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 Open Access
Best proximity points and extension of MizoguchiTakahashi’s fixed point theorems
 Poom Kumam^{1},
 Hassen Aydi^{2},
 Erdal Karapınar^{3} and
 Wutiphol Sintunavarat^{4}Email author
https://doi.org/10.1186/168718122013242
© Kumam et al.; licensee Springer. 2013
 Received: 30 May 2013
 Accepted: 26 September 2013
 Published: 7 November 2013
Abstract
In this paper, we introduce a multivalued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for nonself mappings. We also establish a best proximity point for such type contraction mappings in the context of metric spaces. Later, we characterize this result to investigate the existence of best proximity point theorems in uniformly convex Banach spaces. We state some illustrative examples to support our main theorems. Our results extend, improve and enrich some celebrated results in the literature, such as Nadler’s fixed point theorem, Mizoguchi and Takahashi’s fixed point theorem.
MSC:41A65, 46B20, 47H09, 47H10.
Keywords
 best proximity points
 multivalued contraction
 cyclic contraction
 $\mathcal{MT}$function (or ℛfunction)
Dedication
Dedicated to Prof. W Takahashi on the occasion of his 70th birthday
1 Introduction
It is evident that the fixed point theory is one of the fundamental tools in nonlinear functional analysis. The celebrated Banach contraction mapping principle [1] is the most known and crucial result in fixed point theory. It says that each contraction in a complete metric space has a unique fixed point. This theorem not only guarantees the existence and uniqueness of the fixed point but also shows how to evaluate this point. By virtue of this fact, the Banach contraction mapping principle has been generalized in many ways over the years (see e.g., [2–5]).
where $(X,d)$ is a metric space, and A, B are subsets of X. A best proximity point represents an optimal approximate solution to the equation $Tx=x$ whenever a nonselfmapping T has no fixed point. It is clear that a fixed point coincides with a best proximity point if $d(A,B)=0$. Since a best proximity point reduces to a fixed point if the underlying mapping is assumed to be selfmappings, the best proximity point theorems are natural generalizations of the Banach’s contraction principle.
In 1969, Fan [6] introduced the notion of a best proximity and established a classical best approximation theorem. More precisely, if $T:A\to B$ is a continuous mapping, then there exists an element $x\in A$ such that $d(x,Tx)=d(Tx,A)$, where A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B. Subsequently, many researchers have studied the best proximity point results in many ways (see in [7–14] and the references therein).
In the same year, Nadler [15] gave a useful lemma about Hausdorff metric. In paper [15], the author also characterized the celebrated Banach fixed point theorem in the context of multivalued mappings.
Lemma 1.1 (Nadler [15])
If $A,B\in CB(X)$ and $a\in A$, then for each $\u03f5>0$, there exists $b\in B$ such that $d(a,b)\le H(A,B)+\u03f5$.
Theorem 1.2 (Nadler [15])
for all $x,y\in X$, then T has at least one fixed point, that is, there exists $z\in X$ such that $z\in Tz$.
The theory of multivalued mappings has applications in many areas such as in optimization problem, control theory, differential equations, economics and many branches in analysis. Due to this fact, a number of authors have focused on the topic and have published some interesting fixed point theorems in this frame (see [16–19] and references therein). Following this trend, in 1989, Mizoguchi and Takahashi [17] proved a generalization (Theorem 1.3 below) of Theorem 1.2; see Theorem 2 in Alesina et al. [20]. Theorem 2 is a partial answer of Problem 9 in Reich [21]. See also [22–24].
Theorem 1.3 (Mizoguchi and Takahashi [17])
for all $t\in [0,\mathrm{\infty})$. Then T has at least one fixed point, that is, there exists $z\in X$ such that $z\in Tz$.
Remark 1.4 In original statement of Mizoguchi and Takahashi [17], the domain α is $(0,\mathrm{\infty})$. However both are equivalent, because $d(x,y)=0$ implies that $H(Tx,Ty)=0$.
Remark 1.5 We obtain that if $\alpha :[0,\mathrm{\infty})\to [0,1)$ is a nondecreasing function or a nonincreasing function, then α is a $\mathcal{MT}$function. Therefore, the class of $\mathcal{MT}$functions is a rich class, and so this class has been investigated heavily by many authors.
In 2007, Eldred et al. [25] claimed that Theorem 1.3 is equivalent to Theorem 1.2 in the following sense:
 (i)
M is Tinvariant, that is, $Tx\subseteq M$ for all $x\in M$,
 (ii)
T satisfies (1.1) for all $x,y\in M$.
