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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 263621, 16 pages
http://dx.doi.org/10.1155/2011/263621
Research Article

Some Common Fixed Point Theorems in Partial Metric Spaces

Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 19 June 2011; Revised 29 August 2011; Accepted 22 September 2011

Academic Editor: James Buchanan

Copyright © 2011 Erdal Karapınar and Uğur Yüksel. 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

Many problems in pure and applied mathematics reduce to a problem of common fixed point of some self-mapping operators which are defined on metric spaces. One of the generalizations of metric spaces is the partial metric space in which self-distance of points need not to be zero but the property of symmetric and modified version of triangle inequality is satisfied. In this paper, some well-known results on common fixed point are investigated and generalized to the class of partial metric spaces.

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 . Different approaches in this area have been reported including applications of mathematical techniques to computer science [37].

In [2], Matthews discussed some properties of convergence of sequences 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 e.g., [816]) have focused on this subject and generalized some fixed point theorems from the class of metric spaces to the class of partial metric spaces.

The definition of partial metric space is given by Matthews (see e.g., [1]) as follows.

Definition 1.1. Let be a nonempty set and let satisfy 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 (usual) metrics on . It is clear that and are equivalent. Each partial metric on generates a topology on with a base of the family of open -balls , where for all and . A basic example of partial metric is , where . One can easily deduce that .

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

We give same topological definitions on partial metric spaces.

Definition 1.3 (see e.g., [1, 2, 13]). (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 .

The following lemmas will be frequently used in the proofs of the main results.

Lemma 1.4 (see e.g., [1, 2, 13]). (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. Let be a PMS. Therefore, (A)if , then ; (B)if , then .
This follows immediately from the definition and can easily be verified by the reader.

Lemma 1.6 (See e.g., [15]). Assume as in a PMS such that . Then for every .

In this paper, we extend some common fixed point theorems for two self-mappings without commuting property from the class of usual metric spaces (see e.g., [17]) to the class of partial metric spaces (see Theorems 2.6 and 2.2). We also consider some common fixed point theorems with commuting property on partial metric spaces (see Theorem 2.7, Corollary 2.8).

2. Main Results

We first recall the definition of a common fixed point of two self-mappings.

Definition 2.1. Let be a PMS and two self-mappings on . A point is said to be a common fixed point of and if .

In the sequel, we give the first results about a common fixed point theorem. We prove the existence and uniqueness of a common fixed point of two self-mappings under certain conditions. Notice that here the operators need not commute with each other.

Theorem 2.2. Suppose that is a complete PMS and , are self-mappings on . If there exists an such that for any , where then there exists such that .

Proof. Let . Define the sequence in a way that and and inductively If there exists a positive integer such that , then is a fixed point of and hence a fixed point of . Indeed, since , then Also, due to (2.1) we have where Thus we have (2.5) implies . Since , then , which yields that . Notice that is the fixed point of . As a result, is the common fixed point of and . A similar conclusion holds if for some positive integer . Therefore, we may assume that for all . If is odd, due to (2.1), we have where In view of (PM4), we have
Thus, (2.8) turns into
If , then since , the inequality (2.7) yields a contradiction. Hence, and by (2.7) we have
If is even, the same inequality (2.11) can be obtained analogously.
We get that is a nonnegative, nonincreasing sequence of real numbers. Regarding (2.11), one can observe that
Consider now Hence, regarding (2.12), 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 the sequence is convergent in to, say, .
Again by Lemma 1.4, Since is a Cauchy sequence in , we have . We assert that . Without loss of generality, we assume that . Now observe that Analogously, Taking into account (2.16), the expression (2.17) yields Inductively, we obtain Due to (2.12), the expression (2.19) turns into Regarding , by simple calculations, one can observe that Therefore, from (2.15), we have
We assert that . On the contrary, assume . Then . Let be a subsequence of and hence of . Due to (2.1), we have where Letting and taking into account (2.22), the expression (2.24) implies that Thus, Since , we have . By Remark 1.5, we get . Analogously, if we choose a subsequence of , we obtain . Hence .

Remark 2.3. We notice that Theorem 2.2 can be obtained from Theorem 2.1 in [18] or Theorem 5 in [19] by simple manipulations.
However, explicit proof of Theorem 2.2 has a crucial role in the proofs of Proposition 2.5, Theorems 2.6 and 2.7.
These results cannot be obtained from the mentioned papers [18, 19].

Example 2.4. Let and . Then is a complete PMS. Clearly, is not a metric. Suppose such that and . Without loss of generality, assume . Then where Thus, all conditions of Theorem 2.2 are satisfied, and 0 is the common fixed point of and .

