- Open Access
Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems
© Kumam et al.; licensee Springer. 2013
- Received: 30 May 2013
- Accepted: 26 September 2013
- Published: 7 November 2013
In this paper, we introduce a multi-valued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for non-self 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.
- best proximity points
- multi-valued contraction
- cyclic contraction
- -function (or ℛ-function)
Dedicated to Prof. W Takahashi on the occasion of his 70th birthday
It is evident that the fixed point theory is one of the fundamental tools in nonlinear functional analysis. The celebrated Banach contraction mapping principle  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 is a metric space, and A, B are subsets of X. A best proximity point represents an optimal approximate solution to the equation whenever a non-self-mapping T has no fixed point. It is clear that a fixed point coincides with a best proximity point if . Since a best proximity point reduces to a fixed point if the underlying mapping is assumed to be self-mappings, the best proximity point theorems are natural generalizations of the Banach’s contraction principle.
In 1969, Fan  introduced the notion of a best proximity and established a classical best approximation theorem. More precisely, if is a continuous mapping, then there exists an element such that , 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  gave a useful lemma about Hausdorff metric. In paper , the author also characterized the celebrated Banach fixed point theorem in the context of multi-valued mappings.
Lemma 1.1 (Nadler )
If and , then for each , there exists such that .
Theorem 1.2 (Nadler )
for all , then T has at least one fixed point, that is, there exists such that .
The theory of multi-valued 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  proved a generalization (Theorem 1.3 below) of Theorem 1.2; see Theorem 2 in Alesina et al. . Theorem 2 is a partial answer of Problem 9 in Reich . See also [22–24].
Theorem 1.3 (Mizoguchi and Takahashi )
for all . Then T has at least one fixed point, that is, there exists such that .
Remark 1.4 In original statement of Mizoguchi and Takahashi , the domain α is . However both are equivalent, because implies that .
Remark 1.5 We obtain that if is a nondecreasing function or a nonincreasing function, then α is a -function. Therefore, the class of -functions is a rich class, and so this class has been investigated heavily by many authors.
In 2007, Eldred et al.  claimed that Theorem 1.3 is equivalent to Theorem 1.2 in the following sense:
M is T-invariant, that is, for all ,
T satisfies (1.1) for all .
Very recently, Suzuki  gave an example which says that Mizoguchi-Takahashi’s fixed point theorem for multi-valued mappings is a real generalization of Nadler’s result. In his remarkable paper, Suzuki also gave a very simple proof of Mizoguchi-Takahashi’s theorem.
On the other hand, Kirk-Srinavasan-Veeramani  introduced the concept of a cyclic contraction.
Let A and B be two nonempty subsets of a metric space , and let be a mapping. Then T is called a cyclic map if and . In addition, if T is a contraction, then T is called cyclic contraction.
The authors  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 multi-valued cyclic contraction pair, which is an extension of Mizoguchi-Takahashi’s contraction mappings for non-self version and establish a best proximity point of such mappings in metric spaces via property UC∗ due to Sintunavarat and Kumam . 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 well-known results in literature.
for every , where is the distance from a to .
H is a metric on .
If and is given, then for every , there exists such that .
We notice that a best proximity point reduces to a fixed point for a multi-valued mapping if the underlying mapping is a self-mapping.
- (i)strictly convex if the following implication holds for all :
- (ii)uniformly convex if for each ϵ with , there exists such that the following implication holds for all :
It is easy to see that a uniformly convex Banach space X is strictly convex, but the converse is not true.
Definition 2.3 
Let A and B be nonempty subsets of a metric space . The ordered pair is said to satisfy the property UC if the following holds:
If and are sequences in A, and is a sequence in B such that and , then .
Example 2.4 
Every pair of nonempty subsets A, B of a metric space such that .
Every pair of nonempty subsets A, B of a uniformly convex Banach space X such that A is convex.
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 
Let A and B be nonempty subsets of a metric space . The ordered pair satisfies the property UC∗ if has property UC, and the following condition holds:
- (ii)For every , there exists such that
for all ,
Every pair of nonempty subsets A, B of a metric space such that .
Every pair of nonempty closed subsets A, B of uniformly convex Banach space X such that A is convex (see Lemma 3.7 in ).
In this section, we investigate the existence and convergence of best proximity points for generalized multi-valued cyclic contraction pairs and obtain some new results on fixed point theorems for such mappings. We begin by introducing the notion of multi-valued cyclic contraction.
for all and .
Note that if is a generalized multi-valued cyclic contraction, then is also a generalized multi-valued cyclic contraction. Here, we state the main results of this paper on the existence of best proximity points for a generalized multi-valued 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 and satisfy the property UC∗. Let and . If is a generalized multi-valued 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 .
for all .
Hence we get . Analogously, we also obtain .
Case 2. We will show that T or S have best proximity points in A and B, respectively, under the assumption of . Suppose, to the contrary, that for all , and for all , .
for all . So we derive and for all .
for all and .
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 , then Theorem 3.2 yields existence of a fixed point in of two multi-valued non-self mappings S and T. Moreover, if and , then Theorem 3.2 reduces to Mizoguchi-Takahashi’s fixed point theorem .
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 and . If is a generalized multi-valued 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 with Euclidean norm. Let and . Then A and B are nonempty closed and convex subsets of X and . Since A and B are convex, we have and satisfy the property UC∗.
for all .
Let be defined by for all . Next, we show that is a generalized multi-valued cyclic contraction pair with for all .
Therefore, all assumptions of Corollary 3.4 are satisfied, and then T has a best proximity point in A, that is, a point . Moreover, S also has a best proximity point in B, that is, a point .
Then A and B are nonempty closed and convex subsets of X and . Since A and B are convex, we have and satisfy the property UC∗.
for all .
Let define by for all . Next, we show that is a generalized multi-valued cyclic contraction pair with mapping for all .
Therefore, all assumptions of Corollary 3.4 are satisfied, and then T has a best proximity point in A that is a point . Furthermore, S also has a best proximity point in B that is a point .
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?
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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