Abstract
In this paper we define and study three extensions of the notion of Ćirić quasicontraction to the context of partial metric spaces. For such mappings, we prove fixed point theorems. Among other things, we generalize a recent result of Altun, Sola and Simsek, of Ilić et al., of Matthews, and the main result of Ćirić is also recovered. The theory is illustrated by some examples.
MSC:15A09, 15A24.
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1 Introduction and preliminaries
Let be a complete metric space. A map , such that for some constant and for every , there is the inequality
is called a quasicontraction. Let us remark that Ćirić [1] (see also [2, 3]) introduced and studied quasicontractions as one of the most general types of contractive maps. The well-known Ćirić result is that every quasicontraction T possesses a unique fixed point.
There exist many generalizations of the concept of metric spaces in the literature. In particular, Matthews [4] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces have been studied by many other authors.
References [5–31] are some works in this line of research. The existence of several connections between partial metrics and topological aspects of domain theory were pointed out in, e.g., [11, 24, 32–34].
In this paper we study fixed point results about certain extensions of the notion of Ćirić quasicontraction to the setting of partial metric spaces, and we give some generalized versions of the fixed point theorem of Matthews as well as the main result of Ilić et al. [35]. The theory is illustrated by some examples.
Throughout this paper, the letters ℝ, , ℚ, ℕ denote the sets of real numbers, nonnegative real numbers, rational numbers and positive integers, respectively.
Let us recall [4] that a mapping , where X is a nonempty set, is said to be a partial metric on X if for any , the following four conditions hold true:
-
(P1)
if and only if ,
-
(P2)
,
-
(P3)
,
-
(P4)
.
The pair is then called a partial metric space. A sequence of elements of X is called p-Cauchy if the limit exists and is finite. The partial metric space is called complete if for each p-Cauchy sequence , there is some such that
If is a partial metric space, then , , is a metric on X, converges to with respect to if and only if (1.2) holds, and is a complete partial metric space if and only if is a complete metric space (see [4, 21]).
A sequence in a partial metric space is called 0-Cauchy [27] if . We say that is 0-complete if every 0-Cauchy sequence in X converges, with respect to p, to a point such that . Note that every 0-Cauchy sequence in is Cauchy in , and that every complete partial metric space is 0-complete. A paradigm for partial metric spaces is the pair where and for , which provides an example of a 0-complete partial metric space which is not complete.
2 Auxiliary results
In this section we define three extensions of the notion of Ćirić quasicontraction to the context partial metric spaces and establish a few auxiliary results that will be used in the next, main section.
Definition 2.1 Let be a partial metric space and be a mapping. If for some and all , there holds
where , then we shall say that T is a -quasicontraction. If for all , the stronger condition
is satisfied, we shall call T a -quasicontraction. If the even stronger condition
holds for all , then we shall say that T is a p-quasicontraction.
Lemma 2.1 Let be a partial metric space, be a -quasicontraction, and n, m and k be integers such that .
If for all , then for some , there must hold
Proof This is an easy induction on k. For this follows from (2.1) and the assumptions made. Now suppose that the assertion holds for some and all , and let be such that we have for all .
From and (2.1) we see that there must hold for some if , or for some if ; in either case, we have .
There cannot be any such that because this would imply . Thus, by the induction hypothesis, there are such that , whence . □
Lemma 2.2 Let be a partial metric space, be a -quasicontraction, and . Then
-
(1)
for all , there holds
(2.4) -
(2)
if we put , then for some we have
(2.5)
Proof (1) Fix and put .
It is easy to see that there must be some with . Indeed, take any such that . If , we are done. Otherwise , so for some . Therefore either , i.e., for any , or , thus .
Now take any such that . If , then follows trivially. If , then for some ,
so , where l is the least integer such that and . So it must either be or . The latter possibility in the case gives directly, and if , then, since the minimality of l implies , we must have for some that and again .
(2) By part (1) we have . If , then there is nothing to prove. Thus let . Suppose to the contrary that for all ,
Take such that and such that . By (2.6) we may assume that . Also by (2.6) there must be some such that and for all . Denote by the least such integer.
Since for all and since , Lemma 2.1 implies that for some , we have . But , a contradiction. □
Lemma 2.3 Let be a partial metric space, be a -quasicontraction, and . Then
Proof Notice first that by Lemma 2.2. The lemma easily follows from the following two claims:
-
(1)
for each , there is some such that for all , we have ;
and
-
(2)
if , then
(2.8)
To prove (1), fix any . From , we see that there is some such that and such that for all we have .
