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Abstract and Applied Analysis
Volume 2012, Article ID 793862, 18 pages
http://dx.doi.org/10.1155/2012/793862
Research Article

Common Fixed Points of Weak Contractions in Cone Metric Spaces

1College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
2Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Beograd, Serbia
3Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
4Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 21 January 2012; Accepted 4 March 2012

Academic Editor: D. Anderson

Copyright © 2012 Hui-Sheng Ding 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

Results on common fixed points of mappings in cone metric spaces under weak contractive conditions (B. S. Choudhury and N. Metiya (2010)) are unified and generalized. Also, cone metric versions of some other related results on weak contractions are proved. Examples show that our results are different than the existing ones.

1. Introduction

The idea to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric, goes back to the mid-20th century (see, e.g., Kurepa [1], Kreĭn and Rutman [2], Kantorovič [3]). Fixed point theory in -metric and -normed spaces was developed by Perov [4], Vandergraft [5], and others. For more details we refer the reader to survey papers of Zabrejko [6] and Proinov [7]. In 2007, Huang and Zhang [8] reintroduced such spaces under the name of cone metric spaces and gave definitions of convergent and Cauchy sequences in the terms of interior points of the underlying cone, proving some fixed point theorems in such spaces. After that, fixed-points in cone metric spaces have been a subject of intensive research (see [9] for a survey of these results, and also [1012]).

Fixed point results under so-called weak contractive conditions were first obtained in [13, 14]. They were generalized by various authors (see, e.g., [15]), in particular, using a pair of control functions and [1621]. Note, however, that it was shown in [22] that in a certain sense the usage of function is superfluous. Weak contractions in cone metric spaces were treated in [2325], results from [24] being the most general ones.

In this paper we generalize and unify the results on weak contractions in cone metric spaces from [24]. Examples show that these generalizations are proper. Further, we extend theorems from [16] and some related results to the case of cone metric spaces and give examples of applications of the obtained results.

2. Preliminaries

Let be a real Banach space with as the zero element, and let be a subset of with the interior . The subset is called a cone if (a) is closed, nonempty, and ; (b) , , and imply ; (c) . For the given cone , a partial ordering with respect to is introduced in the following way: if and only if . If , we write .

If , the cone is called solid (we will always assume that the given cone has this property). It is called normal if there is a number , such that, for all , implies or, equivalently, if , and imply . The cone is called regular if every decreasing sequence in which is order-bounded from below is convergent, that is, if whenever is a sequence in such that for some , then there exists such that , . Finally, is called strongly minihedral if every subset of has an infimum in , provided it is order-bounded from below. It is well known that every strongly minihedral cone is regular and every regular cone is normal. The converses are not true [26].

Let be a nonempty set. Suppose that a mapping satisfies for all and if and only if ; for all ; for all . Then is called a cone metric on , and is called a cone metric space [8]. The concept of a cone metric space is obviously more general than that of a metric space.

Sometimes the following additional property of the cone metric will be needed:(), for all with .

For definitions of notions such as convergent and Cauchy sequences, completeness, and so forth, we refer to [8] and for a survey of fixed point results in such spaces to [9].

Definition 2.1 (see [24]). Let be a cone metric space over a solid cone . Denote by the class of functions satisfying the following conditions:(1) if and only if ;(2) for ;(3)for all and , either or holds.Denote by the class of functions satisfying the following conditions:(4) is strictly increasing; that is, if and only if ;(5) if and only if .Let two mappings and two arbitrary points be given. The following four sets of vectors will be used: If is the identity mapping, we will write for .
Let be two self-maps on a nonempty set . Recall that a point is called a coincidence point of the pair and is its point of coincidence if . The pair is said to be weakly compatible if for each , implies . A classical result of Jungck states that if, two weakly compatible maps have a unique point of coincidence , then is their unique common fixed point.
Roughly speaking, there are two types of common fixed point results with weak contractive conditions. Those of the first type use conditions with on the left-hand side and some element of the -set on the right-hand one. The other use conditions with on the left-hand side and some element of the -set on the right-hand side. An example of the first type is the following results in cone metric spaces that were proved by Choudhury and Metiya.

Theorem 2.2 (see [24], Theorems  3.1, 3.2, and 3.3). Let be a cone metric space over a regular cone such that holds. Let be such that one of the following inequalities holds for all : where and are continuous. If and is complete, then and have a unique point of coincidence in (and so they have a unique common fixed point in if the pair is weakly compatible).

