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Best proximity results: optimization by approximate solutions
Fixed Point Theory and Applications volume 2016, Article number: 79 (2016)
Abstract
In this paper we utilize a generalized weakly contractive mapping to establish some best proximity point results which are global optimization results for finding the minimum distances between two sets. Amongst many approaches to this problem, we adopt the approach where the problem is treated as that of finding global optimal approximate solution of the fixed point equation for the generalized weak contraction mapping. We use three control functions to define such mappings. The results are obtained in metric spaces with a partial ordering defined therein. There is a blending of analytic and order theoretic approaches in the proofs. The uniqueness is obtained by imposing some order theoretic conditions additionally. There are several corollaries. An illustration of the main theorem through an example is given which also shows that the corollaries are properly contained in the main theorem.
Introduction and mathematical preliminaries
The program of the paper is to find best proximity pairs between two subsets of a metric space with a partial ordering. There are several works which utilize for that purpose nonself mappings in the following manner. Let A and B be two nonintersecting subsets of a metric space \((X, d)\). A mapping \(S \colon A \longrightarrow B\) realizes the best proximity pair \((x, Sx)\) if \(d(x, Sx) = d(A, B)\). In that case the point x is called a best proximity point of S and the problem of finding such a point is designated as best proximity point problem. This area of research has attracted attention in recent time which has resulted into the publication of a good number of papers as, for instances, those which are noted in [1–16].
The problem has two aspects. Primarily, it is a global minimization problem, where the quantity \(d(x, Sx)\) is minimized over \(x\in A\) subject to the condition that the minimum value is \(d(A, B)\). When this global minimum is attained at a point z, then we have a best proximity point for which \(d(z, Sz) = d(A, B)\). Another aspect is that it is an extension of the idea of fixed point to which it reduces in the cases where \(A \cap B\) is nonempty. This is the reason that fixed point methodologies are applicable to these category of problems. More elaborately, the problem can be treated as that of finding a global optimal approximate solution of the fixed point equation \(x = Sx\) even when the exact solution is nonexistent for \(A \cap B = \emptyset\), which is the case of interest here. We adopt the latter approach in this paper.
We use a generalized weak contraction in our results. Weak contraction was studied in partially ordered metric spaces by Harjani and Sadarangani [17]. In a recent result by Choudhury et al. [18], a generalization of the above result to a coincidence point theorem has been made using three control functions. More specifically, here we utilize a generalized weak contraction mapping defined with the help of three control functions for the purpose of obtaining the desired minimum distance. The above mentioned mapping is assumed to be defined from one set A to the other set B. Then under suitable conditions, by applying fixed point methodologies, we obtained a best proximity point of the above mentioned mapping which realizes the minimum distance. Several metric and order theoretic concepts are utilized in our results. The main result has four corollaries and an illustrative example. Separate order theoretic condition are imposed to ensure the uniqueness of the best proximity point in the main result. It is also shown that the corollaries are properly contained in the main theorem.
The following are the requisite mathematical concepts for the discussions in this paper.
Throughout the paper \((X, d)\) denotes a metric space, ⪯ a partial order on X and \(A, B \subseteq X\). We use the following notations:
It is to be noted that if \((A, B)\) is a nonempty, weakly compact, and convex pair in a Banach space X, then \(A_{0}\) and \(B_{0}\) are nonempty [5, 11]. If a mapping \(S:A\cup B \longrightarrow A\cup B\) is cyclic relatively nonexpansive mapping, and \((A, B)\) is a nonempty, weakly compact, and convex pair in a Banach space X then further we have \(S(A_{0})\subseteq B_{0}\).
Definition 1.1
(Pproperty [16])
Let A and B be two nonempty subsets of a metric space \((X, d)\) with \(A_{0}\neq\emptyset\). Then the pair \((A, B)\) is said to have the Pproperty if, for any \(x_{1}, x_{2}\in A_{0}\) and \(y_{1}, y_{2}\in B_{0}\),
In [1], Abkar and Gabeleh show that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach spaces has the Pproperty. Some nontrivial examples of a nonempty pair of subsets which satisfies the Pproperty are given in [1].
Lemma 1.1
([10])
Let \((A, B)\) be a pair of nonempty closed subsets of a complete metric space \((X, d)\) such that \(A_{0}\) is nonempty and \((A, B)\) has the Pproperty. Then, \((A_{0}, B_{0})\) is a closed pair of subsets of X.
Definition 1.2
A mapping \(S \colon A \longrightarrow A\) is said to be increasing if for all \(x, y\in A\),
Definition 1.3
([4])
A mapping \(S \colon A \longrightarrow B\) is called proximally increasing if for all \(v_{1}, v_{2}, y_{1}, y_{2}\in A\),
In case of selfmapping the above definition reduces to the definition of increasing mapping.
Definition 1.4
A mapping \(S \colon A \longrightarrow B\) is called proximally increasing on \(A_{0}\) if for all \(v_{1}, v_{2}, y_{1}, y_{2}\in A_{0}\),
Definition 1.5
The partially ordered metric space \((X, d, \preceq)\) is called regular if it has the following properties:

(i)
if \(\{z_{n}\}\) is any nondecreasing sequence in X converging to z, then \(z_{n} \preceq z\) for any \(n \geq0\);

(ii)
if \(\{z_{n}\}\) is any nonincreasing sequence in X converging to z, then \(z_{n} \succeq z\) for any \(n \geq0\).
Main results
Let Γ and Λ denote the following classes of functions:

\(\Gamma=\{\eta\colon[0, \infty) \longrightarrow[0, \infty), \eta \mbox{ is continuous and monotonic increasing}\}\);

\(\Lambda=\{\xi\colon[0, \infty) \longrightarrow[0, \infty), \xi \mbox{ is bounded on any bounded interval in } [0, \infty)\}\).
Now, we discuss some properties of some special type of functions in Λ.
Let \(\Theta= \{\theta\in\Lambda: \underline{\lim}\, \theta(z_{n}) > 0, \mbox{whenever } \{z_{n}\}\mbox{ is any sequence of nonnegative real}\mbox{ }\mbox{numbers converging to }l> 0\}\).
We note that Θ is nonempty. For an illustration, we define \(\theta_{1}\) on \([0,\infty)\) by \(\theta_{1}(x)=e^{2x}\), \(x\in[0, \infty)\). Then \(\theta_{1}\in\Theta\). Here we observe that \(\theta_{1}(0)=1 > 0\). On the other hand, if \(\theta_{2}(x)=x^{3}\), \(x\in[0, \infty)\), then \(\theta_{2}\in\Theta\) and \(\theta_{2}(0)= 0\).
Also, for any \(\theta\in\Theta\), it is clear that \(\theta(x)> 0\) for \(x>0\); and \(\theta(0)\) need not be equal to 0.
Let \(\Upsilon= \{\varphi\in\Lambda: \overline{\lim}\, \varphi (z_{n}) < l, \mbox{whenever } \{z_{n}\}\mbox{ is any sequence of nonnegative real}\mbox{ }\mbox{numbers converging to }l > 0\}\).
It follows from the definition that, for any \(\varphi\in\Upsilon\), \(\varphi(y) < y\) for all \(y > 0\).
Theorem 2.1
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty subsets of X such that \(A_{0}\) is nonempty and closed. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Assume that there exist \(\eta\in\Gamma\) and \(\xi, \theta\in\Lambda\) such that

(i)
for \(x, y\in[0, \infty)\), \(\eta(x)\leq\xi(y) \Longrightarrow x\leq y\),

(ii)
\(\eta(z)  \overline{\lim}\, \xi(z_{n}) + \underline{\lim}\, \theta(z_{n}) > 0\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z> 0\),

