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Best proximity points and extension of MizoguchiTakahashi’s fixed point theorems
Fixed Point Theory and Applications volume 2013, Article number: 242 (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.
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]).
Investigation of the existence and uniqueness of a fixed point of nonself mappings is one of the interesting subjects in fixed point theory. In fact, given nonempty closed subsets A and B of a complete metric space $(X,d)$, a contraction nonselfmapping $T:A\to B$ does not necessarily yield a fixed point $Tx=x$. In this case, it is very natural to investigate whether there is an element x such that $d(x,Tx)$ is minimum. A notion of best proximity point appears at this point. A point x is called best proximity point of $T:A\to B$ if
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])
Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$. If there exists $r\in [0,1)$ such that
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])
Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$. Assume that
for all $x,y\in X$, where $\alpha :[0,\mathrm{\infty})\to [0,1)$ is $\mathcal{MT}$function (or ℛfunction), i.e.,
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:
If a mapping $T:X\to CB(X)$ satisfies (1.2), then there exists a nonempty complete subset M of X satisfying the following:

(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
In this section, we recall some basic definitions and elementary results in literature. Throughout this paper, we denote by ℕ the set of all positive integers, by ℝ the set of all real numbers and by ${\mathbb{R}}_{+}$ the set of all nonnegative real numbers. We denote by $CB(X)$ the class of all nonempty closed bounded subsets of a metric space $(X,d)$. The Hausdorff metric induced by d on $CB(X)$ is given by
for every $A,B\in CB(X)$, where $d(a,B)=inf\{d(a,b):b\in B\}$ is the distance from a to $B\subseteq X$.
Remark 2.1 The following properties of the Hausdorff metric induced by d are well known:

(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 $d(a,b)\le qH(A,B)$.
Definition 2.2 Let A and B be nonempty subsets of a metric space $(X,d)$ and let $T:A\to {2}^{B}$ be a multivalued mapping. A point $x\in A$ is said to be a best proximity point of a multivalued mapping T if it satisfies the condition that
We notice that a best proximity point reduces to a fixed point for a multivalued mapping if the underlying mapping is a selfmapping.
A Banach space X is said to be

(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]
The following are examples of a pair of nonempty subsets $(A,B)$ satisfying the property UC.

(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:
If $\{{x}_{n}\}$ and $\{{z}_{n}\}$ are sequences in A, and $\{{y}_{n}\}$ is a sequence in B satisfying

(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 $d({x}_{n},{z}_{n})\to 0$.
Example 2.6 The following are examples of a pair of nonempty subsets $(A,B)$ satisfying the property UC^{∗}.

