Abstract and Applied Analysis

Volume 2013, Article ID 303626, 7 pages

http://dx.doi.org/10.1155/2013/303626

## Fixed Point Results in Quasi-Cone Metric Spaces

School of Mathematical Sciences Faculty of Science and Technology, University Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 23 November 2012; Accepted 22 February 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Fawzia Shaddad and Mohd Salmi Md Noorani. 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

The main aim of this paper is to prove fixed point theorems in quasi-cone metric spaces which extend the Banach contraction mapping and others. This is achieved by introducing different kinds of Cauchy sequences in quasi-cone metric spaces.

#### 1. Introduction and Preliminaries

The Banach contraction principle is a fundamental result in fixed point theory. Due to its importance, several authors have obtained many interesting extensions and generalizations (see, e.g., [1–12]).

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric: it can be regarded as an “asymmetric metric.” In fact, quasi-metric space is more comprehensive than metric space. As metric space is important and has numerous applications, Huang and Zhang [13] have announced the concept of the cone metric spaces, replacing the set of real numbers by an ordered Banach space. They have proved some fixed point theorems for contractive-type mappings on cone metric spaces, whereas Rezapour and Hamlbarani [14] omitted the assumption of normality in cone metric spa ces, which is a milestone in developing fixed point theory in cone metric spaces. Since then, numerous authors have started to generalize fixed point theorems in cone metric spaces in many various directions. For some recent results (see, e.g., [15–25]) and for a current survey of the latest results in cone metric spaces, see Janković et al. [26].

Very recently, some authors generalized the contractive conditions in the literature by replacing the constants with functions. Using these generalizations, they have proved the existence and uniqueness of the fixed point in cone metric spaces; for more details see [27, 28]. Because quasi-metric space is more general than metric space and is a subject of intensive research in the context of topology and theoretical computer science, Abdeljawad and Karapinar [29] and Sonmez [30] have given a definition of quasi-cone metric space which extends the quasi-metric space.

In this paper, we also introduce the concept of a quasi-cone metric space which is somewhat different from that of Abdeljawad and Karapinar [29] and Sonmez [30]. Then we establish four kinds of Cauchy sequences in this space according to Reilly et al. [31]. Furthermore, we extend and generalize the Banach contraction principle and some results in the literature to this space. We support our results by examples. In this paper we do not impose the normality condition for the cones, and the only assumption is that the cone is, solid; that is . Now we recall some known notions, definitions, and results which will be used in this work.

*Definition 1. *Let be a real Banach space and be a subset of . is called a cone if and only if (i)is closed, , ; (ii)for all , where ; (iii) and .

For a given cone , we define a *partial ordering * with respect to by the following: for , we say that if and only if . Also, we write for , where denotes the *interior* of . The cone is called *normal* if there is a number such that for all
The least positive number satisfying this is called the normal constant of . The cone is called regular if every increasing sequence which is bounded above is convergent; that is, if is a sequence such that
for some , then there is such that as . Equivalently, the cone is regular if every decreasing sequence which is bounded below is convergent (for details, see [13]). In this paper, we always suppose that is a real Banach space, is a cone in with , and is a partial ordering with respect to .

*Definition 2 (see [13]). * Let be a nonempty set. Suppose the mapping satisfies (*d*1) for all , and if and only if ; (*d*2) for all ; (*d*3) for all .

Then, is called a cone metric on , and is called a cone metric space.

Now, we state our definition which is more general than cone metric space.

*Definition 3. *Let be a nonempty set. Suppose the mapping satisfies(*q*1) for all ; (*q*2) if and only if ; (*q*3) for all .

Then, is called a quasi-cone metric on , and is called a quasi-cone metric space.

*Remark 4. *Note that in [30] Sonmez defined the quasi-cone metric space as follows.

A quasi-cone metric space on a nonempty is a function such that for all ; (1), (2).

A quasi-cone metric space is a pair such that is a nonempty set and is a quasi-cone metric on .

In fact, it has not mentioned that takes value in , but in this paper we require this condition.

*Remark 5. *Abdeljawad and Karapinar’s definition of quasi-cone metric space [29] is as follows.

Let be a nonempty set. Suppose that the mapping satisfies the following: (*q*1) for all ; (*q*2); (*q*3) for all .

Then is said to be a quasi-cone metric on , and the pair is called a quasi-cone metric space.

The following example indicates that our definition is more general than the one given in [29].

*Example 6. *Let , , , and defined by
Then satisfies our definition of a quasi-cone metric space but not the definition in [29] because if then or .

*Remark 7. *Note that any cone metric space is a quasi-cone metric space.

#### 2. Necessary Facts and Statements

By considering the established notions in metric spaces [31], we introduce the appropriate generalization in cone metric spaces.

*Definition 8. *Let be a quasi-cone metric space. A sequence in is said to be (a)-Cauchy or bi-Cauchy if for each , there is such that for all ; (b)left (right) Cauchy if for any , there is such that (, resp.) for all ; (c)weakly left (right) Cauchy if for each , there is such that (, resp.) for all ; (d)left (right) -Cauchy if for every , there exist and such that (, resp.) for all .

