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New existence theorems of coincidence points approach to generalizations of Mizoguchi-Takahashi’s fixed point theorem

Ing-Jer Lin* and Tai-Hung Chen

Author Affiliations

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 824, Taiwan

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Fixed Point Theory and Applications 2012, 2012:156  doi:10.1186/1687-1812-2012-156

 Received: 24 May 2012 Accepted: 30 August 2012 Published: 19 September 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we first establish some new existence theorems of coincidence points and common fixed points for -functions. By applying our results, we obtain some generalizations of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle. Some examples illustrating our results are also given. Our results generalize and improve some main results in the literature and references therein.

Keywords:
coincidence point; common fixed point; τ-function; -function; -metric; -function; Mizoguchi-Takahashi’s fixed point theorem; Nadler’s fixed point theorem; Banach contraction principle

1 Introduction

In recent years, the celebrated Banach contraction principle (see, e.g., [1]) always plays an essential role in various fields of applied mathematical analysis. The Banach contraction principle has been employed to solve the problems in Banach spaces such as the existence of solutions for nonlinear integral equations and nonlinear differential equations. Also, it has been applied to study the convergence of algorithms in computational mathematics. Additionally, many generalizations of the Banach contraction principle in various different directions have been investigated by several authors in the past; see [1-22]. Because of the importance of the Banach contraction principle, we begin with the theorem as follows.

Theorem BCP (Banach [1])

Letbe a complete metric space andbe a selfmap. Assume that there exists a nonnegative numbersuch that

ThenThas a unique fixed point inX. Moreover, for each, the iterative sequenceconverges to the fixed point.

In 1969, Nadler [2] first gave a famous generalization of the Banach contraction principle for multivalued maps, which is as important as the Banach contraction principle.

Letbe a complete metric space andbe ak-contraction; that is, there exists a nonnegative numbersuch that

whereis the class of all nonempty closed bounded subsets ofX. Then there existssuch that.

In 1989, Mizoguchi and Takahashi [3] proved a generalization of Nadler’s fixed point theorem which also gave a partial answer to Problem 9 in Reich [4-6]. It is worth mentioning that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is difficult. Recently, Suzuki [7] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem MT (Mizoguchi and Takahashi [3])

Letbe a complete metric space andbe a multivalued map. Assume that

whereαis a function fromintosatisfyingfor all. Then there existssuch that.

Subsequently, in 2007, Berinde and Berinde [8] proved the following interesting fixed point theorem. That is a generalization of Mizoguchi-Takahashi’s fixed point theorem.

Theorem BB (Berinde and Berinde [8])

Letbe a complete metric space, be a multivalued map, and. Assume that

whereαis a function fromintosatisfyingfor all. Then there existssuch that.

It is obvious that if we take in Berinde and Berinde’s fixed point theorem, we can obtain Mizoguchi-Takahashi’s fixed point theorem.

Very recently, Du [9] has used a -metric and an -function to establish some new fixed point theorems for nonlinear multivalued contractive maps and generalize the Banach contraction principle, Nadler’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems for nonlinear multivalued contractive maps in complete metric spaces; see [9] for more detail.

In this paper, we first establish some new existence results of coincidence points and common fixed points for -functions. By applying our results, we can obtain some generalizations of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle. Our results generalize and improve some main results in the literature and references therein.

2 Preliminaries

Throughout this paper, we denote the set of positive integers by . Let be a metric space. For each and , let . Also, we denote the class of all nonempty subsets of X by , the family of all nonempty closed subsets of X by , and the family of all nonempty closed and bounded subsets of X by . A function defined by

is said to be the Hausdorff metric on induced by the metric d on X.

Let be a selfmap and be a multivalued map. A point is called

(i) a fixed point of f if ;

(ii) a fixed point of T if ;

(iii) a coincidence point of f and T in X if ;

(iv) a common fixed point of f and T if .

In [9], Sajath and Vijayaraju proved the following theorem.

Theorem 2.1[10]

Letbe a metric space, andbe a function such thatfor every. Ifandsatisfy

(a) ;

(b) ;

(c) is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Remark 2.1 In fact, the condition (a) in Theorem 2.1 should be corrected as

(a) with .

Moreover, it is worth mentioning that the proof of Theorem 2.1 is not correct.

The following is the definition of a τ-function which was introduced and studied by Lin and Du.

Definition 2.1[9,11-17]

Let be a metric space. A function is said to be a τ-function if the following conditions hold:

(τ1) for all ;

(τ2) If and in X with such that for some , then ;

(τ3) For any sequence in X with , if there exists a sequence in X such that , then ;

(τ4) For , and imply .