Very recently, Suzuki [26] gave an example which says that MizoguchiTakahashi’s fixed point theorem for multivalued mappings is a real generalization of Nadler’s result. In his remarkable paper, Suzuki also gave a very simple proof of MizoguchiTakahashi’s theorem.
On the other hand, KirkSrinavasanVeeramani [27] introduced the concept of a cyclic contraction.
Let A and B be two nonempty subsets of a metric space $(X,d)$, and let $T:A\cup B\to A\cup B$ be a mapping. Then T is called a cyclic map if $T(A)\subseteq B$ and $T(B)\subseteq A$. In addition, if T is a contraction, then T is called cyclic contraction.
The authors [27] give a characterization of Banach contraction mapping principle in complete metric spaces. After this initial paper, a number of papers has appeared on the topic in literature (see, e.g., [27–39]).
In this paper, we introduce the notion of a generalized multivalued cyclic contraction pair, which is an extension of MizoguchiTakahashi’s contraction mappings for nonself version and establish a best proximity point of such mappings in metric spaces via property UC^{∗} due to Sintunavarat and Kumam [40]. Further, by applying the main results, we investigate best proximity point theorems in a uniformly convex Banach space. We also give some illustrative examples, which support our main results. Our results generalize, improve and enrich some wellknown results in literature.
2 Preliminaries
for every $A,B\in CB(X)$, where $d(a,B)=inf\{d(a,b):b\in B\}$ is the distance from a to $B\subseteq X$.
 (i)
H is a metric on $CB(X)$.
 (ii)
If $A,B\in CB(X)$ and $q>1$ is given, then for every $a\in A$, there exists $b\in B$ such that $d(a,b)\le qH(A,B)$.
We notice that a best proximity point reduces to a fixed point for a multivalued mapping if the underlying mapping is a selfmapping.
 (i)strictly convex if the following implication holds for all $x,y\in X$:$\parallel x\parallel =\parallel y\parallel =1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}x\ne y\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\parallel \frac{x+y}{2}\parallel <1;$
 (ii)uniformly convex if for each ϵ with $0<\u03f5\le 2$, there exists $\delta >0$ such that the following implication holds for all $x,y\in X$:$\parallel x\parallel \le 1,\phantom{\rule{2em}{0ex}}\parallel y\parallel \le 1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\parallel xy\parallel \ge \u03f5\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\parallel \frac{x+y}{2}\parallel <1\delta .$
It is easy to see that a uniformly convex Banach space X is strictly convex, but the converse is not true.
Definition 2.3 [41]
Let A and B be nonempty subsets of a metric space $(X,d)$. The ordered pair $(A,B)$ is said to satisfy the property UC if the following holds:
If $\{{x}_{n}\}$ and $\{{z}_{n}\}$ are sequences in A, and $\{{y}_{n}\}$ is a sequence in B such that $d({x}_{n},{y}_{n})\to d(A,B)$ and $d({z}_{n},{y}_{n})\to d(A,B)$, then $d({x}_{n},{z}_{n})\to 0$.
Example 2.4 [41]
 (i)
Every pair of nonempty subsets A, B of a metric space $(X,d)$ such that $d(A,B)=0$.
 (ii)
Every pair of nonempty subsets A, B of a uniformly convex Banach space X such that A is convex.
 (iii)
Every pair of nonempty subsets A, B of a strictly convex Banach space, where A is convex and relatively compact and the closure of B is weakly compact.
Definition 2.5 [40]
Let A and B be nonempty subsets of a metric space $(X,d)$. The ordered pair $(A,B)$ satisfies the property UC^{∗} if $(A,B)$ has property UC, and the following condition holds:
 (i)
$d({z}_{n},{y}_{n})\to d(A,B)$.
 (ii)For every $\u03f5>0$, there exists $N\in \mathbb{N}$ such that$d({x}_{m},{y}_{n})\le d(A,B)+\u03f5$
for all $m>n\ge N$,
then $d({x}_{n},{z}_{n})\to 0$.
 (i)
Every pair of nonempty subsets A, B of a metric space $(X,d)$ such that $d(A,B)=0$.
 (ii)
Every pair of nonempty closed subsets A, B of uniformly convex Banach space X such that A is convex (see Lemma 3.7 in [42]).
3 Best proximity point for multivalued mapping theorems
In this section, we investigate the existence and convergence of best proximity points for generalized multivalued cyclic contraction pairs and obtain some new results on fixed point theorems for such mappings. We begin by introducing the notion of multivalued cyclic contraction.
for all $x\in A$ and $y\in B$.
Note that if $(T,S)$ is a generalized multivalued cyclic contraction, then $(S,T)$ is also a generalized multivalued cyclic contraction. Here, we state the main results of this paper on the existence of best proximity points for a generalized multivalued cyclic contraction pair, which satisfies the property UC^{∗} in metric spaces.
