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Abstract and Applied Analysis
Volume 2011, Article ID 561245, 13 pages
http://dx.doi.org/10.1155/2011/561245
Research Article

Fixed Points of Geraghty-Type Mappings in Various Generalized Metric Spaces

1Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia
2Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia

Received 11 June 2011; Accepted 10 September 2011

Academic Editor: Allan C. Peterson

Copyright © 2011 Dušan Ðukić 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

Fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in the frame of partial metric spaces, ordered partial metric spaces, and metric-type spaces. Examples are given showing that these results are proper extensions of the existing ones.

1. Introduction

Let denote the class of real functions satisfying the condition An example of a function in may be given by for and . In an attempt to generalize the Banach contraction principle, M. Geraghty proved in 1973 the following.

Theorem 1.1 (see [1]). Let be a complete metric space, and let be a self-map. Suppose that there exists such that holds for all . Then has a unique fixed point and for each the Picard sequence converges to when .

Recently, A. Amini-Harandi and H. Emami extended this result to partially ordered metric spaces as follows.

Theorem 1.2 (see [2]). Let be a complete partially ordered metric space. Let be an increasing self-map such that there exists with . Suppose that there exists such that (1.2) holds for all with . Assume that either is continuous or is such that Then, has a fixed point in . If, moreover, then the fixed point of is unique.

Similar results were also obtained in [3, 4].

In recent years several authors have worked on domain theory in order to equip semantics domain with a notion of distance. In particular, Matthews [5] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, and obtained, among other results, a nice relationship between partial metric spaces and so-called weightable quasimetric spaces. He showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification. Subsequently, several authors (see, e.g., [6, 7]) studied fixed point theorems in partial metric spaces, as well as ordered partial metric spaces (see, e.g., [8, 9]).

Huang and Zhang introduced cone metric spaces in [10], replacing the set of real numbers by an ordered Banach space as the codomain for a metric. Cone metric spaces over normal cones inspired another generalization of metric spaces that were called metric-type spaces by Khamsi [11] (see also [12]; note that, in fact, spaces of this kind were used earlier under the name of -spaces by Czerwik [13]).

In the present paper, we extend Theorems 1.1 and 1.2 to the frame of partial metric spaces, ordered partial metric spaces, and metric type spaces. Examples are given to distinguish new results from the existing ones.

2. Notation and Preliminary Results

2.1. Partial Metric Spaces

The following definitions and details can be seen in [59, 14, 15].

Definition 2.1. A partial metric on a nonempty set is a function  such that, for all  (p1),(p2),(p3),(p4). A partial metric space is a pair such that is a nonempty set and is a partial metric on .

It is clear that, if , then from (p1) and (p2) . But if , may not be 0.

Each partial metric on generates a topology on which has as a base the family of open -balls ,, where for all and . A sequence in converges to a point , with respect to , if . This will be denoted as , or .

If is a partial metric on , then the function given by is a metric on . Furthermore, if and only if

Example 2.2. (1) A basic example of a partial metric space is the pair , where for all . The corresponding metric is (2) If is a metric space and is arbitrary, then defines a partial metric on and the corresponding metric is .

Other examples of partial metric spaces which are interesting from a computational point of view may be found in [5, 15].

Remark 2.3. Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and implies . For example, if and for , then for , for each and so, for example, and when .

Definition 2.4 (see [8]). Let    be a partial metric space. Then one has the following(1)A sequence in is called a Cauchy sequence if exists (and is finite).(2)The space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Lemma 2.5 (see [5, 6]). Let be a partial metric space. (a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .(b)The space is complete if and only if the metric space is complete.

Definition 2.6. Let    be a nonempty set. Then    is called an ordered partial metric space if:(i) is a partial metric space and (ii)   is a partially ordered set.
The space is called regular if the following holds: if is a nondecreasing sequence in with respect to such that as , then for all .

2.2. Some Auxiliary Results

Assertions similar to the following lemma (see, e.g., [16]) were used (and proved) in the course of proofs of several fixed point results in various papers.

Lemma 2.7. Let be a metric space, and let be a sequence in such that If is not a Cauchy sequence, then there exist and two sequences and of positive integers such that the following four sequences tend to when :

As a corollary we obtain the following.