(iii)
for all \(x, y, u, v \in A_{0}\)
$$\left . \textstyle\begin{array}{r@{}} x \preceq y, \\ d(u, Sx)= d(A, B),\\ d(v, Sy)= d(A, B) \end{array}\displaystyle \right \}\quad\Rightarrow\quad\eta \bigl(d(u, v)\bigr) \leq\xi\bigl(M(x, y, u, v)\bigr)  \theta \bigl(M(x, y, u, v) \bigr), $$where \(M(x, y, u, v) = \max \{d(x, y), \frac{d(x, u) + d(y, v)}{2}, \frac{d(y, u) + d(x, v)}{2} \}\).
Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
It follows from the definition of \(A_{0}\) and \(B_{0}\) that for every \(x\in A_{0}\) there exists \(y\in B_{0}\) such that \(d(x, y) = d(A, B)\) and conversely, for every \(y'\in B_{0}\) there exists \(x'\in A_{0}\) such that \(d(x', y') = d(A, B)\). Since \(S(A_{0})\subseteq B_{0}\), for every \(x \in A_{0}\) there exists a \(y \in A_{0}\) such that \(d(y, Sx) = d(A, B)\).
By the hypothesis of the theorem there exist \(x_{0}, x_{1} \in A_{0}\) for which \(x_{0} \preceq x_{1}\) and
Now, \(x_{1}\in A_{0}\) and \(S(A_{0}) \subseteq B_{0}\) imply the existence of a point \(x_{2} \in A_{0}\) such that
As S is proximally increasing on \(A_{0}\), we get \(x_{1} \preceq x_{2}\). In this way we obtain a sequence \(\{x_{n}\}\) in \(A_{0}\) such that for all \(n \geq0\),
and
By the hypothesis (iii), \(x_{n} \preceq x_{n + 1}\), \(d(x_{n + 1}, Sx_{n}) = d(A, B)\) and \(d(x_{n + 2}, Sx_{n + 1}) = d(A, B)\) imply that
where
By the triangular inequality, \(\frac{d(x_{n}, x_{n+2})}{2}\leq\frac {d(x_{n}, x_{n+1}) + d(x_{n+1}, x_{n+2})}{2}\). So, it follows that
Let \(U_{n} = d(x_{n}, x_{n + 1})\), for all \(n \geq0\).
Case 1: \(M(x_{n }, x_{n + 1}, x_{n + 1}, x_{n + 2}) = d(x_{n}, x_{n +1})\). Then by (2.5),
that is,
which implies that \(\eta(U_{n+1})\leq\xi(U_{n})\). Then it follows by the hypothesis (i) of the theorem that \(U_{n+1} \leq U_{n}\), for all \(n \geq0\).
Case 2: \(M(x_{n }, x_{n + 1}, x_{n + 1}, x_{n + 2}) = \frac{d(x_{n}, x_{n+1}) + d(x_{n+1}, x_{n+2})}{2} = \frac{U_{n} + U_{n + 1}}{2} = V_{n}\) (say). Then it follows from (2.5) that
which implies that \(\eta(U_{n+1})\leq\xi(V_{n})= \xi (\frac {U_{n+1} + U_{n}}{2} )\). Again by the hypothesis (i) the theorem it follows that \(U_{n+1} \leq \frac{U_{n} + U_{n + 1}}{2}\), that is, \(U_{n+1} \leq U_{n}\), for all \(n \geq0\).
From Case 1 and Case 2, we conclude that \(\{U_{n}\}\) is a monotone decreasing sequence of nonnegative real numbers. As \(\{U_{n}\}\) is bounded below by zero, there exists an \(t \geq0\) such that
Then it follows that
Taking the limit supremum in both sides of the inequality (2.6), using (2.8), the continuity of η, and the property of ξ and θ, we obtain
Since \(\overline{\lim}\, ( \theta(U_{n})) =  \underline{\lim}\, \theta(U_{n})\), it follows that
that is,
which, by the hypothesis (ii) and (2.8), is a contradiction unless \(t = 0\).
Arguing similarly as above, from (2.7) and (2.8), we have
which, by the hypothesis (ii) and (2.9), is a contradiction unless \(t = 0\). Hence
Next we show that \(\{x_{n}\}\) is a Cauchy sequence.
Suppose that \(\{x_{n}\}\) is not a Cauchy sequence. Then there exist \(\delta> 0\) and two sequences \(\{m(k)\}\) and \(\{n(k)\}\) of positive integers such that for all positive integers k, \(n(k) > m(k) > k\) and \(d(x_{m(k)}, x_{n(k)})\geq\delta\). Assuming that \(n(k)\) is the smallest such positive integer, we get
Now,
From the above inequality and (2.10), it follows that
Again,
and
The above two inequalities imply that
From the above inequality, (2.10) and (2.11), we have
Again,
and
The above two inequalities imply that
From the above inequality, (2.10) and (2.11), we have
Similarly, we can prove that
By the construction of the sequence \(\{x_{n}\}\), we have
which, by the hypothesis (iii), imply that
where
From (2.10), (2.11), (2.13), and (2.14), it follows that
Taking the limit supremum in both sides of the inequality (2.15), using (2.12), (2.16), the continuity of η, and the property of ξ and θ, we obtain
As \(\overline{\lim}\, ( \theta(M(x_{m(k)}, x_{n(k)}, x_{m(k) + 1}, x_{n(k) + 1}))) =  \underline{\lim}\, \theta(M(x_{m(k)}, x_{n(k)}, x_{m(k) + 1}, x_{n(k) + 1}))\), it follows that
that is,
which, by the hypothesis (ii) and (2.16), is a contradiction. Therefore, \(\{x_{n}\}\) is a Cauchy sequence in \(A_{0}\). Since \(A_{0}\) is a closed subset of complete metric space \((X, d)\), there exists \(a \in A_{0} \) such that

Suppose that S is continuous.
Taking \(n\longrightarrow\infty\) in (2.4) and using the continuity of S, we have \(d(a, Sa) = d(A, B)\); that is, a is a best proximity point of S.

