#### Abstract

We prove some common fixed point theorems for -contractions in 0-complete partial metric spaces. Our results extend, generalize, and unify several known results in the literature. Some examples are included which show that the generalization is proper.

#### 1. Introduction and Preliminaries

In 1994, Matthews [1] introduced the notion of partial metric spaces, as a part of the study of denotational semantics of dataflow network. In partial metric space, the usual distance was replaced by partial metric, with an interesting property “nonzero self-distance” of points. Also, the convergence of sequences in this space was defined in such a way that the limit of a convergent sequence need not be unique. Matthews showed that the Banach contraction principle is valid in partial metric space and can be applied in program verification. Later, several authors generalized the result of Matthews (see, e.g., [2–29]). O'Neill [22] 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 [21] generalized it by omitting small self-distance axiom. The partial metric defined by Heckmann is called weak partial metric.

Romaguera [23] introduced the notion of 0-Cauchy sequence, 0-complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0-completeness. The notion of 0-complete partial metric spaces is more general than the complete partial metric space, as shown by an example in [23].

Recently, Wardowski [30] introduces a new concept of -contraction and proves a fixed point theorem which generalizes the Banach contraction principle in a different way than the known results of the literature on complete metric spaces. In this paper, we consider a more generalized type of -contraction and prove some common fixed point theorems for such type of mappings in 0-complete partial metric spaces. The results of this paper generalize and extend the results of Wardowski [30], Altun et al. [17], Ćirić [31, 32], and some well-known results in the literature. Some examples are given which show that the results of this paper are proper generalizations of known results.

First we recall some definitions and properties of partial metric space [1, 22, 23, 25, 33].

*Definition 1. *A partial metric on a nonempty set is a function ( stands for nonnegative reals) such that for all , , and (P1) if and only if ; (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 . Also every metric space is a partial metric space, with zero self-distance.

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

*Definition 2. *A mapping is continuous if and only if, whenever a sequence in converges with respect to to a point , the sequence converges with respect to to .

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 induced metric. In further discussion until specified, will represent induced metric space.

*Example 4 (see [1]). * If is defined by , for all , then is a partial metric space.

*Example 5. * Let be a partial metric space; then is a partial metric space, where for all and is the metric induced by .

For some more examples of partial metric spaces, we refer to [14] and the references therein.

Let be a partial metric space. (1)A sequence in converges to a point if and only if .(2)A sequence in is called the 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 .(4)A sequence in is called 0-Cauchy sequence if . The space is said to be 0-complete if every 0-Cauchy sequence in converges with respect to to a point such that .

Lemma 6 (see [1, 23, 25, 33]). *Let be a partial metric space and any sequence in .*(i)* is a Cauchy sequence in if and only if it is a Cauchy sequence in metric space .*(ii)* is complete if and only if the metric space is complete. Furthermore, if and only if .*(iii)*Every 0-Cauchy sequence in is Cauchy in .*(iv)*If is complete, then it is 0-complete. *

The converse assertions of (iii) and (iv) do not hold. Indeed, the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an easy example of a 0-complete partial metric space which is not complete. Also, it is easy to see that every closed subset of a 0-complete partial metric space is 0-complete.

The proof of the following lemma is easy, and for detail we refer to [34] and the references therein.

Lemma 7. *Assume as in a partial metric space such that . Then for all .*

Analogous to [30] we have following definitions.

*Definition 8. *Let be a mapping satisfying: (F1) is strictly increasing, that is, for such that implies ; (F2)for each sequence of positive numbers if and only if ; (F3)there exists such that .

For examples of function , we refer to [30]. We denote the set of all functions satisfying properties (F1)–(F3) by .

Wardowski in [30] defined the -contraction as follows.

Let be a metric space. A mapping is said to be an -contraction if there exists and such that, for all , , we have The following lemma will be useful in proving our main result.

Lemma 9. *Let be a partial metric space and two mappings. Suppose there exist and such that, for all , , one has
**
If has a fixed point , then is a unique common fixed point of and , and .*

*Proof. * Let be a fixed point of . Suppose ; then by (2) we obtain
a contradiction (as ). Therefore we have ; that is, . Thus is a common fixed point of and . Again, if , then by a similar process as above we obtain
a contradiction (as ). Therefore, . For uniqueness, let be another common fixed point of and ; that is, . If , then by (2) we obtain
a contradiction. Therefore, ; that is, . Thus a common fixed point is unique.

Now we can state our main results.

#### 2. Main Results

The following theorem extends and generalizes the results of [30] in partial metric spaces.

