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 Open Access
Best proximity point theorems via fixed point theorems for multivalued mappings
 Pinya Ardsalee^{1} and
 Satit Saejung^{1, 2}Email author
https://doi.org/10.1186/s1366301605165
© Ardsalee and Saejung 2016
 Received: 11 August 2015
 Accepted: 28 February 2016
 Published: 11 March 2016
Abstract
It is well known that the concept of a best proximity point includes that of a fixed point as a special case. In this paper, we show that the best proximity point theorems of Basha and Shahzad (Fixed Point Theory Appl. 2012:42, 2012) and of FernándezLeón (J. Nonlinear Convex Anal. 15(2):313324, 2014) can be regarded as a fixed point theorem for multivalued mappings which is modified as regards the results of Mizoguchi and Takahashi (J. Math. Anal. Appl. 141(1):177188, 1989) and of Kada et al. (Math. Jpn. 44(2):381391, 1996).
Keywords
 best proximity point
 multivalued mapping
 metric space
MSC
 47H10
 47H09
 54E50
1 Introduction
Let X be any nonempty set and \(T:X\to X\) be a given mapping. A point \(x\in X\) such that \(x=Tx\) is called a fixed point of T. Many problems can be reformulated to the problem of finding a fixed point of a certain mapping. If T is not a selfmapping, it is plausible that the equation \(x=Tx\) has no solution. In this situation, we may find an element \(x\in X\) which is close to Tx in some sense.
Basha [5] proposed the following result for the existence of a best proximity point of a nonselfmapping.
Theorem 1
([5], Theorem 3.1)

A and B are closed;

B is approximatively compact with respect to A;

T is a proximal contraction, that is, there exists \(\alpha\in [0,1)\) such that, for all \(u,v,x,y\in A\),implies$$d(u,Tx)=d(A,B)=d(v,Ty) $$$$d(u,Tx)+d(Tx,Ty)+d(Ty,v)\leq\alpha d(x,y). $$
 (a)
there exists a unique element \(x\in A\) such that \(d(x,Tx)=d(A,B)\);
 (b)
if \(\{x_{n}\}\) is a sequence in \(A_{0}\) satisfying \(d(x_{n+1},Tx_{n})=d(A,B)\) for all \(n\geq0\), then \(\lim_{n\to\infty}x_{n}=x\).
It is clear that Theorem 1 extends Banach’s contraction principle in the setting that \(A=B=X\). By the way, there are plenty of papers which had generalized this result (for example, see [1, 2, 6]).
Basha and Shahzad [1] introduced the following two concepts of contractiveness for nonselfmappings.
Definition 2
([1])
 (a)a generalized proximal contraction of the first kind if there exist nonnegative numbers α, β, γ with \(\alpha+2\beta+2\gamma<1\) such that the conditionimplies$$d(u,Tx)=d(A,B)=d(v,Ty) $$$$d(u,v)\leq\alpha d(x,y)+\beta d(x,u)+\beta d(y,v)+\gamma d(x,v)+\gamma d(y,u); $$
 (b)a generalized proximal contraction of the second kind if there exist nonnegative numbers α, β, γ with \(\alpha+2\beta+2\gamma<1\) such that the conditionimplies$$d(u,Tx)=d(A,B)=d(v,Ty) $$$$\begin{aligned} d(Tu,Tv)\leq{}&\alpha d(Tx,Ty)+\beta d(Tx,Tu)+\beta d(Ty,Tv)\\ &{}+\gamma d(Tx,Tv)+\gamma d(Ty,Tu). \end{aligned}$$
Remark 3
Every proximal contraction is a generalized proximal contraction of the first kind.
In this paper, we show that the problem of finding a best proximity point recently established by FernándezLeón [2] and Basha and Shahzad [1] reduces to a problem of finding a fixed point of a multivalued mapping. Recall that \(x\in X\) is a fixed point of a multivalued mapping \(T:X\to2^{X}\setminus\{\varnothing\}\) if \(x\in Tx\). There are many conditions guaranteeing the existence of a fixed point of a multivalued mapping. Two of the classical works in this research are due to Nadler [7] and Caristi [8]. The interested reader is referred to [9], Chapter 5, for more discussion.
2 Main results
By studying the works of [4] and [3], we obtain the following fixed point theorem for a multivalued mapping.
Theorem 4
Proof
2.1 Results for a generalized proximal contraction of the first kind
We show that the following result of FernándezLeón [2] is a consequence of our Theorem 4.
Theorem 5
([2], Proposition 3.5)

