#### Abstract

We consider generalized Berinde-type contractions in the context of partial metric spaces. Such contractions are also known as generalized almost contractions in the literature. In this paper, we extend, generalize, and enrich the results in this direction. Some examples are presented to illustrate our results.

#### 1. Introduction and Preliminaries

Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle [2] can be generalized to the partial metric context for applications in program verifications. Later, there have been several recent extensive researchs on (common) fixed points for different contractions on partial metric spaces, see [3–28].

First, we recall some basic concepts and notations.

*Definition 1. *A partial metric on a nonempty set is a function such that for all : (),(), (),
(). A partial metric space is a pair such that is a nonempty set and is a partial metric on .

*Example 2 (see [1]). *Let and defined on by for all . Then is a partial metric space.

*Example 3 (see [20, 26]). *Let and be a metric space and a partial metric space, respectively. Functions () given by
define partial metrics on , where is an arbitrary function and .

*Example 4 (see [1]). *Let and define . Then is a partial metric space.

*Example 5 (see [1]). *Let and define by
Then is a partial metric space.

*Remark 6. *It is clear that, if , then from and , we get . On the other hand, may not be even if .

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

If is a partial metric on , then the functions , given by are equivalent metrics on .

*Definition 7 (see [1]). *Let be a partial metric space. (1) A sequence in is called a Cauchy sequence in if exists and is finite.(2) is called complete if every Cauchy sequence converges with respect to to a point such that .

Lemma 8 (see [1]). *Let be a partial metric space. *(1)* is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .*(2)* A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
*

Lemma 9 (see [20]). *Let be a convergent sequence in a partial metric space such that and with respect to . If
**
then .*

Lemma 10 (see [20]). *Let and be two sequences in a partial metric space such that
**
then . In particular, for every . *

Lemma 11 (see [3]). *Let be a partial metric space and , with respect to , with . Then for all . *

The concept of almost contractions was introduced by Berinde [29, 30] on metric spaces. Other results on almost contractions could be found in [31–34]. Recently, Altun and Acar [35] characterized this concept in the setting of partial metric space and proved some fixed point theorems using these concepts. Very recently, Turkoglu and Ozturk [27] established a fixed point theorem for four mappings satisfying an almost generalized contractive condition on partial metric spaces. In this paper, we generalize the results given in [27, 35] by presenting some fixed point results for self mappings involving some almost generalized contractions in the setting of partial metric spaces. Also, we give some illustrative examples making our results proper.

#### 2. Main Results

We start to this section by defining some sets of auxiliary functions. Let denote all functions such that if and only if . We denote by and be subsets of such that Let a partial metric space. We consider the following expressions: for all .

Our first result is the following.

Theorem 12. *Let be a complete partial metric space. Let be a self mapping. Suppose there exist , and such that for all **
Then has a unique fixed point, say . Also, one has . *

* Proof. *Let . We construct a sequence in in a way that for all . Suppose that for some . So we have , that is, is the fixed point of .

From now on, assume that for all . By (9), we have
where
Since , we get . Hence, it follows that
which yields that
Since is nondecreasing, then
where
Due to , we have
Hence, the expression (15) turns into

If for some ,
then by (12)
so . By (), we get , which is a contradiction with respect to for all . Thus
so from (14)
Thus, the sequence is non-increasing and so there exists such that
Suppose that . Taking in inequality (12) and using (20), we get
By continuity of and lower semicontinuity of , we get , so , that is, , a contradiction. We conclude that
We will show that is a Cauchy sequence in the partial metric space . From Lemma 8, we need to prove that is a Cauchy sequence in the metric space . Suppose to the contrary that is not a Cauchy sequence in the metric space . Then, there is a such that for an integer there exist integers such that
By definition of , we have for each , so (25) gives us
For every integer , let be the least positive integer exceeding satisfying (26) then
Now, using (26), (27), and the triangular inequality (which still holds for the partial metric ), we obtain
Then by (24) it follows that
Also, by the triangle inequality, we have
From (24) and (29) we get
Similarly, by triangle inequality
and from (24), (29), and (31) we get
Having
so referring to (24), we get
Moreover
Thus, from (24), (29), (31), and (34), we get
From (9), we have
where

By (36), we get
and referring to (33), (38) and letting , we get
so , which is a contradiction with respect to . Thus we proved that is a Cauchy sequence in the metric space .

Since is complete, then from Lemma 8, is a complete metric space. Therefore, the sequence converges to some in , that is,
Again, from Lemma 8,
On the other hand, thanks to (24) and the condition from Definition 1,
so it follows that
Now, we show that . Assume this is not true, then from (9) we obtain
where
Thanks to (46), it is obvious that . Therefore, using (24) and again (46), we deduce that
Also
because (24) and (45) give . Now, taking the upper limit as , we obtain using the properties of and
so , that is, , so . We conclude that has a fixed point and .

