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Fixed point theorems for contracting mappings in partial metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 185 (2014)
Abstract
The paper deals with a class of contracting mappings that includes the Boyd-Wong and the Matkowski classes. Some of the new fixed point theorems obtained here unify and extend well-known results also in the case of metric spaces.
MSC:47H09, 47H10, 54H25, 54E99.
Introduction
In the simplest case, the condition with a contracting mapping has the form , where is a metric space, f is a mapping on X, and is such that , . Boyd and Wong [1] assumed that φ is upper semicontinuous from the right (i.e., , ); Matkowski in [2] assumed that φ is nondecreasing and such that for each . We assume that all sequences such that , , converge to zero. It appears that the classes of Boyd-Wong’s and Matkowski’s mappings are included in this new class (the problem of for Boyd-Wong mappings is meaningless for contractions). The main results are Theorems 21, 22, 23, and theorems extending the well-known classical results: Theorem 30 (Matkowski’s theorem) and Theorem 31 (covering the theorems of Romaguera and Boyd-Wong).
Let us recall the notions of a partial metric space due to Matthews [[3], Definition 3.1] and of a dualistic partial metric due to Oltra and Valero [4] and O’Neill [5].
Definition 1 A dualistic partial metric is a mapping such that
If p is nonnegative, then it is a partial metric.
If p is a dualistic partial metric on X, then defined by
is a quasi-metric [[3], Theorem 4.1] ( iff , ).
An open ball for , is defined by
The family of open balls generates topology on X. It is accepted that the (dualistic) partial metric space is equipped with the topology .
It is known (see, e.g., [4]) that a metric d can be defined by a dualistic partial metric p as follows:
In this paper it is understood that q, d are defined by (5), (7) respectively for a (dualistic) partial metric p.
For dualistic partial metric spaces, it is accepted (see, e.g., [[4], p.19]) that is called a Cauchy sequence in if , and is complete if for every Cauchy sequence in , there exists such that .
The following notions are useful.
Definition 2 [[6], Definition 2.1]
Let be a mapping. The kernel of p is the set .
Definition 3 (cp. [[7], Definition 2.1])
A dualistic partial metric space is 0-complete if for every sequence in X such that , there exists in .
In fact Definition 3 is too abstract. The condition in means that (see (6), (2)). If, in addition, , then in (see [[4], Lemma 2.2] or [[6], Proposition 1.4]).
Corollary 4 A dualistic partial metric space is 0-complete iff every sequence such that converges in to a point . If p is a metric, then 0-completeness is identical with completeness.
There exist 0-complete partial metric spaces which are not complete [8]. Some criterions of 0-completeness can be found in [[9], Section 4].
Proposition 5 Let be a partial metric space. Then converges in to iff and iff we have in .
Proof A sequence converges to x in iff (see [[4], Lemma 2.2] or [[6], Proposition 1.4]). Assume . Then, for nonnegative p, condition (2) yields and , i.e., and in . We also have
and consequently, , i.e., and converges in to x. □
Let be the family of all subsets of Y. We say that is a (multivalued) mapping if for all .
Now, let us investigate the concept of a 0-closed graph.
Definition 6 Let be a dualistic partial metric space. A mapping has a 0-closed graph if for all sequences , in X the following condition is satisfied:
Proposition 5 and [[6], Proposition 1.4] yield the following.
Corollary 7 For all sequences , with , condition (8) can be replaced by
If p is nonnegative, then (8) is equivalent to (9).
Clearly, if has a closed graph in , then (9) is satisfied as Kerp with topology induced by p is a closed metric subspace of [[6], Lemma 2.2].
For a partial metric space , a nonempty set , and , let us adopt
and
The family generates a topology , and we get a topological space . Assume that p is nonnegative and
holds. Then, for all sequences , such that and , , there exist , with . We have
and consequently, in (Proposition 5). If in , then holds.
For a partial metric space and nonempty , let us adopt
If p is a metric, then P is the Hausdorff metric of the metric p, whenever A and C are nonempty closed and bounded subsets of X. In general, P is not a partial metric (see [[10], Proposition 2.2(i), Proposition 2.3(h3)]).
Clearly, for and if (10) holds, then is continuous on Kerp.
Corollary 8 Let be a partial metric space. Assume that for each , the mapping satisfies (10) and in . Then F has a 0-closed graph (and conditions (8), (9) are equivalent).
Proposition 9 Let be a dualistic partial metric space. If has a 0-closed graph, then is closed in .
Proof If , , then for , and each such that , condition (8) yields , i.e., is closed in Kerp (which with topology induced by p is a closed metric subspace of [[6], Lemma 2.2]). □
Now, let us investigate a ‘contraction’ condition.
