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.
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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
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 \(y_{1},y_{2} \in B\),
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
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
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
-
(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
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
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
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
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
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
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
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
and
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
From (13), (14) and (15), we have
and
As \(u_{2}\in T(x_{1},y_{1})\subseteq B_{0} \), there exists \(x_{2}\ne x_{1}\in A_{0} \) such that
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
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
From (16), (17) and (20) we have
Since ψ is strictly increasing, we have
Put
We also have
and
Since T is an α-proximal admissible, then \(\alpha (x_{1},x_{2}) \ge1 \) and \(\alpha(y_{1},y_{2}) \ge1 \). Thus we have
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
and
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
From (23), (24) and (25) we have
and
As \(u_{3}\in T(x_{2},y_{2})\in B_{0} \), there exists \(x_{3}\ne x_{2}\in A_{0} \) such that
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
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
From (26), (27) and (30) we have
and
Since ψ is strictly increasing, we have
Put
We also have
and
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
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
and
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
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
Since \((A,B) \) satisfies the weak P-property, from (35), (37) and (38) we have
Thus, from (36) we have
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
and
respectively. For \(m>n>h \), using the triangular inequality, we obtain
and
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
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,
and
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
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
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
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
and
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
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,
and
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
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
and define \(\alpha: A\times A \to[0,\infty) \) by
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
and
If \(x,x',y,y'\in \{(\frac{1}{2},a) : 0 \le a\le1 \}^{2}\), then we have
for otherwise
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
and \(\alpha: A\times A \to[0,\infty) \) by
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
for otherwise
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.
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The author is grateful to Lampang Rajabhat University for financial support during the preparation of this manuscript and to the referees for useful suggestions.
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Nantadilok, J. Coupled best proximity point theorems for α-ψ-proximal contractive multimaps. Fixed Point Theory Appl 2015, 30 (2015). https://doi.org/10.1186/s13663-015-0280-y
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DOI: https://doi.org/10.1186/s13663-015-0280-y