Theorem 3.2 Let A and B be nonempty closed subsets of a complete metric space X such that $(A,B)$ and $(B,A)$ satisfy the property UC^{∗}. Let $T:A\to CB(B)$ and $S:B\to CB(A)$. If $(T,S)$ is a generalized multivalued cyclic contraction pair, then T has a best proximity point in A, or S has a best proximity point in B.
Proof We consider two cases separately.
for all $t\in [0,\mathrm{\infty})$.
for all $n\in \mathbb{N}$.
Hence we get $d(z,Tz)=d(A,B)$. Analogously, we also obtain $d(z,Sz)=d(A,B)$.
Case 2. We will show that T or S have best proximity points in A and B, respectively, under the assumption of $d(A,B)>0$. Suppose, to the contrary, that for all $a\in A$, $d(a,Ta)>d(A,B)$ and for all ${b}^{\prime}\in B$, $d(S{b}^{\prime},{b}^{\prime})>d(A,B)$.
for all $t\in [d(A,B),\mathrm{\infty})$. So we derive ${lim\hspace{0.17em}sup}_{x\to {t}^{+}}\beta (x)<1$ and $\alpha (t)<\beta (t)$ for all $t\in [d(A,B),\mathrm{\infty})$.
for all $a\in A$ and $b\in Ta$.
which is a contradiction. Therefore, (3.16) holds.
From (3.26) and (3.30), we have a contradiction. Therefore, T has a best proximity point in A or S has a best proximity point in B. This completes the proof. □
Remark 3.3 If $d(A,B)=0$, then Theorem 3.2 yields existence of a fixed point in $A\cap B$ of two multivalued nonself mappings S and T. Moreover, if $A=B=X$ and $T=S$, then Theorem 3.2 reduces to MizoguchiTakahashi’s fixed point theorem [17].
Note that every pair of nonempty closed subsets A, B of a uniformly convex Banach space such that A is convex satisfies the property UC^{∗}. Therefore, we obtain the following corollary.
Corollary 3.4 Let A and B be nonempty closed convex subsets of a uniformly convex Banach space $X,T:A\to CB(B)$ and $S:B\to CB(A)$. If $(T,S)$ is a generalized multivalued cyclic contraction pair, then T has a best proximity point in A or S has a best proximity point in B.
Next, we give some illustrative examples of Corollary 3.4.
Example 3.5 Consider the uniformly convex Banach space $X=\mathbb{R}$ with Euclidean norm. Let $A=[1,2]$ and $B=[2,1]$. Then A and B are nonempty closed and convex subsets of X and $d(A,B)=2$. Since A and B are convex, we have $(A,B)$ and $(B,A)$ satisfy the property UC^{∗}.
for all $y\in B$.
Let $\alpha :[d(A,B),\mathrm{\infty})\to [0,1)$ be defined by $\alpha (t)=\frac{1}{2}$ for all $t\in [d(A,B),\mathrm{\infty})=[2,\mathrm{\infty})$. Next, we show that $(T,S)$ is a generalized multivalued cyclic contraction pair with $\alpha (t)=\frac{1}{2}$ for all $t\in [2,\mathrm{\infty})$.
Therefore, all assumptions of Corollary 3.4 are satisfied, and then T has a best proximity point in A, that is, a point $x=1$. Moreover, S also has a best proximity point in B, that is, a point $y=1$.
Then A and B are nonempty closed and convex subsets of X and $d(A,B)=2$. Since A and B are convex, we have $(A,B)$ and $(B,A)$ satisfy the property UC^{∗}.
for all $x,y\ge 0$.
Let $\alpha :[d(A,B),\mathrm{\infty})\to [0,1)$ define by $\alpha (t)=\frac{1}{2}$ for all $t\in [d(A,B),\mathrm{\infty})=[2,\mathrm{\infty})$. Next, we show that $(T,S)$ is a generalized multivalued cyclic contraction pair with mapping $\alpha (t)=\frac{1}{2}$ for all $t\in [2,\mathrm{\infty})$.
Therefore, all assumptions of Corollary 3.4 are satisfied, and then T has a best proximity point in A that is a point $(0,0)$. Furthermore, S also has a best proximity point in B that is a point $(2,0)$.
Open problems

In Theorem 3.2, can we replace the property UC^{∗} by a more general property?

In Theorem 3.2, can we drop the property UC^{∗}?

Can we extend the result in this paper to another spaces?
Declarations
Acknowledgements
Poom Kumam was supported by the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No. MRG5580213).
Authors’ Affiliations
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