Abstract

The notion of asymptotically regular mapping in partial metric spaces is introduced, and a fixed point result for the mappings of this class is proved. Examples show that there are cases when new results can be applied, while old ones (in metric space) cannot. Some common fixed point theorems for sequence of mappings in partial metric spaces are also proved which generalize and improve some known results in partial metric spaces.

1. Introduction

Matthews [1] introduced partial metric spaces as a part of the study of denotational semantics of data flow networks. In partial metric space, the usual metric was replaced by partial metric, with a property that the self-distance of any point may not be zero. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation. Partial metric has applications in the branches of science where the size of data point is represented by its self-distance. The fixed point of a contraction mapping in partial metric space has zero self-distance; that is, fixed point is a total object. Every metric space is a partial metric space with zero self-distance that is, partial metric spaces are the generalization of metric spaces.

O’Neill [2] generalized the concept of partial metric space a bit further by admitting negative distances. The partial metric defined by O’Neill is called dualistic partial metric. Heckmann [3] generalized it by omitting small self-distance axiom. The partial metric defined by Heckmann is called weak partial metric.

Banach contraction principle ensures the existence and uniqueness of a fixed point of a contractive self-map of metric space and has many applications in applied sciences. The fixed point result of Matthews is the generalization of the following Banach contraction principle.

Let be a complete metric space and let be a self-map on . If there exists such that for all , then has a unique fixed point in .

The fixed point result of Matthews is generalized by several authors for single self map in partial metric spaces (see, e.g., [4–6]). Almost all contractive conditions in these papers imply the asymptotic regularity of the mapping under consideration.

The purpose of this paper is to prove some common fixed point theorems for a sequence of self maps on partial metric spaces and generalize the result of Matthews. The notion of asymptotically regular mapping in partial metric spaces is introduced and a fixed point result for the mappings of this class is also proved.

2. Definitions and Preliminaries

First, we recall some definitions and properties of partial metric spaces.

Definition 1 (see [1]). A partial metric on a nonempty set is a function ( stands for nonnegative reals) 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 .

Example 2. Let be defined by , for all , and then is a partial metric space.

Each partial metric on generates a topology on which has a base the family of open balls , where for every and .

Theorem 3 (see [1]). For each partial metric the pair , where for all is a metric space.

Here, is called induced metric space, and is metric induced by .

Throughout this paper, we suppose that is induced metric space and is metric induced by .

Let be partial metric space. Then, (1)a sequence in converges to a point if and only if ,(2)a sequence in is called Cauchy sequence if there exists and is finite, (3) is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Lemma 4 (see [1, 7]). Let be partial metric space, and then(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in metric space .(b) is complete if and only if the metric space is complete. Furthermore, if and only if

Definition 5. A self map on a partial metric space is said to be asymptotically regular at a point in , if where denotes the th iterate of at .

Note that Banach type contractions, generalized contractions, and so forth are asymptotically regular at every point of space.

Example 6. Let be defined by , for all , and then is a partial metric space. The mapping is defined by , for every . Then is asymptotically regular at every point of .

3. Fixed Point Theorems

First, we prove the fixed point result for asymptotically regular mappings.

Theorem 7. Let be a complete partial metric space and let be a self map on , satisfying the following condition: for all , where are nonnegative functions such that for arbitrarily fixed and for all where are functions such that and are continuous at . If is asymptotically regular at some , and then has a unique fixed point in with .

Proof. Let for every . If there exists such that , then is a fixed point of . Suppose that for every . We show that is a Cauchy sequence.
Let denote and then from (3) we obtain Using (3), we obtain where . As Therefore from (7), it follows that Using (4) and (5), we obtain Since is asymptotically regular at and and are continuous at zero, therefore from the above inequality we obtain Hence, is a Cauchy sequence in , and by Lemma 4, it is a Cauchy sequence in . Using completeness of and Lemma 4, it follows that is complete and converges in . Thus, for some .
Again, from Lemma 4, and (11), we have To prove that , let us consider the following inequalities: where . As Therefore, (13) gives Using (4) and (5), we obtain From (12) and the above inequality, it follows that This contradiction shows that , that is, . Thus, is a fixed point of .
Let be another fixed point of and . From (3), we obtain where . The above inequality with (5) gives This contradiction proves uniqueness.

Taking , , , , in the above theorem, we obtain following corollary.

Corollary 8. Let be a complete partial metric space and let be a self map on , satisfying the following condition: for all , where are nonnegative reals, such that . If is asymptotically regular at some , then has a unique fixed point in with .

Again taking and in Theorem 7, we obtain following corollary.

Corollary 9. Let be a complete partial metric space and be a self map on , satisfying following condition: for all , where are nonnegative reals, such that . If is asymptotically regular at some , then has a unique fixed point in with .

Taking in above the corollary, we obtain the following result.

Corollary 10. Let be a complete partial metric space and let be a self map on , satisfying the following condition: for all , where . If is asymptotically regular at some , then has a unique fixed point in with .

The following example shows that the assumption of asymptotic regularity in above theorems cannot be dropped.

Example 11. Let , then , where is a complete partial metric space. Define a self map on , as follows: Take then satisfies the contractive condition of Corollary 10, but is not asymptotically regular at any point of , and has no fixed point in .
Results similar to the above corollaries are available in usual metric spaces (see, e.g., [8]). In the following we illustrate the existence of self map which satisfies contractive condition of Corollary 10, in partial metric space but not in usual metric space.

Example 12. Let , then , where is a complete partial metric space. Define a self map on , as follows: Note that satisfies all the conditions of Corollary 10, with , and has a unique fixed point . But does not satisfy the contractive condition for all , in usual metric space with (which is also the induced metric); for example, if we take , then there is no such that and . Therefore, results of usual metric spaces cannot be applied.

Now, we will prove some common fixed point theorems.

Theorem 13. Let be a complete partial metric space and be a sequence of self maps on satisfying the following.
There exist with such that for all Then, all the mappings of sequence have a unique common fixed point in with .

Proof. Let and for every . We show that is a Cauchy sequence.
If there exists , such that , then from (25) we obtain As we have as , it follows that .
Similarly, it can be seen that for every . Thus, is a Cauchy sequence.
Assume that for every . Denote , then from (25) it follows that and so Using, symmetry of , we obtain It follows from (29) and (30) that where , which implies , where .
For , we obtain As , it follows that as , which implies Hence is a Cauchy sequence in , and by Lemma 4, it is a Cauchy sequence in . Using completeness of and Lemma 4, it follows that is complete and converges in . Thus for some .
Again from Lemma 4, and (33) we have To prove that for any arbitrary fixed , let us consider the following inequalities: Using (34), it follows that . Thus, is a common fixed point of all the maps of sequence .
Let be another common fixed point of all the maps of sequence and .
For any , from (25), we obtain Using (p2), we have This contradiction proves uniqueness.