Proposition 2.5. Suppose is a complete PMS and , are self-mappings on . If there exists an such that for any and some positive integers , where then there exists such that .

Proof. Let . As in the proof of the previous theorem, we define a sequence in a way that and ; we get inductively If there exists a positive integer such that , then is a fixed point of and hence a fixed point of . A similar conclusion holds if for some positive integer . Therefore, we may assume that for all .
If is odd, due to (2.29), we have where By means of (PM4), we have Thus, (2.33) becomes
If , then since , the inequality (2.32) yields a contradiction. Hence, and by (2.32), we have
If is even, analogously we obtain the same inequality (2.36).
We obtain that is a nonnegative, nonincreasing sequence of real numbers. Regarding (2.36), one has
Consider now Hence, regarding (2.37), we have . Moreover, which implies that is a Cauchy sequence in , that is, as . Since is complete, by Lemma 1.4, is complete and the sequence is convergent in to, say, .
By Lemma 1.4, Since is a Cauchy sequence in , we have We claim that . Following the steps (2.16)–(2.22) in the proof of Theorem 2.2, we conclude the result. Thus,
Thus, letting in view of (2.37), (2.41), the expression (2.42) yields that . Therefore, from (2.40) we have
We assert that . Assume the contrary, that is, , then . Let be a subsequence of and hence of . Due to (2.29), we have
Letting and taking into account (2.43), the expression (2.44) implies that Since , we have . By Remark 1.5, we get . Analogously, if we choose a subsequence of , we obtain . Hence .

The following theorem is a generalization of a common fixed point theorem that requires no commuting criteria (see e.g., [17]).

Theorem 2.6. Suppose is a complete PMS and are self-mappings on . If there exists an such that for any and some positive integers , where then and have a unique common fixed point .

Proof. Due to Proposition 2.5, we have
We claim that is a common fixed point of and . From (2.46) and (2.48), it follows that where Due to (PM3), we have . Hence Regarding the assumption and the expression (2.49), we get which implies that , and by Remark 1.5, we obtain . Analogously, one can show that . Hence, .
For the uniqueness of the common fixed point , assume the contrary. Suppose is another common fixed point of and . Then, where Therefore, . Since , one has which yields by Remark 1.5. Hence, is a unique common fixed point of and .

Theorem 2.7. Let be a complete PMS. Suppose that , and are self-mappings on , and and are continuous. Suppose also that and are commuting pairs and that If there exists an , and such that for any , in , where then , , , and have a unique common fixed point in .

Proof. Fix . Since and , we can choose , in such that and . In general, we can choose , in such that
We claim that the constructive sequence is a Cauchy sequence. If there exists a positive integer such that , then for all . Therefore, is a Cauchy sequence and we proved claim. Thus, we may assume that for all .
By (2.55) and (2.57), where
Due to (PM4), we have Hence But if , then by (2.58) which implies . Thus, , and consequently, or, equivalently, Analogously, one can show that Indeed, from (2.55) and (2.57), where If , then by (2.66), we have a contradiction. Thus which proves (2.65).
Thus, we conclude that , for all .
By elementary calculation, regarding , we conclude that is a Cauchy sequence. Since is complete, converges to a point . Consequently, the subsequences , , , and converge to .
Regarding that , and , are commuting pairs and the continuity of and , the sequences tend to , and the sequences tend to , as .
Thus, where
Since , one has , that is, . Analogously, one obtains To conclude the proof, consider where Letting in (2.72) and having in mind (2.70), we get that . Due to (2.71), we have . Thus, we have
We assert that is unique. Suppose on the contrary that there is another common fixed point of , , , and . Then , where Since , Therefore, , and by Remark 1.5, we have . Hence is the unique common fixed point of , , , and .

Regarding the relation between Theorems 2.2 and 2.6, one concludes the following corollary in view of Theorem 2.7.

Corollary 2.8. Let be a complete PMS. Suppose that , and are self-mappings on , and and are continuous. Suppose also that , and are commuting pairs and also and commutes each other and If there exists , and such that for any , in , where then , , , and have a unique common fixed point in .

Proof. Due to Theorem 2.7, Following the steps of the proof of Theorem 2.7 with , , we get (2.70) which is equivalent to (2.79). Thus, , , , and have a unique common fixed point in .
We claim that By (2.79), where
Hence, (2.81) is equivalent to which yields , that is, . Analogously, one can get . Indeed, By (2.79), where Hence, (2.83) is equivalent to which yields , that is, . Hence, Combining (2.80) and (2.85), we obtain .

Acknowledgment

The authors express their gratitude to the referees for constructive and useful remarks and suggestions.

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