*Remark 9. *These notions in quasi-cone metric space are related in the following way: (i)-Cauchy left (right) Cauchy weakly left (right) Cauchy left (right) -Cauchy; (ii)a sequence is -Cauchy if and only if it is both left and right Cauchy.

In this paper, we use the notion of left Cauchy.

*Definition 10. *Let be a quasi-cone metric space. Let be a sequence in . We say that the sequence left converges to if . One denotes this by

We will utilize the word converges instead of left converges for simplicity.

*Example 11. *Let , , , and defined by
where . is a quasi-cone metric on . Considering a sequence , then is left Cauchy and is convergent to due to
On the other hand, it is not right Cauchy.

*Definition 12. *A quasi-cone metric space is called left complete if every left Cauchy sequence in converges.

*Definition 13. *Let be a quasi-cone metric space. A function is called (1) continuous if for any convergent sequence in with , the sequence is convergent and ;(2) contractive if there exists some such that
and if , then is nonexpansive.

The following example shows that there exists a contractive function in quasi-cone metric space which is not continuous.

*Example 14. *Let , , , and defined by
and defined by
Then is a quasi-cone metric space and is a contractive map but not continuous due to .

#### 3. Fixed Point Theorems

In this section, we prove some fixed point results in quasi-cone metric space. Also, we generalize the contractive conditions in the literature by replacing the constants with functions. First, we state the following useful lemma.

Lemma 15. *Let be a quasi-cone metric space and a sequence in . Suppose there exist a sequence of nonnegative real numbers such that , in which
**
for some , and for all . Then the sequence is left Cauchy sequence in . *

*Proof. * For , we get
Let and choose such that where . Since , there exists a natural number such that for all , also . Since is open, therefore ; that is . Thus, for and so
Thus, is a left Cauchy sequence.

We are now in a position to state the main fixed point theorem in the context of quasi-cone metric spaces. We will need the notion of Hausdorff in quasi-cone metric space. A quasi-cone metric space is Hausdorff if for each pair of distinct points of , there exist neighborhoods and of and , respectively, that are disjoint.

Theorem 16. *Let be a left complete Hausdorff quasi-cone metric space and let be a continuous function. Suppose that there exist functions which satisfy the following for : *(1)* and ; *(2)*; *(3)*. **
Then, has a unique fixed point. *

*Proof. * Let be arbitrary and fixed, and we consider the sequence for all . If we take and in (3) we have
So,
where . Thus, by Lemma 15, is left Cauchy in . Because of completeness of and continuity of , there exists such that and . Since is Hausdorff, .*Uniqueness*. Let be another fixed point of , then
Therefore, due to . Similarly, . Hence, .

Corollary 17. *Let be a left complete Hausdorff quasi-cone metric space and let be a continuous function. Suppose that there exist functions which satisfy the following for : *(1)* and ; *(2)*; *(3)*. **
Then, has a unique fixed point. *

*Proof. * We can prove this result by applying Theorem 16 to and .

Corollary 18. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and
**
for all and with . Then, has a unique fixed point. *

*Proof. * We can prove this result by applying Theorem 16 to and .

The following corollaries generalize some results of [14] in cone metric spaces to quasi-cone metric spaces.

Corollary 19. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and
**
for all and . Then, has a unique fixed point. *

Corollary 20. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and
**
for all and . Then, has a unique fixed point. *

Corollary 21. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and
**
for all and with . Then, has a unique fixed point. *

The next corollary is a generalization of Banach contraction principle.

Corollary 22. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and
**
for all and . Then, has a unique fixed point. *

The example in [31] shows that the Hausdorff condition is necessary for quasi-metric spaces and is so for quasi-cone metric spaces. Now, we present two examples. The first one fulfills Theorem 16in which is normal. The second example satisfies Corollary 18 without normality of .

*Example 23. *Let , , such that
where . Suppose and . Then for all , we have the following.(1), , , , and . (2). (3)Condition of Theorem 16 is satisfied. For , we have
due to
and for it is trivial.

Therefore, is a fixed point.

*Example 24. *Let , , , and defined by
where . Suppose . If we take , and , then all the assumptions of Corollary 18 are satisfied. Thus, is a fixed point.

Theorem 25. *Let be a left complete Hausdorff quasi-cone metric space, and let be a continuous function and for all **
where , and . Then, has a unique fixed point. *

*Proof. * Let be arbitrary and fixed and for all . If we take and in , we have
Rewriting this inequality as
implies that
Since , we have
Therefore, we obtain
due to , and , and we get . Therefore,
Thus, by Lemma 15, is left Cauchy in . Because of completeness of and continuity of , there exists such that and . Since is Hausdorff, .*Uniqueness*. Let be another fixed point. Putting and in , we obtain
Hence,
Similarly, applying with and , we have
Adding up the above two inequalities, we get
Subsequently, we obtain
Thus,
Hence, due to . Therefore, and .

#### Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant ERGS/1/2011/STG/UKM/01/13.

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