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Journal of Operators
Volume 2013 (2013), Article ID 923843, 5 pages
Suzuki-Type Generalization of Chatterjea Contraction Mappings on Complete Partial Metric Spaces
1Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
2Kirikkale University, Faculty of Arts and Sciences, Department of Mathematics, Kirikkale, 71450 Yahsihan, Turkey
Received 18 February 2013; Revised 21 May 2013; Accepted 22 May 2013
Academic Editor: Lingju Kong
Copyright © 2013 Mohammad Imdad and Ali Erduran. 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.
Motivated by Suzuki (2008), we prove a Suzuki-type fixed point theorem employing Chatterjea contraction on partial metric spaces.
1. Introduction and Preliminaries
Historically, the idea of a complete metric space has interesting and important applications in classical analysis especially in existence and uniqueness theories on one hand while on the other hand Banach fixed point theorem  is one of the most useful results in nonlinear analysis. In the recent past, many authors considered the equivalence of existence results on fixed points (of mappings) by proving suitable equivalence theorems ascertaining the completeness of the underlying metric space. Kirk  proved that a metric space is complete if and only if every Caristi type mapping has a unique fixed point. Similarly, Subrahmanyam  showed that a metric space is complete if and only if every Kannan mapping has a unique fixed point. However, Banach contraction condition does not characterize the metric completeness of the underlying space. Indeed, Connell  gave an example of an incomplete metric space on which every contraction map has a fixed point. Despite this fact, Suzuki obtained Banach contraction principle that characterizes the metric completeness of the space using a different type of contraction. Thereafter, many authors proved different generalizations by proving different types of fixed point theorems in complete metric space, for example, Suzuki  proved Kannan version of a Suzuki-type generalized result wherein authors discussed contraction mappings and Kannan mappings from a different point of view while Popescu  attempted Chatterjea version on complete metric space. In this continuation, Altun and Erduran proved a Suzuki-type fixed point theorem using an implicit function on complete metric space  wherein authors obtained unified and generalized results of Suzuki-type. Finally, Kikkawa and Suzuki proved multivalued version of Suzuki-type results which generalize classical results of Markin  and Nadler . For further details on this theme, one can be referred to [7, 10–15].
Evidently, contractions are always continuous, but Kannan mappings are not necessarily continuous. Also, one may notice that a contraction mapping is not necessarily Kannan mapping, and a Kannan mapping is not essentially a contraction mapping so that both conditions are not essentially comparable but Kannan contractions are relatively stronger than Banach contraction in certain sense. However, Chatterjea contraction is obtained by interchanging the roles of variables , in Kannan contraction. Therefore, Chatterjea contraction is similar to Kannan contraction enabling one to infer that the mappings satisfying a Kannan contraction also satisfy the corresponding Chatterjea contraction.
In 1992, Matthews  introduced partial metric spaces wherein the distance of a point from itself may not always be zero. After introducing the idea of partial metric spaces, Matthews proved the partial metric version of Banach fixed point theorem. Thereafter, many authors further studied partial metric spaces and their topological properties (e.g., [17–22]) which are followed by Valero , Oltra and Valero , and Altun et al.  wherein authors obtained some generalizations of the core result of Matthews . In a noted paper, Romaguera  extended Kirk's characterization of metric completeness  to partial metric spaces.
Recently, many authors proved several fixed point results in partial metric spaces. Out of such results, Suzuki-type (cf. ) generalizations are noted ones. Very recently, Paesano and Vetro  proved an analogous fixed point result for a self-mapping defined on a partial metric space (also on a partially ordered metric space) which generalizes certain noted results of Ran and Reurings  besides offering a characterization of partial metric -completeness in terms of fixed points. This result can be viewed as an extension of Suzuki-type (cf. ) characterization of metric completeness.
Inspired by Paesano and Vetro , we prove Suzuki-type fixed point result for Chatterjea contraction mappings in -complete partial metric spaces wherein we show that constant is the best for every which substantiates the genuineness and utility of our result.
In what follows, we give some relevant definitions and results from the existing literature which are relevant to our subsequent discussion.
Definition 1 (see ). A partial metric on a nonempty set is a function such that for all , , : (i) , (ii) , (iii) , (iv) . A partial metric space is a pair such that is a nonempty set and is a partial metric on .
