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Common fixed point of the generalized Mizoguchi-Takahashi’s type contractions
Fixed Point Theory and Applications volume 2014, Article number: 195 (2014)
In this work, we introduce a new class of contractive multi-valued mappings, called generalized Mizoguchi-Takahashi’s contractions, which is an extension of many known results in the literature. Finally, a partial answer to the conjecture which was introduced by Rouhani and Moradi is given.
In 1922, Banach established the most famous fundamental fixed point theorem (the so-called the Banach contraction principle ) which has played an important role in various fields of applied mathematical analysis. It is known that the Banach contraction principle has been extended in many various directions by several authors (see [2–7]). An interesting direction of research is the extension of the Banach contraction principle of multi-valued maps, known as Nadler’s fixed point theorem , Mizoguchi-Takahashi’s fixed point theorem ; see M Berinde and V Berinde , Ćirić , Reich , Daffer and Kaneko , Rhoades , Rouhani and Moradi , Amini-Harandi [1, 7], Moradi and Khojasteh , and Du  and references therein.
Let us recall some basic notations, definitions, and well-known results needed in this paper. Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively. Let be a metric space. For each and , let . Denote by the class of all nonempty subsets of X, by the family of all nonempty closed subsets of X, and by the family of all nonempty, closed, and bounded subsets of X. A function defined by
is said to be the Hausdorff metric on induced by d on X where for each . A point v in X is a fixed point of a map T if (when is a single-valued map) or (when is a multi-valued map). The set of fixed points of T is denoted by and the set of common fixed points of two multi-valued mappings T, S is denoted by .
Definition 1 [, Du]
A function is said to be an -function (or ℛ-function) if for all .
It is evident that if is a non-decreasing function or a non-increasing function, then φ is an -function. So the set of -functions is a rich class.
Theorem 1 [, Mizoguchi and Takahashi]
Let be a complete metric space, be a -function and be a multi-valued map. Assume that
for all . Then .
A mapping is said to be a weak contraction if there exists such that
for all , where
Two multi-valued mappings are called generalized weak contractions if there exists such that
Also two mappings are called generalized φ-weak contractive if there exists a map with and for all such that
for all .
The concepts of weak and φ-weak contractive mappings were defined by Daffer and Kaneko  in 1995.
In 2010 Rouhani and Moradi  introduced an extension of Daffer and Kaneko’s result for two multi-valued weak contraction mappings of a complete metric space X into without assuming to be l.s.c.
Theorem 2 Let be a complete metric space. Suppose that are contraction mappings in the sense that, for some ,
for all (i.e., weak contractions). Then there exists a point such that and .
The paper is organized as follows. In Section 2, we first introduce the Generalized Mizoguchi-Takahashi’s Contraction (GMT for short) as an extension of Mizoguchi and Takahashi’s type and of work by Daffer and Kaneko. Section 3 is dedicated to the study of some new fixed point theorems, which generalize and improve Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and some well-known results. Furthermore, we give a partial answer to the conjecture introduced by Rouhani and Moradi (see [, Theorem 4.1 and Section 5]). Consequently, some of our results in this paper are original in the literature, and we obtain many results in the literature as special cases.
2 Main result
Definition 2 A function is called a GMT function if the following conditions hold:
(ϑ 1) for all ;
(ϑ 2) for any bounded sequence and any non-increasing sequence , we have
We denote the set of all GMT functions by .
Here, we give simple examples of manageable functions.
Example 1 [, Example B]
Let be an -function, then is a GMT-function.
Example 2 Let for all . Define
It is clear that . Also for any bounded sequence and any non-increasing sequence , if , then for some . We have
Otherwise, and since g is continuous, we get
which means that (ϑ 2) holds. Hence .
Definition 3 Let be a metric space. The mapping is called a WGMT-contraction if there exists such that
for each .
Theorem 3 Let be a complete metric space and let and suppose there exists such that
for each . Then T, S have a common fixed point.