Lemma 2.8. Let be a partial metric space, and let be a sequence in such that If is not a Cauchy sequence in , then there exist and two sequences and of positive integers such that the following four sequences tend to when :

Proof. Suppose that is a sequence in satisfying (2.7) such that is not Cauchy. According to Lemma 2.5, it is not a Cauchy sequence in the metric space , either. Applying Lemma 2.7 we get the sequences tending to some when . Using definition (2.1) of the associated metric and (2.7) (which by (p2) implies that also ), we get that the sequences (2.8) tend to when .

2.3. Property (P)

Let be a nonempty set and a self-map. As usual, we denote by the set of fixed points of . Following Jeong and Rhoades [17], we say that the map has property () if it satisfies for each . The proof of the following lemma is the same as in the metric case [17, Theorem  1.1].

Lemma 2.9. Let be a partial metric space, and let be a selfmap such that . Then has property () if holds for some and either (i) for all   or (ii) for all .

2.4. Metric Type Spaces

Definition 2.10 (see [11]). Let    be a nonempty set,    a real number, and let a function  satisfy the following properties:(a) if and only if ;(b) for all ;(c) for all . Then is called a metric type space.

Obviously, for , metric type space is simply a metric space.

The notions such as convergent sequence, Cauchy sequence, and complete space are defined in an obvious way.

A metric type space may satisfy some of the following additional properties:(d) for arbitrary points ;(e)function is continuous in two variables, that is, (The last condition is in the theory of symmetric spaces usually called “property ’’.)

Condition (d) was used instead of (c) in the original definition of a metric type space by Khamsi [11].

Note that weaker version of property (e):(e′) and (in ) implies that is satisfied in an arbitrary metric type space. It can also be proved easily that the limit of a sequence in a metric type space is unique. Indeed, if and (in ) and , then for sufficiently large , which is impossible.

3. Results

3.1. Results in Partial Metric Spaces

Theorem 3.1. Let be a complete partial metric space, and let be a self-map. Suppose that there exists such that holds for all . Then has a unique fixed point and for each the Picard sequence converges to when .

Proof. Let be arbitrary, and let for . Consider the following two cases:(1) for some ;(2) for each .Case 1. Under this assumption we get that and it follows that . By induction, we obtain that for all and so for all . Hence, is a Cauchy sequence, converging to which is a fixed point of .Case 2. We will prove first that in this case the sequence is decreasing and tends to 0 as .
For each we have that Hence, is decreasing and bounded from below, thus converging to some . Suppose that . Then, it follows from (3.3) that where from, passing to the limit when , we get that . Using property (1.1) of the function , we conclude that , that is, , a contradiction. Hence, is proved.

In order to prove that is a Cauchy sequence in , suppose the contrary. As was already proved, as , and so, using (p2), as . Hence, using (2.1), we get that as . Using Lemma 2.8, we obtain that there exist and two sequences and of positive integers such that the following four sequences tend to when : Putting in the contractive condition and , it follows that Hence, and . Since , it follows that , which is in contradiction with .
Thus is a Cauchy sequence, both in and in . Since these spaces are complete, it follows that sequence converges in the metric space , say . Again from Lemma 2.5, we have Moreover since is a Cauchy sequence in the metric space , we have and so, by the definition of , we have . Then (3.8) implies that and We will prove that is a fixed point of .
By (p4), and using the contractive condition, we get that Thus, and .
Assume that are two fixed points of . Then a contradiction. Hence the fixed point of is unique. The theorem is proved.

Remark 3.2. It follows from Lemma  1, of the paper [18] of Jachymski, that under conditions of Theorem 3.1 there exists a continuous and nondecreasing function such that for all and for all .
On the other hand, Romaguera [19] recently obtained a partial metric extension of the celebrated Boyd and Wong fixed point theorem, from which it follows that if is a complete partial metric space and is a map satisfying   for all , with a function with the aforementioned properties, then has a unique fixed point. Hence, combining Jachymski’s and Romaguera’s results, an alternative proof of Theorem 3.1 is obtained.

Theorem 3.3. If satisfies conditions of Theorem 3.1, then it has property (P).