On the other hand, in the case of metric spaces, the following result was proved by Đorić.

Theorem 2.3 (see [16], Theorem  2.1). Let be a complete metric space, be continuous, and be lower semicontinuous (here ). Let be two self-maps satisfying the inequality for all , where . Then and have a unique common fixed point in .

In this paper we generalize and unify results of Theorem 2.2 (we will call the conditions that we use “weak contractive conditions of the first type”). Examples show that these generalizations are proper. Further, we extend Theorem 2.3 and some related results to the case of cone metric spaces (the respective conditions will be called “weak contractive conditions of the second type”) and give examples of applications of the obtained results.

3. Auxiliary Results

We will make use of the following result of Choudhury and Metiya.

Lemma 3.1 (see [24]). Let be a cone metric space over a regular cone such that holds and suppose that there exists (see Definition 2.1). If is a sequence in such that is decreasing, then converges either to or to .

Note that is not supposed to be continuous. It is easy to show that without the existence of function the conclusion of Lemma 3.1 may fail to hold.

The following result is a cone metric version of [21, lemma  2.1].

Lemma 3.2. Let be a cone metric space over a regular cone such that holds and suppose that there exists (see Definition 2.1). Let be a sequence in such that is decreasing w.r.t. and that If is not a Cauchy sequence, then there exists and two sequences and of positive integers such that the following five sequences tend to when :

Proof. Suppose that is not a Cauchy sequence. Then there exists such that for each there exist with and . Hence, by property (2.4) of function , holds for . Therefore, there exist sequences and of positive integers such that (the last inequality is obtained by taking the smallest possible ). Now we have Letting and using assumption (3.1) and the normality of the cone, we obtain that Further, where , imply that The other three limits can be obtained similarly.

4. Weak Contractions of the First Type in Cone Metric Spaces

Theorem 4.1. Let be a cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be two selfmaps such that and let one of these subsets of be complete. Suppose that for all there exists such that holds true. Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Remark 4.2. Theorem 4.1 remains true if condition (4.2) is replaced by for some continuous (see Definition 2.1). The proof is essentially the same, and so, for the sake of simplicity, we stay within the given version. The same remark applies to all other results in the rest of the paper. See also paper [22] where it is shown that practically each weak contractive condition with function can be replaced by an equivalent condition without .

Proof. Starting from arbitrary and using the assumption , construct a Jungck sequence satisfying for . If for some , then and and have a point of coincidence. Suppose, further, that for . Putting , in (4.2) we obtain that where The case is impossible, since it would imply and (using properties of the function ), which is already excluded. In all other cases we get that , and, more precisely, Indeed, the right-hand inequality is trivial in the case when , and in the case , then and .
We have proved that the sequence is decreasing w.r.t. and so Lemma 3.1 implies that it converges to some , where either or . But, if , then (4.6) implies that also as . Hence, passing to the limit in (4.4) we get that and , a contradiction. Thus, .
Let us prove that is a Cauchy sequence in . Suppose that it is not. It follows from monotonicity of the sequence and that neither is a Cauchy sequence. Lemma 3.2 implies that there exist sequences and of positive integers such that the sequences (3.2) all tend to for some . Using (4.6) and putting , in (4.2) we get that Letting we get that . Properties of function imply that , a contradiction. Hence, is a Cauchy sequence.
By the assumption, there exists for some . Let us prove that . Putting , in (4.2) we get that where In other words, at least one of four possible inequalities holds for infinitely many . Hence, passing to the limit, we obtain that or or . By the properties of it follows that and ; hence is a point of coincidence for the pair .
To prove that this point of coincidence is unique, assume that there is another such that for some . Then where In both cases we get that , that is, the point of coincidence is unique.

Obviously, the theorem in [24, Theorem  3.1] (Theorem 2.2 with condition (2.2)) is a special case of Theorem 4.1.

Remark 4.3. The previous theorem can be modified so that continuity of is substituted by its lower semicontinuity; however, in this case it has to be assumed that the cone is strongly minihedral. For details see [10]. The same applies to other assertions to the end of the paper.

