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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 308921, 11 pages
http://dx.doi.org/10.1155/2012/308921
Research Article

Coincidence Points for Expansive Mappings under c-Distance in Cone Metric Spaces

1Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, H. Sousse 4002, Tunisia
2Department of Mathematics, Atilim University, 06836 İncek, Ankara, Turkey
3Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 10 March 2012; Accepted 27 April 2012

Academic Editor: Charles Chidume

Copyright © 2012 Hassen Aydi 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 establish some fixed (common fixed) and coincidence point results for mappings verifying some expansive type contractions in cone metric spaces with the help of the concept of a c-distance. Our results generalize, extend, and unify several well-known comparable results in the literature. Some examples are also presented.

1. Introduction and Preliminaries

Huang and Zhang [1] reintroduced the notion of cone metric spaces and established fixed point theorems for mappings on this space. After that, many fixed point theorems have been proved in normal or nonnormal cone metric spaces by some authors (see e.g., [126] and references contained therein).

We need to recall some basic notations, definitions, and necessary results from literature. Let be the set of nonnegative real numbers. Let be a real Banach space and is the zero vector of .

Definition 1.1 (see [1]). A nonempty subset of is called a cone if the following conditions hold:(i) is closed and ,(ii),(iii).
Given a cone , a partial ordering with respect to is naturally defined by if and only if , for . We will write to indicate that but , while will stand for , where denotes the interior of . A cone is said solid if is non-empty.

Definition 1.2 (see [1]). Let be a non-empty set and satisfies(i) if and only if ,(ii) for all ,(iii) for all .Then, the mapping is called a cone metric on and the pair is called a cone metric space.

Definition 1.3 (see [1]). Let be a cone metric space, is a sequence in and .(i)If for every with there is such that for all , then is said to be convergent to . This limit is denoted by or as .(ii)If for every with , there is such that for all , then is called a Cauchy sequence in .(iii)If every Cauchy sequence in is convergent in , then is called a complete cone metric space.

Definition 1.4. Let be a cone metric space and let be a given mapping. We say that is continuous on if for every sequence is , we have If is continuous on each point , then we say that is continuous on .

In 2011, Cho et al. [11] and Wang and Guo [27] introduced a new concept of -distance in cone metric spaces, which is a cone version of -distance of Kada et al. [16] and proved some fixed point theorems for some contractive-type mappings in partially ordered cone metric spaces using the -distance. For other results, see [13, 24].

Definition 1.5 (see [11, 27]). Let be a cone metric space. Then, a function is called a -distance on if the following are satisfied:(q1) for all ,(q2) for all ,(q3) for each and , if for some , then whenever is a sequence in converging to a point ,(q4) for all with , there exists with such that and imply .

Remark 1.6 (see [11, 27]). The -distance is a -distance on if we take is a metric space, , , and is replaced by the following condition. For any , is lower semicontinuous. Moreover, holds whenever is lower semicontinuous. Thus, if is a metric space, and , then every -distance is a -distance. But the converse is not true in general case. Therefore, the -distance is a generalization of the -distance.

Example 1.7 (see [11, 27]). Let be a cone metric space and let be a normal cone. Define a mapping by for all . Then, is a -distance.

Example 1.8 (see [11, 27]). Let be a cone metric space and let be a normal cone. Define a mapping by for all , where is a fixed point in X. Then, is a -distance.

Example 1.9. Let be a cone metric space and let be a normal cone. Define a mapping by for all , where is a fixed point in X. Then is a -distance.

Remark 1.10 (see [11, 27]). (1) does not necessarily hold for all .
(2) is not necessarily equivalent to for all .

Lemma 1.11 (see [11, 27]). Let be a cone metric space and let be a -distance on . Let and be a sequences in and . Suppose that is a sequence in converging to . The following hold. (1)If and , then . (2)If and , then converges to . (3)If for all , then is a Cauchy sequence in . (4)If , then is a Cauchy sequence in .

Let be two selfmaps on a nonempty set . Recall that a point is called a coincidence point of the pair if . The point is called a point of coincidence. If , then is said a common fixed point of and .

The purpose of this paper is to give some common fixed and coincidence point theorems for mappings verifying some expansive type contractions on cone metric spaces via a -distance. Also, some examples are presented.

2. Main results

First, we present the following useful lemma, which is a variant of (2.2) in Lemma 1.11.

Lemma 2.1. Let be a cone metric space and let be a -distance on . Let be a sequence in . Suppose that and are tow sequences in converging to . If and , then .

Proof. Let be arbitrary. Since , so there exists such that for all . Similarly, there exists such that for all . Thus, for all , we have Take , so by , we get that for each , hence .

Now, we present coincidence point results in the frame work of cone metric spaces in terms of a -distance. Note that (2.2) is called an expansive type contraction.

Theorem 2.2. Let be a cone metric space and let be a -distance on . Let be two functions such that there are three nonnegative real numbers , , and with such that Assume the following hypotheses: (1) and , (2), (3) is a complete subset of . Then and have a coincidence point, say . One has .
Also, if , then the point of coincidence is unique.

