Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces

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.

Matthews [15] 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.[4] 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.

The following definitions and details can be seen, e.g., in [3,4,11,15,16,21].

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 .

If is continuous at (with respect to ), then for each sequence in X, we have

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 .

If p is a partial metric on X, then the function given by

is a metric on X. Furthermore, if and only if

Example 1

(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 .

Other examples of partial metric spaces which are interesting from a computational point of view may be found in [9,15].

Definition 2 Let be a partial metric space. Then:

1. A sequence in is called a Cauchy sequence if exists (and is finite).

2. The space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .

3. [18] 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 .

Lemma 1Letbe a partial metric space.

(a) is a Cauchy sequence inif and only if it is a Cauchy sequence in the metric space.

(b) The spaceis complete if and only if the metric spaceis complete.

(c) Every 0-Cauchy sequence inis Cauchy in.

(d) 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 ([18])

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 [18], 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.

Let be a partial metric space and be two selfmaps. When constructing various contractive conditions, usually one of the following sets is used:

Then, the contractive condition takes the form

where . Mappings f satisfying (1.2) with for all (in metric case) are usually called g-quasicontractions (see Ćirić [5] and Das and Naik [7]).

(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 [17], (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.

Theorem 1Letbe a 0-complete partial metric space and letbe two selfmaps such that, and one of these two subsets ofXis closed. If there existssuch that the condition

holds for all, where

thenfandghave a unique point of coincidence. If, moreover, fandgare weakly compatible, then they have a unique common fixed pointusuch 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.

Proof For arbitrary , and using that , choose a Jungck sequence in X by

Denote by the nth 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 .

Claim 1.

Indeed, let . Then

Since the points , , , belong to the set , it follows that

Hence, there exists such that . Since, by (p₄),

we have

i.e., . Taking the supremum in this inequality, the proof of Claim 1 is obtained.

Claim 2. Let . Then

Similarly as in (2.2), we have that

Since , , , , we have

for some . Now, similarly,

which, together with (2.4), gives

for some . Continuing the process, we obtain that

and Claim 2 is proved.

It follows that as , i.e., is a 0-Cauchy sequence. Since is 0-complete, there exists , such that , (we have supposed that gX is closed, and hence 0-complete) and

Now, we prove that also . We have

Since , and tend to 0 as , and since

if we suppose that , we get a contradiction

Hence, and so .

Suppose that there exists , and such that . Then

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. □

Remark 2 If u is the unique common fixed point of f and g obtained as a limit of a Jungck sequence as in the previous proof, then the following error estimate holds

Since p is not continuous in general, this cannot be obtained directly from (2.3). Instead, notice that for

Passing to the limit when , we get that

According to the well-known classification of Rhoades [17] (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 [13], 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:

1. ;

2. ;

3. .

Thenfandghave a unique point of coincidence. If, moreover, fandgare weakly compatible, then they have a unique common fixed pointuandholds.

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.

Example 3 Consider the set and the function given by and . Obviously, p is a partial metric on X, not being a metric (since for and ). Define a selfmap f on X by

Then f is not a (Banach)-contraction since

and there is no such that . We will check that f is an -quasicontraction with . If , then and (2.1) trivially holds. Let, e.g.; then we have the following three cases:

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.

Example 4 Let be equipped with the partial metric p defined by for . Let be given by

By Example 2, the space is 0-complete (but not complete). Take . The contractive condition (2.1) for (say) takes the form

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.

In this section, we prove a version of Sehgal-Guseman theorem ([10,20], see also [17]) for 0-complete partial metric spaces.

Theorem 2Letbe a 0-complete partial metric space and let. Suppose that there existssuch that for eachthere issatisfying

for every. Thenfhas a unique fixed point. Moreover, and every Picard sequenceconverges toz.

Proof We first note that, similarly as in the metric case, the following can be proved:

Under the assumptions of the theorem,

In particular, is a finite real number for each.

Let be arbitrary. Construct the sequence in the following way:

We will prove that this is a 0-Cauchy sequence.

If for some n, then it easily follows that this sequence is eventually constant, and hence a 0-Cauchy one. Suppose further that for each n. Condition (3.1) implies that

Repeating this procedure n times, we get that

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 .

It follows from condition (3.1) that

as . Hence, in . Further we have

The first and third summand on the right-hand side tend to 0 when . For the second summand we have

as . Thus, and so . If z and u were two distinct fixed points of , then (3.1) would imply that

a contradiction.

Now, we easily get that

and it must be , i.e., z is a (unique) fixed point of f.

In order to prove that f is a Picard operator, let be arbitrary and be the corresponding Picard sequence. Each , can be uniquely written in the form

and when . Let z be the (unique) fixed point of f whose existence has just been proved. Then

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.

The following theorem is a partial metric version of an interesting result obtained by B. E. Rhoades [17], Theorem 4].

Theorem 3Letbe a 0-complete partial metric space. Letbe two mappings such thatand one of these subsets ofis closed. Suppose that there exist decreasing functions, , such thatfor eachand satisfying

for all. Thenfandghave a unique point of coincidence. If, moreover, fandgare weakly compatible, thenfandghave a unique common fixed point, sayz, 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.

Suppose that for each . Using (4.1) (and putting temporarily , ), we obtain that

for each . Also,

Adding up the last two relations, we obtain

where

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

where is fixed.

Now we prove that is a 0-Cauchy sequence in the usual way: for it is

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 .

Put , in the contractive condition. We obtain (writing temporarily ) that

Taking into account that all ’s are bounded in , passing to the limit in the last inequality, we obtain that

i.e., . Since , it follows that , , and f and g have a point of coincidence z.

Suppose that is another point of coincidence for f and g. Then (4.1) implies that and also that

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 [17], Theorem 4].

Remark 4 Taking appropriate choices of fg and in Theorem 3, one can easily get the results of Reich [17], (7), (8)], Hardy-Rogers [17], (18)] and Ćirić [17], (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., [1], Theorem 8], [2], Theorem 5], [3], Theorems 1 and 2], [6], Theorem 2.1], [14], Theorem 5], [19], 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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