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Abstract and Applied Analysis
Volume 2012, Article ID 287127, 10 pages
http://dx.doi.org/10.1155/2012/287127
Research Article

A Generalized Meir-Keeler-Type Contraction on Partial Metric Spaces

1Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, Tunisia
2Department of Mathematics, Atilim University, Ankara, 06836 Incek, Turkey
3Department of Mathematics, Azarbaidjan University of Tarbiat Moallem, Azarshahr, Tabriz, Iran

Received 12 November 2011; Revised 16 February 2012; Accepted 17 February 2012

Academic Editor: Irena Rachůnková

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 introduce a generalization of the Meir-Keeler-type contractions, referred to as generalized Meir-Keeler-type contractions, over partial metric spaces. Moreover, we show that every orbitally continuous generalized Meir-Keeler-type contraction has a fixed point on a 0-complete partial metric space.

1. Introduction

In 1992, Matthews introduced the notion of a partial metric space which is a generalization of usual metric space [1]. The main motivation behind the idea of a partial metric space is to transfer mathematical techniques into computer science. This is mostly apparent in the research areas of computer domains and semantics, which have many applications (see, e.g., [210]). Following this initial work, Matthews generalized the Banach contraction principle in the context of complete partial metric spaces. He proved that a self-mapping on a complete partial metric space has a unique fixed point if there exists such that for all . After Matthews’ innovative approach, many authors conducted further studies on partial metric spaces and their topological properties (see, e.g., [24, 6, 1141]).

A partial metric is a function satisfying the following conditions:

(P1) ,

(P2) if , then ,

(P3) ,

(P4) ,

for all . Then is called a partial metric space.

Example 1.1 (see [42]). Let and be a metric space and partial metric space, respectively. Mappings () defined by induce partial metrics on , where is an arbitrary function and .

Each partial metric on generates a topology on with the family of open -balls as a base, where for all . Similarly, a closed -ball is defined as .

In [1, page 187], Matthews gave the characterization of convergence in partial metric space as follows: a sequence in a partial metric space converges to with respect to if and only if .

Now we recall some basic concepts and useful facts on completeness of partial metric spaces. A sequence in a partial metric space is called Cauchy whenever exists (and is finite) [1, Definition  5.2].

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that [1, Definition  5.3].

In [35], Romaguera introduced the concepts 0-Cauchy sequence in a partial metric space and 0-complete partial metric space as follows.

Definition 1.2. A sequence in a partial metric space is called 0-Cauchy if . A partial metric space is said to be 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that . In this case, is said to be a 0-complete partial metric on .

Notice that each 0-Cauchy sequence is also a Cauchy sequence in a partial metric space. In particular, each complete partial metric is a 0-complete partial metric on . But the converse is not true. The following example shows that there exists a 0-complete partial metric which is not complete.

Example 1.3 (see [35, 39]). Let be the partial metric space, where and represent the set of rational numbers and the partial metric , respectively.

A self-mapping on a partial metric space is continuous at if and only if for every , there exists such that (see, e.g., [15]).

It is quite natural to consider characterizations of continuity of mappings in partial metric spaces. For example, Samet et al. [43] proved the following.

Lemma 1.4. Let be a partial metric space. is continuous if given a sequence and such that ; then, .

Very recently, Samet et al. [43] also observed the relationship between the continuity of a mapping in a partial metric space and in a metric space.

Lemma 1.5. Consider endowed with the partial metric defined by for all . Let be a nondecreasing function. If is continuous with respect to the standard metric for all , then is continuous with respect to the partial metric .

In 1971, Ćirić [44] introduced orbitally continuous maps on metric spaces as follows.

Definition 1.6. Let be a metric space. A mapping on is orbitally continuous if implies for each .

Recently, Karapınar and Erhan [28] renovated the definition above in the context of partial metric spaces in the following way.

Definition 1.7. Let be a partial metric space, and let be a self-map. One says that is orbitally continuous whenever implies that for each .

It is clear that continuous mappings are orbitally continuous.

We would like to point out the close relationship between metrics and partial metrics. In fact, if is a partial metric on , then the function given by is a metric on . Moreover,

Lemma 1.8 (see, e.g., [1, 15]). Let be a partial metric space.
(a) A sequence is Cauchy if and only if is a Cauchy sequence in the metric space ;
(b) is complete if and only if the metric space is complete.

In 1969, Meir and Keeler [45] published their celebrated paper in which an interesting and general contraction condition for self-maps in metric spaces was considered.

