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Journal of Applied Mathematics
Volume 2011, Article ID 327941, 11 pages
http://dx.doi.org/10.1155/2011/327941
Research Article

Some New Fixed-Point Theorems for a ()-Pair Meir-Keeler-Type Set-Valued Contraction Map in Complete Metric Spaces

Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu, Taiwan

Received 25 July 2011; Revised 7 October 2011; Accepted 25 October 2011

Academic Editor: George Jaiani

Copyright © 2011 Chi-Ming Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain some new fixed point theorems for a ()-pair Meir-Keeler-type set-valued contraction map in metric spaces. Our main results generalize and improve the results of Klim and Wardowski, (2007).

1. Introduction and Preliminaries

Let be a metric space, a subset of , and a map. We say is contractive if there exists such that, for all , The well-known Banach’s fixed-point theorem asserts that if , is contractive and is complete, then has a unique fixed point in . It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping is called a quasi-contraction if there exists such that for any . In 1974, Ćirić [2] introduced these maps and proved an existence and uniqueness fixed-point theorem.

Throughout this paper, by we denote the set of all real numbers, while is the set of all natural numbers. Let be a metric space. Let denote a collection of all nonempty closed subsets of and a collection of all nonempty closed and bounded subsets of .

The existence of fixed points for various multivalued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler Jr. [3] extended the famous Banach contraction principle from single-valued mapping to multivalued mapping and proved the below fixed-point theorem for multivalued contraction.

Theorem 1.1 (see [3]). Let be a complete metric space, and let be a mapping from into . Assume that there exists such that where denotes the Hausdorff metric on induced by ; that is, , for all and . Then has a fixed point in .

In 1989, Mizoguchi-Takahashi [4] proved the following fixed-point theorem.

Theorem 1.2 (see [4]). Let be a complete metric space, and let be a map from into . Assume that for all , where satisfies for all . Then has a fixed point in .

In 2006, Feng and Liu [5] gave the following theorem.

Theorem 1.3 (see [5]). Let be a complete metric space, and let be a multivalued map. If there exist , such that for any , there is satisfying the following two conditions:(i),(ii). Then has a fixed point in provided that the mapping defined by , , is lower semicontinuous; that is, if for any and , , then .

In 2007, Klim and Wardowski [6] proved the following fixed point theorem.

Theorem 1.4 (see [6]). Let be a complete metric space, and let be a multivalued map. Assume that the following conditions hold:(i)the mapping defined by , , is lower semicontinuous;(ii)there exist and such that Then has a fixed point in .

Recently, Pathak and Shahzad [7] introduced a new class of mapping and generalized the results of Klim and Wardowski [6]. Suppose that , denote the class of functions satisfying the following conditions:(1) is nondecreasing on ;(2) for all ;(3) is subadditive in ; that is, .

The following theorem was introduced in Pathak and Shahzad [7].

Theorem 1.5 (see [7]). Let be a complete metric space and suppose that . Assume that the following conditions hold:(i)the mapping defined by , , is lower semicontinuous,(ii)there exists such that (iii)there exists satisfying the following condition: Then has a fixed point in .

Later, Kamran and Kiran [8] improved some results of Pathak and Shahzad [7] by allowing to have values in closed subsets of . They proved that the function is positive homogenous in , that is,(4) for all , ,

and denote by the class of functions satisfying condition (4). They proved the following theorem.

Theorem 1.6 (see [8]). Let be a complete metric space and suppose that is a function from to such that Suppose that . Assume that the following condition holds: where . Then(i)for each , there exists an orbit of and such that ;(ii) is a fixed point of if and only if the function is -orbitally lower semicontinuous at .

2. Main Results

In this section, we first recall the notion of the Meir-Keeler-type function (see [9]). A function is said to be a Meir-Keeler-type function, if, for each , there exists such that for with , we have . We now define a new stronger Meir-Keeler-type function, as follows.

Definition 2.1. One calls the stronger Meir-Keeler-type function, if, for each , there exists such that for with , there exists such that .

Remark 2.2. It is clear that, if the function satisfies for all , then is also a stronger Meir-Keeler-type function.

Example 2.3. (1) If with , then is a stronger Meir-Keeler-type function.
(2) If , then is a stronger Meir-Keeler-type function.

Definition 2.4. Let , be two functions where . Then the mappings are called a -pair Meir-Keeler-type function, if, for each , there exists such that, for with , there exists such that .

Remark 2.5. It is clear that if the functions , satisfy for all , then are also a -pair Meir-Keeler-type function.

Example 2.6. If , and , then are a -pair Meir-Keeler-type function.

