B) \quad\textit{and}\\ &D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) = \operatorname{dist}(A,B). \end{aligned} $$

Proof

From condition (iii), there exist elements $(x_{0},y_{0})$, $(x_{1},y_{1})$ in $A_{0}\times A_{0}$ and $u_{1}\in T(x_{0},y_{0})$, $v_{1}\in T(y_{0},x_{0})$ such that

$$ \begin{aligned} &d(x_{1},u_{1}) = \operatorname{dist}(A,B),\qquad \alpha(x_{0},x_{1})\ge1 \quad\mbox{and}\\ &d(y_{1},v_{1}) = \operatorname{dist}(A,B),\qquad \alpha(y_{0},y_{1})\ge1. \end{aligned} $$

(12)

Assume that $u_{1}\notin T(x_{1},y_{1})$, $v_{1}\notin T(y_{1},x_{1})$; for otherwise $(x_{1},y_{1})$ is the coupled best proximity point. From condition (iv), we have

$$ \begin{aligned}[b] 0 &< d \bigl(u_{1},T(x_{1},y_{1}) \bigr) \le H \bigl(T(x_{0},y_{0}),T(x_{1},y_{1}) \bigr) \\ & \le\alpha(x_{0},x_{1})H \bigl(T(x_{0},y_{0}),T(x_{1},y_{1}) \bigr) \\ & \le\psi \bigl(d(x_{0},x_{1}) \bigr) \end{aligned} $$

(13)

and

$$ \begin{aligned}[b] 0 &< d \bigl(v_{1},T(y_{1},x_{1}) \bigr) \le H \bigl(T(y_{0},x_{0}),T(y_{1},x_{1}) \bigr) \\ & \le\alpha(y_{0},y_{1})H \bigl(T(y_{0},x_{0}),T(y_{1},x_{1}) \bigr) \\ & \le\psi \bigl(d(y_{0},y_{1}) \bigr). \end{aligned} $$

(14)

For $q,q' > 1$, it follows from Lemma 1.5 that there exist $u_{2}\in T(x_{1},y_{1})$ and $v_{2}\in T(y_{1},x_{1})$ such that

$$ \begin{aligned} &0 < d(u_{1},u_{2}) < qd \bigl(u_{1},T(x_{1},y_{1}) \bigr)\quad\mbox{and}\\ &0 < d(v_{1},v_{2}) < q'd \bigl(v_{1},T(y_{1},x_{1}) \bigr) . \end{aligned} $$

(15)

From (13), (14) and (15), we have

$$ 0 < d(u_{1},u_{2}) < qd \bigl(u_{1},T(x_{1},y_{1}) \bigr) \le q\psi \bigl(d(x_{0},x_{1}) \bigr) $$

(16)

and

$$ 0 < d(v_{1},v_{2}) < q'd \bigl(v_{1},T(y_{1},x_{1}) \bigr) \le q'\psi \bigl(d(y_{0},y_{1}) \bigr). $$

(17)

As $u_{2}\in T(x_{1},y_{1})\subseteq B_{0}$, there exists $x_{2}\ne x_{1}\in A_{0}$ such that

$$ d(x_{2},u_{2}) = \operatorname{dist}(A,B), $$

(18)

and as $v_{2}\in T(y_{1},x_{1})\subseteq B_{0}$, there exists $y_{2}\ne y_{1}\in A_{0}$ such that

$$ d(y_{2},v_{2}) = \operatorname{dist}(A,B); $$

(19)

for otherwise $(x_{1},y_{1})$ is the coupled best proximity point. As $(A,B)$ satisfies the weak P-property, from (12), (18) and (19) we have

$$ \begin{aligned} &0 < d(x_{1},x_{2}) \le d(u_{1},u_{2}) \quad\mbox{and}\\ &0 < d(y_{1},y_{2}) \le d(v_{1},v_{2}). \end{aligned} $$

(20)