(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.
Definition 3.1 Let A and B be nonempty subsets of a metric space $X,T:A\to {2}^{B}$ and $S:B\to {2}^{A}$. The ordered pair $(T,S)$ is said to be a generalized multivalued cyclic contraction if there exists a function $\alpha :[d(A,B),\mathrm{\infty})\to [0,1)$ with
for each $t\in [d(A,B),\mathrm{\infty})$ such that
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.
Case 1. Suppose that $d(A,B)=0$. Define the function $\beta :[d(A,B),\mathrm{\infty})\to [0,1)$ by
for $t\in [d(A,B),\mathrm{\infty})=[0,\mathrm{\infty})$. Then we obtain that
for all $t\in [0,\mathrm{\infty})$.
Now, we will construct the sequence $\{{x}_{n}\}$ in X. Let ${x}_{0}\in A$ be an arbitrary point. Since $T{x}_{0}\in CB(B)$, we can choose ${x}_{1}\in T{x}_{0}$. If ${x}_{1}={x}_{0}$, we have ${x}_{0}\in T{x}_{0}$, and then ${x}_{0}$ is a best proximity point of T. Also, it follows from (3.1) with $x={x}_{0}$ and $y={x}_{1}$ that $T{x}_{0}=S{x}_{1}$. This implies that ${x}_{1}\in S{x}_{1}$. Therefore, ${x}_{1}$ is a best proximity point of S, and we finish the proof. Otherwise, if ${x}_{0}\ne {x}_{1}$, by Lemma 1.1, there exists ${x}_{2}\in S{x}_{1}$ such that
If ${x}_{2}={x}_{1}$, we have ${x}_{1}\in S{x}_{1}$, and then ${x}_{1}$ is a best proximity point of S. Also, it follows from (3.1) with $x={x}_{2}$ and $y={x}_{1}$ that $T{x}_{2}=S{x}_{1}$. This implies that ${x}_{2}\in T{x}_{2}$. Therefore, ${x}_{2}$ is a best proximity point of T, and we finish the proof. Otherwise, if ${x}_{2}\ne {x}_{1}$, by Lemma 1.1, there exists ${x}_{3}\in T{x}_{2}$ such that
By repeating this process, we can find ${x}_{n}$ such that
for all $n\in \mathbb{N}$.
Thus, for fixed ${x}_{0}\in A$, we can define a sequence $\{{x}_{n}\}$ in X satisfying
such that
for $n\in \mathbb{N}$. Therefore, $\{d({x}_{n},{x}_{n+1})\}$ is a strictly decreasing sequence in ${\mathbb{R}}_{+}$. So $\{d({x}_{n},{x}_{n+1})\}$ converges to some nonnegative real number ρ. Since ${lim\hspace{0.17em}sup}_{s\to {\rho}^{+}}\beta (s)<1$ and $\beta (\rho )<1$, there exist $r\in [0,1)$ and $\eta >0$ such that $\beta (s)\le r$ for all $s\in [\rho ,\rho +\eta ]$. We can take $\nu \in \mathbb{N}$ such that
for all $n\in \mathbb{N}$ with $n\ge \nu $. Then since
for $n\in \mathbb{N}$ with $n\ge \nu $, we have
that is, $\{{x}_{n}\}$ is a Cauchy sequence. Since X is complete, $\{{x}_{n}\}$ converges to some point $z\in X$. Clearly, the subsequences $\{{x}_{2n}\}$ and $\{{x}_{2n1}\}$ converge to the same point z. Since A and B are closed, we derive that $z\in A\cap B$. We consider that
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 ${b}^{\prime}\in B$, $d(S{b}^{\prime},{b}^{\prime})>d(A,B)$.
Next, we define a function $\beta :[d(A,B),\mathrm{\infty})\to [0,1)$ by
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 each $a\in A$ and $b\in Ta$, we have
Therefore,
and then we get
Since $(T,S)$ is a generalized multivalued cyclic contraction pair, by (3.2), we conclude
for all $a\in A$ and $b\in Ta$.
Similarly, we obtain that for each ${b}^{\prime}\in B$ and ${a}^{\prime}\in S{b}^{\prime}$, we have
Next, we will construct the sequence $\{{x}_{n}\}$ in $A\cup B$. Let ${x}_{0}$ be an arbitrary point in A and ${x}_{1}\in T{x}_{0}\subseteq B$. From (3.3), there exists ${x}_{2}\in S{x}_{1}\subseteq A$ such that
Since ${x}_{1}\in B$ and ${x}_{2}\in S{x}_{1}$, from (3.4), we can find ${x}_{3}\in T{x}_{2}$ such that
Analogously, we can define the sequence $\{{x}_{n}\}$ in $A\cup B$ such that
and
for all $n\in \mathbb{N}$. Since $\beta (d({x}_{n1},{x}_{n}))<1$ and $d(A,B)<d({x}_{n1},{x}_{n})$ for all $n\in \mathbb{N}$, we get