Definition 2.7. Let be a metric space, let , be two functions where , and let be a set-valued map. Then is called a -pair Meir-Keeler-type set-valued contraction map, if the following conditions hold:(C1)for each , there exists such that for with , there exists such that (C2)for all , there exists such that

In this paper, we obtain some new fixed-point theorems for a -pair Meir-Keeler-type set-valued contraction map in metric spaces. Our main results generalize and improve the results of Klim and Wardowski [6]. We now state our main theorem as follows.

Theorem 2.8. Let be a complete metric space, and let be a -pair Meir-Keeler-type set-valued contraction map. Then has a fixed point in provided the mapping defined by , , is lower semicontinuous.

Proof. Given and by (C2), there exists such that . Since is a -pair Meir-Keeler type set-valued contraction map, there exists such that Continuing this process, we can choose a sequence with such that, for all , Therefore, we can deduce that, for all , Thus, the sequence is decreasing and bounded below. Then there exists such that Hence, there exists and such that, for all , By the condition (C1), we have that there exists such that So for each with , by (2.6), we can deduce that Take with . Then we get and so we conclude that since . Thus, is a Cauchy sequence in . Since is complete, there exists such that as .
Since , , , is lower semicontinuous, we have The closeness of implies .

The following is a simple example for Theorem 2.8, and it generalize the result of Klim and Wardowski [6].

Example 2.9. Let be a metric space with the standard metric . Let be defined by Let , be defined by Then is a -pair Meir-Keeler-type set-valued contraction map, and is a fixed point of .
In particular, if we let , then this example satisfies all of the conditions of Theorem 1.4 (that was introduced in Klim and Wardowski [6]).

Using Example  3.1 in [6] and Example  1 in [10], we get the following another example for Theorem 2.8.

Example 2.10. Let be a metric space with the standard metric . Let be defined as in Example 3.1 of Klim and Wardowski [6]: Let be defined as in Example 1 of Ćirić [10]: and let be defined by Clearly, a function is lower semicontinuous. Then are a -pair Meir-Keeler-type function, and is a -pair Meir-Keeler-type set-valued contraction map. Moreover, by Theorem 2.8, we have that is a fixed point of .

If we let be closed, then we also have the following fixed result.

Theorem 2.11. Let be a complete metric space, and let be a -pair Meir-Keeler-type set-valued contraction map and closed. Then has a fixed point in .

Proof. Following the proof of Theorem 2.8, we get that is a Cauchy sequence in . Since is complete, there exists such that as . Since is closed and , we have that .

The following is a simple example for Theorem 2.11.

Example 2.12. Let be a metric space with the metric for all . Let be defined by Let , be defined by Then is a -pair Meir-Keeler-type set-valued contraction map and closed, and is a fixed point of .

Applying Theorem 2.8 and Remark 2.5, we are easy to get the following result.

Theorem 2.13. Let be a complete metric space, let , be two functions where , and let be a set-valued contraction map. Suppose the following conditions hold:(1)for each , (2)for all , there exists such that Then has a fixed point in provided the mapping defined by , is lower semicontinuous.

The following is a simple example for Theorem 2.13.

Example 2.14. Let be a metric space with the metric for all . Let be defined as in Example 3.1 of Klim and Wardowski [6]: Let be defined as in Example 1 of Ćirić [10]: and let be defined by Clearly, a function is lower semicontinuous. Clearly, . We also conclude the following.
Case 1. If , then , and satisfy the condition (2) of Theorem 2.13.
Case 2. If , then (resp., ), and also satisfy the condition (2) of Theorem 2.13.
Thus, by Theorem 2.13, we have that is a fixed point of .

Using Example 2.10, we also get the following example for Theorem 2.13.

Example 2.15. Let be a metric space with the standard metric . Let be defined as Let be defined as and be defined by Clearly, , and satisfies all of the conditions of Theorem 2.13. So, we have that is a fixed point of .

If we let the function be for all and let the function , , be a stronger Meir-Keeler-type function; that is for if, for each , there exists such that, for with , there exists such that , then, by Theorem 2.8, it is easy to get the following theorem.

Theorem 2.16. Let be a complete metric space, let , be a stronger Meir-Keeler-type function, and let be a set-valued contraction map. Suppose that, for all , there exists such that Then has a fixed point in provided that the mapping defined by , , is lower semicontinuous.

The following is a simple example for Theorem 2.16.

Example 2.17. Let be a metric space with the standard metric . Let be defined by Let be defined by Then a stronger Meir-Keeler-type function, and is a fixed point of .

Acknowledgments

The authors would like to thank the referees whose helpful comments and suggestions led to many improvements in this paper. This research was supported by the National Science Council of Taiwan.

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