From (16), (17) and (20) we have

$$ \begin{aligned} &0 < d(x_{1},x_{2}) \le d(u_{1},u_{2}) < qd \bigl(u_{1},T(x_{1},y_{1}) \bigr)\le q\psi \bigl(d(x_{0},x_{1}) \bigr)\quad \mbox{and}\\ &0 < d(y_{1},y_{2}) \le d(v_{1},v_{2}) < q'd \bigl(v_{1},T(y_{1},x_{1}) \bigr)\le q'\psi \bigl(d(y_{0},y_{1}) \bigr). \end{aligned} $$

(21)

Since ψ is strictly increasing, we have

$$ \begin{aligned} &\psi \bigl(d(x_{1},x_{2}) \bigr) < \psi \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr)\quad\mbox{and}\\ &\psi \bigl(d(y_{1},y_{2}) \bigr) < \psi \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). \end{aligned} $$

Put

$$ \begin{aligned} &q_{1} = \psi \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) /\psi \bigl(d(x_{1},x_{2})\bigr),\\ &q'_{1} = \psi \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr) /\psi \bigl(d(y_{1},y_{2})\bigr). \end{aligned} $$

We also have

$$\alpha(x_{0},x_{1})\ge1,\qquad d(x_{1},u_{1}) = \operatorname{dist}(A,B) \quad\mbox{and}\quad d(x_{2},u_{2}) = \operatorname{dist}(A,B) $$

and

$$\alpha(y_{0},y_{1})\ge1,\qquad d(y_{1},v_{1}) = \operatorname{dist}(A,B) \quad\mbox{and}\quad d(y_{2},v_{2}) = \operatorname{dist}(A,B). $$

Since T is an α-proximal admissible, then $\alpha (x_{1},x_{2}) \ge1$ and $\alpha(y_{1},y_{2}) \ge1$. Thus we have

$$ \begin{aligned} &d(x_{2},u_{2}) = \operatorname{dist}(A,B), \qquad\alpha(x_{1},x_{2})\ge1 \quad\mbox{and}\\ &d(y_{2},v_{2}) = \operatorname{dist}(A,B),\qquad \alpha(y_{1},y_{2})\ge1. \end{aligned} $$

(22)

Assume that $u_{2}\notin T(x_{2},y_{2})$ and $v_{2}\notin T(y_{2},x_{2})$; for otherwise $(x_{2},y_{2})$ is the coupled best proximity point. From condition (iv) we have

$$ \begin{aligned}[b] 0 &< d \bigl(u_{2},T(x_{2},y_{2}) \bigr) \le H \bigl(T(x_{1},y_{1}),T(x_{2},y_{2}) \bigr) \\ & \le\alpha(x_{1},x_{2})H \bigl(T(x_{1},y_{1}),T(x_{2},y_{2}) \bigr) \\ & \le\psi \bigl(d(x_{1},x_{2}) \bigr) \end{aligned} $$

(23)

and

$$ \begin{aligned}[b] 0 &< d \bigl(v_{2},T(y_{2},x_{2}) \bigr) \le H \bigl(T(y_{1},x_{1}),T(y_{2},x_{2}) \bigr) \\ & \le\alpha(y_{1},y_{2})H \bigl(T(y_{1},x_{1}),T(y_{2},x_{2}) \bigr) \\ & \le\psi \bigl(d(y_{1},y_{2}) \bigr). \end{aligned} $$

(24)

For $q_{1},q'_{1}>1$, it follows from Lemma 1.5 that there exist $u_{3}\in T(x_{2},y_{2})$ and $v_{3}\in T(y_{2},x_{2})$ such that

$$ \begin{aligned} &0 < d(u_{2},u_{3}) < q_{1}d \bigl(u_{2},T(x_{2},y_{2}) \bigr),\\ &0 < d(v_{2},v_{3}) < q'_{1}d \bigl(v_{2},T(y_{2},x_{2}) \bigr). \end{aligned} $$

(25)

From (23), (24) and (25) we have

$$ \begin{aligned}[b] 0 &< d(u_{2},u_{3}) < q_{1}d \bigl(u_{2},T(x_{2},y_{2}) \bigr) \\ & \le q_{1}\psi \bigl(d(x_{1},x_{2}) \bigr) \\ & = \psi \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \end{aligned} $$

(26)

and

$$ \begin{aligned} 0 &< d(v_{2},v_{3}) < q'_{1}d \bigl(v_{2},T(y_{2},x_{2}) \bigr) \\ & \le q'_{1}\psi \bigl(d(y_{1},y_{2}) \bigr) \\ & = \psi \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). \end{aligned} $$

