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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 190862, 11 pages
http://dx.doi.org/10.1155/2012/190862
Research Article

Common Fixed Point Results for Four Mappings on Partial Metric Spaces

1Faculty of Science and Arts, University of Amasya, Amasya, Turkey
2Department of Mathematics, Faculty of Science, University of Gazi, Teknikokullar, 06500 Ankara, Turkey
3Department of Mathematics, Faculty of Science and Arts, University of Artvin Coruh, Seyitler Yerleskesi, 08000 Artvin, Turkey

Received 23 March 2012; Revised 8 June 2012; Accepted 7 September 2012

Academic Editor: Paul Eloe

Copyright © 2012 A. Duran Turkoglu and Vildan Ozturk. 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 fixed point results for four mappings which satisfy almost generalized contractive condition on partial metric space and we support the results with an example.

1. Introduction and Preliminaries

Partial metric spaces, introduced by Matthews [1, 2], are a generalization of the notion of the metric space in which in definition of metric, the condition is replaced by the condition .

In [1], Matthews discussed some properties of convergence of sequence and proved the fixed point theorems for contractive mapping on partial metric spaces: any mapping of a complete partial metric space into itself that satisfies, where , the inequality for all , has a unique fixed point. Recently, many authors (see [315]) have focused on this subject and generalized some fixed point theorems from the class of metric spaces.

The definition of partial metric space is given by Matthews (see [2]) as follows.

Definition 1.1. Let be a nonempty set and let satisfy(PM1) , (PM2) , (PM3) , (PM4) ,for all , and , where . Then the pair is called a partial metric space (in short PMS) and is called a partial metric on .

Let be a PMS. Then, the functions given by are ordinary equivalent metrics on . Each partial metric on generates a topology on with a base of the family of open -balls , where for all and .

Example 1.2 (see [1, 2]). Let and define Then is a partial metric space.

We give same topological definitions on partial metric spaces.

Definition 1.3 (see [1, 2, 4]). (i) A sequence in a PMS converges to if and only if .(ii) A sequence in a PMS is called a Cauchy sequence if and only if exists (and finite).(iii) A PMS is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .(iv) A mapping is said to be continuous at if for every , there exists such that .

Lemma 1.4 (see [1, 2, 4]). (A) A sequence is Cauchy in a PMS if and only if is Cauchy in a metric space . (B) A PMS is complete if and only if the metric space is complete. Moreover, where is a limit of in .

Remark 1.5 (see [11]). Let be a PMS. Therefore,(A) if , then ;(B) if , then .

Lemma 1.6 (see [10]). Assume as in a PMS such that . Then for every .

On the other hand, Kannan [16] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. Afterward Sessa [17] introduced the notion of weakly commuting maps, which generalized the concept of commuting maps. Then Jungck generalized this idea, first to compatible mappings [18] and then to weakly compatible mappings [19].

A pair of self-mappings on is said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). A point is called point of coincidence of a family , , of self-mappings on if there exists a point such that for all .

The concept of almost contraction property was given to as follows by Berinde.

Definition 1.7 (see [20, 21]). Let be a metric space. A map is called an almost contraction if there exist a constant and some such that for all Berinde called this as “weak contraction” in [20], then he renamed it as “almost contraction” in [21, 22], also Berinde [21] proved some fixed point theorems for almost contraction in complete metric space. Definition 1.7 is a special case of the following definition (choose , is the identity map on ).

Definition 1.8 (see [7]). Let be a metric space. A map is called an almost contraction with respect to a mapping if there exist a constant and some such that for all ,

Babu et al. [23] considered the class of mappings that satisfy “condition (B).”

Let be a metric space. A map is said to satisfy “condition (B)” if there exist a constant and some such that for all ,

Afterward, Berinde [21], Abbas and Ilić [24], and Ćirić et al. [7] generalized the above definition and proved some fixed point results.

In recent paper, Altun and Acar [25] introduced the notion of weak contraction in the sense of Berinde in partial metric space.

Definition 1.9 (see [25]). Let be a partial metric space. A map is called -weak contraction if there exist a and some such that for all .

In this paper, we give a fixed point theorem for four mappings satisfying almost generalized contractive condition in [26] on partial metric spaces.

2. Main Results

Theorem 2.1. Let be a complete partial metric space and , , and be self maps on , with and . If there exists and with such that for any , , where,

If and are weakly compatible and one of , and is a complete subspace of , then , , , and have a common fixed point.

Proof. Let be an arbitrary point in . Since , we can find such that and also, as , there exist such that . In general, is chosen such that and such that , we obtain a sequences in such that Suppose for some . Thus, and have a coincidence point. Due to (2.1), we have where So, Therefore, by , we have , that is, . So, and have a coincidence point.
Suppose now that for all . From (2.1), we obtain where Due to (2.7), we have Due to PM4, we have Hence, . If , then by (2.7) Since , the inequality (2.9) yields a contradiction. Hence, , then by (2.7) we have Thus, one can observe that Consider now Hence, regarding (2.13), we have Moreover, After standard calculation, we obtain that is a Cauchy sequence in , that is, as , . Since is complete, by Lemma 1.4, is complete and sequence is convergent in to say . From Lemma 1.4, Since is a Cauchy sequence in , we have We assert that . Without loss of generality, we assume that , Similarly, Taking into account (2.20), the expression (2.19) yields Inductively, we obtain Due to (2.13), Regarding , we can observe that .
Since in , ,  ,  ,   converge to .
Now we show that is the fixed point for maps and . Assume that is complete, there exists such that . We will show that . On the contrary, assume that .
From, (2.1) we have where Since and . We get Since , we get . Therefore, . Since the maps and are weakly compatible, we have . We will also show that . From (2.1), we have where Since and , then Since , . By Remark 1.5, we get .
Similarly, we show that is also fixed point of and . Hence, .
The proofs for the cases in which , , or is complete are similar.
Last, we show is unique. Suppose on the contrary that there is another common fixed point of , , , and . Then where Thus, Therefore, and Remark 1.5. So, is the unique common fixed point os , , , and .

Example 2.2. Let endowed with the partial metric given by for all . It is clear that is a complete partial metric space. Define the mappings , , , by We have . For , , Then, the contractive condition (2.1) is satisfied for every , . Moreover, is weakly compatible. So all conditions of Theorem 2.1 are satisfied. We deduce the existence and uniqueness of a common fixed point of and . Here, 0 is the unique common fixed point.

Corollary 2.3. Let is complete PMS and and be self maps on , with . If there exists and such that where, for every , . If is weakly compatible and one of and is a complete subspace of , then and have a common fixed point.

Remark 2.4. It is easy to see that for every map , is weakly compatible, where is identity map on , so by taking in Theorem 2.1 we have the following results.

Corollary 2.5. Let is complete PMS and and be self maps on . If there exists and such that for every , , where Then and have a common fixed point.

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