Common fixed point results are obtained in 0-complete partial metric spaces under various contractive conditions, including g-quasicontractions and mappings with a contractive iterate. In this way, several results obtained recently are generalized. Examples are provided when these results can be applied and neither corresponding metric results nor the results with the standard completeness assumption of the underlying partial metric space can.
MSC: 47H10, 54H25.
Keywords:fixed point; common fixed point; partial metric space; 0-complete space; quasicontraction
1 Introduction and preliminaries
Matthews  introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Subsequently, several authors (see, e.g., [1-3,6,8,12-14,16,17,20,22]) derived fixed point theorems in partial metric spaces. See also the presentation by Bukatin et al. where the motivation for introducing non-zero distance (i.e., the ‘distance’ p where need not hold) is explained, which is also leading to interesting research in foundations of topology.
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and . A sequence in converges to a point (in the sense of ) if . This will be denoted as () or .
3.  a sequence in is called 0-Cauchy if . The space is said to be 0-complete if every 0-Cauchy sequence in X converges (in ) to a point such that .
The converse assertions of (c) and (d) do not hold as the following easy example shows.
Example 2 ()
Recall that Romaguera proved in , Theorem 2.3] that a partial metric space is 0-complete if and only if every -Caristi mapping on X has a fixed point.
It is easy to see that every closed subset of a 0-complete partial metric space is 0-complete.
Then, the contractive condition takes the form
(Common) fixed point results in partial metric spaces using conditions of mentioned type in the case were obtained in various papers. We prove in Section 2 a common fixed point theorem for g-quasicontractions in 0-complete spaces that contains as special cases several other results. In Section 3 a partial metric extension of Sehgal-Guseman result for mappings having a contractive iterate is obtained. Finally, in Section 4 we deduce a partial metric version of (common) fixed point theorem under the condition , (19)] of B. E. Rhoades.
Examples are provided when these results can be applied and neither corresponding metric results nor the results with the standard completeness assumption of the underlying partial metric space can.
2 Quasicontractions in partial metric spaces
Similarly as in (2.2), we have that
which, together with (2.4), gives
and Claim 2 is proved.
which is possible only if , and hence . Thus, we have proved that the point of coincidence of f and g is unique. By a well-known result, if f and g are weakly compatible, it follows that f and g have a unique common fixed point. □
According to the well-known classification of Rhoades  (which obviously holds for partial as well as for standard metric), Theorem 1 implies several other (common) fixed point results, e.g., those of Banach, Kannan, Chatterjea, Bianchini, Hardy-Rogers and Zamfirescu. We state the last one which was obtained in , Theorem 4.2] in the special case .
Corollary 1Letbe a 0-complete partial metric space, and letbe such thatand one of these two subsets ofXis closed. Suppose that there existα, β, γ, withand, such that for all, at least one of the following conditions hold:
We give an easy example of a partial metric space, which is not a metric space, and a selfmap in it which is a quasicontraction and not a contraction.
Then f is not a (Banach)-contraction since
Thus, the conditions of Theorem 1 are satisfied and the existence of a common fixed point of f and (which is b) follows. The same conclusion cannot be obtained by Banach-type fixed point results from [15,21].
We present another example showing the use of Theorem 1. It also shows that there are situations when standard completeness of the p-metric as well as usual metric arguments cannot be used to obtain the existence of a fixed point.
3 Mappings with a contractive iterate
Proof We first note that, similarly as in the metric case, the following can be proved:
Under the assumptions of the theorem,
We will prove that this is a 0-Cauchy sequence.
Repeating this procedure n times, we get that
It follows from condition (3.1) that
Now, we easily get that
Example 5 Let and f be as in Example 3. We have seen that f is not a contraction in the partial metric space . However, and f satisfies condition (3.1) of Theorem 2 with for each since for each . As we have seen, f has a unique fixed point b.
4 Partial metric version of a theorem of Rhoades
The following theorem is a partial metric version of an interesting result obtained by B. E. Rhoades , Theorem 4].
Proof Suppose, e.g., that gX is closed. Take an arbitrary and, using that , construct a Jungck sequence defined by , . Let us prove that this is a 0-Cauchy sequence. If for some n, then as in the proof of Theorem 1, one proves that the sequence becomes eventually constant, and thus convergent.
Adding up the last two relations, we obtain
It is easy to see that monotonicity of all ’s implies that β is also a decreasing function and that for each . In particular, and so the sequence is strictly decreasing (and bounded from below). It follows that there exists and for each n. Then for each n, and hence
The proof is similar if the subset fX of X is closed.
By a well-known result, if f and g are weakly compatible, it follows that f and g have a unique common fixed point. □
Remark 3 Taking to be a standard metric space and , we obtain a shorter proof of , Theorem 4].
Remark 4 Taking appropriate choices of fg and in Theorem 3, one can easily get the results of Reich , (7), (8)], Hardy-Rogers , (18)] and Ćirić , (21)] in the setting of partial metric spaces.
Remark 5 We finally note that, in a similar way, several other fixed point results in partial metric spaces obtained recently (e.g., , Theorem 8], , Theorem 5], , Theorems 1 and 2], , Theorem 2.1], , Theorem 5], , Theorems 3 and 4]) can be proved with a (strictly) weaker assumption of 0-completeness instead of completeness.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
The second and the fourth author are thankful to the Ministry of Science and Technological Development of Serbia.
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