(27)

As $u_{3}\in T(x_{2},y_{2})\in B_{0}$, there exists $x_{3}\ne x_{2}\in A_{0}$ such that

$$ d(x_{3},u_{3}) = \operatorname{dist}(A,B); $$

(28)

and as $v_{3}\in T(y_{2},x_{2})\in B_{0}$, there exists $y_{3}\ne y_{2}\in A_{0}$ such that

$$ d(y_{3},v_{3}) = \operatorname{dist}(A,B); $$

(29)

for otherwise $(x_{2},y_{2})$ is the coupled best proximity point. As $(A,B)$ satisfies the weak P-property, from (22), (28) and (29) we have

$$ \begin{aligned} &0 < d(x_{2},x_{3}) \le d(u_{2},u_{3}),\\ &0 < d(y_{2},y_{3}) \le d(v_{2},v_{3}). \end{aligned} $$

(30)

From (26), (27) and (30) we have

$$ \begin{aligned}[b] 0 &< d(x_{2},x_{3}) < q_{1}d \bigl(u_{2},T(x_{2},y_{2}) \bigr) \\ & \le q_{1}\psi \bigl(d(x_{1},x_{2}) \bigr) \\ & = \psi \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \end{aligned} $$

(31)

and

$$ \begin{aligned} 0 &< d(y_{2},y_{3}) < q'_{1}d \bigl(v_{2},T(y_{2},x_{2}) \bigr) \\ & \le q'_{1}\psi \bigl(d(y_{1},y_{2}) \bigr) \\ & = \psi \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). \end{aligned} $$

(32)

Since ψ is strictly increasing, we have

$$ \psi \bigl(d(x_{2},x_{3}) \bigr) < \psi^{2} \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \quad \mbox{and}\quad \psi \bigl(d(y_{2},y_{3}) \bigr) < \psi^{2} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). $$

(33)

Put

$$ \begin{aligned} &q_{2} = \psi^{2} \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) /\psi \bigl(d(x_{2},x_{3})\bigr),\\ &q'_{2} = \psi^{2} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr) /\psi \bigl(d(y_{2},y_{3})\bigr). \end{aligned} $$

We also have

$$\alpha(x_{1},x_{2})\ge1,\qquad d(x_{2},u_{2}) = \operatorname{dist}(A,B)\quad \mbox{and}\quad d(x_{3},u_{3}) = \operatorname{dist}(A,B) $$

and

$$\alpha(y_{1},y_{2})\ge1,\qquad d(y_{2},v_{2}) = \operatorname{dist}(A,B) \quad\mbox{and}\quad d(y_{3},v_{3}) = \operatorname{dist}(A,B). $$

Since T is an α-proximal admissible, then $\alpha (x_{2},x_{3}) \ge1$ and $\alpha(y_{2},y_{3}) \ge1$, respectively. Thus we have

$$ \begin{aligned} &d(x_{3},u_{3}) = \operatorname{dist}(A,B),\qquad \alpha(x_{2},x_{3})\ge1 \quad\mbox{and}\\ &d(y_{3},v_{3}) = \operatorname{dist}(A,B),\qquad \alpha(y_{2},y_{3})\ge1. \end{aligned} $$

(34)

Continuing in the same process, we get sequences $\{x_{n} \}$, $\{y_{n} \}$ in $A_{0}$ and $\{u_{n} \}$, $\{v_{n} \}$ in $B_{0}$, where $u_{n}\in T(x_{n-1},y_{n-1})$ and $v_{n}\in T(y_{n-1},x_{n-1})$ for each $n\in \mathbb{N}$, such that

$$ \begin{aligned} &d(x_{n+1},u_{n+1}) = \operatorname{dist}(A,B),\qquad \alpha (x_{n},x_{n+1})\ge 1\quad\mbox{and}\\ &d(y_{n+1},v_{n+1}) = \operatorname{dist}(A,B), \qquad\alpha (y_{n},y_{n+1})\ge1, \end{aligned} $$

