Open Access

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

Fixed Point Theory and Applications20152015:30

https://doi.org/10.1186/s13663-015-0280-y

Received: 11 November 2014

Accepted: 4 February 2015

Published: 24 February 2015

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.

Keywords

proximal contractive multivalued mapping best proximity point coupled fixed point coupled best proximity point

MSC

47H09 47H10

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 [16]). 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 [918]. 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, [1941]. 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:
  1. (a)

    ψ is monotone nondecreasing;

     
  2. (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)
  1. (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:
  1. (i)

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

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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:
  1. (i)

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

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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:
  1. (i)

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

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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:
  1. (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;

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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:
  1. (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;

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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:
  1. (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;

     
  2. (ii)

    T is an α-proximal admissible map;

     
  3. (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)
     
  4. (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.

Declarations

Acknowledgements

The author is grateful to Lampang Rajabhat University for financial support during the preparation of this manuscript and to the referees for useful 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, Faculty of Science, Lampang Rajabhat University

References

  1. Arvanitakis, AD: A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc. 131(12), 3647-3656 (2003) View ArticleMATHMathSciNetGoogle Scholar
  2. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969) View ArticleMATHMathSciNetGoogle Scholar
  3. Choudhury, BS, Das, KP: A new contraction principle in Menger spaces. Acta Math. Sin. 24(8), 1379-1386 (2008) View ArticleMATHMathSciNetGoogle Scholar
  4. Mongkolkeha, C, Sintunavarat, W, Kumam, P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011, Article ID 93 (2011) View ArticleMathSciNetMATHGoogle Scholar
  5. Sintunavarat, W, Kumam, P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011, Article ID 3 (2011) View ArticleMathSciNetMATHGoogle Scholar
  6. Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861-1869 (2008) View ArticleMATHMathSciNetGoogle Scholar
  7. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal., Theory Methods Appl. 75(4), 2154-2165 (2012) View ArticleMATHMathSciNetGoogle Scholar
  8. Karapınar, E, Samet, B: Generalized α-ψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012) MathSciNetMATHGoogle Scholar
  9. Asl, JH, Rezapour, S, Shahzad, N: On fixed points of α-ψ-contractive multifunctions. Fixed Point Theory Appl. 2012, Article ID 212 (2012) View ArticleMathSciNetMATHGoogle Scholar
  10. Mohammadi, B, Rezapour, S, Shahzad, N: Some results on fixed points of \((\alpha\mbox{-}\psi)\)-Ćirić generalized multifunctions. Fixed Point Theory Appl. 2013, Article ID 24 (2013) View ArticleMathSciNetMATHGoogle Scholar
  11. Ali, MU, Kamran, T: On \((\alpha^{*},\psi) \)-contractive multi-valued mappings. Fixed Point Theory Appl. 2013, Article ID 137 (2013) View ArticleMathSciNetMATHGoogle Scholar
  12. Amiri, P, Rezapour, S, Shahzad, N: Fixed points of generalized \((\alpha\mbox{-}\psi) \)-contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 108(2), 519-526 (2014) View ArticleMATHMathSciNetGoogle Scholar
  13. Minak, G, Altun, I: Some new generalizations of Mizoguchi-Takahashi type fixed point theorem. J. Inequal. Appl. 2013, Article ID 493 (2013) View ArticleMathSciNetMATHGoogle Scholar
  14. Ali, MU, Kamran, T, Sintunavarat, W, Katchang, P: Mizoguchi-Takahashi’s fixed point theorem with α, η functions. Abstr. Appl. Anal. 2013, Article ID 418798 (2013) MathSciNetMATHGoogle Scholar
  15. Chen, CM, Karapınar, E: Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. J. Inequal. Appl. 2013, Article ID 410 (2013) View ArticleMathSciNetMATHGoogle Scholar
  16. Ali, MU, Kamran, T, Karapınar, E: \((\alpha,\psi,\xi) \)-Contractive multivalued mappings. Fixed Point Theory Appl. 2014, Article ID 7 (2014) View ArticleMathSciNetMATHGoogle Scholar