Let Φ be a class of mappings such that , ; iff and .
Proposition 10 Let be a partial metric space. Assume that satisfy
for . Then ; if H has a 0-closed graph, then holds.
Proof For and nonnegative p, conditions (2), (12) yield
i.e., and . What is more, from our inequality it follows that there exist such that . Now, (8) for , yields . □
Propositions 9, 10 yield the following.
Theorem 11 Let be a 0-complete partial metric space, and let ℱ be a family of mappings with 0-closed graphs. Assume that ; some and at least all different satisfy (12). Then all members of ℱ have the same set of fixed points; this set is closed in and contained in Kerp.
The previous result becomes a little more interesting if a mapping has a fixed point.
Let us assume that the following condition is satisfied for a partial metric space , P defined by (11), and :
Then, for any , we have
i.e., condition (13) yields condition (12). The subsequent two propositions enable us to strengthen condition (12).
Proposition 12 Let be a partial metric space, and let be compact in . Then, for any , there exists such that .
Proof Let be a sequence in C such that
There exists a subsequence of and such that in , i.e., [[4], Lemma 2.2]
Now, from
we get
□
Corollary 13 Let be a partial metric space, and let be mappings with H compact valued in . Then condition (12) is equivalent to
Condition (14) extends the idea of α-step mappings [[11], Definition 17].
The next example shows that even a ‘good’ mapping φ in condition (14) does not guarantee the existence of a fixed point.
Example 14 Let us consider a continuous mapping defined by for , and for . An easy computation proves that φ is increasing on and, therefore, φ is nondecreasing on . Let us consider and , . For , we have
Now, for , we obtain
i.e., (14) is satisfied for , , . Still it is clear that has no fixed point.
It is a good idea suggested by [12] to gather together the properties of φ. Let us recall that Φ is a class of mappings such that , ; and iff and .
Proposition 15 Assume that . Then every sequence such that , (in particular , ) is nonincreasing; if, in addition, φ is nondecreasing, then , holds.
Proof From , , it follows that . Similarly, for nondecreasing φ, we get . □
Let us present some subclasses of .
Let consist of mappings for which every sequence such that , converges to zero.
Proposition 16 We have . For , the sequence is nonincreasing and it converges to 0, . If a mapping satisfies
then .
Proof Assume that . Suppose that for . Then , , is a good counterexample (the sequence does not converge to 0). Therefore holds. Suppose . Then, for , , , we obtain a divergent sequence such that , . Consequently, every satisfies . The sequence is nonincreasing, , and it converges to 0 as . Assume . Then any sequence such that (), is nonincreasing and therefore it converges, say, to . Suppose . Then (15) yields
a contradiction. □
Let consist of mappings φ upper semicontinuous from the right, call them Boyd-Wong mappings. Proposition 16 yields .
In turn, let consist of nondecreasing mappings such that , (Matkowski mappings). It is well known [[13], Lemma] that . Moreover, , , yields (Proposition 15) and hence . Consequently, holds. Let us note that φ from Example 14 belongs to .
The following is a kind of the reverse condition.
Proposition 17 Let be a sequence convergent to zero and such that , , for a . Then there exists a mapping such that , and , , .
Proof Let us adopt and , . Clearly, ψ is nondecreasing and continuous from the right as . Therefore, ψ is also upper semicontinuous. The sequence is nonincreasing as , . Let us adopt . For , we have
and the case of is trivial as then . Consequently, we get (if ), (if ), and so on. Finally, we obtain , (the possible case of some ). □
The following modification of φ is useful.
Proposition 18 Assume that is a mapping. Then ψ defined by
belongs to Φ iff ; iff ; φ satisfies (15) iff ψ satisfies (15). In addition, if φ is nondecreasing, then ψ is increasing.
Proof It is clear that holds iff . If φ is nondecreasing, then for we obtain
i.e., ψ is increasing. The remaining part of the proof is also trivial. □
The next two propositions can be helpful in proving fixed point theorems.
Proposition 19 Assume that is a partial metric space, and let . If mappings satisfy (12) and H has a 0-closed graph, then for , where ψ is defined by (16), condition (14) holds.
Proof For , there exist and such that . If , then (2) yields , i.e., x is a fixed point of G, and by Proposition 10, . □
Proposition 20 Let be a partial metric space, let ℱ be a nonempty family of at most two mappings with 0-closed graphs, and let . Assume that all members of ℱ either satisfy (12) and φ has property (15) or satisfy (14). Then there exists a sequence such that , , , and .
