A Generalization of a Greguš Fixed Point Theorem in Metric Spaces

Abstract

We first introduce a new class of contractive mappings in the setting of metric spaces and then we present certain Greguš type fixed point theorems for such mappings. As an application, we derive certain Greguš type common fixed theorems. Our results extend Greguš fixed point theorem in metric spaces and generalize and unify some related results in the literature. An example is also given to support our main result.

1. Introduction and Preliminaries

Let be a Banach space and let be a closed convex subset of . In 1980 Greguš [1] proved the following result.

Theorem 1. Let be a mapping satisfying the inequality for all , where , , and . Then has a unique fixed point.

Fisher and Sessa [2], Jungck [3], and Hussain et al. [4] obtained common fixed point generalizations of Theorem 1. In recent years, many theorems which are closely related to Greguš’s Theorem have appeared (see [1–21]). Very recently, Moradi and Farajzadeh [21] extended Greguš fixed point theorem in complete metric spaces.

Theorem 2 ([21, Theorem 2.4]). Let be a complete metric space and let be a mapping such that, for all , where , , , , and . Then has a unique fixed point.

Let and be self-maps of . A point is a coincidence point (resp., common fixed point) of and if (resp., ). The pair is called (1) commuting if for all ; (2) weakly commuting [2] if, for all , ; (3) compatible [3] if whenever is a sequence such that for some in ; (4) weakly compatible if they commute at their coincidence points, that is, if whenever . Clearly, commuting maps are weakly commuting, and weakly commuting maps are compatible. References [2, 3] give examples which show that neither implication is reversible.

The purpose of this paper is to define and to investigate a class of new generalized contractive mappings (not necessarily continuous) on metric spaces. We will prove certain fixed point and common fixed results which are generalizations of the above mentioned theorems.

2. Fixed Point Results

We denote by the set of all nonnegative real numbers and by the set of all functions satisfying the following conditions:() is continuous,() is nondecreasing in , and ,(), for each ,(), for each ,() for each .

Example 3. If for , where , , and , then .

Example 4. If for , where , then .

Example 5. Let . Then it is easy to see that where for each .

Now we are ready to state our main result.

Theorem 6. Let be a complete metric space and let be a mapping satisfying for each , where . Then has a unique fixed point.

Proof. We first show that . If for some , then is a fixed point of and we are done. So, we may assume that for each . From (4), (), and (), we have and so Now let be a sequence such that From (4), (6), and (), we get for each . From (4), (6), (8), and (), we have for each . From (6) and (7), we get and so by (7) From (7), (9), (11), and (), we obtain Hence by () Now, let Notice that, by (13), for each . We show that On the contrary, assume that there are sequences and with satisfying From (4), (14), and (), we have for each . From (14), (16), (17), and , we get which contradicts (). Thus (15) holds. Hence is a decreasing sequence of closed nonempty sets with and so, by Cantor’s intersection theorem, We show that is a fixed point of . Since , there exists such that for all . Now for each , we have Since is continuous, from (20), and hence, by , ; that is, . To prove the uniqueness, note that if is a fixed point of , then and hence .

Remark 7. Note that, to prove Theorem 2, we may assume that and (see the proof of Theorem 2.4 in [21]). Thus, by Example 3, Theorem 6 is a generalization of the above mentioned Theorem 2 of Moradi and Farajzadeh.
If we take as in Example 4, from Theorem 6 we get the main result of Ćirić [10].

The following corollary improves Theorem 2.4 in [6].

Corollary 8. Let be a complete metric space and let be a mapping satisfying for each , where and . Then has a unique fixed point.

As an easy consequence of the axiom of choice, [13, page 5], AC5: for every function , there is a function such that and for every , ], we obtain the following lemma (see also [22]).

Lemma 9. Let be a nonempty set and let be a mapping. Then, there exists a subset such that and is one-to-one.

As an application of Theorem 6, we now establish a common fixed point result.

Theorem 10. Let be a metric space and let , be mappings satisfying for each , where . Suppose that and is complete subspace of . Then and have a unique coincidence point. Further, if and are weakly compatible, then they have a unique common fixed point.

Proof. By Lemma 9, there exists such that and is one-to-one. We define a mapping by for all . As is one-to-one on and , is well defined. Thus, it follows from (23) and (24) that for all . Thus the function satisfies all conditions of Theorem 6, so has a unique fixed point . As , there exists such that . Thus which implies that and have a unique coincidence point. Further if and are weakly compatible, then they have a unique common fixed point.

If we take as in Example 5, then from Theorem 10 we obtain the following result which extends many related results in the literature (see [16, 17]).

Theorem 11. Let be a metric space and let , be mappings satisfying for each , where and . Suppose that and is a complete subspace of . Then and have a unique coincidence point. Further, if and are weakly compatible, then they have a unique common fixed point.

Corollary 12. Let be a metric space and let , be mappings satisfying for each , where . Suppose that and is a complete subspace of . Then and have a unique coincidence point. Further if and are weakly compatible, then they have a unique common fixed point.

As a linear continuous operator defined on a closed subset of a normed space is closed operator, we obtain the following new common fixed point results as corollaries to Theorem 11.

Corollary 13 (see Fisher and Sessa [2]). Let and be two weakly commuting mappings on a closed convex subset of a Banach space into itself satisfying the inequality for all , where and . If is linear and nonexpansive on and , then and have a unique common fixed point in .

Corollary 14 (see Jungck [3]). Let and be compatible self-maps of a closed convex subset of a Banach space . Suppose that is continuous and linear and that . If and satisfy inequality (28), then and have a unique common fixed point in .

Now, we illustrate our main result by the following example.

Example 15. Let and let , , , and . Then is a complete metric space. Let be given by , , , and . Then it is straightforward to show that for each . Then by Corollary 8, has a unique fixed point ( is the unique fixed point of ).
Now, we show that does not satisfy the condition of Theorem 2 of Moradi and Farajzadeh. On the contrary, assume that there exist nonnegative numbers , , , such that for all . Let and . Then from (30), we have which yield a contradiction. Thus we cannot invoke the above mentioned theorem of Moradi and Farajzadeh (Theorem 2), to show the existence of a fixed point of .

Remark 16. The technique of proof of Theorem 6 is in line with the proof of Theorem 2.4 in [23]. Therefore the reader interested in fixed point results for generalized contractions/nonexpansive mappings in the general setup of uniformly convex metric spaces is referred to [23].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, Marwan A. Kutbi and N. Hussain acknowledge with thanks DSR, KAU, for financial support. A. Amini-Harandi was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran, and by a Grant from IPM (no. 92470412).