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Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces
Journal of Inequalities and Applications volume 2018, Article number: 130 (2018)
Abstract
In this paper, our aim is to ascertain the distance between two sets iteratively in two simultaneous ways with the help of a multivalued coupling define for this purpose. We define the best proximity points of such couplings that realize the distance between two sets. Our main theorem is deduced in metric spaces. As an application, we obtain the corresponding results in uniformly convex Banach spaces using the geometry of the space. We discuss two examples.
Introduction
In this paper, our intention is to discuss a procedure for finding the distance between two given subsets of a metric space by using a multivalued coupling between two sets. The aforesaid coupling is a multivalued coupled cyclic type mapping between two sets we define. There are several works for determining the minimum distance mentioned, where nonselfmappings, both cyclic and noncyclic, have been utilized. Such problems are known as best proximity point problems for the concern mappings and were introduced by Eldered et al. [16] in 2006. This line of research developed through works like [1, 6, 7, 9, 12–14, 23, 24, 27, 30, 31] in subsequent times. Multivalued mappings have been considered in fixed and proximity point problems in works like [2, 8, 18, 20, 26]. Essentially, this is a kind of global optimization problem, which, in the approach mentioned, is treated as the problem of finding an optimal approximate solution of a fixed point equation (or a fixed point inclusion in the case of a multivalued mappings) although the exact solution may not exist, which is actually the case where the two sets are disjoint. This situation is of interest for the present problem. Within the above framework, we take a new approach to this problem by defining a multivalued coupling between two sets. We define a multivalued coupled proximity point for these couplings, which simultaneously realize two best proximity pairs between two sets. We discuss the existence of such points under a contractive condition. We apply our theorem to obtain the corresponding result in uniformly convex Banach spaces. Uniformly convex Banach spaces constitute a category of Banach spaces including Hilbert spaces and in which interest has been widespread. We use some bit of Banach space geometry to obtain our result. Further, we illustrate our results with examples. Open problems are discussed at the end of the paper.
Mathematical preliminaries
Throughout this paper, \((X, d)\) stands for a general metric space. Let A and B be two subsets of X. A pair of points \((a, b)\) with \(a\in A\) and \(b\in B\) is said to be a best proximity pair if \(d(a, b)=d(A, B)\) where \(d(A, B)= \operatorname{inf} \{d(x, y): x\in A \text{ and } y\in B\} \). If \(T: A \longrightarrow B\) is a nonselfmapping for which there exists \(z\in A\) such that \(d(z, Tz)= d(A, B)\), that is, where \((z, Tz)\) is a best proximity pair, we say that z is a best proximity point of T, and the problem of finding such a point is known as the best proximity point problem. It is a global optimization problem in that it seeks to find a point \(z\in A\) that minimizes the value of \(d(z, Tz)\) over \(z \in A\) subject to the condition that the minimum is \(d(A, B)\). On the other hand, it is an extension of the fixed point problem and reduces to that problem in the special case where \(A\cap B\neq\emptyset\). The following are the concepts from setvalued analysis which we use in this paper. We consider the following classes of subsets of the metric space X:
and
For \(A, B \in B(X)\), the functions D and δ are defined as follows:
If \(A= \{a\}\), then we write \(D(A, B)= D(a, B)\) and \(\delta(A, B)= \delta(a, B)\). In addition, if \(B= \{b\}\), then \(D(A, B)= d(a, b)\) and \(\delta(A, B)= d(a, b)\). Obviously, \(D(A, B)\leq\delta(A, B)\). For all \(A, B, C\in B(X)\), the definition of \(\delta(A, B)\) yields the following:
\(\delta(A, B)= \delta(B, A)\), \(\delta(A, B)\leq\delta(A, C) + \delta(C, B)\), \(\delta(A, B)= 0\) iff \(A = B = \{a\}\), \(\delta (A, A)= \operatorname{diam} A\) [17].