Proof If satisfies (15) and (12) holds, then for ψ as in (16) and , condition (14) is satisfied (Proposition 19), and (Propositions 18, 16). Thus, it is sufficient to consider the case of and condition (14). Let be arbitrary, , and let be such that . If is defined, then is such that , and, similarly, satisfies . Thus, for , we have , , and converges to zero as . □
As was shown in Example 14, conditions (12), (14) are too weak to guarantee the existence of a fixed point, even for . The next two theorems, with stronger assumptions, are general results.
Theorem 21 Let be a 0-complete partial metric space, and let be a mapping with 0-closed graph. Assume that for a there exists a sequence in X such that
and . Then in is a fixed point of F, and .
Proof Clearly, converges to zero as . Therefore, is a Cauchy sequence (see (2)) and it converges in , say, to (X is 0-complete). There exist such that . Now, from
and condition (8) it follows that . □
Theorem 21, with , (for ), is an extension of the Nadler theorem on multivalued contractions [[14], Theorem 5].
Now, Theorem 21 and Theorem 11 yield the following.
Theorem 22 Let be a 0-complete partial metric space, and let ℱ be a family of mappings with 0-closed graphs. Assume that ; some and at least all different satisfy (12). If for an there exists a sequence such that and (17) holds, then all members of ℱ have the same nonempty set of fixed points; this set is closed in and contained in Kerp.
A simple consequence of Theorem 21 is the following one.
Theorem 23 Let be a 0-complete partial metric space, and let be a mapping with a 0-closed graph (e.g., is continuous). Assume that for a and the following condition is satisfied:
Then in is a fixed point of f, and .
Proof For (and ), we have (see (2))
and condition (17) holds. □
The next proposition shows that Theorem 23 is related to the well-known theorem of Matkowski [[2], Theorem 1.2, p.8].
Proposition 24 Let be a partial metric space, and let be a bounded mapping satisfying
for a nondecreasing mapping . Then condition (18) holds.
Proof Let us adopt and , . For each , we have
as φ is nondecreasing, and we get (18). □
Let be a partial metric space, and let be a mapping. Let us recall the conditions used by Romaguera in [15]:
for
and
for
For nondecreasing φ, Ćirić’s condition (21) is more general than (20), and (20) is more general than condition (19). All these conditions are used to prove that f has a fixed point.
Let us note that for conditions (19), (20), (21) can be arbitrary as on the right-hand side of any of these inequalities (for ) means that x is a fixed point of f.
Many fixed point theorems use more sophisticated conditions than (19), (20), (21) (see, e.g., [16], [17]) or the spaces under consideration have a richer structure (see, e.g., [9], [18]). We are interested in extending the most classical results.
The subsequent two lemmas are proved for condition (21), and the reasonings for (20) and (19) as well can be easily deduced. The next lemma (for condition (21)) has much in common with Romaguera’s Lemma 1 and Lemma 2 [15].
Lemma 25 Let be a partial metric space, and let be a mapping satisfying condition (19), (20), or (21) for a . Then, for any , the condition
is satisfied, and .
Proof For notational simplicity, let us adopt , . For , condition (21) has the following form:
We have
and therefore,
holds, i.e., (Lemma 1 [15]). This last equality and condition (21) for yield
i.e., (). This contradiction proves that must hold, and then condition (21) yields (see Romaguera’s Lemma 2 [15]). Now, it is clear that for arbitrary and the sequence , where , , converges to zero as and holds. □
The next lemma is also helpful in proving fixed point theorems.
Lemma 26 Let be a 0-complete partial metric space, and let be a mapping satisfying condition (19), (20), or (21) for a . If for , holds, then converges in to a unique fixed point of f, and this point belongs to Kerp.
Proof From it follows that converges to a point in (Corollary 4). We get
Suppose . Then from
and it follows that
for large n. Now we get
a contradiction. Clearly, means that (see (2)), and (see (7)). If x, y are fixed points of f, then (see (2), (21)) means that . □
The next result extends Romaguera’s Theorem 4 [15], and consequently, an earlier celebrated result due to Matkowski [[2], Theorem 1.2, p.8].
Theorem 27 Let be a 0-complete partial metric space, and let be a mapping satisfying condition (19) or (20) for a such that
holds (e.g., if φ is nondecreasing). Then f has a unique fixed point; if , then and in , .