Let . If for some , then . Otherwise, as , by Lemma 2.1 there are such that .
To prove (2) suppose . Take any positive and let be as claimed to exist in (1). We will show that for any , it must be that . Indeed, suppose that for some this were not true. By (2) of Lemma 2.2, there is an integer l such that and . Let m be the least such integer. Then by our assumption , so, by the choice of m, we must have . Hence . Since by definition of and m it must be , the last inequality now yields , i.e., , a contradiction. □
Lemma 2.4 Let be a partial metric space, be a -quasicontraction and be such that
Then .
Proof Denote and and notice that
implies, using , that
Now simply take the limit as in the above inequality. □
3 Main results
In this paper we study three extensions of Ćirić quasicontraction to a partial metric space. For such mappings, we prove fixed point theorems. Among other things, we generalize a recent result of Altun, Sola and Simsek, and we give some generalized versions of the fixed point theorem of Matthews, and the main result of Ćirić is also recovered. The theory is illustrated by some examples.
Theorem 3.1 Let be a complete partial metric space and be a -quasicontraction. Then
-
(1)
for each , the sequence converges with respect to to some point such that ;
-
(2)
there is some such that and .
Proof By Lemmas 2.3 and 2.4 and completeness of , for each , there must be some with such that
Also notice that we must have
Denote . Using the construction described bellow, we will find a specific such that and later on we prove that we must actually have for that particular point u. This will complete the proof.
Use (2.7) and (3.1) to find for each some and such that
Set and .
Now
gives, using and similarly ,
So , and thus there is some such that
Let us show that
Set and . From
it follows, using , that
so we must have .
We now prove by induction on that
For , this is just (3.3).
Now suppose that (3.5) holds for some . We have
From this, using
and
and
and
we deduce that
The last inequality, in view of the induction hypothesis and (3.3), immediately gives
We finally prove . By (3.4) we only need to show . For a proof by contradiction, assume this is not the case. Then .
Set . By (3.1) and , there must be some such that . Let m be the least such j. By our assumption, we must have . Then for all . Thus by Lemma 2.1 there must be some such that . So
We have , which follows by (3.3) and (3.5) after taking the limit as in . But now yields , i.e., , a contradiction. □
Theorem 3.2 Let be a complete partial metric space and be a -quasicontraction. Then there is a unique fixed point of T. For each , the sequence converges with respect to to some point such that , and there is the equality .
Proof By Theorem 3.1 it remains to verify the uniqueness of the fixed point, so let be such that and . By (2.2) we have
Hence either or and the assertion follows. □
Theorem 3.3 Let be a 0-complete partial metric space and be a p-quasicontraction. Then there is a unique fixed point of T. Furthermore, we have and for each the sequence converges to u with respect to .
Proof By Lemma 2.3 we have . Also by Lemma 2.2.
If we had , then there would be some positive and such that for all . Thus by (2.3) there would be some such that
i.e., , a contradiction.
So and thus by 0-completeness of there is some such that . But by Lemma 2.4 we have so . The argument for uniqueness of the fixed point is standard. □
Remark 3.1 Recently a very interesting paper by Haghi, Rezapour and Shahzad [36] showed up in which the authors associated to each partial metric space a metric space by setting and if and proved that is 0-complete if and only if is complete. They then proceeded to demonstrate how using the associated metric d some of the fixed point results in partial metric spaces can easily be deduced from the corresponding known results in metric spaces.
Let us point out that these considerations can apply neither to -quasicontractions nor to -quasicontractions, since the terms and on the right-hand side of (2.1) and on the right-hand side of (2.2) do not get multiplied by α. Thus Theorems 3.1 and 3.2 cannot follow from the result of Ćirić they generalize.
On the other hand, using the approach of Haghi, Rezapour and Shahzad, we now show how Theorem 3.3 can be directly deduced from Ćirić’s result [1].
Given the assumptions of Theorem 3.3, let d be defined as said above. So is a complete metric space (see Proposition 2.1 of [36]). Observe that we have for all . For , set
and also and . We thus have that
for all . We check that for all it holds , so that the main result from [1] can immediately be applied. Since in the case the inequality trivially holds, suppose . So .
Since , it suffices to show that . Let be such that . If , then . If , then, since , it follows that .