for all $n\in \mathbb{N}$. Therefore, $\{d({x}_{n1},{x}_{n})\}$ is a strictly decreasing sequence in ${\mathbb{R}}_{+}$ and bounded below. So the sequence $\{d({x}_{n1},{x}_{n})\}$ converges to some nonnegative real number d. Since ${lim\hspace{0.17em}sup}_{x\to {d}^{+}}\beta (x)<1$ and $\beta (d)<1$, there exist ${d}_{0}\in [0,1)$ and $\u03f5>0$ such that $\beta (s)\le {d}_{0}$ for all $s\in [d,d+\u03f5]$. Now, we can take ${N}_{0}\in \mathbb{N}$ such that
for all $n\ge {N}_{0}$. From (3.7), we have
for all $n\ge {N}_{0}$. By the same consideration, we obtain
for all $n\ge {N}_{0}$. Since ${d}_{0}\in [0,1)$, we get
From (3.11), we conclude that
and
Since $\{{x}_{2n}\}$ and $\{{x}_{2n+2}\}$ are two sequences in A, and $\{{x}_{2n+1}\}$ is sequence in B with $(A,B)$ satisfies the property UC^{∗}, we derive that
Since $(B,A)$ satisfies the property UC^{∗}, and by (3.11), we find that
Next, we show that for each $\u03f5>0$, there exists $N\in \mathbb{N}$ such that for all $m>n\ge N$, we have
Suppose, to the contrary, that there exists ${\u03f5}_{0}>0$ such that for each $k\ge 1$, there is ${m}_{k}>{n}_{k}\ge k$ such that
Further, corresponding to ${n}_{k}$, we can choose ${m}_{k}$ in such a way that it is the smallest integer with ${m}_{k}>{n}_{k}\ge k$ satisfying (3.17). Then we have
and
From (3.18), (3.19) and the triangle inequality, we have
Using the fact that ${lim}_{k\to \mathrm{\infty}}d({x}_{2{m}_{k}},{x}_{2({m}_{k}1)})=0$. Letting $k\to \mathrm{\infty}$ in (3.20), we have
From (3.8), (3.9) and $(T,S)$ is a generalized multivalued cyclic contraction pair, we get
Letting $k\to \mathrm{\infty}$ in (3.22) and using (3.14), (3.15) and (3.21), we have
which is a contradiction. Therefore, (3.16) holds.
Since (3.12) and (3.16) hold, by using property UC^{∗} of $(A,B)$, we have $d({x}_{2n},{x}_{2m})\to 0$. Therefore, $\{{x}_{2n}\}$ is a Cauchy sequence. By the completeness of X and since A is closed, we get
for some $p\in \overline{A}=A$. But
for all $n\in \mathbb{N}$. From (3.11) and (3.23),
Since
for all $n\in \mathbb{N}$. By (3.23) and (3.24), we get
In a similar mode, we can conclude that the sequence $\{{x}_{2n1}\}$ is a Cauchy sequence in B. Since X is complete, and since B is closed, we have
for some $q\in \overline{B}=B$. Since
for all $n\in \mathbb{N}$. It follows from (3.11) and (3.27) that
Since
for all $n\in \mathbb{N}$, then by (3.27) and (3.28), we have
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^{∗}.
Let $T:A\to CB(B)$ and $S:B\to CB(A)$ be defined as
for all $x\in A$ and
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 $t\in [2,\mathrm{\infty})$.
For each $x\in A$ and $y\in B$, we have
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 $y=1$.
Example 3.6 Consider the uniformly convex Banach space $X={\mathbb{R}}^{2}$ with Euclidean norm. Let
and
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^{∗}.
Let $T:A\to CB(B)$ and $S:B\to CB(A)$ be defined as
and
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 $t\in [2,\mathrm{\infty})$.
For each $(0,x)\in A$ and $(2,y)\in B$, we have
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?
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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).
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Kumam, P., Aydi, H., Karapınar, E. et al. Best proximity points and extension of MizoguchiTakahashi’s fixed point theorems. Fixed Point Theory Appl 2013, 242 (2013). https://doi.org/10.1186/168718122013242
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Keywords
 best proximity points
 multivalued contraction
 cyclic contraction
 $\mathcal{MT}$function (or ℛfunction)