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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 908423, 15 pages
http://dx.doi.org/10.1155/2012/908423
Research Article

Some Fixed and Periodic Points in Abstract Metric Spaces

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia
2Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan
3Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Beograd, Serbia
4Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 23 July 2012; Accepted 16 September 2012

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Abd Ghafur Bin Ahmad et al. 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 generalize in this paper some results on common fixed points of two, respectively four, contractive-type mappings in abstract metric spaces by removing condition of normality of the cone in their formulations. Further, some results about periodic points of self-maps are extended to the setting of abstract metric spaces.

1. Introduction and Preliminaries

Ordered normed spaces, cones, and topical functions have applications in applied mathematics, for instance, in using Newton's approximation method [14] and in optimization theory [5, 6]. -metric and -normed spaces were introduced in the mid-20th century ([2], see also [3, 4]) by replacing an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Huang and Zhang [7] reintroduced such spaces under the name of cone metric spaces, but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. In such a way, nonnormal cones can be used as well (although they used only normal cones), paying attention to the fact that Sandwich Theorem and continuity of the metric may not hold. These and other authors (e.g., [814]) proved some fixed point theorems for contractive-type mappings in cone metric spaces, as well as topological vector-space-valued cone metric spaces (e.g., [15, 16]).

The following definitions and results will be needed in the sequel (see, e.g., [2, 3, 5, 17, 18]).

Let be a real topological vector space. A subset of is called a cone if (a) is closed, nonempty and ; (b) , , imply that ; (c) .

Given a cone , we define the partial ordering with respect to by if and only if . We will write for , where stands for the interior of and use for ( and ). If , then is called a solid cone [3]. Note that the notation for an interior point of a positive cone was first used by Kreĭn and Rutman [19].

The cone in the topological vector space is called normal if has a base of neighborhoods of consisting of order-convex subsets (see [16]). In the case of a normed space, this is equivalent to the condition that there is a number such that, for all , implies . Equivalently, the cone is normal if For details see [5].

Example 1.1 (see [3]). Let with and . This cone is nonnormal. Consider, for example, and . Then , and , but ; hence does not converge to zero. It follows by (1.1) that is a nonnormal cone.

Definition 1.2 (see [4, 15, 16]). Let be a nonempty set and a topological vector space with a cone . Suppose that a mapping satisfies the following: for all and if and only if ; for all ; for all . The function is called an abstract metric and is called an abstract metric space (or a topological vector-space-valued cone metric space or a -metric space); we will use the first mentioned term.

The concept of an abstract metric space is obviously more general than that of a metric space. If is a Banach space then abstract metric space becomes a cone metric space of [7]. For new results in cone metric spaces see [2026].

Definition 1.3. Let be an abstract metric space. We say that a sequence in is(i)a Cauchy sequence if, for every in with , there is an such that for all , ;(ii)a convergent sequence if, for every in with , there is an such that for all , for some fixed .
An abstract metric space is said to be complete if every Cauchy sequence in is convergent in .

Let be an abstract metric space. The following properties are often used, particularly in the case when the underlying cone is nonnormal. The only assumption is that the cone is solid. For details about these properties see, for example, [11]. If where and , then .If for each , , then .If and , then .If , , and , then there exists such that, for all we have . (Note that, in general, the converse is not true. Indeed, in Example 1.1, , but for sufficiently large.)

In generalizing some theorems of Huang-Zhang [7], Abbas and Rhoades proved the following result in abstract metric spaces over normal cones.

Theorem 1.4 (see [9]). Let be a complete cone metric space over a normal cone. Suppose that are two self-maps satisfying for all , where , and . Then and have a unique common fixed point in . Moreover, any fixed point of is a fixed point of and conversely.

Sing et al. extended this result of Abbas-Rhoades to four maps. They proved the following theorem.

Theorem 1.5 (see [27]). Let be a complete cone metric space over a normal cone. Suppose that the mappings , , , and are four selfmaps on such that and and satisfying for all , where and . Suppose that the pairs and are weakly compatible. Then , , , and have a unique common fixed point.

In 1977, Rhoades proved the following interesting result.

Theorem 1.6 (see [28]). Let be a complete metric space. Let , and suppose that there exist decreasing functions , , such that for each and satisfying for all , . Then has a unique fixed point and for each the sequence converges to .

We generalize in this paper Theorems 1.4 and 1.5 by removing normality condition in their formulations. An example will show that these generalizations are proper. Further, some results of Abbas and Rhoades about periodic points of selfmaps from [29] are extended to abstract metric spaces. Theorem 1.6 is also presented in this new setting, with a slightly shorter proof.