The δdistance has all the properties of a metric except one. It has been used in works like [4, 5, 10]. We use this concept in our theorem.
Let A and B be two nonempty subsets of a metric space \((X, d)\), and let \(T:A \longrightarrow CB(B)\) be a multivalued mapping. A point \(x^{\ast}\in A\) is called a best proximity point of T if \(D(x^{\ast }, Tx^{\ast})= \operatorname{inf} \{d(x^{\ast}, y): y\in Tx^{\ast}\}= \operatorname{dist}(A, B)\) [2]. This is a natural generalization of its singlevalued counterpart described above. A fixed point x of a multivalued mapping T is given by the inclusion relation \(x\in Tx\). Now there need not be a fixed point of the multivalued mapping in general. Here the task in the best proximity point problem is to find a global minimum of the function \(x\mapsto D(x, Tx)\) by constraining an approximate solution of the inclusion relation \(x\in Tx\) to satisfy \(D(x, Tx)= \operatorname{dist}(A, B)\).
Next, we give the following:
Definition 2.1
Let \((X, d)\) be a metric space, and let A, B be two nonempty subsets of X. A mapping \(F\colon X\times X \longrightarrow X\) is said to be a coupling with respect to A and B if \(F(x, y)\in B\) for \((x, y)\in A \times B\) and \(F(x, y)\in A \) for \((x, y)\in B \times A\).
The following definition is a multivalued generalization of Definition 2.1.
Definition 2.2
Let \((X, d)\) be a metric space, and let A, B be two nonempty subsets of X. A multivalued mapping \(F\colon X\times X\longrightarrow N(X)\) is said to be a multivalued coupling with respect to A and B if \(F(x, y)\subseteq B \) for \((x, y)\in A \times B\) and \(F(x, y)\subseteq A \) for \((x, y)\in B \times A\).
Our purpose here is to realize the minimum distance between two sets A and B through a coupled best proximity point for coupling, which we define further.
Definition 2.3
Let \((X, d)\) be a metric space, and let A, B be two nonempty subsets of X. Let \(F\colon X\times X\longrightarrow X\) be a coupling with respect to A and B. An element \((x, y)\in A \times B\) is called a coupled best proximity point if \(d(x, F(x, y))=d(A, B)\) and \(d(y, F(y, x))=d(A, B)\), where \(d(A, B)= \inf \{d(x, y): x \in A \mbox{ and } y \in B\}\).
Definition 2.4
Let \((X, d)\) be a metric space, and let A, B be two nonempty subsets of X. Let \(F\colon X\times X\longrightarrow N(X)\) be a multivalued coupling with respect to A and B. An element \((x, y)\in A \times B\) is called a coupled best proximity point if \(D(x, F(x, y))=d(A, B)\) and \(D(y, F(y, x))=d(A, B)\), where \(d(A, B)= \inf \{d(x, y): x \in A\mbox{ and }y \in B\}\).
Note that proximity points for coupled mapping have been defined in [21]. Our definition is for both singlevalued and multivalued couplings, which is different from the abovementioned concept. Also, the present work is in a different course from that of [21].
Here the problem of finding coupled best proximity points for multivalued couplings is viewed as that of finding simultaneous optimal approximate solutions of the coupled fixed point inclusions \(x \in F(x, y)\) and \(y \in F(y, x)\) for a multivalued coupling F in the set \(A\times B\) such that the solution satisfies \(D(x, F(x, y))= D(y, F(y, x))=d(A, B)\). In general, the exact solutions may not exist. This is surely the case where \(d(A, B)\neq0\), which is of interest here. From another viewpoint, it is the problem of simultaneous minimization of \(D(x, F(x, y))\) and \(D(y, F(y, x))\) for \(x\in A\), \(y\in B\) such that the minimum values at the point of optimality are the global minimum \(d(A, B)\). It is to be noted that the pairs \((x, F(x, y))\) and \((y, F(y, x))\) are in general different. Thus the distance between two sets is obtained through two different pairs of points, that is, in the process, we obtain two best proximity pairs. Also, the coupled best proximity point may be not unique. The following illustration elucidates these points.