\(A_{0}\) is closed;

T is a generalized proximal contraction of the first kind.
 (a)
there exists a unique element x in A such that \(d(x,Tx)=d(A,B)\);
 (b)
if \(\{x_{n}\}\) is a sequence in \(A_{0}\) satisfying \(d(x_{n+1},Tx_{n})=d(A,B)\) for each \(n\geq0\), then \(\lim_{n\to\infty }x_{n}= x\).
Proof
2.2 Results for a generalized proximal contraction of the second kind
The following result of FernándezLeón [2] is also a consequence of our Theorem 4.
Theorem 6
([2], Proposition 3.10)

\(T(A_{0})\) is closed;

T is a generalized proximal contraction of the second kind.
 (a)
there exists \(x\in A\) such that \(d(x,Tx)=d(A,B)\);
 (b)
if there is \(\widehat{x}\in A\) such that \(d(\widehat {x},T\widehat{x})=d(A,B)\), then \(T\widehat{x}=Tx\);
 (c)
if \(\{x_{n}\}\) is a sequence in \(A_{0}\) satisfying \(d(x_{n+1},Tx_{n})=d(A,B)\) for each \(n\geq0\), then \(\lim_{n\to\infty }Tx_{n}= Tx\).
Proof
Definition 8
([1])
Let \((X,d)\) be a metric space. Let A and B be nonempty subsets of X. The set B is said to be approximatively compact with respect to A if every sequence \(\{y_{n}\}\) of B satisfying the condition that \(\lim_{n\to\infty}d(x,y_{n})=d(x,B)\) for some x in A has a convergent subsequence.
We show that the following theorem of Basha and Shahzad [1] is also a consequence of our Theorem 4.
Theorem 9
([1], Theorem 3.4)

A, B are closed;

A is approximatively compact with respect to B;

T is continuous;

T is a generalized proximal contraction of the second kind.
 (a)
there is an element x in A such that \(d(x,Tx)=d(A,B)\);
 (b)
if there exists \(\widehat{x}\in A\) such that \(d(\widehat {x},T\widehat{x})=d(A,B)\), then \(T\widehat{x}=Tx\);
 (c)
if \(\{x_{n}\}\) is a sequence in \(A_{0}\) satisfying \(d(x_{n+1},Tx_{n})=d(A,B)\) for each \(n\geq0\), then \(\lim_{n\to\infty }Tx_{n}= Tx\).
Proof
We define the mappings \(S:T(A_{0})\to2^{T(A_{0})}\setminus\{ \varnothing\}\) and \(F:T(A_{0})\to[0,\infty)\) as the ones in the proof of Theorem 6. It follows that the condition (2.1) in Theorem 4 holds.
For a generalized proximal contraction of the first kind, the closedness of \(A_{0}\) is more general than the condition that B is approximatively compact with respect to A (see Proposition 3.3 of [2]). Hence Proposition 3.5 of [2] (see our Theorem 5) is a generalized version of Theorem 3.1 of [1]. However, this is not the case for a generalized proximal contraction of the second kind. The following example is applicable in Theorem 9 but not in Theorem 6. That is, there is a continuous generalized proximal contraction of the second kind \(T:A\to B\) such that \(T(A_{0})\) is not closed but A is approximatively compact with respect to B.
Example 10
Declarations
Acknowledgements
The authors would like to thank Professor Charles Chidume and the four referees for their comments. The first author is thankful to the Development and Promotion of Science and Technology Talents Project (DPST) for financial support. The second author is partially supported by the Research Center for Environmental and Hazardous Substance Management, Khon Kaen University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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