Let be a τ-function. Define .

The following results are crucial and useful in this paper.

Lemma 2.1[9,11,12,14-17]

Letbe a metric space andbe any function satisfying (τ3). Ifis a sequence inXwith, thenis a Cauchy sequence in X.

Recently, Du [5,6] first introduced the concepts of -functions and -metrics as follows.

Definition 2.2[9,13]

Let be a metric space. A function is called a -function if it is a τ-function on X with for all .

Remark 2.3 From , if p is a -function, then if and only if .

Definition 2.3[9,13]

Let be a metric space and p be a -function. For any , define a function by

where ; then is said to be a -metric on CB(X) induced by p.

Clearly, any Hausdorff metric is a -metric, but the reverse is not true.

Definition 2.4[9,15-22]

A function is said to be an -function (or an -function) if for all .

Lemma 2.2[9]

Letbe an-function. Thendefined byis also an-function.

Theorem D[22]

Letbe a function. Then the following statements are equivalent.

(a) φis an-function.

(b) For each, there existandsuch thatfor all.

(c) For each, there existandsuch thatfor all.

(d) For each, there existandsuch thatfor all.

(e) For each, there existandsuch thatfor all.

(f) For any nonincreasing sequencein, we have.

(g) φis a function of a contractive factor[19]; that is, for any strictly decreasing sequencein, we have.

It is obvious that if a function is nondecreasing or nonincreasing, then it is an -function.

3 New coincidence point theorems and a common fixed point theorem

In this section, we generalize Theorem 2.1 which is one of the main results in [10]. Please notice that our proof is quite different from the proof of Theorem 2.1 in [10].

Theorem 3.1Letbe a metric space, be a-function, be a-metric oninduced bypandbe an-function. Ifandsatisfy

(i) , ;

(ii) ;

(iii) is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Proof By Lemma 2.2, we can define an -function by . Then and for all . Let . By (ii), there exists such that . If , we have which means that is a coincidence point of T and f in X and we finish the proof. Otherwise, if , since p is a -function, . By (i), we have

Hence there exists such that . By (ii) again, there exists such that . Therefore,

By induction, we can obtain a sequence in X satisfying and

(3.1)

Since for all , the inequality (3.1) implies the sequence is strictly decreasing in . Since κ is an -function, by Theorem D, we have

Let . Then and for all . For any , we have from (3.1) that

(3.2)

Let for all . We claim that . Put , . For with , by (3.2), we have

(3.3)

Since , and hence

By Lemma 2.1, is a Cauchy sequence in . By the completeness of , there exists such that as . From and (3.3), we have

(3.4)

So, for each , we have

(3.5)

Therefore, there exists such that for each , which implies . Then, by , we have . Moreover, since as and

we get

which means that as . Since for all and is closed, , i.e., is a coincidence point of f and T. The proof is completed. □

Remark 3.1 In Theorem 3.1, if (the identity map), then we obtain Mizoguchi-Takahashi’s fixed point theorem. So Theorem 3.1 is a generalization of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle.

Here, we give a simple example illustrating Theorem 3.1.

Example 3.1 Let with the metric , . Let , and , . Let be defined by

for all x, and . It is easy to see that p is a -function and φ is an -function.

Clearly, and is a complete subspace of X. We claim that , . Indeed, we consider the following two possible cases:

Case 1. If , we have and , then

and

Case 2. If , similarly, we have

and

By Cases 1 and 2, we verify that , . Therefore, all the assumptions of Theorem 3.1 are satisfied. So, we can apply Theorem 3.1 to show that f and T have a coincidence point in X. Actually, 0 is a coincidence point of f and T since .

The following result follows immediately from Theorem 3.1.

Corollary 3.1Letbe a metric space, be a-function, be a-metric oninduced bypandbe a nondecreasing or nonincreasing function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX,

thenTandfhave a coincidence point inX.

In Theorem 3.1, if , then and we have the following corollary.

Corollary 3.2Letbe a metric space andbe an-function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Corollary 3.3Letbe a metric space andbe a nondecreasing or nonincreasing function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Theorem 3.2Letbe a metric space, be a-function, be a-metric oninduced bypandbe an-function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX;

(iv) ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Proof Following the same argument as in the proof of Theorem 3.1, we can construct two sequences and satisfying

(a) for all ;

(b) is a Cauchy sequence in X and ;

(c) there exist such that

as ;

;

, where , .