Example 2.1
Let \(X= {\mathbb {R}}^{2}\) with the metric d defined as \(d((x_{1}, y_{1}), (x_{2}, y_{2}))= \vert x_{1}  x_{2} \vert + \vert y_{1}  y_{2} \vert \). Let \(A=\{0\}\times\mathbb {R}\) and \(B=\{1\}\times\mathbb {R}\). Then A and B are nonempty subsets of X, and \(d(A, B)=1\). Consider \(F\colon X\times X \longrightarrow N(X)\) defined as
Here F is a coupling with respect to A and B. For any \(c\neq0\), the pair \(((0, c), (1, 0)) \in A \times B\) is a coupled best proximity point of F.
We discuss an application of our main result in uniformly convex Banach spaces.
Definition 2.5
([3])
A Banach space X is said to be uniformly convex if for every ϵ satisfying \(0 < \epsilon\leq2\), there is \(\delta(\epsilon )> 0\) such that, for all \(x, y \in X\),
We use the following results in our application.
Lemma 2.1
([16])
Let A be a nonempty closed and convex subset, and let B be a nonempty closed subset of a uniformly convex Banach space. Let \(\{x_{n}\} \) and \(\{z_{n}\}\) be sequences in A, and let \(\{y_{n}\}\) be a sequence in B satisfying:

(i)
\(\Vert z_{n}y_{n} \Vert \longrightarrow d(A, B)\) as \(n \longrightarrow\infty\),

(ii)
for every \(\epsilon> 0\), there exists \(N_{0}\) such that \(\Vert x_{m}y_{n} \Vert \leq d(A, B)+ \epsilon\) for all \(m > n \geq N_{0}\).
Then, for every \(\epsilon> 0\), there exists \(N_{1}\) such that \(\Vert x_{m}z_{n} \Vert \leq\epsilon\) for all \(m > n \geq N_{1}\).
Lemma 2.2
([16])
Let A be a nonempty closed and convex subset, and let B be a nonempty closed subset of a uniformly convex Banach space. Let \(\{x_{n}\} \) and \(\{z_{n}\}\) be sequences in A, and let \(\{y_{n}\}\) be a sequence in B satisfying:

(i)
\(\Vert x_{n}y_{n} \Vert \longrightarrow d(A, B)\) as \(n \longrightarrow\infty\),

(ii)
\(\Vert z_{n}y_{n} \Vert \longrightarrow d(A, B)\) as \(n \longrightarrow\infty\).
Then \(\Vert x_{n}z_{n} \Vert \longrightarrow0\) as \(n \longrightarrow\infty\).
Lemma 2.1 and Lemma 2.2 from [16] are generalized in the general case of a function modular space [25] as far as any uniformly convex Banach space is a function modular space generated by a function modular ρ that satisfies \((UC1)\) and has \(\Delta_{2}\)property. Therefore, Lemma 2.1 and Lemma 2.2 from [16] are particular cases of the results from [22] and [32] enriching the knowledge on the geometry of function modular spaces.
Main results
Theorem 3.1
Let A and B be two nonempty closed subsets of a complete metric space \((X, d)\). Let \(F\colon X\times X\longrightarrow B(X)\) be a coupling with respect to A and B satisfying the following inequality:
where \(x, v\in A\), \(y, u\in B\), and \(k\in(0, 1)\).

(1)
Then there exist two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in A and B, respectively, such that
$$ \underset{n \longrightarrow\infty}{\lim}d(x_{n}, y_{n+1}) = d(A, B)\quad \textit{and}\quad \underset{n \longrightarrow\infty}{\lim }d(y_{n}, x_{n+1}) = d(A, B). $$ 
(2)
Further, if \(\{x_{n}\}\) and \(\{y_{n}\}\) are Cauchy sequences, then F has a coupled best proximity point.