Proof Let be arbitrary and . Since choose . If we have nothing to prove, because
Thus . If , then and and so is the common fixed point of T, S. Hence without loss of generality, we can assume that . Also if , then and since we have and and again is the common fixed point of T, S. Thus we can assume that . Therefore, by any choice of , we can assume that
(note that ), there exists such that such that
By the above argument, we can assume that
Choose . Taking
(note that ), there exists such that and
By induction, if is known to satisfy , , , , and
then, by taking
one can obtain with such that
Hence by induction, we can establish a sequence in X satisfying, for each ,
By (7), we have
By combining (11) and (12), we get
By repeating the above argument (replacing S by T) one can easily verified that
It means that and . Hence for each we have
which means that the sequence is strictly decreasing in . So
By (12) and (15), we have
which means that is a bounded sequence. By (ϑ 2), we have
Now, we claim . Suppose . Then, by (18) and taking the limsup in both sides of (13), we get
a contradiction. Hence we prove
To complete the proof it suffices to show that is a Cauchy sequence in X.
For each , let
Then for all . By (13), we obtain
From (18), we have , so there exist and , such that
For any , since for all and , taking into account (20) and (21), we conclude that
Put , . For with , we have from the last inequality
Since , and hence
So is a Cauchy sequence in X and converges to . If
and where is the cardinal number of S, then and we have . If one deduces that there exists such that for all with . It means that and for each . Hence and for each and so
Suppose that , then
By (23) one can choose such that for each . Now by taking , for each we have
For each , let
Then for all . By (24), we obtain
Since we have is bounded and is a fixed sequence. From (), we have , so there exist and , such that
Taking the limit in both sides of (25) we have
Thus and this is a contradiction. So we have and then . The same argument can be applied for S and one can easily verify that and so T, S have a common fixed point . □
In the following, an example is given covering our result.
Example 3 Suppose that and let be defined as follows:
and for all . For any bounded sequence and any non-increasing sequence , we have
First suppose and , then
In the other case, suppose and , then
Other cases are easily verified as the above arguments. Henceforth, T is a WGMT-contraction and enjoys all conditions of Theorem 3. Also, T, S have a common fixed point .
Here we deduce some of the known and unknown results by Theorem 3.
Corollary 1 Let be a complete metric space and be two multi-valued mappings such that
where . Then T, S have at least a common fixed point in X.
Proof It suffices to take and apply Theorem 3. □
Corollary 2 Let be a complete metric space and be two multi-valued mappings such that
where be an -function, then T has a fixed point in X.
Proof It suffices to take and apply Theorem 3. □
Corollary 3 [, Generalized weak contraction]
Let be a complete metric space and be two multi-valued mappings such that
where be a function such that and . Then T has a fixed point in X.
Proof It suffices to take and apply Theorem 3. □
4 A partial answer to a known conjecture
In 2010, Rouhani and Moradi  proved the following theorem.
Corollary 4 Let be a complete metric space and let and be two mappings such that, for all ,
(i.e. generalized φ-weak contractions) where is l.s.c. with , and for all . Then there exists a unique point such that .
Motivated by the above the authors extended Rhoades’s theorem by assuming φ to be only l.s.c., as well as Zhang and Song’s  theorem to the case where one of the mappings is multi-valued. They also asserted the following: ‘Future directions to be pursued in the context of this research include the investigation of the case where both mappings in Zhang and Song’s theorem are multi-valued.’ By research in the literature such as [1, 7] and specially  (see [, Problem 3.2]) one deduces the following problem, a conjecture in the literature.
Problem (A) Let be a complete metric space and let be two mappings such that, for all ,
(i.e. generalized φ-weak contractions) where is l.s.c. with , and , for all . Then do T and S have a common fixed point?
Definition 4 We say that is a weak l.s.c. function if for each bounded sequence ,
The collection of all weak l.s.c. functions is denoted by .
In the following theorem a partial solution to Problem (A) is given as an application of Theorem 3.
Corollary 5 Let be a complete metric space and let be two mappings such that, for all ,
(i.e. generalized φ-weak contractions) where , with , and . Then .
Proof Define , for all . Since for each bounded sequence , , thus . Thus,
It means that . Also
Applying Theorem 3 yields . □
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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Javahernia, M., Razani, A. & Khojasteh, F. Common fixed point of the generalized Mizoguchi-Takahashi’s type contractions. Fixed Point Theory Appl 2014, 195 (2014). https://doi.org/10.1186/1687-1812-2014-195
- multi-valued mapping
- Hausdorff metric
- Mizoguchi-Takahashi’s type