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 MizoguchiTakahashi’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; τfunction; function; metric; function; MizoguchiTakahashi’s fixed point theorem; Nadler’s fixed point theorem; Banach contraction principle1 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 [122]. 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.
Theorem NA (Nadler [2])
Letbe a complete metric space andbe akcontraction; 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 [46]. It is worth mentioning that the primitive proof of MizoguchiTakahashi’s fixed point theorem is difficult. Recently, Suzuki [7] gave a very simple proof of MizoguchiTakahashi’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 MizoguchiTakahashi’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 MizoguchiTakahashi’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, MizoguchiTakahashi’s fixed point theorem, BerindeBerinde’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 MizoguchiTakahashi’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
(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
(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
Moreover, it is worth mentioning that the proof of Theorem 2.1 is not correct.
The following is the definition of a τfunction which was introduced and studied by Lin and Du.
Let be a metric space. A function is said to be a τfunction if the following conditions hold:
(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 ;
The following results are crucial and useful in this paper.
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.
Let be a metric space. A function is called a function if it is a τfunction on X with for all .
Remark 2.3 From , if p is a function, then if and only if .
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.
A function is said to be an function (or an function) if for all .
Lemma 2.2[9]
Letbe anfunction. Thendefined byis also anfunction.
Theorem D[22]
Letbe a function. Then the following statements are equivalent.
(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 afunction, be ametric oninduced bypandbe anfunction. Ifandsatisfy
(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
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
Let for all . We claim that . Put , . For with , by (3.2), we have
By Lemma 2.1, is a Cauchy sequence in . By the completeness of , there exists such that as . From and (3.3), we have
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 MizoguchiTakahashi’s fixed point theorem. So Theorem 3.1 is a generalization of MizoguchiTakahashi’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 afunction, be ametric oninduced bypandbe a nondecreasing or nonincreasing function. Ifandsatisfy
(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 anfunction. Ifandsatisfy
(iii) is a complete subspace ofX,
thenTandfhave a coincidence point inX.
Corollary 3.3Letbe a metric space andbe a nondecreasing or nonincreasing function. Ifandsatisfy
(iii) is a complete subspace ofX,
thenTandfhave a coincidence point inX.
Theorem 3.2Letbe a metric space, be afunction, be ametric oninduced bypandbe anfunction. Ifandsatisfy
(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
(b) is a Cauchy sequence in X and ;
By (c) and (iv), we have . Then
Therefore, there exists such that
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 MizoguchiTakahashi’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 afunction, be ametric oninduced bypandbe a nondecreasing or nonincreasing function. Ifandsatisfy
(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 anfunction. Ifandsatisfy
(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
(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 WeiShih Du for their valuable suggestions and comments.
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