(35)

and

$$ \begin{aligned} &d(u_{n+1},u_{n+2}) < \psi^{n} \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \quad\mbox{and}\\ &d(v_{n+1},v_{n+2}) < \psi^{n} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). \end{aligned} $$

(36)

As $u_{n+2}\in T(x_{n+1},y_{n+1}) \in B_{0}$, there exists $x_{n+2}\ne x_{n+1}\in A_{0}$ such that

$$ d(x_{n+2},u_{n+2}) = \operatorname{dist}(A,B) $$

(37)

and as $v_{n+2}\in T(y_{n+1},x_{n+1}) \in B_{0}$, there exists $y_{n+2}\ne y_{n+1}\in A_{0}$ such that

$$ d(y_{n+2},v_{n+2}) = \operatorname{dist}(A,B). $$

(38)

Since $(A,B)$ satisfies the weak P-property, from (35), (37) and (38) we have

$$d(x_{n+1},x_{n+2}) \le d(u_{n+1},u_{n+2}) \quad\mbox{and}\quad d(y_{n+1},y_{n+2}) \le d(v_{n+1},v_{n+2}). $$

Thus, from (36) we have

$$ \begin{aligned} &d(x_{n+1},x_{n+2}) < \psi^{n} \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \quad\mbox{and}\\ &d(y_{n+1},y_{n+2}) < \psi^{n} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr). \end{aligned} $$

(39)

Now, we shall prove that $\{x_{n} \}$ and $\{ y_{n} \}$ are Cauchy sequences in A. Let $\epsilon> 0$ be fixed. Since $\sum_{n=1}^{\infty}\psi^{n} (q\psi (d(x_{0},x_{1}) ) ) < \infty$ and $\sum_{n=1}^{\infty}\psi^{n} (q'\psi (d(y_{0},y_{1}) ) ) <\infty$, there exist some positive integers $h=h(\epsilon)$ and $h'=h'(\epsilon)$ such that

$$\sum_{k \ge h}^{\infty}\psi^{k} \bigl(q \psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) < \epsilon $$

and

$$\sum_{k \ge h'}^{\infty}\psi^{k} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr) < \epsilon, $$

respectively. For $m>n>h$, using the triangular inequality, we obtain

$$ \begin{aligned}[b] d(x_{n},x_{m}) &\le \sum_{k=n}^{m-1}d(x_{k},x_{k+1}) \le\sum_{k=n}^{m-1}\psi^{k} \bigl(q \psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) \\ &\le\sum_{k \ge h}^{\infty}\psi^{k} \bigl(q\psi \bigl(d(x_{0},x_{1}) \bigr) \bigr) < \epsilon \end{aligned} $$

(40)

and

$$ \begin{aligned}[b] d(y_{n},y_{m}) &\le \sum_{k=n}^{m-1}d(y_{k},y_{k+1}) \le\sum_{k=n}^{m-1}\psi^{k} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr) \\ &\le\sum_{k \ge h'}^{\infty}\psi^{k} \bigl(q'\psi \bigl(d(y_{0},y_{1}) \bigr) \bigr) < \epsilon, \end{aligned} $$

(41)

respectively. Hence $\{x_{n} \}$ and $\{y_{n} \}$ are Cauchy sequences in A. Similarly, one can show that $\{u_{n} \}$ and $\{v_{n} \}$ are Cauchy sequences in B. Since A and B are closed subsets of a complete metric space, there exists $(x^{*},y^{*})$ in $A\times A$ such that $x_{n}\to x^{*}$, $y_{n}\to y^{*}$ as $n\to \infty$ and there exist $u^{*}$, $v^{*}$ in B such that $u_{n}\to u^{*}$, $v_{n}\to v^{*}$ as $n\to\infty$. By (37) and (38) we conclude that

$$ \begin{aligned} &d\bigl(x^{*},u^{*}\bigr) = \operatorname{dist}(A,B) \quad\mbox{as } n\to\infty \quad\mbox{and}\\ &d\bigl(y^{*},v^{*}\bigr) = \operatorname{dist}(A,B) \quad \mbox{as } n\to\infty. \end{aligned} $$