Theorem 10. *Let be a 0-complete partial metric space and two mappings. Suppose there exist and such that, for all , , one has
**
If (i) or is continuous or (ii) is continuous, then and have a unique common fixed point , and .*

*Proof. * Let be arbitrary. Define a sequence by , and for all . If for any , for example, let , then it follows from (6) that
that is, . As, , we get a contradiction. Therefore, we must have ; that is, . Similarly, we obtain , and so . Thus, is a common fixed point of and . Now, we assume that for all ; then it follows from (6) that
If , then it follows from the above inequality that , a contradiction (as ). Therefore, we must have . So, setting from the above inequality we obtain that
Again, using (6) we obtain
If , then it follows from the above inequality that , a contradiction (as ). Therefore, we must have . So from the above inequality we obtain that
Using (9) and (11) we obtain
Similarly
It follows from (12) and (13) that . As by (F2) we have
Again, by (F3), there exists such that
From (12) and (13) we have
Using (14) and (15) in the above inequalities we obtain
It follows from above that there exists such that for all ; that is,
Let with ; then it follows from (18) that
As , therefore the series converges; so it follows from the above inequality that ; that is, the sequence is a 0-Cauchy sequence in . Therefore, by 0-completeness of , there exists such that
We will prove that is a common fixed point of and . We consider two cases.*Case 1*. Suppose is continuous. Using continuity of , (20), and Lemma 7 we obtain
We claim that . Suppose . If for each there exists such that and with , then by Lemma 7 and (20) we have , and so, by Lemma 9, is a unique common fixed point of and .

Now suppose there exists such that for all . Then, since , there exists such that .

For any we have
Using (20) and (21), there exists such that and for all ; therefore, it follows from the above inequality that
As by (F1) we have for all , a contradiction. Therefore, we must have ; that is, , and by Lemma 9, is a unique common fixed point of and . Similarly, if is continuous, then is a unique common fixed point of and .*Case 2.* Now suppose that is continuous. We can assume that there exists such that for all ; otherwise we get (similar as in previous case).

For any , we obtain from (6) that
From (20), there exists such that
for all . Therefore, it follows from (24) that
Using continuity of and letting in the above inequality we obtain
a contradiction (as ). Therefore we must have ; that is, . Again by Lemma 9 is a unique common fixed point of and .

The following corollaries are immediate consequences of Theorem 10.

Corollary 11. *Let be a 0-complete partial metric space and two mappings. Suppose there exist and such that, for all , , one has
**
If (i) or is continuous or (ii) is continuous, then and have a unique common fixed point , and .*

Corollary 12. *Let be a 0-complete partial metric space and two mappings. Suppose there exist and such that, for all , , one has
**
If (i) or is continuous or (ii) is continuous, then and have a unique common fixed point , and .*

Taking in Theorem 10 we obtain the following corollary.

Corollary 13. *Let be a 0-complete partial metric space and a mapping. Suppose there exist and such that, for all , , one has
**
If (i) is continuous or (ii) is continuous, then has a unique fixed point , and .*

The following is a fixed point result for -contraction in 0-complete partial metric space and follows from Corollary 13.

Corollary 14. *Let be a 0-complete partial metric space and a mapping. Suppose there exist and such that, for all , , one has
**
If (i) is continuous or (ii) is continuous, then has a unique fixed point , and .*

The following are some examples which illustrate the above results and that the generalizations are proper.

*Example 15. *Let , and let be defined by for all . Then is a 0-complete partial metric space, but it is not complete partial metric space. Indeed, the induced metric space , where for all , is not a complete metric space. Define by
We note that satisfies the condition (31) of Corollary 14 with for all , , and is unique fixed point of with . On the other hand, the metric version of Corollary 14 is not applicable because is not complete metric space. Also, this example shows that the class of -contraction in partial metric spaces is wider than that in metric spaces. Indeed, for , , there is no and such that , where is usual as well as the metric induced by .

The following example illustrates the case when Corollary 11 is applicable, while Corollary 12 is not, as well as the metric versions of Corollary 11 are not applicable.

*Example 16. *Let , and let be defined by
Then, is a 0-complete partial metric space. Note that the metric induced by is given by for all . Define by
Now, by a careful observation one can see that and satisfy the condition (28) of Corollary 11, with for all , , and and have a unique common fixed point, namely, with . While, and do not satisfy the condition (29) of Corollary 12. Indeed, for , , there are no and such that . Again, and do not satisfy the metric versions of Condition (28) of Corollary 11. Indeed, , are the points where the induced metric and usual metric versions of condition (28) of Corollary 11 are not satisfied.