Each partial metric on generates a topology on whose base is the family of open -balls , where for all and .
If is a partial metric on , then the function given by is a metric on .
Definition 2. A mapping is said to be continuous at , if for every , there exists such that .
Definition 3 (see ). Let be a partial metric space and a sequence in . Then,(i) converges to a point if ;(ii) is called a Cauchy sequence if exists and is finite.
Moreover, if , then is said to be a -Cauchy sequence in .
Definition 4. A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point , such that .
A partial metric space is called -complete  if every -Cauchy sequence in converges to some (with respect to ) such that . Therefore, is -complete if and only if every -Cauchy sequence converges with respect to . It is clear that every -Cauchy sequence in is a Cauchy sequence in . Therefore, if is complete, then it is -complete but not conversely (e.g., ).
In general, is not a continuous function in two variables, in the sense that and (in ) imply that , as . However, the following holds:
Lemma 5. Let be a partial metric space and . If as and , then
Lemma 6 (see ). Let be a partial metric space. Then, (a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ;(b) is complete if and only if the metric space is complete. Furthermore, if and only if
Before proving our main result (i.e., Theorem 8), we state the following lemma available in [5, 13] for standard metric spaces. As the proof of the lemma is identical for the partial metrices, therefore, we merely state the lemma (for partial metric spaces).
Lemma 7. Let be a partial metric space and let be mapping on . Let satisfies for some . Then for , either holds.
Now, we prove our main result as follows.
Theorem 8. Define a nonincreasing function from to by Let be a -complete partial metric space and a self-mapping defined on . Let and put . If for all , then has a unique fixed point.
Proof. Since , holds. In view of one of the hypotheses, we can write the following: so that for . Now, fix and define a sequence in by . On using (8), we can have the following: so that for all and so is a -Cauchy sequence. Since is -complete, converges to some point ; that is, Now, we show that Since , there exists such that for all . Owing to and , for all , we obtain the following: and henceforth Now, for with . Let us prove that is a fixed point of . In case , let on contrary that . Then owing to (12), so that which is a contradiction. Therefore, we have . In case , using Lemma 7, we have the following: for . Thus, there exists a subsequence of such that for . On using one of our assumption, we have the following: As , we have . The uniqueness of fixed point is obvious in view of (12). This concludes the proof.
The following theorem shows that is the best constant for every value of .
Theorem 9. Define a function as in Theorem 8. For every , setting , there exists a -complete partial metric space and a mapping on such that has no fixed point but for all .
Proof. Let and define a complete partial metric on by the following:
Then is obviously -complete. Define a mapping on by the following:
Notice that has no fixed point.
Firstly if , then the following cases arise.(i) If , , , and , then . Hence, the right-hand side of (21) trivially holds so that implication is also verified.(ii) If , , then , and hence the left-hand side of (21) holds good as . Also, the right-hand side of (21) is satisfied in view of the following: as there exists such a .(iii) If , , then , and hence the right-hand side of (21) trivially holds so that implication is also realized.(iv) If , then so that the left-hand side of (21) holds good as (as ). The right-hand side of (21) also remains true as .(v) If , then , wherein we examine two cases. If , then the right-hand side of (21) trivially holds and hence so does the implication. If , then the left-hand side of (21) remains true as . The right-hand side of (21) also holds good as
Secondly, if , then we distinguish the following cases.(i) If , , , and , then . Hence, the right-hand side of (21) as well as implication remains true.(ii) If , then , . The left-hand side of (21) holds good as . The right-hand side of (21) remains true as (iii) If , , then , . The left-hand side of (21) remains true as , The right-hand side of (21) is satisfied as follows: if we choose .(iv) If , , then , . The left-hand side of (21) holds good as . The right-hand side of (21) is also satisfied as follows: if we can choose .(v) If , then . If , then the left-hand side of (21) remains true as . The right-hand side of (21) is also verified as follows: If , then left-hand side of (21) holds good as . The right-hand side of (21) remains true as
Thus, in all possible cases the condition (21) is satisfied.
Both the authors are grateful to both the learned referees for their fruitful comments towards the improvement of this paper.
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