The following example shows that there are cases when the existence of a common fixed point can be deduced using Theorem 4.1, but cannot be obtained using the theorem in [24, Theorems  3.1, 3.2, and  3.3] (Theorem 2.2 with either of the conditions (2.2), (2.3), or (2.4)).

Example 4.4. Let , (), and , where is fixed. is obviously a cone metric satisfying property and the cone is regular (even minihedral). Function defined by and for , belongs to the respective class . Consider the mappings defined by: for , for and for . We will show that, taking, for example, , neither of conditions (2.2), (2.3), (2.4) is satisfied; hence neither of Theorems 3.1, 3.2, and 3.3 from [24] can be used to conclude that there exists a common fixed point of and (which is obviously ).
Indeed, for , , we get that . On the other hand . Hence, condition (2.2) is not satisfied. Further, for , , we obtain that and . Hence condition (2.3) is not satisfied. Finally, taking , , we have that , but . Hence, neither (2.4) is satisfied.
On the other hand, condition (4.2) of Theorem 4.1 is satisfied. Indeed, if or , then and the condition is trivially satisfied. If and (or vice versa), take and we obtain that and . Thus, conclusion about the existence of a common fixed point can be deduced from Theorem 4.1.

In the next theorem the set is used instead of . The proof is essentially the same as for Theorem 4.1 and so is omitted.

Theorem 4.5. Let be a cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be two selfmaps such that and let one of these subsets of be complete. Suppose that for all there exists such that holds true. Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

In the following theorem we unify Theorems 3.1, 3.2, and 3.3 of [24] (Theorem 2.2 with conditions (2.2), (2.3), or (2.4)).

Theorem 4.6. Let be a cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be two selfmaps such that and let one of these subsets of be complete. Suppose that for all there exists such that holds true. Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Proof. As usual, form a Jungck sequence by . If , then it can be proved as in Theorem 4.1 that has a point of coincidence. Assume that for all . Then where In each of the three possible cases it is easy to obtain that is a decreasing sequence and that Hence, all these three terms tend to some . Passing to the limit in relation (4.16) we get that , wherefrom it follows that .
That is a Cauchy sequence can be proved using Lemma 3.2 similarly as in the proof of Theorem 4.1. Hence, there exists such that when . Let us prove that is a point of coincidence of the pair .
Putting , in (4.15) we get that where Passing to the limit (more precisely, considering one of the inequalities that holds for infinitely many as in the proof of Theorem 4.1) we get either or and both are possible only if . Hence is a point of coincidence of .
The proof that the point of coincidence is unique is essentially the same as in Theorem 4.1.

The next example illustrates how Theorem 4.6 can be used to prove the existence of a common fixed point, while either Theorem  3.2 or 3.3 of [24] cannot.

Example 4.7. Let , and let , and be as in the previous example. Take function defined by and . Mappings are defined as and .

Condition (2.3) is not satisfied. Indeed, take and to obtain that and . Similarly, condition (2.4) is not satisfied, for taking again and one obtains that , but .

We show that, however, condition (4.15) is satisfied and so Theorem 4.6 can be used to conclude that there exists a common fixed point of and (which is obviously ) (note that this can also be done using condition (2.2)). Indeed, take . In order to prove inequality (4.15) it is enough to consider the first coordinates of respective vectors, that is, we have to prove that holds for all . But, it is an easy consequence of .

Finally, we state (proof can be deduced similarly as for the previous theorems) the following cone metric version of [18, Theorems  3.1 and 4.1] (see also [21, Theorem  3.6]).

Theorem 4.8. Let be a cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be a selfmap such that for all there exist such that . Then has a fixed point.

Note that in this case fixed point of need not be unique. It is enough to consider the identity mapping and take .

5. Weak Contractions of the Second Type in Cone Metric Spaces

In this section we consider weak contractions which we have called “of the second type” (see the end of Section 2).

Theorem 5.1. Let be a complete cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be two mappings such that for all there exists such that Then and have a unique common fixed point.