Since T is continuous and $u_{n}\in T(x_{n-1},y_{n-1})$, we have $u^{*}\in T(x^{*},y^{*})$ and $v_{n}\in T(y_{n-1},x_{n-1})$, we have $v^{*}\in T(y^{*},x^{*})$. Hence,

$$\operatorname{dist}(A,B) \le D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) \le d \bigl(x^{*},u^{*}\bigr) = \operatorname{dist}(A,B) $$

and

$$\operatorname{dist}(A,B) \le D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) \le d \bigl(y^{*},v^{*}\bigr) = \operatorname{dist}(A,B). $$

Therefore, $(x^{*},y^{*})$ is the coupled best proximity point of the mapping T. □

Theorem 2.5

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and let $T : A\times A \to\mathrm{K}(B)$ be a mapping satisfying the following conditions:

• (i) $T(x,y) \subseteq B_{0}$ for each $(x,y)\in A_{0}\times A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $(x_{0},y_{0})$, $(x_{1},y_{1})$ in $A_{0}\times A_{0}$ and $u_{1}\in T(x_{0},y_{0})$, $v_{1}\in T(y_{0},x_{0})$ such that

  $$ \begin{aligned} &d(x_{1},u_{1}) = \operatorname{dist}(A,B), \qquad\alpha(x_{0},x_{1})\ge1 \quad\textit{and}\\ &d(y_{1},v_{1}) = \operatorname{dist}(A,B),\qquad \alpha(y_{0},y_{1})\ge1; \end{aligned} $$

  (42)

• (iv) T is a continuous α-ψ-proximal contraction.

Then there exists an element $(x^{*},y^{*})\in A_{0}\times A_{0}$ such that

$$ \begin{aligned} &D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) = \operatorname{dist}(A,B) \quad\textit{and}\\ &D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) = \operatorname{dist}(A,B). \end{aligned} $$

Theorem 2.6

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and let $\psi\in\Psi$ be a strictly increasing map. Suppose that $T : A\times A \to\operatorname{CL}(B)$ is a mapping satisfying the following conditions:

• (i) $T(x,y) \subseteq B_{0}$ for each $(x,y)\in A_{0}\times A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $(x_{0},y_{0})$, $(x_{1},y_{1})$ in $A_{0}\times A_{0}$ and $u_{1}\in T(x_{0},y_{0})$, $v_{1}\in T(y_{0},x_{0})$ such that

  $$ \begin{aligned}& d(x_{1},u_{1}) = d(A,B),\qquad \alpha(x_{0},x_{1})\ge1 \quad\textit{and}\\ &d(y_{1},v_{1}) = d(A,B),\qquad \alpha(y_{0},y_{1}) \ge1; \end{aligned} $$

  (43)

• (iv) property (C) holds and T is an α-ψ-proximal contraction.

Then there exists an element $(x^{*},y^{*})\in A_{0}\times A_{0}$ such that

$$ \begin{aligned} &D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) = \operatorname{dist}(A,B) \quad\textit{and}\\ &D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) = \operatorname{dist}(A,B). \end{aligned} $$

Proof

Similar to the proof of Theorem 2.4, there exist Cauchy sequences $\{x_{n} \}$ and $\{y_{n} \}$ in A and Cauchy sequences $\{u_{n} \}$ and $\{v_{n} \}$ in B such that

$$ \begin{aligned} &d(x_{n+1},u_{n+1}) = \operatorname{dist}(A,B), \qquad\alpha (x_{n},x_{n+1})\ge1 \quad\mbox{and}\\ &d(y_{n+1},v_{n+1}) = \operatorname{dist}(A,B), \qquad\alpha (y_{n},y_{n+1})\ge1; \end{aligned} $$

(44)

and $x_{n}\to x^{*}\in A$, $y_{n}\to y^{*} \in A$ as $n\to\infty$ and $u_{n}\to u^{*} \in B$, $v_{n}\to v^{*} \in B$ as $n\to\infty$.