Next we suppose that X is regular.
Now \(a \in A_{0}\) and \(S(A_{0}) \subseteq B_{0}\) imply the existence of a point \(p \in A_{0}\) for which
By (2.4), (2.18) and (2.19), we have
which, by the hypothesis (iii) of the theorem, imply that
where
From (2.17), it follows that
Taking the limit supremum in both sides of the inequality (2.20), using (2.17), (2.21), the properties of η, and the property of ξ and θ, we obtain
Arguing similarly as discussed above, we have
which, by the hypothesis (ii) and (2.21), is a contradiction unless \(d(a, p) = 0\); that is, \(p = a\). Then by (2.19) we have \(d(a, Sa) = d(A, B)\); that is, a is a best proximity point of S. □
Theorem 2.2
In addition to the hypotheses of Theorem 2.1, suppose that for every \(x, y \in A_{0}\) there exists \(u \in A_{0}\) such that u is comparable to x and y. Then S has a unique best proximity point.
Proof
By Theorem 2.1, the set of best proximity points of S is nonempty. Suppose \(x, y\in A_{0}\) are two best proximity points of S; that is,
By the assumption, there exists \(u \in A_{0}\), which is comparable with x and y.
Put \(u_{0} = u\). Suppose that
\(S(A_{0}) \subseteq B_{0}\) and \(u_{0} = u\in A_{0}\) imply the existence of a point \(u_{1} \in A_{0}\) for which
Since S is proximally increasing on \(A_{0}\), from (2.22), (2.23), and (2.24) we have
Following this process, we obtain a sequence \(\{u_{n}\}\) in \(A_{0}\) such that for all \(n \geq0\),
which, by the hypothesis (iii) of Theorem 2.1, imply that
where
By the triangular inequality \(\frac{d(u_{n}, u_{n+1})}{2}\leq\frac {d(x, u_{n + 1}) + d(u_{n}, x)}{2}\). Then it follows that
Let \(Q_{n} = d(u_{n}, x)\), for all \(n \geq0\).
Arguing similarly as in the proof of Theorem 2.1 (Case 1 and Case 2), we can prove that \(\{Q_{n}\}\) is a monotone decreasing sequence of nonnegative real numbers and
Similarly, we show that
By the triangle inequality, and using (2.28) and (2.29), we have
which implies that \(d(x, y) = 0\); that is, \(x = y\); that is, the best proximity point of S is unique. □
With the help of Pproperty we have the following theorem which is obtained by an application of Theorem 2.1.
Theorem 2.3
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty and closed subsets of X such that \(A_{0}\) is nonempty and \((A, B)\) satisfies the Pproperty. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Assume that there exist \(\eta\in \Gamma\) and \(\xi, \theta\in\Lambda\) such that

(i)
for \(x, y\in[0, \infty)\), \(\eta(x)\leq\xi(y) \Longrightarrow x\leq y\),

(ii)
\(\eta(z)  \overline{\lim}\, \xi(z_{n}) + \underline{\lim}\, \theta(z_{n}) > 0\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z> 0\),