Proof In view of Lemmas 25, 26 it is sufficient to prove that is a Cauchy sequence for , . Suppose that there are infinitely many such that . Let be the smallest numbers satisfying this inequality. For simplicity let us adopt and , . We have
which for means that
as we have . Now, for , condition (20) yields
and we obtain (from (19) as well)
for large k. Now, , , and condition (22) yield
a contradiction. Therefore, is a Cauchy sequence. □
Let be continuous mappings. Then such that on and on is a member of and φ satisfies conditions (15), (22); clearly, φ does not necessarily belong to or to .
Paesano and Vetro [9] have proved some theorems on coincidences and common fixed points. Maybe Theorem 27 can be extended to that case.
The next result extends Romaguera’s Theorem 3 [15], and consequently, an earlier celebrated result due to Boyd-Wong [[1], Theorem 1]. Let us recall that does not spoil the generality of condition (21). The proof is a modification of the one presented by Romaguera.
Theorem 28 Let be a 0-complete partial metric space, and let be a mapping satisfying condition (21) for a mapping and satisfying (15) (e.g., with φ upper semicontinuous from the right). Then f has a unique fixed point; if , then and in , .
Proof We follow the initial part of the proof of Theorem 27 preceding the sentence with condition (20) (Proposition 16 yields ). For large and , we obtain
and . Now, condition (21) yields
and we obtain (see (15))
a contradiction. Therefore, is a Cauchy sequence. □
The next lemma enables us to extend the preceding two theorems.
Lemma 29 Let be a mapping such that for has a unique fixed point, say, x. Then x is the unique fixed point of f. If, in addition, , , then , holds.
Proof If x is a fixed point of , then means that is a fixed point of and the uniqueness yields . If are fixed points of f, then we get , , and as has a unique fixed point. If holds for each , then we also obtain , which means that . □
Theorem 30 Let be a 0-complete partial metric space, and let be a mapping satisfying condition (19) or (20) with f replaced by for a , and a having property (22) (e.g., with φ nondecreasing). Then f has a unique fixed point; if , then and in , .
Proof Clearly, all the assumptions of Theorem 27 are satisfied for f replaced by . Now, we apply Lemma 29. □
The next theorem is a consequence of Theorem 28 and of Lemma 29.
Theorem 31 Let be a 0-complete partial metric space, and let be a mapping satisfying the condition
for , and a mapping satisfying (15) (e.g., with φ upper semicontinuous from the right). Then f has a unique fixed point; if , then and in , .
Now, we present the respective versions of conditions (19), (20), (21) for a multivalued mapping , where P is defined by (11).
for
and
for
For nondecreasing φ, condition (26) is more general than (25), and (25) is more general than condition (24).
The subsequent two lemmas are proved for condition (26), and the reasonings for (25) and (24) as well can be easily deduced.
At first, let us present the following extension of Lemma 25.
Lemma 32 Let be a partial metric space, and let be a mapping. Then
holds, and condition (26) for any yields , , ( in place of for (25)), and
If and
holds (e.g., F satisfies (28) and is compact valued), then there exists , , such that
Proof For , we have (see (2))
Consequently, we get
as , holds. Clearly,
is satisfied and hence we get (27). Suppose for a . Then (see (26))
means that (as ), and . This contradiction proves (28).
Let , be arbitrary. If , are known, then let be such that (see (29) or (28) and Proposition 12) . If , then for we get . □
Now, let us prove an analog of Lemma 26.
Lemma 33 Let be a 0-complete partial metric space, and let be a mapping with 0-closed graph satisfying condition (24), (25), or (26) for a . If and , then converges in to a fixed point of F, and this point belongs to Kerp.
Proof From it follows that converges to a point in (Corollary 4). We get
Suppose . Then from
and it follows that
for large n. Now we get
a contradiction. Clearly, means that (see (2)). There exist such that , and now,
together with condition (8) yield . □
It should be noted that a partial metric p defines metric δ in the following way: iff , and for . The topology of is clearly larger than the topology of (see (7)). Moreover, is 0-complete iff is complete [19], [[20], Proposition 2.1].
If the proof of a theorem is based on , , then it works for and the theorem is an immediate consequence of the respective result (if known) for metric spaces. Numerous examples can be found in [20].
Let us add that Corollary 4 and Proposition 5 show that for a 0-complete partial metric space , if we prove that , then the remaining part of the proof concerns the metric space (see (7)). Let us also recall that Kerp with its partial metric topology is a closed metric subspace of [[6], Lemma 2.2].
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The work has been supported by the Polish Ministry of Science and Higher Education.
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Pasicki, L. Fixed point theorems for contracting mappings in partial metric spaces. Fixed Point Theory Appl 2014, 185 (2014). https://doi.org/10.1186/1687-1812-2014-185
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DOI: https://doi.org/10.1186/1687-1812-2014-185