From condition (C), there exist subsequences $\{ x_{n_{k}} \}$ of $\{x_{n} \}$, $\{y_{n_{k}} \}$ of $\{ y_{n} \}$ such that $\alpha (x_{n_{k}},x^{*})\ge1$, $\alpha(y_{n_{k}},y^{*})\ge1$ for all k. Since T is an α-ψ-proximal contraction, we have

$$ \begin{aligned}[b] H \bigl(T(x_{n_{k}},y_{n_{k}}),T \bigl(x^{*},y^{*}\bigr)\bigr) & \le\alpha \bigl(x_{n_{k}},x^{*}\bigr)H \bigl(T(x_{n_{k}},y_{n_{k}}),T\bigl(x^{*},y^{*}\bigr)\bigr) \\ & \le\psi \bigl(d\bigl(x_{n_{k}},x^{*}\bigr) \bigr),\quad \forall k, \end{aligned} $$

and

$$ \begin{aligned}[b] H \bigl(T(y_{n_{k}},x_{n_{k}}),T \bigl(y^{*},x^{*}\bigr)\bigr) & \le\alpha \bigl(y_{n_{k}},y^{*}\bigr)H \bigl(T(y_{n_{k}},x_{n_{k}}),T\bigl(y^{*},x^{*}\bigr)\bigr) \\ & \le\psi \bigl(d\bigl(y_{n_{k}},y^{*}\bigr) \bigr),\quad \forall k. \end{aligned} $$

Letting $k\to\infty$ in the above inequality, we get $T(x_{n_{k}},y_{n_{k}})\to T(x^{*},y^{*})$ and $T(y_{n_{k}},x_{n_{k}})\to T(y^{*},x^{*})$, respectively. By the continuity of the metric d, we have

$$ \begin{aligned} &d\bigl(x^{*},u^{*}\bigr) = \lim _{k\to\infty}d(x_{n_{k}+1},u_{n_{k}+1}) = \operatorname{dist}(A,B),\\ &d\bigl(y^{*},v^{*}\bigr) = \lim_{k\to\infty}d(y_{n_{k}+1},v_{n_{k}+1}) = \operatorname{dist}(A,B). \end{aligned} $$

(45)

Since $u_{n_{k}+1} \in T(x_{n_{k}},y_{n_{k}})$, $u_{n_{k}}\to u^{*}$ and $T(x_{n_{k}},y_{n_{k}})\to T(x^{*},y^{*})$, then $u^{*}\in T(x^{*},y^{*})$ and since $v_{n_{k}+1} \in T(y_{n_{k}},x_{n_{k}})$, $v_{n_{k}}\to v^{*}$ and $T(y_{n_{k}},x_{n_{k}})\to T(y^{*},x^{*})$, then $v^{*}\in T(y^{*},x^{*})$. Hence,

$$\operatorname{dist}(A,B) \le D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) \le d \bigl(x^{*},u^{*}\bigr) = \operatorname{dist}(A,B) $$

and

$$\operatorname{dist}(A,B) \le D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) \le d \bigl(y^{*},v^{*}\bigr) = \operatorname{dist}(A,B). $$

Therefore, $(x^{*},y^{*})$ is the coupled best proximity point of the mapping T. □

Theorem 2.7

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $\alpha: A \times A \to[0,\infty)$ and let $T : A\times A \to\mathrm{K}(B)$ be a mapping satisfying the following conditions:

• (i) $T(x,y) \subseteq B_{0}$ for each $(x,y)\in A_{0}\times A_{0}$ and $(A,B)$ satisfies the weak P-property;

• (ii) T is an α-proximal admissible map;

• (iii) there exist elements $(x_{0},y_{0})$, $(x_{1},y_{1})$ in $A_{0}\times A_{0}$ and $u_{1}\in T(x_{0},y_{0})$, $v_{1}\in T(y_{0},x_{0})$ such that

  $$ \begin{aligned} &d(x_{1},u_{1}) = \operatorname{dist}(A,B), \qquad\alpha(x_{0},x_{1})\ge1 \quad\textit{and}\\ &d(y_{1},v_{1}) = \operatorname{dist}(A,B),\qquad \alpha(y_{0},y_{1})\ge1; \end{aligned} $$

  (46)

• (iv) property (C) holds and T is an α-ψ-proximal contraction.