(iii)
for all \(x, y, u, v \in A_{0}\)
$$\left . \textstyle\begin{array}{r@{}} x \preceq y, \\ d(u, Sx)= d(A, B),\\ d(v, Sy)= d(A, B) \end{array}\displaystyle \right \}\quad\Rightarrow \quad\eta \bigl(d(Sx, Sy)\bigr) \leq\xi\bigl(M(x, y, u, v)\bigr)  \theta \bigl(M(x, y, u, v)\bigr), $$where \(M(x, y, u, v) = \max \{d(x, y), \frac{d(x, u) + d(y, v)}{2}, \frac{d(y, u) + d(x, v)}{2} \}\).
Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
By Lemma 1.1, \(A_{0}\) is nonempty and closed. Since \((A, B)\) satisfies the Pproperty, \(d(u, Sx)= d(A, B)\) and \(d(v, Sy)= d(A, B)\) imply that \(d(u, v)= d(Sx, Sy)\). Then condition (iii) of the theorem reduces to the condition (iii) of Theorem 2.1. Therefore, all the conditions of the Theorem 2.1 are satisfied and hence we have the required proof. □
Corollaries and example
Corollary 3.1
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty subsets of X such that \(A_{0}\) is nonempty and closed. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Let there exists \(\xi\in\Lambda\) such that \(\overline{\lim}\, \xi(z_{n}) < z\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z > 0\) and for all \(x, y, u, v \in A_{0}\)
where \(M(x, y, u, v)\) is the same as in Theorem 2.1. Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) such that \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
Let η be the identity mapping and \(\theta (t) = 0\) for all \(t \in[0, \infty)\) in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Corollary 3.2
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty subsets of X such that \(A_{0}\) is nonempty and closed. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Assume that there exist \(\eta\in\Gamma\) and \(\theta\in\Lambda\) such that for any sequence of nonnegative real numbers \(\{z_{n}\}\) with \(z_{n}\longrightarrow z > 0\), \(\underline{\lim}\, \theta(z_{n}) > 0\) and for all \(x, y, u, v \in A_{0}\)
where \(M(x, y, u, v)\) is the same as in Theorem 2.1. Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
The required proof is obtained by considering ξ to be identical with the function η in Theorem 2.1. □
Corollary 3.3
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty subsets of X such that \(A_{0}\) is nonempty and closed. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Let there exists \(\theta\in\Lambda\) such that \(\underline{\lim}\, \theta (z_{n}) > 0\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z > 0\) and for all \(x, y, u, v \in A_{0}\)
where \(M(x, y, u, v)\) is the same as in Theorem 2.1. Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
Let η and ξ be the identity mappings in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Corollary 3.4
Let \((X, d)\) be a complete metric space and ⪯ be a partial order on X. Let \((A, B)\) be a pair of nonempty subsets of X such that \(A_{0}\) is nonempty and closed. Let \(S \colon A \longrightarrow B\) be a mapping with the properties that \(S(A_{0})\subseteq B_{0}\) and S is proximally increasing on \(A_{0}\). Assume that there exists \(k\in[0, 1)\) such that for all \(x, y, u, v \in A_{0}\)
where \(M(x, y, u, v)\) is the same as in Theorem 2.1. Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
We consider that η and ξ are the identity mappings and \(\theta(t)=(1k)t\), where \(0\leq k < 1\), in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Example 3.1
Let \(X = \mathbb {R}^{2}\) (\(\mathbb {R}\) denotes the set of all real numbers) and d be a metric on X defined as \(d(x, y) =  x_{1}  x_{2}+  y_{1}  y_{2}\), for \(x = (x_{1}, y_{1}), y = (x_{2}, y_{2})\in X\). Define a partial order ⪯ on X such that \((x, y) \preceq(u, v)\) if and only if \(x \leq u\) and \(y \leq v\), for all \((x, y), (u, v) \in X\). Let
Let \(S \colon A \rightarrow B\) be defined as
and \(\eta, \xi, \theta\colon[0, \infty) \longrightarrow[0, \infty)\) be defined as follows:
The function S now satisfies all the postulates of Theorems 2.1 and 2.2. Then, by joint applications of Theorems 2.1 and 2.2 we conclude that S must have a best proximity point which is unique. The point can be seen here to be \((0, 1) \in A_{0}\).
Note
In this example A and B are not closed sets. This is an illustration of the fact that the closedness of A and B are not required in our theorem.
Remark 3.1
Corollaries 3.1, 3.2, 3.3, and 3.4 are not applicable to this example and hence Theorem 2.1 is an actual extension of its Corollaries 3.1, 3.2, 3.3, and 3.4.
Conclusions
The present paper is an application of weak inequalities satisfied by nonself mappings. Weak contractions are intermediate to the contractions and nonexpansions which have been generalized in various ways and have been utilized in different types of problem. We make such an application for finding a best proximity pair. The speciality of this paper is that it has been obtained in the most general settings of a metric space without any special assumptions on this space.
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Acknowledgements
The authors gratefully acknowledge the suggestions made by the learned referee.
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Choudhury, B.S., Metiya, N., Maniu, G. et al. Best proximity results: optimization by approximate solutions. Fixed Point Theory Appl 2016, 79 (2016). https://doi.org/10.1186/s1366301605695
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MSC
 47H10
 54H10
 54H25
Keywords
 partially ordered metric space
 best proximity point
 fixed point equation
 approximate solution
 optimization