Proof. By Theorem 3.1, the set of fixed points of is a singleton, . Then also for all . Let for some , and suppose that , that is, . Then We have that (otherwise , which is excluded). It follows that Continuing, we obtain that a contradiction. Hence, and , that is, for each .

Example 3.4. Let , , , for and . The mapping defined by does not satisfy conditions of Theorem 1.1. Indeed, take , and obtain that On the other hand, take with, for example, . Then since for . Hence, satisfies conditions of Theorem 3.1 and thus has a unique fixed point ().

3.2. Results in Ordered Partial Metric Spaces

Theorem 3.5. Let be a complete ordered partial metric space. Let be an increasing self-map (with respect to. ) such that there exists with . Suppose that there exists such that (3.1) holds for all comparable . Assume that either is continuous or is regular. Then, has a fixed point in . The set of fixed points of is a singleton if and only if it is well ordered.

Proof. Take with and, using monotonicity of , form the sequence with Since and are comparable we can apply contractive condition to obtain Proceeding as in the proof of Theorem 3.1 we obtain that , that is a Cauchy sequence in (and in ). Thus, it converges (in and in ) to a point such that Also, it follows as in the proof of Theorem 3.1 that We will prove that is a fixed point of .(i)Suppose that is continuous. We have, by (p4), Passing to the limit when and using continuity of we get that It follows that . Since , using contractive condition, we get that and so and .(ii)If is regular, since is an increasing sequence tending to , we have that for each . So we can apply (p4) and contractive condition to obtain Letting we get Hence, we again obtain that .
Let the set of fixed points of be well ordered, and suppose that there exist two distinct points . Then these points are comparable, and we can apply the contractive condition to obtain a contradiction. Hence, the set is a singleton. The converse is trivial.

Example 3.6. Let , and define the partial order on by Consider the function given as , which is increasing with respect to .
Define first the metric on by , , for and for . Then is a complete partially ordered metric space. The function , defined by , , and , belongs to the class .
Take and . Then Hence, conditions of Theorem 1.2 are not fulfilled and this theorem cannot be used to prove the existence of a fixed point of .
Define now the partial metric on by , , for ; further , , and (it is easy to check conditions (p1)–(p4)). Let us check contractive condition (3.1) of Theorem 3.5: Hence, we can apply Theorem 3.5 to conclude that there is a unique fixed point of (which is ).

A variant of Theorem 1.2 which uses an altering function was obtained in [3, Theorems  2.2, 2.3]. Recall that is called an altering function if it is continuous, increasing and . We state a partial metric version of this result. The proof is omitted since it is similar to the previous one.

Theorem 3.7. Let be a complete ordered partial metric space. Let be an increasing self-map (w.r.t. ) such that there exists with . Suppose that there exist and an altering function such that holds for all comparable . Assume that either is continuous or is regular. Then, has a fixed point in . The set of fixed points of is a singleton if and only if it is well ordered.

3.3. Results in Metric Type Spaces

For the use in metric type spaces (with the given ) we will consider the class of functions , where if and has the property An example of a function in is given by for and .

Theorem 3.8. Let , and let be a complete metric type space. Suppose that a mapping satisfies the condition for all and some . Then has a unique fixed point , and for each the Picard sequence converges to in .

Proof. Using condition (3.32) it is easy to show that the fixed point of in is unique (if it exists) and that is -continuous in the sense that implies that in (for details see [12]).
Let be arbitrary and for . If for some , then it is easy to show that for , and the proof is complete. Suppose that for all . Then, using (3.32), we get that By [12, Lemma  3.1], is a Cauchy sequence in . As this space is complete, converges to some as . Obviously, also and continuity of implies that . Since the limit of a sequence in a metric type space is unique, it follows that .

Example 3.9. Let be equipped with the metric type function given by with . Consider the mapping defined by , , , and the function given by , , and . Then Hence, satisfies all the assumptions of Theorem 3.8 and thus it has a unique fixed point (which is ).

Acknowledgments

The authors are thankful to the referees for very useful suggestions that helped to improve the paper, and they are thankful to the Ministry of Science and Technological Development of Serbia.

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