Then there exists an element $(x^{*},y^{*})\in A_{0}\times A_{0}$ such that

$$ \begin{aligned} &D \bigl(x^{*},T\bigl(x^{*},y^{*}\bigr) \bigr) = \operatorname{dist}(A,B) \quad\textit{and}\\ &D \bigl(y^{*},T\bigl(y^{*},x^{*}\bigr) \bigr) = \operatorname{dist}(A,B). \end{aligned} $$

With a similar idea to the examples in [42], we give the following examples to support our main results.

Example 2.8

Let $X=[0,\infty)\times[0,\infty)$ be a product space endowed with the usual metric d. Suppose that $A= \{(\frac{1}{2},x) : 0 \le x<\infty \}$ and $B= \{(0,x):0\le x<\infty \}$.

Define $T:A\times A \to\operatorname{CL}(B)$ by

$$ T \biggl(\biggl(\frac{1}{2},a\biggr),\biggl( \frac{1}{2},b\biggr) \biggr) = \begin{cases} \{(0,\frac{x}{2}) : 0 \le x\le\max\{a,b\} \}&\mbox{if } a,b\le1,\\ \{(0,x^{2}) : 0 \le x\le\max\{a^{2},b^{2}\} \}&\mbox{if } a,b> 1, \end{cases} $$

(47)

and define $\alpha: A\times A \to[0,\infty)$ by

$$ \alpha(x,y) = \begin{cases} 1&\mbox{if } x,y\in \{(\frac{1}{2},a) : 0 \le a\le1 \},\\ 0&\mbox{otherwise}. \end{cases} $$

Let $\Psi(t)=\frac{t}{2}$ for all $t\ge0$. Note that $A_{0}=A$, $B_{0}=B$, and $T(x,y)\subseteq B_{0}$ for each $(x,y)\in A_{0}\times A_{0}$. Also, the pair $(A,B)$ satisfies the weak P-property.

Let $(x_{0},y_{0}),(x_{1},y_{1})\in \{(\frac{1}{2},x) : 0 \le x\le 1 \}^{2}$; then $T(x_{0},y_{0}), T(x_{1},y_{1}) \subseteq \{(0,\frac{x}{2}) : 0 \le x\le1 \}$. Consider $u_{1}\in T(x_{0},y_{0})$, $u_{2}\in T(x_{1},y_{1})$ and $w_{1},w_{2}\in A$ such that $d(w_{1},u_{1}) =\operatorname{dist}(A,B)$ and $d(w_{2},u_{2})=\operatorname{dist}(A,B)$. Then we have $w_{1},w_{2}\in \{(\frac{1}{2},x) : 0 \le x\le\frac{1}{2} \}$, so $\alpha(w_{1},w_{2})=1$. And, for $v_{1}\in T(y_{0},x_{0})$, $v_{2}\in T(y_{1},x_{1})$ and $w'_{1},w'_{2}\in A$ such that $d(w'_{1},v_{1})=\operatorname{dist}(A,B)$ and $d(w'_{2},v_{2})=\operatorname{dist}(A,B)$. Then we have $w'_{1},w'_{2}\in \{ (\frac{1}{2},x) : 0 \le x\le\frac{1}{2} \}$, so $\alpha (w'_{1},w'_{2})=1$. Therefore, T is an α-proximal admissible map. For $(x_{0},y_{0})= ((\frac{1}{2},1),(\frac{1}{2},1) )\in A_{0}\times A_{0}$ and $u_{1}=(0,\frac{1}{2})\in T(x_{0},y_{0})$, $v_{1}=(0,\frac {1}{4})\in T(y_{0},x_{0})$ in $B_{0}$, we have $(x_{1},y_{1})= ((\frac {1}{2},\frac{1}{2}),(\frac{1}{2},\frac{1}{4}) )\in A_{0}\times A_{0}$ such that

$$d(x_{1},u_{1})=\operatorname{dist}(A,B),\qquad \alpha(x_{0},x_{1})=\alpha \biggl(\biggl( \frac{1}{2},1\biggr),\biggl(\frac{1}{2},\frac{1}{2}\biggr) \biggr)=1 $$

and

$$d(y_{1},v_{1})=\operatorname{dist}(A,B),\qquad \alpha(y_{0},y_{1})=\alpha \biggl(\biggl( \frac {1}{2},1\biggr),\biggl(\frac{1}{2},\frac{1}{4}\biggr) \biggr)=1. $$