Note that it was shown in [15, 30, 31] that some of the fixed point results in abstract metric spaces can be directly reduced to the respective metric results. However, the results of the present paper do not fall into this category, since some of them are new even in the context of metric spaces.

2. Fixed Point Theorems

In this section we will prove Theorems 1.4 and 1.5 by omitting the assumption of normality in the results. We use only the definition of convergence in terms of the relation “”. The only assumption is that is a solid cone, so we use neither continuity of the vector metric , nor Sandwich Theorem. We begin with the following.

Theorem 2.1. Let be an abstract metric space over a solid cone . Suppose that , , , and are four self-maps on such that and and suppose that at least one of these four subsets of is complete. Let for all , where and . Then the pairs and have a unique common point of coincidence. If, moreover, pairs and are weakly compatible, then , , , and have a unique common fixed point.

For definitions of terms like “point of coincidence” and “weakly compatible pair” see, for example, [11].

Remark 2.2. In the papers [9] and [27], the cone is supposed to be normal and solid. In that case the proof is essentially the same as in the setting of usual metric spaces.

We now give the proof of Theorem 2.1.

Proof. Suppose is an arbitrary point, and define the sequence by , , . Now, as in [27], by (2.1), we have which implies that , where . Similarly it can be shown that Therefore, for all , Now, for any , Thus, by properties and and Definition 1.3, is a Cauchy sequence.
Suppose, for example, that is a complete subset of . Then , , for some . Of course, subsequences and also converge to . Let us prove that . Using (2.1) we get that which further implies that Let be given. Since as , choose a natural number such that for all (Definition 1.3) we have that Thus, according to (2.7) we obtain . Therefore, for all . Using property (), it follows that and so . Since , we get that there exists such that . Let us prove that also . By triangle inequality and (2.1), we have which further implies that Now, for given , since as , choose a natural such that for all we have that According to (2.10) we obtain . Therefore, for all . Using property (), it follows that and so . We have proved that is a common point of coincidence for pairs and .
If now these pairs are weakly compatible, then (say) and (say). Moreover, implies that . So we have that . It remains to prove that, for example, . Indeed, implying that . The uniqueness follows from (2.1). The proofs for cases in which , , or is complete are similar and are therefore omitted. The theorem is proved.

We present now two examples showing that Theorem 2.1 is a proper extension of the known results. In both examples, the conditions of Theorem 2.1 are fulfilled, but in the first one (because of nonnormality of the cone) the main theorems from [9, 27] cannot be applied.

Example 2.3 (the case of a nonnormal cone). Let , and let be the set of all real-valued functions on which also have continuous derivatives on . Note that is a vector space over under usual function operations. Let be the strongest vector (locally convex) topology on . Then is a topological vector space which is not normable and is not even metrizable. Let and be defined by . Then is an abstract metric space over a nonnormal solid cone (Example 1.1). Consider the four mappings defined by Clearly and .
For , Now Thus all the conditions of Theorem 2.1 are satisfied with . Note that is the unique common fixed point of , , , and .

Example 2.4 (the case of a normal cone). Let with (this cone is normal; see [5]). Let , and let be defined as . Take the functions , , which map the set into . All the conditions of Theorem 2.1 are fulfilled with , . Obviously, , , , and have the unique common fixed point .

Remark 2.5. Taking and appropriate choices of , , , , and in Theorem 2.1, one easily gets [9, Corollaries 2.2–2.8]. In each of the following cases (1)–(7), is a complete abstract metric space,    a solid cone, and is a selfmap on .(1) Let for all , where and , and and are fixed positive integers. Then has a unique fixed point in .(2) If for all , where and , then has a unique fixed point in .(3) If for all , where for each and , then has a unique fixed point in .(4) If for all , where , then has a unique fixed point in .(5) If for all , where , then has a unique fixed point in .(6) If for all , where , then has a unique fixed point in .(7) If for all , where and , then has a unique fixed point in .

We add an example of a Banach-type contraction on a nonnormal abstract metric space.

Example 2.6. Let , , . An abstract metric on is defined by where is an arbitrary function (e.g., ). It is easy to see that is a complete abstract metric space. Suppose that mapping satisfies for all , where . All the conditions from Remark 2.5(4) hold, and has a unique fixed point in .
This example verifies that Theorem 2.1 is a proper extension of the results from [7]. Indeed, we know (see Example 1.1) that the cone is nonnormal. So, in this example Theorem 1 from [7] cannot be applied.

Corollary 2.7. Let be an abstract metric space and a solid cone. Suppose that the mappings , , and are four selfmaps of such that and , and suppose that at least one of these four subsets of is complete. Let for all , where and and commute. Then , , , and have a unique common fixed point.

