Three extensions of Ćirić quasicontraction on partial metric spaces
© Ilić et al.; licensee Springer. 2013
Received: 18 July 2013
Accepted: 15 October 2013
Published: 19 November 2013
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.
1 Introduction and preliminaries
is called a quasicontraction. Let us remark that Ćirić  (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  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. . 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.
if and only if ,
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  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.
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 .
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 . □
- (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 .
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 .
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. □
for each , there is some such that for all , we have ;
- (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. □
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.
for each , the sequence converges with respect to to some point such that ;
there is some such that and .
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.
Set and .
so we must have .
For , this is just (3.3).
We finally prove . By (3.4) we only need to show . For a proof by contradiction, assume this is not the case. Then .
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 .
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.
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  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 .
for all . We check that for all it holds , so that the main result from  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 ()
the set is nonempty;
there is a unique such that ;
for each , the sequence converges with respect to the metric to u.
Finally, if is a fixed point, then by the preceding discussion , i.e., so . □
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  and ). 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 )
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  is recovered. Theorem 3.3 also implies Corollaries 1-4 of , and the next Hardy and Rogers type  fixed point result. This result, under some extra conditions, was proved as one of the main results, Theorem 2 of .
Then there is a unique such that . Also and for each the sequence converges with respect to the metric to z.
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 .
and define (thus if , then and ). Here ‘’ stands for the domain of the function x. Then is a partial metric space (see ) and a complete one as can easily be verified.
, (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.
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.
This work was supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.
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