Proof
Starting with arbitrary \(x_{0}\in A\) and \(y_{0}\in B\), we construct two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X such that
Then \(\{x_{n}\}\) is a sequence in A, and \(\{y_{n}\}\) is a sequence in B. By (3.1) and (3.2) we have
By (3.1), (3.2), and (3.5) we have
Similarly,
Let us suppose that, for some integer \(n=p\),
and
By (3.1), (3.2), and (3.10) we obtain
Similarly, by (3.1), (3.2), and (3.10) we obtain
Therefore, by induction theorem we can write that, for all \(n \geq1\),
and
Since \(k \in(0, 1)\), taking the limits as \(n \longrightarrow\infty\) in these two inequalities, we have
and
Further, suppose that \(\{x_{n}\}\) and \(\{y_{n}\}\) are Cauchy sequences. Here X is complete, and A and B are closed subsets of X. Since \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences respectively in A and B, there exist \(x \in A\) and \(y \in B\) such that
By (3.13), (3.14), and (3.15) we have
and
Using (3.1), (3.2), and (3.16) we obtain
Taking the limit as \(n \longrightarrow\infty\) in this inequality and using (3.15), we have
which implies that \(D(x, F(x, y)) = d(A, B)\). Similarly, we can prove that \(D(y, F(y, x)) = d(A, B)\). Hence \(D(x, F(x, y)) = d(A, B)\) and \(D(y, F(y, x)) = d(A, B)\), that is, \((x, y)\) is a coupled best proximity point of F. □
Remark 3.1
It is a natural question to ask under what conditions the two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X should converge. We give an answer to this question in the context of uniformly convex Banach spaces for the singlevalued case. It is also possible to deduce a similar result by using UC property [31] in a metric space. We prefer to obtain it in uniformly convex Banach spaces since UC property is a consequence of the structure of the space for certain pairs of subsets.
Remark 3.2
If \(A \cap B \neq\emptyset\), then we obtain a multivalued coupled fixed point result for couplings. There are several recent works in which some coupled fixed point results are obtained [11, 15, 28, 29]. Further, it has been pointed out in [19] that several generalizations of existing results in fixed point theory are not actual generalizations. The fixed point result obtained under the conditions mentioned above is not a direct generalization of any existing result. Since multivalued couplings of the above type have not appeared earlier in the literature and therefore is not relevant to the discussion presented in [19].
Example 3.1
Let \(X= R^{2}\) with the metric d defined as \(d((x_{1}, y_{1}), (x_{2}, y_{2}))= \vert x_{1}  x_{2} \vert + \vert y_{1}  y_{2} \vert \). Let \(A=\{0\}\times[0, \infty)\) and \(B=\{1\}\times(\infty, 0]\). Then A and B are nonempty closed subsets of X, and \(d(A, B)=1\). Let \(F\colon X\times X \longrightarrow B(X)\) be defined as
Let \(x, v \in A\) and \(y, u \in B\). Then \(x= (0, a_{1})\), \(v= (0, a_{2})\), \(y= (1, b_{1})\), and \(u= (1, b_{2})\) for some \(a_{1}, a_{2}\geq0\) and \(b_{1}, b_{2}\leq0\). Then
Since \((a_{1}+a_{2})\geq0\), \((b_{1}+b_{2})\leq0\), it follows that inequality (3.1) holds for all \(\frac{1}{2} \leq k < 1\).
For any \(x_{0} = (0, a)\in A\) and \(y_{0} = (1, b)\in B\), consider \(x_{n}= (0, \frac{ab}{4^{n}})\) and \(y_{n} = (1, \frac{ab}{4^{n}})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) satisfy (1) and (2) of Theorem 3.1. Then, applying Theorem 3.1, we conclude that \(((0, 0), (1, 0))\) is a coupled best proximity point of F.
Application to uniformly convex Banach spaces
In this section, we discuss the implications of our result in the previous section in uniformly convex Banach spaces for the case of singlevalued couplings.
