Journal of Applied Mathematics

Volume 2013 (2013), Article ID 831491, 7 pages

http://dx.doi.org/10.1155/2013/831491

Research Article

On Cyclic Generalized Weakly -Contractions on Partial Metric Spaces

Erdal Karapınar¹ and Vladimir Rakocević²

¹Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey

²University of Nıs, Faculty of Sciences and Mathematics, Visegradska 33, 18000 Nıs, Serbia

Received 23 April 2013; Accepted 29 May 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Erdal Karapınar and Vladimir Rakocević. 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 give new results of a cyclic generalized weakly -contraction in partial metric space. The results of this paper extend, generalize, and improve some fixed point theorems in the literature.

1. Introduction and Preliminaries

The notion of partial metric space [1], represented by the abbreviation PMS, departs from the usual metric spaces due to removing the assumption of self-distance. In other words, in PMS self-distance needs not to be zero. This interesting distance function is defined by Matthews [1], as a generalization metric to study in computer science, in particular, to get a more efficient programs in computer science. In the remarkable publication of Matthews [1], a characterization of the Banach Contraction Principle was given in the context of PMS. Due to its wide application potential [2–6], PMS and its topological properties are considered by many authors [7–25]. Very recently, Haghi et al. [26] proved that some obtained results in the context of PMS can be deduced from earlier results in the setting of usual metric space.

In the sequel, , will represent the set of all real nonnegative numbers and the set of all positive natural numbers, respectively. Moreover, we use the abbreviations MS, CMS, PMS, and CMPS for metric space, complete metric space, partial metric space, and complete partial metric space, respectively. Let be the collection of function which is nondecreasing, continuous together with the property for and . The following definition introduced by Chatterjea [27] to generalize the Banach contraction principle.

Definition 1. Suppose that is an MS. A mapping is said to be a -contraction if there exists such that the following inequality holds:

Moreover, Chatterjea [27] reported that every -contraction has a unique fixed point, where is a complete metric space. Recently, Choudhury [28] introduced a generalization of -contraction inspired by the notion of weak -contraction (see, e.g., [29, 30]).

Definition 2. Suppose that is an MS. A self-mapping on is called a weakly -contractive if

for all , where the mapping is continuous and has the following property:

The notion of weakly -contractive can be also called a weak -contraction. In [28], the author proves that on the setting of CMS, every weak -contraction possesses a unique fixed point.

On other hand, in 2003, Kirk et al. [31] introduce cyclic contraction and give a characterization of the celebrated fixed-point theorem of Banach (known also as the Banach contraction mapping principle) in the set-up cyclic contraction. The authors [31] introduced the notion of cyclic representation in the following way.

Definition 3 (see [31]). Suppose that is an MS and is a self-mapping on . Let be a natural number and let , be nonempty sets. Then, is called a cyclic representation of with respect to if

where .

Kirk et al. [31] prove that a self-mapping , on a cyclic representation of , possesses a fixed point if

where is a CMS and is a function, upper semicontinuous from the right and for .

Recently, Păcurar and Rus [32] generalize the result of Kirk et al. [31] via the notion of cyclic -contraction. Following the paper of Păcurar and Rus [32], the notion of cyclic weak--contraction was introduced by Karapınar [33]. Let be the collection of function which is nondecreasing, continuous together with the property for and .

Definition 4 (see [33]). Suppose that is an MS and is a self-mapping on . Let be a natural number and let , , be nonempty closed sets. Assume that is a cyclic representation of with respect to . A mapping is said to be a cyclic weaker -contraction if there exists such that

for any , , , where .

The author [33] shows that a self-mapping , on a cyclic representation of , possesses a fixed point if is a cyclic weaker -contraction on a CMS .

In the last decade, the existence and uniqueness of a fixed point of various cyclic contractions in the context of PMS have been investigated and improved by several authors, see, for example, [7, 11].

In this paper, we derive some fixed-point result on certain cyclic contractions in the setup of complete PMS. Presented results of the paper extend, improve, and generalize some recent results on the topic in the literature. Among them, we list a few of them as follows: [7, 11, 13, 17, 28, 34].

For the sake of completeness, we call up some basic definitions and essential results in PMS. For more details, see, for example, [1, 7, 8, 17, 22].

Definition 5. Let be a nonempty set. A function is called partial metric if the following conditions hold:

(),(),(),(),

for all . A pair is called partial metric space.

It is evident that if , then due to assumptions () and (). However, if , then need not be . It is also known that a PMS generates a topology which is . We say that a sequence is convergent to a point in if , denoted as () or , with respect to the corresponding topology. We underline the simple fact that a limit of a sequence in a PMS need not be unique. Notice also that the function need not be continuous; that is, and need not yield .

There is strong correlation between partial metric and metric. For example, a mapping given by

forms a metric on , where is a partial metric. It is called the corresponding metric of partial metric.

Example 6. Let . The pair is an elementary example of a PMS, where for all . Notice that the corresponding metric is

Example 7. The mapping forms a partial metric on . Note also that the corresponding metric is , where is a metric space and is arbitrary.

For the further nontrivial examples of PMS, they can be found in [1–6].