Proof. Let . Since , we can choose such that . Again since , we can choose such that . Continuing this process, we can construct a sequence in such that for all .
By (2.2), we have Therefore, Set . By hypotheses, we have . Also, By induction, we get that Let . By , we have Since , so by Lemma 1.11(3), the sequence is Cauchy in , which is complete, hence there exists such that . Thus, as , that is, We claim that . Recall that and , so by and as , we get that From (2.2), we have Since , so it follows that By (2.9), we obtain Set and . Since , so as . Thus, by (2.9), (2.12), and Lemma 2.1, get that .
Using (2.2), we get Since , so . Now, we prove that if , then the point of coincidence is unique.
Let and be two points of coincidence of and , that is, there exist such that and . By the above, we have . By (2.2), the following holds: If , we conclude that .
Let be arbitrary. Take . Since , then by the condition , we get that . Thus, . This completes the proof.

Now, we state the following corollaries.

Corollary 2.3. Let be a cone metric space and let be a -distance on . Let be two functions. Assume there exists such that Assume the following hypotheses: (1), (2) is a complete subset of . Then, and have a coincidence point, say . One has and is the unique point of coincidence of and .

Proof. It follows by taking in Theorem 2.2.

Corollary 2.4. Let be a cone metric space and let be a -distance on . Let be two functions. Assume there exist two nonnegative real numbers and , with such that Assume the following hypotheses: (1), (2), (3) is a complete subset of . Then, and have a coincidence point, say . One has and is the unique point of coincidence of and .

Proof. It follows by taking in Theorem 2.2.

Corollary 2.5. Let be a complete cone metric space and let be a -distance on . Let be a surjective function. Assume there are three nonnegative real numbers , and with such that If and , then has a fixed point, say . One has . Also if , then the fixed point is unique.

Proof. It follows by taking , the identity on , in Theorem 2.2. Note that when is surjective, so , that is, the hypothesis (2) in Theorem 2.2 holds.

The next result is similar to Theorem 2.2, except that the contractive condition (2.2) is replaced by Note that this contractive condition is studied since it is different to (2.2) because of Remark 1.10.(2). Its proof is essentially the same as for Theorem 2.2 and so is omitted.

Theorem 2.6. Let be a cone metric space and let be a -distance on . Let be two functions such that there are three nonnegative real numbers , , and with such that Assume the following hypotheses: (1) and , (2), (3) is a complete subset of . Then, and have a coincidence point say . One has . If , is the unique point of coincidence of and .

Remark 2.7. Let be a cone metric space and let be a normal cone. Take in Theorem 2.2 or Theorem 2.6 the -distance defined by for all . Then, the inequalities (2.2) and (2.19) correspond to the contractive condition given in Theorem 2.1 of Shatanawi and Awawdah [28]. Thus, our results (Theorems 2.2 and 2.6) extend and generalize the results in [28].

Remark 2.8. Some similar results as above corollaries could be derived from Theorem 2.6.

Now, we present the following example.

Example 2.9. Let with and . Let and let be defined by . Then, is a cone metric space. Let, further, be defined by . It is easy to check that is a -distance. Consider the mappings defined by Take and (we have , and ). For all , we have All hypotheses of Corollary 2.3 are satisfied, and is a coincidence point of and . Also, and is the unique point of coincidence of and .

Next, we present a common fixed point theorem for two maps involving some expansive type contractions given by the conditions (2.22).

Theorem 2.10. Let be a complete cone metric space and let be a -distance on . Let be two mappings. Suppose that and satisfy the following inequalities: for all and some nonnegative real numbers , , and with and . If and are continuous and surjective, then and have a common fixed point.

Proof. Let be an arbitrary point in . Since is surjective, there exists such that . Also, since is surjective, there exists such that . Continuing this process, we construct a sequence in such that and for all . Now, for , we have Thus, we have By , we have . Hence, we get that Therefore, On other hand, we have Thus, Again, using , we have Hence, Let Then, by combining (2.26) and (2.30), we have Repeating (2.32) -times, we get Thus, for , we have By assumption, we get that . By Lemma 1.11(3), is a Cauchy sequence in the complete cone metric space . Then, there exists such that as . Since and as , so clearly, the fact that and are continuous and uniqueness of limit yields that , that is, is a common fixed point of and .

Corollary 2.11. Let be a complete cone metric space and let be a -distance on . Let be a continuous surjective mapping satisfying for all and some nonnegative real numbers and with . Then, has a fixed point.

Proof. It follows from Theorem 2.10 by taking and .

Remark 2.12. Corollary 4.1 of [29], Theorem 4 of [30], and Corollary 2.8 of [28] are particular cases of Corollary 2.11.

We give the following examples illustrating our result obtained by Theorem 2.10.

Example 2.13. Let with , , and . Let be defined by . Then, obviously, is a cone metric space and is a -distance. Consider two mappings defined by and . For every , take and . We have and the conditions (2.22) hold. So and have a (unique) common fixed point, which is .

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