If $x,x',y,y'\in \{(\frac{1}{2},a) : 0 \le a\le1 \}^{2}$, then we have

$$\alpha(x,y)H \bigl(T\bigl(x,x'\bigr),T\bigl(y,y'\bigr) \bigr) = \frac{|x-y|}{2} = \frac {1}{2}d(x,y)=\psi \bigl(d(x,y) \bigr), $$

for otherwise

$$\alpha(x,y)H \bigl(T\bigl(x,x'\bigr),T\bigl(y,y'\bigr) \bigr) \le \psi \bigl(d(x,y) \bigr). $$

Hence, T is an α-ψ-proximal contraction. Moreover, if $\{x_{n}\}$ is a sequence in A such that $\alpha(x_{n},x_{n+1})=1$ for all n and $x_{n}\to x\in A$ as $n\to\infty$, then there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\alpha (x_{n_{k}},x)=1$ for all k. Therefore, all the conditions of Theorem 2.6 hold and T has the coupled best proximity point.

Example 2.9

Let $X=[0,\infty)\times[0,\infty)$ be endowed with the usual metric d. Let $a>1$ be any fixed real number, $A= \{(a,x) : 0 \le x<\infty \}$ and $B= \{(0,x):0\le x<\infty \}$. Define $T : A\times A \to\operatorname{CL}(B)$ by

$$ T \bigl((a,x),(a,y) \bigr) = \bigl\{ \bigl(0,b^{2}\bigr) : 0 \le b \le\max\{x,y\} \bigr\} , $$

(48)

and $\alpha: A\times A \to[0,\infty)$ by

$$ \alpha \bigl((a,x),(a,y) \bigr) = \begin{cases} 1&\mbox{if } x=y=0,\\ \frac{1}{a(x+y)}&\mbox{otherwise}. \end{cases} $$

(49)

Let $\psi(t)=\frac{t}{a}$ for all $t\ge0$. Note that $A_{0}=A$, $B_{0}=B$ and $T(x,y)\in B_{0}$ for each $x,y\in A_{0}$. If $w_{1}=(a,y_{1}),w'_{1}=(a,y'_{1}), w_{2}=(a,y_{2}), w'_{2}=(a,y'_{2})\in A$ with either $y_{1}\ne0$ or $y_{2}\ne0$ or both are nonzero, we have

$$\begin{aligned} \alpha(w_{1},w_{2})H \bigl(T\bigl(w_{1},w'_{1} \bigr),T\bigl(w_{2},w'_{2}\bigr) \bigr) & = \frac {1}{a(y_{1}+y_{2})}\bigl\vert y_{1}^{2}-y_{2}^{2} \bigr\vert \\ & = \frac{1}{a}|y_{1}-y_{2}| \\ & = \psi \bigl(d(w_{1},w_{2}) \bigr) \end{aligned}$$

for otherwise

$$\alpha(w_{1},w_{2})H \bigl(T\bigl(w_{1},w'_{1} \bigr),T\bigl(w_{2},w'_{2}\bigr) \bigr) = 0 = \psi \bigl(d(w_{1},w_{2}) \bigr). $$

For $x_{0}=(a,\frac{1}{2a}), y_{0}=(a,\frac{1}{3a}) \in A_{0}$ and $u_{1}=(0,\frac{1}{4a^{2}}) \in T(x_{0},y_{0})$ such that $d(x_{1},u_{1})=a=\operatorname{dist}(A,B)$ and $\alpha(x_{0},x_{1}) = \frac{4a}{1+2a} > 1$. And for $x_{1}=(a,\frac{1}{3a}), y_{1}=(a,\frac{1}{9a^{2}}) \in A_{0}$ and $v_{1}=(0,\frac{1}{9a^{2}}) \in T(x_{1},y_{1})$ such that $d(y_{1},v_{1})=a=\operatorname{dist}(A,B)$ and $\alpha(y_{0},y_{1}) = \frac{9a}{1+3a} > 1$. Furthermore, one can see that the remaining conditions of Theorem 2.4 also hold. Therefore, T has the coupled best proximity point.