International Journal of Analysis

1. Introduction

The celebrated Banach contraction principle has been generalized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics.

Cyclic representations and cyclic contractions were introduced by Kirk et al. [1] and further used by several authors to obtain various fixed point results. See, for example, papers [2–9]. Note that while a classical contraction has to be continuous, cyclic contractions might not be.

On the other hand, Matthews [10] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not be equal to zero. Several authors obtained many useful fixed point results in these spaces–-we just mention [11–27]. Several results in ordered partial metric spaces have been obtained as well [28–36]. Some results for cyclic contractions in partial metric spaces have been very recently obtained in [37–41].

Khan et al. [42] addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function. This idea was further used in many papers, such as Choudhury [43] where generalized control functions were used. This approach has been very recently used in [44, 45] to obtain fixed point results in partial metric spaces.

In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a generalized control function (with two variables) for mappings on -complete partial metric spaces (Romaguera [16]). We obtain fixed point theorems for such mappings, thus generalizing several known results. Various examples are presented showing how the obtained results can be used and that they are proper extensions of the known ones.

2. Preliminaries

In 2003, Kirk et al. introduced the following notion of cyclic representation.

Definition 1 (see [1]). Let be a nonempty set, , and let be a self-mapping. Then is a cyclic representation of with respect to if(a) are nonempty subsets of ;(b), , .

They proved the following fixed point result.

Theorem 2 (see [1]). Let be a complete metric space, , and let be a cyclic representation of with respect to . Suppose that satisfies the following condition: where and is a function, upper semicontinuous from the right and for . Then, has a fixed point .

In 2010, Păcurar and Rus introduced the following notion of cyclic weaker -contraction.

Definition 3 (see [2]). Let be a metric space, , and let be closed nonempty subsets of and . An operator is called a cyclic weaker -contraction if(1) is a cyclic representation of with respect to ;(2)there exists a continuous, nondecreasing function with for and such that for any , , where .

They proved the following result.

Theorem 4 (see [2]). Suppose that is a cyclic weaker -contraction on a complete metric space . Then, has a fixed point .

This was generalized by Karapınar in [3].

Khan et al. introduced the following notion.

Definition 5 (see [42]). A function is called an altering distance function if the following properties are satisfied: (a) is continuous and nondecreasing, (b).

Choudhury introduced a generalization of Chatterjea type contraction as follows.

Definition 6 (see [43]). A self-mapping , on a metric space , is said to be a weakly -contractive (or a weak Chatterjea type contraction) if for all , where is a continuous function such that

In [43], the author proved that every weak Chatterjea type contraction on a complete metric space has a unique fixed point.

The following definitions and details can be seen, for example, in [10, 12, 13, 15, 16].

Definition 7. A partial metric on a nonempty set is a function such that for all :  (p₁) ,  (p₂) ,  (p₃) ,  (p₄) . The pair is called a partial metric space.

It is clear that, if , then from and . But if , may not be .

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 (in the sense of ) if . This will be denoted as () or . 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 imply .

Example 8 (see [10]). (1) A paradigmatic example of a partial metric space is the pair , where for all .
(2) Let and let . Then is a partial metric space.

Definition 9. Let be a partial metric space. Then consider the following.(1) A sequence in is called a Cauchy sequence if exists (and is finite). The space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .(2)(see [16]) A sequence in is called -Cauchy if . The space is said to be -complete if every -Cauchy sequence in converges (in ) to a point such that .

Lemma 10. Let be a partial metric space. (a) (see [46, 47]) If as , then as for each . (b) (see [16]) If is complete, then it is -complete.

The converse assertion of (b) does not hold as the following easy example shows.

Example 11 (see [16]). The space with the partial metric is -complete, but is not complete. Moreover, the sequence with for each is a Cauchy sequence in , but it is not a -Cauchy sequence.

It is easy to see that every closed subset of a -complete partial metric space is -complete.

3. Main Results

In this section, we will prove some fixed point theorems for self-mappings defined on a -complete partial metric space and satisfying certain cyclic weak contractive condition involving a generalized control function. To achieve our goal, we introduce the new notion of a cyclic contraction.

Definition 12. Let be a partial metric space, , and let be nonempty subsets of and . An operator is called a cyclic contraction under weak contractive condition if: (NZ1) is a cyclic representation of with respect to ; (NZ2) for any (with ), where and is a lower semicontinuous mapping such that if and only if .

Our main result is the following.

Theorem 13. Let be a -complete partial metric space, let be nonempty closed subsets of , and let . Suppose that is a cyclic contraction as defined in Definition 12. Then, has a unique fixed point such that . Moreover, . Each Picard sequence , converges to in topology .

Proof. Let be an arbitrary point of . Then there exists some such that . Now and, similarly, for , where . In the case for some , it is clear that is a fixed point of .
Without loss of the generality, we may assume that From the condition (NZ1), we observe that for all , there exists such that .
Putting and in (NZ2) condition, we have By , we have Therefore, By (8) and (10), we have If , then from (11), we have which is a contradiction (it was used that since ). Hence, , and , which is excluded. Therefore, we have and hence By (13), we have that is a nonincreasing sequence of positive real numbers. Thus, there exists such that Passing to the limit as in (13) and using (14) and lower semicontinuity of , we have thus, and hence . Therefore
Next, we claim that is a -Cauchy sequence in the space . Suppose that this is not the case. Then there exists for which we can find two sequences of positive integers and such that for all positive integers Using (17) and (), we get Thus we have Passing to the limit as in the above inequality and using (16), we obtain On the other hand, for all , there exists such that . Then (for large enough, ) and lie in different adjacently labelled sets and for certain .
Using () and (20), we get that is, Using (16), we have Again, using (), we get Passing to the limit as in the pervious inequality, and using (24) and (22), we get Similarly, we have by () Passing to the limit as , and using (16) and (22), we obtain Similarly, we have by () Using (NZ2), we obtain for all . Passing to the limit as in the last inequality (and using the lower semicontinuity of the function ), we obtain which implies that ; that is a contradiction since . We deduce that is a -Cauchy sequence.
Since is -complete and is closed, it follows that the sequence converges to some , that is: We shall prove that From condition (NZ1), and since , we have . Since is closed, from (32), we get that . Again, from the condition (NZ1), we have . Since is closed, from (32), we get that . Continuing this process, we obtain (33) and .
Now, we shall prove that is a fixed point of . Indeed, from (33), since for all , there exists such that ; applying (NZ2) with and , we obtain for all . Passing to the limit as in (34), and using (32), we get which is impossible unless , so that is, is a fixed point of .
We claim that there is a unique fixed point of . Assume on the contrary that and with . By supposition, we can replace by and by in (NZ2) to obtain a contradiction. Hence , that is, . We conclude that has only one fixed point in . The proof is complete.