Proof. By Theorem 2.1, we obtain such that The result then follows from the fact that since so that by property (). Again implies that . And hence is the unique common fixed point of , , , and .

3. Periodic Point Theorems

It is clear that if is a map which has a fixed point , then is also a fixed point of for every . However the converse is not true. For example, consider and defined by , . Then has a unique fixed point at , but every even iterate of is the identity map, which has each point of as its fixed point. On the other hand, if , , then every iterate of has the same fixed point as . If a map satisfies for each , where stands for the set of all fixed points of , then it is said to have property [29]. We will say that have property if for each .

The next result is a generalization of the corresponding result in metric spaces (see [29, Theorem 1.1]). It will be deduced also without using normality of the cone.

Theorem 3.1. Let be an abstract metric space over a solid cone , and let be such that and that holds for some and either (i) for all or (ii) for all , . Then has property .

Proof. We will always assume that , since the statement for is trivial. Let . Suppose that satisfies (i). Then According to property () it follows that , that is, . Suppose that satisfies (ii). If , then there is nothing to prove. Suppose, if possible, that . Then, similarly as in case (i) we get that . In order to use (3.1) we need that . But, if this is not the case, then and so , a contradiction. Hence, applying (3.1) we obtain that Repeating the same argument several times we finally obtain, similarly as in case (i), that , which again implies since , a contradiction.

Corollary 3.2. Let be an abstract metric space over a solid cone . Suppose that a mapping satisfies for all , where and . Then has property .

Proof. From Remark 2.5(2), . We will prove that satisfies the condition (i) of Theorem 3.1. Indeed, which implies that , where . Hence, has property .

The method of proof of the following result differs to the one from [29] (see also [9, Theorem 3.2]).

Theorem 3.3. Let be a complete abstract metric space over a solid cone . Suppose that mappings satisfy (2.1) (with ). Then and have property .

Proof. By Theorem 2.1, we have that , where is the unique common fixed point of and . So for each . Let , where is arbitrary. Then, we obtain wherefrom it follows that , where .
Further, we have that Indeed, which implies (3.7). Hence, Since , according to property () it follows . Hence , which implies that and have property .

Corollary 3.4. Let be a complete abstract metric space over a solid cone . Suppose that the mapping satisfies one of the conditions (3)–(6) of Remark 2.5. Then has property .

Remark 3.5. In the paper [9], the space is supposed to be a complete cone metric space over a normal and solid cone . Hence, our Theorems 3.1, 3.3, and Corollary 3.2 are proper extensions of Theorems  3.1, 3.2.  and  3.3 from [9].

In the following result the cone is regular, hence also normal (for the definition see, e.g., [5]).

Theorem 3.6. Let be an abstract metric space over a regular cone . Let be two mappings such that and one of these subset of is complete. Suppose that there exist decreasing functions , , such that for each and satisfying for all , . Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Proof. Suppose, for example, that is complete. Take an arbitrary and, using that , construct a Jungck sequence defined by , . Let us prove that this is a Cauchy sequence. If for some , then it is easy to prove that the sequence becomes eventually constant and so convergent.
Suppose that for each . Using (3.10), we obtain that for each . Also, Adding the last two relations (and putting temporarily , ) we obtain where It is easy to see that monotonicity of all 's implies that is also a decreasing function and that for each . In particular, and so the sequence is strictly decreasing (and bounded from below).
Since the cone is regular, there exists and for each . Then for each , and hence where is a fixed scalar belonging to .
Now we prove that is a Cauchy sequence in the usual way: for it is Thus, by properties and and Definition 1.3, is a Cauchy sequence in and so there is such that when . We will prove that .
Put , in the contractive condition. We obtain (writing temporarily ) that Taking into account that all 's are bounded in and that the abstract metric is continuous (because the cone is normal), passing to the limit in the last vector inequality, we obtain that that is, . Since , it follows that and and have a point of coincidence .
Suppose that is another point of coincidence for and . Then (3.10) implies that Since , the last relation is possible only if . So, the point of coincidence is unique.
If is weakly compatible, then [8, Proposition 1.4] implies that and have a unique common fixed point. The proof for the case in which is complete is similar and is therefore omitted.

Remark 3.7. Taking , , , we obtain a shorter proof of Theorem 1.6 (i.e., [28, Theorem 4]).

Remark 3.8. Taking appropriate choices of , and , in Theorem 3.6, one can easily get the results of Reich (see relations (7), (8) in [28]), Hardy-Rogers (see relation (18) in [28]) and Ćirić (see relation (21) in [28]) in the setting of abstract metric spaces.

Acknowledgments

The authors (the first and the second) would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant OUP-UKM-FST-2012. The fourth and fifth authors are thankful to the Ministry of Science and Technological Development of Serbia.

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