Definition 1.9. Let be a metric space, and let be a self-map on . Then is called a Meir-Keeler-type contraction whenever for each there exists such that

Many authors have discussed several variations, generalizations, and modifications of that condition both in metric spaces and other related structures (see, e.g., [4649]). Following this trend, we introduce a generalized Meir-Keeler-type contraction on partial metric spaces. In this paper, we show an orbitally continuous self-mapping on a 0-complete partial metric spaces satisfying that generalized Meir-Keeler-type contraction has a unique fixed point.

2. Main Results

We start this section by recalling the following two lemmas ([13]), which will be frequently used in the proofs of the main results.

Lemma 2.1. Let be a partial metric space. Then
(a) if , then ,
(b) if , then ,
(c) if with , then for all .

We introduce the definition of a generalized Meir-Keeler-type contraction.

Definition 2.2. Let be a partial metric space and a self-map on . Then is called a generalized Meir-Keeler-type contraction whenever for each there exists such that where .

Remark 2.3. Note that if is a generalized Meir-Keeler-type contraction, we have If , it follows from (2.2) that . On the other hand, if , we get the strict inequality by (2.1).

Now, we are ready to state and prove our main results.

Proposition 2.4. Let be a partial metric space and a generalized Meir-Keeler-type contraction. Then, for all .

Proof.. Take , and set . Define for all . If for some , then by Lemma 2.1. Then, for all . In this case, the proposition follows. In the rest of the proof, we assume that for every . As a consequence, we have for every . By Remark 2.3,
Since is strictly positive for each , we find that by the use of Remark 2.3 again. Notice that the case where is not possible. Hence we derive that for every . Thus, is a decreasing sequence which is bounded below by 0. Hence, it converges to some , that is, In particular, we have Notice that .
We claim that . Suppose, to the contrary, that . Regarding (2.8) together with the assumption that is generalized Meir-Keeler-type contraction, for this , there exists and a natural number such that This is a contradiction since .

Theorem 2.5. Let be a 0-complete partial metric space, and let be an orbitally continuous generalized Meir-Keeler-type contraction. Then, has a unique fixed point, say . Moreover, for all and .

Proof.. Take , and set . Define for all . We claim that . If this is not the case, then there exist a and a subsequence of such that
For the same above, there exists such that which implies that . Set and for all . By Proposition 2.4, one can choose a natural number such that for all . Let . We have . If , then by using (P4) we derive which contradicts with assumption (2.10). Therefore, there are values of such that and . Now if , then This is a contradiction because of (2.11). Hence, there are values of with such that . Choose the smallest integer with such that . Thus, we find . So we see that Now, we can choose a natural number satisfying such that Therefore, we obtain the inequalities Thus, we have Now, inequalities (2.16)–(2.18) imply that . Hence, the fact that is a generalized Meir-Keeler-type contraction yields . By using (P4), we obtain We combine the inequality above with (2.15) and (2.17) to conclude which is a contradiction. Therefore, our claim is proved. So is a 0-Cauchy sequence. Since is 0-complete, then by Definition 1.2, the sequence converges with respect to to some such that . Thus Now, we will show that is a fixed point of .
Since is orbitally continuous and , we get that On the other hand, from Lemma 2.1, we have which follows from the fact that converges to in with , where . Combining this with (2.22), we get that .
We aim to show that . Assume that . Then, we obtain . By (2.2), we have a contradiction. This implies by Lemma 2.1.
Finally, we show that has a unique fixed point. If there exists such that and , then we get . Since is a generalized Meir-Keeler-type contraction, we derive which is a contradiction. Thus, we find that . So by Lemma 2.1 we conclude that . In particular, has a unique fixed point.

We state two examples to illustrate our results.

Example 2.6. Let be the set equipped with the partial metric . Clearly, is a 0-complete partial metric space. Consider defined by . Given , we will show that there exists such that (2.1) holds for all . Without loss of generality, take . Then, it is easy to show that Thus, taking , we get that (2.1) holds. Also, by Lemma 1.5, the mapping is continuous, and hence it is orbitally continuous. All hypotheses of Theorem 2.5 are satisfied and is the unique fixed point of .

Example 2.7. Let be the interval equipped with the partial metric . Consider defined by Take . Given , we have the two following cases.Case 1 . (). We have Case 2 . (( and ) or ()). We have In each case, it suffices to take in order that (2.1) holds. Again, by Lemma 1.5, the mapping is continuous, and hence it is orbitally continuous. All hypotheses of Theorem 2.5 are satisfied and is the unique fixed point of .

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