By (c) and (iv), we have . Then

Therefore, there exists such that

(3.6)

Since

we have . By (3.6), . By , we have . Since as and

we have , which implies . Since is closed and for all , we get . Therefore, , which means that is a common fixed point of f and T in X. The proof is completed. □

Remark 3.2 Theorem 3.2 also generalizes and improves Mizoguchi-Takahashi’s fixed point theorem.

Example 3.2 In Example 3.1, we have shown that 0 is a coincidence point of f and T. Clearly, . So, all the assumptions of Theorem 3.2 are satisfied. By Theorem 3.2, we know that f and T have a common fixed point in X. Actually, 0 is a common fixed point of f and T since .

Similarly, we have the following corollary.

Corollary 3.4Letbe a metric space, be a-function, be a-metric oninduced bypandbe a nondecreasing or nonincreasing function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX;

(iv) ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Corollary 3.5Letbe a metric space andbe an-function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX;

(iv) ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Corollary 3.6Letbe a metric space andbe a nondecreasing or nonincreasing function. Ifandsatisfy

(i) ;

(ii) ;

(iii) is a complete subspace ofX;

(iv) ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The first author made 80% contribution: problem design, coordination, discussion, revision of the important part, and submission of this paper. The second author made 20% contribution: discussion, responsibility for the important results and typing of this paper.

Acknowledgements

The authors wish to express their hearty thanks to Professor Wei-Shih Du for their valuable suggestions and comments.

References

1. Takahashi, W: Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000)

2. Nadler, SB Jr..: Multi-valued contraction mappings. Pac. J. Math.. 30, 475–488 (1969)

3. Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl.. 141, 177–188 (1989). Publisher Full Text

4. Reich, S: Some problems and results in fixed point theory. Contemp. Math.. 21, 179–187 (1983)

5. Daffer, PZ, Kaneko, H, Li, W: On a conjecture of S. Reich. Proc. Am. Math. Soc.. 124, 3159–3162 (1996). Publisher Full Text

6. Jachymski, J: On Reich’s question concerning fixed points of multimaps. Boll. Unione Mat. Ital., A. 9, 453–460 (1995)

7. Suzuki, T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl.. 340, 752–755 (2008). Publisher Full Text

8. Berinde, M, Berinde, V: On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl.. 326, 772–782 (2007). Publisher Full Text

9. Du, W-S: Some new results and generalizations in metric fixed point theory. Nonlinear Anal.. 73, 1439–1446 (2010). Publisher Full Text

10. Sajath, ZMI, Vijayaraju, P: Coincidence and common fixed points of hybrid mappings. Int. Math. Forum. 5(42), 2063–2069 (2010)

11. Lin, L-J, Du, W-S: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl.. 323, 360–370 (2006). Publisher Full Text

12. Lin, L-J, Du, W-S: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal.. 68, 1246–1262 (2008). Publisher Full Text

13. Du, W-S: Fixed point theorems for generalized Hausdorff metrics. Int. Math. Forum. 3, 1011–1022 (2008)

14. Du, W-S: Critical point theorems for nonlinear dynamical systems and their applications. Fixed Point Theory Appl.. 2010, Article ID 246382. doi:10.1155/2010/246382 (2010)

15. He, Z, Du, W-S, Lin, I-J: The existence of fixed points for new nonlinear multivalued maps and their applications. Fixed Point Theory Appl.. 2011, Article ID 84 (2011)

16. Du, W-S: On generalized weakly directional contractions and approximate fixed point property with applications. Fixed Point Theory Appl.. 2012, Article ID 6 (2012)

17. Du, W-S, Zheng, S-X: Nonlinear conditions for coincidence point and fixed point theorems. Taiwan. J. Math.. 16, 857–868 (2012)

18. Du, W-S: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasiordered metric spaces. Fixed Point Theory Appl.. 2010, Article ID 876372 (2010)

19. Du, W-S: Nonlinear contractive conditions for coupled cone fixed point theorems. Fixed Point Theory Appl.. 2010, Article ID 190606 (2010)

20. Du, W-S: New cone fixed point theorems for nonlinear multivalued maps with their applications. Appl. Math. Lett.. 24(2), 172–178 (2011). Publisher Full Text

21. Du, W-S: On approximate coincidence point properties and their applications to fixed point theory. J. Appl. Math.. 2012, Article ID 302830. doi:10.1155/2012/302830 (2012)

22. Du, W-S: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol. Appl.. 159(1), 49–56 (2012). Publisher Full Text