Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces
© di Bari et al.; licensee Springer 2012
Received: 17 February 2012
Accepted: 2 July 2012
Published: 19 July 2012
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.
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.
Definition 1 A partial metric on a nonempty set X is a function such that for all : (p1) = ,; (p2) = ,; (p3) = ,; (p4) = ..
The pair is called a partial metric on X.
It is clear that, if , then from (p1) and (p2) . But if , may not be 0.
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 .
Remark 1 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 .
- (1)A paradigmatic example of a partial metric space is the pair , where for all . The corresponding metric is
- (2)If is a metric space and is arbitrary, then
defines a partial metric on X and the corresponding metric is .
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 X converges, with respect to , to a point such that .
 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 .
is a Cauchy sequence inif and only if it is a Cauchy sequence in the metric space.
The spaceis complete if and only if the metric spaceis complete.
Every 0-Cauchy sequence inis Cauchy in.
Ifis complete, then it is 0-complete.
The converse assertions of (c) and (d) do not hold as the following easy example shows.
Example 2 ()
The space with the partial metric is 0-complete, but is not complete (since and is not complete). Moreover, the sequence with for each is a Cauchy sequence in , but it is not a 0-Cauchy sequence.
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.
(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
then f and g have a unique point of coincidence. If, moreover, f and g are weakly compatible, then they have a unique common fixed point u such that.
Recall that is called a coincidence point of and y is their point of coincidence if . If f and g commute at their coincidence points, they are called weakly compatible.
Denote by the n th orbit of and by its orbit. Also, denote by the diameter of a nonempty set . Note that implies that A is a singleton, but the converse is not true.
If for some , then it is easy to prove (using properties (p2) and (p4) of the partial metric, and the contractive condition (2.1)) that , i.e., . Hence, in this case, is a 0-Cauchy sequence in .
Suppose now that for each .
i.e., . Taking the supremum in this inequality, the proof of Claim 1 is obtained.
and Claim 2 is proved.
Hence, and so .
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 .
Then f and g have a unique point of coincidence. If, moreover, f and g are weakly compatible, then they have a unique common fixed point u andholds.
Proof Let the assumption of corollary hold and denote . Then for all , condition (2.1) of Theorem 1 is satisfied and the conclusion follows. □
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.
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.
and it is satisfied for all since . Hence, all the conditions of Theorem 1 are satisfied and f and g have a unique common fixed point ().
Since is not complete, nor is the space , where is the Euclidean metric, the existence of a (common) fixed point cannot be deduced using known results.
3 Mappings with a contractive iterate
for every. Then f has a unique fixed point. Moreover, and every Picard sequenceconverges to z.
Proof We first note that, similarly as in the metric case, the following can be proved:
In particular, is a finite real number for each.
We will prove that this is a 0-Cauchy sequence.
as since by (3.2).
Now, using standard arguments, it is easy to show that as . Hence, is a 0-Cauchy sequence. Since the space is 0-complete, there exists satisfying , , with .
and it must be , i.e., z is a (unique) fixed point of f.
Now, using what was previously proved, we obtain that and f is a Picard operator. □
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].
for all. Then f and g have a unique point of coincidence. If, moreover, f and g are weakly compatible, then f and g have a unique common fixed point, say z, with.
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.
where is fixed.
It follows that is a 0-Cauchy sequence. Since this space is 0-complete, there exists (i.e., , ) such that , (we have supposed that gX is closed, and hence 0-complete) and . We will prove that .
i.e., . Since , it follows that , , and f and g have a point of coincidence z.
Since , the last relation is possible only if and hence . So, the point of coincidence is unique.
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 f g 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 second and the fourth author are thankful to the Ministry of Science and Technological Development of Serbia.
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