Now if (so ) is another fixed point of (with ), then by (46),
Hence, using (9) we obtain
that is, , which is a contradiction. The proof of Theorem 12 is completed.

As a consequence of Theorem 12, we may state the following corollaries.

First, taking in Theorem 12, we have the following.

Corollary 13. *Let be a complete partial metric space. Let be a self mapping. Suppose there exist and such that for all **
Then has a unique fixed point, say . Also, one has . *

Corollary 14. *Let be a complete partial metric space. Let be a self mapping. Suppose there exist and such that for all **
Then has a unique fixed point, say . Also, one has . *

* Proof. *It follows by taking and in Theorem 12.

Denote by the set of functions satisfying the following hypotheses: is a Lebesgue-integrable mapping on each compact subset of , (2) for every , we have .

We have the following result.

Corollary 15. *Let be a complete partial metric space. Let be a self mapping. Suppose there exist and such that for all **
Then has a unique fixed point, say . Also, one has . *

* Proof. *It follows from Theorem 12 by taking

Taking in Corollary 15, we obtain the following result.

Corollary 16. *Let be a complete partial metric space. Let be a self mapping. Suppose there exist such that for all **
Then has a unique fixed point, say . Also, one has . *

Now, let be the set of functions satisfying the following hypotheses:() is nondecreasing() converges for all .

Note that if , is said a -comparison function. It is easily proved that if is a -comparison function, then for any . Our second main result is as follows.

Theorem 17. *Let be a complete partial metric space. Let be a mapping such that there exist and such that for all **
Then has a unique fixed point, say . Also, one has . *

* Proof. *Let . Let in such that for all .

If for some , the proof is completed. Assume that for all .

From (59)
As explained in the proof of Theorem 12, we may get
Therefore
If for some , we have . So from (62), we obtain that
a contradiction. Thus, for all , we have
Using (62) and (64), we get that
By induction, we get
for all . By triangle inequality, we have for
Keeping in mind that is a -comparison function, then and so is a Cauchy sequence in with . Since is complete then converges, with respect to , to a point such that
Now we claim that . Suppose the contrary, then . By (59), we have
where
By (68), we have
Therefore
which is a contradiction. That is . Thus, we obtained that is a fixed point for and .

Now if (so ) is another fixed point of , then by (68),
Hence, using (59) we obtain
which is a contradiction. Thus and the proof of Theorem 17 is completed.

Taking in Theorem 17, we have the following.

Corollary 18. *Let be a complete partial metric space. Let be a mapping such that there exists such that for all **
Then has a unique fixed point, say . Also, one has . *

Taking where in Corollary 18, we obtain the Ćirić fixed point theorem [36] in the setting of metric spaces (by considering is a metric).

Corollary 19. *Let be a complete metric space. Let be a mapping such that there exists such that for all **
Then has a unique fixed point. *

*Remark 20. *Corollary 14 generalizes Theorem 10 (with ) of Turkoglu and Ozturk [27]. Corollary 18 improves Theorem 1 of Altun et al. [4] by assuming that is not continuous.

#### 3. Examples

We give in this section some examples making effective our obtained results.

*Example 21. *Let and for all . Then is a complete partial metric space. Consider defined by
Take and for all . Note that and . Take , then
for all . Thus, (9) holds. Applying Theorem 12, has a unique fixed point, which is .

*Example 22. *Let and . Let be defined as follows:
By simple calculation, we get that
Hence, we derive that
For , and all conditions of Theorem 12 are satisfied. Notice that is the unique fixed point of .

*Example 23. *Let and be defined by . Define by
and let defined by
By induction, we have for all , so it is clear that is a -comparison function. Now we show that (59) is satisfied for all . It suffices to prove it for . Consider the following six cases.*Case *1. Let , then
*Case *2. Let , then
*Case *3. Let , then
*Case *4. Let and then
*Case *5. Let and , then
*Case *6. Let and then
Since, for all
then (59) is verified. Applying Theorem 17, has a unique fixed point, which is .

All presented theorems involve generalized almost contractive mappings which have a unique fixed point. But, one of the main features of Berinde contractions is the fact that they do possess more that one fixed point. In this direction, Altun and Acar [35] proved the following result.

Theorem 24. *Let a complete partial metric space. Given satisfying
**
for all . Then, has a fixed point. *

The following example illustrates Theorem 24 where we have two fixed points.

*Example 25. *Let . A partial metric is defined by
Define the mapping by
It is easy to show that (91) is satisfied. Applying Theorem 24, has a fixed point. Note that has two fixed points which are and .