Remark 3.2 Even though the results of Haghi et al. can deduce the same fixed point as the corresponding partial metric fixed point result, using the partial metric version computers evaluate faster since many nonsense terms are omitted. This is very important in computer science due to its cost and explains the vast body of partial metric fixed point results found in literature.
Now we give corollaries of the above theorems.
Corollary 3.1 ([35])
Let be a complete partial metric space, and be a given mapping. Suppose that for each , the following condition holds:
Then
-
(1)
the set is nonempty;
-
(2)
there is a unique such that ;
-
(3)
for each , the sequence converges with respect to the metric to u.
Proof Put and . Note that (3.6) implies for all . So, from and , , we see that . Thus . By Theorem 3.1, there is some with and . This means that . Now let be arbitrary. Since , for all , we have , and so . Also, from using , it follows
Hence .
Finally, if is a fixed point, then by the preceding discussion , i.e., so . □
Corollary 3.2 Let be a complete partial metric space, and be a given mapping. Suppose that for each , the following condition holds:
Then there is a unique such that . Furthermore, and for each , the sequence converges with respect to the metric to z.
As a corollary we obtain the already mentioned result of Matthews (see also Corollary 2 of [8] and [21]). Let us remark that the result of Matthews is for a complete partial metric space, but it is true for a 0-complete partial metric space.
Corollary 3.3 (Matthews [4])
Let be a 0-complete partial metric space, and be a given mapping. Suppose that for each , the following condition holds:
Then there is a unique such that . Also and for each the sequence converges with respect to the metric to z.
Remark 3.3 In the case is a metric, by Theorem 3.3, the main result of Ćirić [1] is recovered. Theorem 3.3 also implies Corollaries 1-4 of [8], and the next Hardy and Rogers type [37] fixed point result. This result, under some extra conditions, was proved as one of the main results, Theorem 2 of [8].
Corollary 3.4 Let be a 0-complete partial metric space, , , and be a given mapping. Suppose that for each , the following condition holds:
Then there is a unique such that . Also and for each the sequence converges with respect to the metric to z.
Example 3.1 Let and define by
Then is a complete partial metric space. Define by
If , then it is easy to see that and . Given any , we have that holds for all if , i.e., for all if . On the other hand, we have that
and
Thus T is a -quasicontraction on X which is not a p-quasicontraction. By Theorem 3.2, there is a unique fixed point . Also we have .
Example 3.2 Denote by the set of all sequences and for by the set of all n-tuples of positive integers. Put . For , set
and define (thus if , then and ). Here ‘’ stands for the domain of the function x. Then is a partial metric space (see [4]) and a complete one as can easily be verified.
Define by , where:
-
and ;
-
, (this condition is vacuous if ) and if in addition , then .
Note that taking, e.g., and , we have and so . Thus the contractive condition of Corollary 3.1 is not satisfied. Nevertheless, there is a unique fixed point of T - the sequence defined by for all . We will show that T is a p-quasicontraction. Consider arbitrary .
Case 1. There is a nonnegative integer i with such that . Denote by k the least such nonnegative integer. Thus simply means that and if , then for all with , we must have .
If , then so .
If , then (because in this case we must have but ). Hence .
Case 2. for all and (meaning and x is the restriction of y to the set ).
If , then .
If , then , and so .
Case 3. for all and . This reduces to the previous case.
To illustrate the role condition (2.3) plays in ensuring the uniqueness of the fixed point, we modify a bit the definition of the operator T to obtain the operator determined by if and only if
-
;
-
and .
has infinitely many fixed points - these are exactly the sequence and the restrictions of to the sets . Due to the existence of infinitely many fixed points, cannot be a -quasicontraction. We verify that is a -quasicontraction.
Given arbitrary , we distinguish three cases exactly as we did with the operator T.
Case 1. There is a nonnegative integer i with such that . This is handled exactly as in the corresponding case with the operator T: if k is the least such nonnegative integer, then or according to whether or , respectively.
Case 2. for all and . If , then , and so . If , then , so .
Case 3. for all and . This reduces to the previous case.
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This work was supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.
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Ilić, D., Pavlović, V. & Rakočević, V. Three extensions of Ćirić quasicontraction on partial metric spaces. Fixed Point Theory Appl 2013, 303 (2013). https://doi.org/10.1186/1687-1812-2013-303
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DOI: https://doi.org/10.1186/1687-1812-2013-303