  17. Ali, MU, Kamran, T, Karapınar, E: A new approach to \((\alpha\mbox{-}\psi) \)-contractive nonself multivalued mappings. J. Inequal. Appl. 2014, Article ID 71 (2014) View ArticleMathSciNetGoogle Scholar
  18. Ali, MU, Kiran, Q, Shahzad, N: Fixed point theorems for multi-valued mappings involving α-function. Abstr. Appl. Anal. 2014, Article ID 409467 (2014) MathSciNetGoogle Scholar
  19. Jleli, M, Samet, B: Best proximity points for \((\alpha\mbox{-}\psi) \)-proximal contractive type mappings and applications. Bull. Sci. Math. 137(8), 977-995 (2013) View ArticleMATHMathSciNetGoogle Scholar
  20. Abkar, A, Gabeleh, M: Best proximity points for asymptotic cyclic contraction mappings. Nonlinear Anal. 74(18), 7261-7268 (2011) View ArticleMATHMathSciNetGoogle Scholar
  21. Abkar, A, Gabeleh, M: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 151(2), 418-424 (2011) View ArticleMATHMathSciNetGoogle Scholar
  22. Abkar, A, Gabeleh, M: The existence of best proximity points for multivalued non-self mappings. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2), 319-325 (2012) View ArticleMathSciNetMATHGoogle Scholar
  23. Alghamdi, MA, Shahzad, N: Best proximity point results in geodesic metric spaces. Fixed Point Theory Appl. 2012, Article ID 234 (2012) View ArticleMathSciNetMATHGoogle Scholar
  24. Al-Thagafi, MA, Shahzad, N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal., Theory Methods Appl. 70(3), 1209-1216 (2009) View ArticleMATHMathSciNetGoogle Scholar
  25. Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal., Theory Methods Appl. 70(10), 3665-3671 (2009) View ArticleMATHMathSciNetGoogle Scholar
  26. Al-Thagafi, MA, Shahzad, N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008, Article ID 457069 (2008) View ArticleMathSciNetMATHGoogle Scholar
  27. Derafshpour, M, Rezapour, S, Shahzad, N: Best proximity points of cyclic φ-contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 37(1), 193-202 (2011) MATHMathSciNetGoogle Scholar
  28. Di Bari, C, Suzuki, T, Vetro, C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal., Theory Methods Appl. 69, 3790-3794 (2008) View ArticleMATHMathSciNetGoogle Scholar
  29. Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001-1006 (2006) View ArticleMATHMathSciNetGoogle Scholar
  30. Markin, J, Shahzad, N: Best proximity points for relatively u-continuous mappings in Banach and hyperconvex spaces. Abstr. Appl. Anal. 2013, Article ID 680186 (2013) View ArticleMathSciNetGoogle Scholar
  31. Rezapour, S, Derafshpour, M, Shahzad, N: Best proximity points of cyclic ϕ-contractions on reflexive Banach spaces. Fixed Point Theory Appl. 2010, Article ID 946178 (2010) View ArticleMathSciNetMATHGoogle Scholar
  32. Basha, SS, Shahzad, N, Jeyaraj, R: Best proximity point theorems for reckoning optimal approximate solutions. Fixed Point Theory Appl. 2012, Article ID 202 (2012) View ArticleMathSciNetMATHGoogle Scholar
  33. Vetro, C: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal., Theory Methods Appl. 73, 2283-2291 (2010) View ArticleMATHMathSciNetGoogle Scholar
  34. Zhang, J, Su, Y, Cheng, Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013, Article ID 83 (2013) View ArticleMathSciNetMATHGoogle Scholar
  35. Mongkolkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 155, 215-226 (2012) View ArticleMATHMathSciNetGoogle Scholar
  36. Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, Article ID 93 (2012) View ArticleMathSciNetMATHGoogle Scholar
  37. Nashine, HK, Vetro, C, Kumam, P: Best proximity point theorems for rational proximal contractions. Fixed Point Theory Appl. 2013, Article ID 95 (2013) View ArticleMathSciNetMATHGoogle Scholar
  38. Cho, YJ, Gupta, A, Karapınar, E, Kumam, P, Sintunavarat, W: Tripled best proximity point theorem in metric spaces. Math. Inequal. Appl. 16, 1197-1216 (2013) MATHMathSciNetGoogle Scholar
  39. Mongkolkeha, C, Kongban, C, Kumam, P: Existence and uniqueness of best proximity points for generalized almost contractions. Abstr. Appl. Anal. 2014, Article ID 813614 (2014) View ArticleMathSciNetGoogle Scholar
  40. Kumam, P, Salimi, P, Vetro, C: Best proximity point results for modified α-proximal c-contraction mappings. Fixed Point Theory Appl. 2014, Article ID 99 (2014) View ArticleMathSciNetGoogle Scholar
  41. Pragadeeswarar, V, Marudai, M, Kumam, P, Sitthithakerngkiet, K: The existence and uniqueness of coupled best proximity point for proximally coupled contraction in a complete ordered metric space. Abstr. Appl. Anal. 2014, Article ID 274062 (2014) View ArticleMathSciNetMATHGoogle Scholar
  42. Ali, MU, Kamran, T, Shahzad, N: Best proximity point for α-ψ-proximal contractive multimap. Abstr. Appl. Anal. 2014, Article ID 181598 (2014) MathSciNetGoogle Scholar

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