Theorem 4.1
Let A and B be two nonempty closed subsets of a Banach space X. Let \(F\colon X\times X\longrightarrow X\) be a coupling with respect to A and B satisfying the following inequality:
where \(x, v\in A\), \(y, u\in B\), and \(k\in(0, 1)\).

(1)
Then, for arbitrary \((x_{0}, y_{0})\in A \times B\), the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) respectively in A and B constructed as
$$ x_{n+1} = F(y_{n}, x_{n}) \quad\textit{and}\quad y_{n+1} = F(x_{n}, y_{n})\quad \textit {for all } n\geq0 $$(4.2)satisfy
$$ \underset{n\longrightarrow\infty}{\lim} \Vert x_{n} y_{n+1} \Vert = d(A, B) \quad\textit{and}\quad \underset{n \longrightarrow\infty}{\lim} \Vert y_{n} x_{n+1} \Vert = d(A, B). $$ 
(2)
Further, if \(\{x_{n}\}\) and \(\{y_{n}\}\) are Cauchy sequences, then F has a coupled best proximity point.
Proof
We know that, for every \(x\in X\), \(\{x\}\in B(X)\). We define the mapping \(T\colon X\times X \longrightarrow B(X)\) as \(T(x, y) = \{F(x, y)\} \text{ for } x, y \in X\). Then all the conditions of the theorem reduce to the conditions of Theorem 3.1, and hence by Theorem 3.1, we have the results in (1) and (2) of the theorem. □
Remark 4.1
It is a natural question to ask for circumstances under which both the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge so that Theorem 4.1 is applicable. An answer is given further by showing, by an application of Lemma 2.2, that this is the case if the coupling is between two closed and convex subsets of a uniformly convex Banach space. In general, Lemma 2.2 is not satisfied in a metric space. If the result of the lemma is valid with respect to subsets A and B in a metric space, then \((A, B)\) is said to satisfy the UCproperty with respect to the pair \((A, B)\). The UCproperty was introduce by Suzuki et al. [31]. We do not use this property in metric spaces; the lemma is rather used for an application in uniformly convex Banach spaces. So we omit any further reference of the UCproperty.
Lemma 4.1
Let \(F\colon X\times X \longrightarrow X\), where X is a normed linear space, be a coupling with respect to two subsets A and B of X. Suppose that F satisfies (4.1). Then, for arbitrary \((x_{0}, y_{0})\in A \times B\), the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) constructed as in (4.2) satisfy \(\underset{n \longrightarrow \infty}{\lim} \Vert x_{n}y_{n} \Vert \longrightarrow d(A, B)\).
Proof
It is clear from (4.2) that \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in A and B, respectively. Using (4.1) and (4.2), for \(n\geq1\), we have
By repeated application of (4.3) we get
Since \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in A and B, respectively, we have
It follows from (4.4) and (4.5) that
Since \(k\in(0, 1)\), taking the limit as \(n \longrightarrow\infty\) in this inequality, we obtain \(\underset{n \longrightarrow\infty}{\lim } \Vert x_{n}y_{n} \Vert = d(A, B)\). □
Remark 4.2
Lemma 4.1 is also generally valid in metric spaces. It is not necessary to assume X to be a normed linear space. We use this result only in the theorem for uniformly convex Banach spaces. For this reason, in particular, we assume that X is a normed linear space.
Theorem 4.2
Let A and B be two nonempty closed and convex subsets of a uniformly convex Banach space, and let \(F\colon X\times X \longrightarrow X\) be a coupling with respect to A and B. Suppose that F satisfies (4.1). Then F has a coupled best proximity point.