Proof. Let us prove first that the common fixed point of and is unique (if it exists). Suppose that are two distinct common fixed points of and . Then (5.2) implies that where . Checking both possible cases and using the properties of function , we readily obtain that , that is, .
In order to prove the existence of a common fixed point, proceed this time constructing a Jungck sequence by , , for arbitrary . Consider the two possible cases.
Suppose that for some . Then and it follows that the sequence is eventually constant, and so convergent. Indeed, let, for example, (in the case the proof is similar). Then, putting , in (5.2), we get that there exists such that . Consider the three possible cases:(1°); it trivially follows that .(2°); it follows that and by the properties of function that .(3°); since , it follows that wherefrom which is only possible if .Suppose now that for all . Putting , in (5.2), we get that there exists such that . Consider the three possible cases:(1°); it follows that and .(2°); it follows that which is impossible.(3°); it follows that By the properties of function we obtain that and .
Hence, in any possible case, and, similarly, . Thus, the sequence is decreasing; moreover,
We prove now that Indeed, passing to the limit in (5.11) when (and using regularity of the cone), we obtain that and () for some . If , then passing to the limit in we obtain that and by the properties of function . Hence, (5.12) holds.
We next prove that is a Cauchy sequence. According to monotonicity of and (5.12), it is sufficient to show that the subsequence is a Cauchy sequence. Suppose that this is not the case. Applying Lemma 3.2 we obtain that there exist and two sequences of positive integers and such that the sequences all tend to when .
Now, from (5.11) and the obtained limits, we have that for any . Letting , utilizing (5.15) and the obtained limits, we get which is a contradiction if . This shows that is a Cauchy sequence and hence is a Cauchy sequence.
Since the space is complete, there exists such that . Then also and (). Putting and in (5.2), we get , where
So, in this case we have four possibilities:(1°);(2°);(3°);(4°).Passing to the limit when in these four relations, we obtain one of the next three inequalities: In each of the cases it easily follows that .
Now, putting in (5.2), one gets , where and in each of the possible three cases it easily follows that . Hence, is a common fixed point of and .

Putting in Theorem 5.1, one obtains

Corollary 5.2. Let be a complete cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be such that for all there exists such that Then has a unique common fixed point.

Note that putting and in Theorem 5.1 and Corollary 5.2 we obtain as corollaries [16, Theorems  2.1 and 2.2], [15, Theorem  2.1], and [21, Theorem  4.1 and Corollary  4.2].

Adapting an example from [16] we give an example when Theorem 5.1 (modified to use a function according to Remark 4.2) can be used to deduce the existence of a common fixed point.

Example 5.3. Let , and let , and be as in Examples 4.4 and 4.7. Take defined by and for ; take defined by for (they satisfy the conditions of Definition 2.1). Consider the mappings given as and . Condition reduced to the first coordinates of respective vectors, has the form which was checked to be true in [16]. Hence, the existence of a common fixed point () of mappings and follows from Theorem 5.1.

The next is a kind of Hardy-Rogers-type result with weak condition. It can be considered as a cone metric version of results from [19, 21]. For the sake of simplicity we take only one mapping and for denote where , , .

Theorem 5.4. Let be a complete cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let and suppose that for all , holds. Then has a unique fixed point.

Proof. The given condition (5.23) and properties of function imply that for each . Starting with arbitrary construct the Picard sequence by . Condition (5.24) implies that wherefrom and, similarly, Adding up, one obtains that where . It follows that is a decreasing sequence which (by the regularity of cone ) tends to some . In order to prove that , put and in (5.23) to obtain where Similarly, On the other hand, (5.24) implies that In the case when , passing to the limit when , we obtain that ; the same conclusion is obtained if (or ). Hence, passing to the limit in (5.29), we get that , wherefrom .
As in some previous proofs, in order to obtain that is a Cauchy sequence, suppose that it is not the case and using Lemma 3.2 deduce that there exist and two sequences and of positive integers such that and the sequences all tend to . Putting and in (5.23) gives Here when . Since also when , we obtain that implying that (because ).
Thus, the sequence converges to some in the complete metric space . In order to prove that , suppose the contrary and put and in (5.24). It follows that Passing to the limit when gives that a contradiction, since .
The proof that the fixed point of is unique is standard.

In a similar way one can obtain a version of the previous theorem containing two selfmaps and (see [21, Theorem  5.2]).

At the end, we again state a cone metric version of a result from [18, Theorems  3.2 and 4.2] (see also [21, Theorem  3.7]).

Theorem 5.5. Let be a cone metric space over a regular cone such that holds and suppose that there exists a continuous function . Let be two selfmaps such that for all there exist such that . Then and have a common fixed point.

Here also common fixed point of and need not be unique.

Acknowledgments

The first author acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), the Foundation of Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. The second and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.

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