Definition 8. Let be a PMS. Then

(1)a sequence is called a Cauchy if the limit of as exists (and is finite). If every Cauchy sequence in converges to a point such that , then the space is called complete,(2)let be a sequence in . If , then the sequence is called -Cauchy. Analogously, if every -Cauchy sequence in converges to a point such that , then the space is called -complete and denoted by -CPMS [22].

This lemma can be found in some recent publication on the topic, see, for example, [2–6].

Lemma 9. Let be a PMS. Then

(a)a sequence is Cauchy in if and only if it is a Cauchy sequence in the metric space ,(b)a PMS is complete if and only if the metric space is complete. Furthermore, if and only if

Lemma 10. Let be a PMS.

(a)If as , then as for each [8, 9, 18].(b)If is complete, then it is -complete [22].

The converse assertion of (b) does not hold; for the counter examples, see [22]. Note that every closed subset of a -CPMS is -complete.

Let be the class of functions which is lower semicontinuous and satisfying .

In what follows we introduce the notion of a cyclic generalized weakly -contraction in PMS.

Definition 11. Assume that is a PMS and is a natural number. Suppose that are closed nonempty subsets of and is a cyclic representation of with respect to ; a mapping is said to be a cyclic generalized weakly -contraction if

for any , , , where and .

In this paper, we establish a fixed point theorem for cyclic generalized weakly -contractions in the frame of CMPS.

2. Main Results

We present the fundamental result of this paper as follows.

Theorem 12. Assume that is a -CPMS and is a cyclic generalized weakly -contraction. Then, the mapping has a unique fixed point , and .

Proof. Take ; that is, there is some with . Since implies that , we find such that . By using the same argument, we construct the sequence , where . Consequently, for , there exists such that and . We suppose that for all . Indeed, if for some , then we conclude that ; that is, is the desired fixed point of . Consequently, the proof is completed.

Due to (10), we derive that

for all . As a result, we find that

for all . We set . On the occasion of the facts above, is a nonincreasing sequence of nonnegative real numbers. Consequently, there exists such that

We will prove that . Suppose, to the contrary, that . From (14) and (15) we derive that

for any . Letting in (18), we have

This yields that

On the other hand, by (13) we have

Letting in inequality (21), we get that

Since , we get .

Due to , we have . Hence, . Then, by (20) we conclude that .

Hence, we have

We assert that the sequence is Cauchy. To reach this goal, the standard techniques in the literature will be used (see, e.g., [17]). For the sake of completeness, we explicitly prove that is Cauchy. First assert that

for each there is such that if with , then .

Suppose, to the contrary, that there is such that for all if with , then

We examine the case . So, taking into account, we can choose with in a way that it is the smallest integer satisfying and . Hence, , by using the triangular inequality

Letting in (25) and keeping in mind , we obtain

Again, by

Taking (23) and (26) into account, we get

as in (26).

By we have the following inequalities:

Letting in (29) we derived that

Again by () we have

Letting in (31) we derived that

Since and lie in distinct adjacently labeled sets and for certain , keeping in mind that is a cyclic generalized weakly -contraction, we have

Taking into account (23), (26), (28), (30), (32), and the lower semicontinuity of , letting in the inequality above, we find that

which is a contradiction. Hence, holds.

We are ready to show that the sequence is Cauchy. Fix . Due to the assumptions, one can find such that if with ,

Since , we also find such that

for any . Assume that and . Consequently, there is a with . Therefore, for . Thus, we obtain for ,

By (35) and (36) together with the last inequality, we find that

which yields that the sequence is Cauchy. Regarding that is arbitrary, we conclude that is a -Cauchy sequence.

Taking into account that is closed in , we observe that is also -complete. Thus, there exists such that in ; equivalently

Now, we assert that is a fixed point of . First, we observed that the sequence has infinite terms in each for , since and as is a cyclic representation of with respect to . Assume that and . We consider a subsequence of with . Notice that such subsequence exists due to the above-mentioned comment. By applying the contractive condition, we find

Letting and by using , together with the lower semicontinuity of , we get

So which yields that . We will prove the uniqueness of to complete the proof. Suppose, on the contrary, that are distinct fixed points of . We observe that , since is cyclic mapping and are fixed points of . Due to mentioned contractive condition, we derive that

that is,

This gives us ; that is, .

Corollary 13. Suppose that is a -CPMS, , are nonempty closed subsets of . Let be and let . be a cyclic representation of with respect to .

If there exists such that

for any , , , where , then, has a fixed point and .

Proof. Let . Hence, it suffices to take the function defined by . It is evident that satisfies the following conditions:

(1) if and only if , and (2) is lower semi-continuous.

The results follow when we apply Theorem 12.

Theorem 14. Suppose that is a -CPMS. If the mapping satisfies

for any , where , then, it has a unique fixed point with .

Proof. It is sufficient to take for in Theorem 12.

Remark 15. Let us remark that if in Definition 11 we consider the following condition

instead of (10), then by following the lines in the proof of Theorem 12, we obtain the same conclusions in our results.

Example 16. Let and . It is clear that is a 0-complete partial metric space. Fix and define , where for . Let and let be defined as

respectively. Then, all the conditions of Theorem 12 are satisfied. Hence, has a unique fixed point, namely, 0.

Remark 17. Notice that we get the same results if we replace -CPMS with CPMS.

Acknowledgment

Vladimir Rakocević is supported by Grant no. 174025 of the Ministry of Science, Technology, and Development, Serbia.

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