Proof
Let \(x_{0}\in A\) and \(y_{0}\in B\). We construct two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) respectively in A and B that satisfy (4.2), that is,
By Theorem 4.1 we have
By Lemma 2.2 we have
Again, by Theorem 4.1 we have
By Lemma 2.2 we have
Next, we prove that \(\{x_{2n}\}\), \(\{y_{2n}\}\), \(\{x_{2n+1}\}\), and \(\{ y_{2n+1}\}\) are Cauchy sequences. We further consider the case of the sequences \(\{x_{2n}\}\) and \(\{y_{2n}\}\). From (4.6) and (4.7) we have that
We first establish that, given \(\epsilon>0\), we can find a positive integer N such that, for all \(m, n > N\),
In view of Lemma 4.1, we can find a positive integer \(p_{0}\) such that, for all \(n\geq p_{0}\),
If (4.9) is not valid, then for some \(\epsilon>0\), in view of this inequality, for all \(p\geq p_{0}\), there exist \(n(p)> m(p)> p\) for which
and
Then
Taking the limit as \(p\longrightarrow\infty\) and using (4.8), we obtain
Again, for all \(p\geq p_{0}\),
and
Taking the limit as \(p \longrightarrow\infty\) in the last two inequalities and using (4.8) and (4.10), we conclude that
Again,
Taking the limit as \(p \longrightarrow\infty\) in this inequality, we have
Since \(0 < k < 1\), it is a contradiction. Therefore, (4.9) holds. Combining this fact with Lemma 4.1, by Lemma 2.1 we conclude that \(\{x_{2n}\}\) and \(\{y_{2n}\}\) are a Cauchy sequences in A and B, respectively. The sets A and B being closed, there exist \(x\in A\) and \(y\in B\) such that
Proceeding exactly as in the previous case, we have that there exist \(\overline{x}\in A\) and \(\overline{y}\in B\) for which
Then using Lemma 4.1, (4.12), and (4.13), we conclude that
Again, using Theorem 4.1(1) with (4.12) and (4.13), we conclude that
Using Lemma 2.2 with (4.14) and (4.15), we conclude that
From (4.12) and (4.13) it follows that
Applying Theorem 4.1, we can prove that \(d(A, B)= \Vert x  F(x, y) \Vert = \Vert y  F(y, x) \Vert \), that is, \((x, y)\) is a coupled best proximity of F. □
Example 4.1
Let us consider \(X= \mathbb {R}\) and \(\Vert x \Vert = \vert x \vert \) for \(x\in\mathbb {R}\). If \(A= (\infty, 1]\) and \(B= [1, \infty)\), then A and B are closed convex subsets of X. Let \(F\colon X\times X\rightarrow X\) be defined as
Then inequality (4.1) is satisfied. Here, X being a uniformly convex Banach space, we can apply Theorem 4.2 to conclude that there is a coupled proximity point of F.
Conclusion
In Sect. 4, we have shown that the coupling has a coupled best proximity point in uniformly convex Banach spaces automatically by the properties of the space. Particularly, the result is valid in Hilbert spaces. It remains to be investigated under what other conditions the coupled best proximity points exist for such couplings. Also, a feature of the present work is that the results are obtained without any continuity assumption on the coupling.
Regarding error estimates, the first such results were obtained for cyclic contraction map in [33] and for coupled best proximity points for cyclic contractive mappings in [21]. It may also be possible to obtain error estimates for coupled best proximity points of coupling through the iterated sequence (4.2) in Theorem 4.1 by following the ideas of [21, 33]. Such problems of error estimation arising out of the considerations of couplings as in the present case can be taken up in future works as open problems.
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The authors gratefully acknowledge the suggestions made by the learned referees.
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The work is supported by the Science and Engineering Research Board, Government of India, under Research Project No. PDF/2016/000353. The support is gratefully acknowledged.
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Choudhury, B.S., Maity, P., Metiya, N. et al. Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces. J Inequal Appl 2018, 130 (2018). https://doi.org/10.1186/s1366001817200
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MSC
 54H10
 54H25
 47H10
Keywords
 Metric spaces
 Coupling
 Best proximity pair
 Optimal approximate solution
 Uniformly convex Banach space