Advances in Pure Mathematics
Vol.09 No.05(2019), Article ID:92461,8 pages
10.4236/apm.2019.95020
Relation Contractive Selfmaps Involving Cauchy Sequences
Maria Luigia Diviccaro1, Salvatore Sessa1,2
1Dipartimento di Architettura, Università degli Studi di Napoli Federico II, Napoli, Italy
2Dipartimento di Architettura, Centro Interdipartimentale di Ricerca “A. Calza Bini”, Università degli Studi di Napoli Federico II, Napoli, Italy
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 4, 2019; Accepted: May 14, 2019; Published: May 17, 2019
ABSTRACT
We obtain two generalizations of a known theorem of A. Alam and M. Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702) showing that some standard proofs can be obtained involving only Cauchy sequences of the successive approximations. Suitable examples prove the effective generalization of our results in metric spaces not necessarily complete.
Keywords:
Cauchy Sequence, d-Self-Closed Relation, Relation Contraction, Relation Preserving
1. Introduction
Fixed point theorems involving contraction conditions under preserving relations are known in literature (cf. [1] - [11] ). These theorems involve usual sequences of successive approximations in complete metric spaces as assumed in the paper of Alam and Imdad [12] . Historically speaking, it is well known that any real number is the sup (resp., inf) of Cauchy increasing (resp., decreasing) sequence of rational numbers. Hence the study of Cauchy sequences is interesting per se; indeed in other contexts such concept is introduced, for instance, in metric spaces and related recent generalizations known like b-metric spaces [13] [14] [15] , partial metric spaces [8] , Yoneda spaces [9] [16] and modular metric spaces [17] (we also suggest the good survey [18] as further deepening). Our aim is to prove that extensive theorems can be obtained by considering only Cauchy sequences in metric spaces not necessarily complete. Here fixed point theorems involving Cauchy sequences of Jungck type [6] [19] are not considered in order to make compact the results involving one only contraction.
2. Preliminaries
We start with some known definitions [1] .
Definition 1. Let X be a nonempty set and be a binary relation (eventually partial) defined on X. A sequence of X is called -preserving if for every .
From now on we consider such binary relations and we write simply .
Definition 2. Let be a nonempty metric space and is called d-self-closed if whenever is -preserving and converging to a point , then there exists a subsequence of such that either or for every .
Definition 3. (cf. [20] ). Let X be a nonempty set and . For , a -path of length k (where ) in X from x to y is a finite sequence , , of points of X satisfying the following conditions:
1) and ;
2) for each .
Notice that a path of length k involves elements of X, although they are not necessarily distinct. In [19] , generalizing a famous theorem of [7] , the following theorem was established.
Theorem 1. Let be a partially ordered set and there exists a metric . Let T be a selfmap of X such that
1) T is monotone non-decreasing;
2) there exists a point such that ;
3) if is a non-decreasing Cauchy sequence in X, then converges to and for every n;
4) there exists such that for all with ;
then T has a fixed point such that .
In [1] , generalizing many theorems contained in the references therein cited, the following theorem was established (not including the case T continuous which we consider later).
Theorem 2. Let be a complete metric space, and T be a selfmap of X such that
1) there exists in X a point such that ;
2) is T-closed, that is implies ;
3) is d-self-closed;
4) there exists such that for all pair .
Then T has a fixed point. Moreover, if there exists a -path from x to y for all , then this fixed point is unique.
3. Unification of Theorems 1 and 2
Now we unify Theorem 1 and 2 with the following:
Theorem 3. Let be a metric space, and T be a selfmap of X. Suppose that
1) there exists in X a point such that ;
2) is T-closed;
3) for any sequence -preserving, , which is Cauchy and converging to a point , there exists a subsequence of such that either or for every ;
4) there exists such that for all .
Then T has a fixed point z in X and there exists a sequence such that either or for every . Moreover,
5) if there exists a -path from x to y for all , then this fixed point is unique.
Proof. Let otherwise the thesis is trivial. Put and for every , so we have
Because of properties 1) and 2), the sequence is -preserving. In virtue of property 4), we have that
and hence is a Cauchy sequence. By property 3), converges to a point z in X and there exists a subsequence of such that either or for all . This implies that
by property 4) and passing , we have and therefore z is a fixed point of T. By setting for every , we have if or if for every because of property 2). If property 5) holds, then it is a routine to prove that the fixed point is unique (cf., e.g. [2] ).
Remark 1. Theorem 1 is generalized from Theorem 3 by defining the non-decreasing order “ ” as relation . Theorem 2 is generalized from Theorem 3 because the condition 3) of Theorem 2, i.e. Definition 2, is restricted only to Cauchy sequences and moreover the hypothesis that X is complete does not appear in Theorem 3 as well.
In the following example Theorem 3, inspired to Example 1 of [19] , holds while Theorem 2 is not applicable.
Example 1. Let endowed with the Euclidean metric d. Then is a non-complete metric space and define as if and , , and if . Let and define as if and if . It is immediate to verify that property 1) holds since , moreover properties 2) and 3) hold trivially. Additionally we have that
thus property 4) holds. Also property 5) holds because there exists at least an -path of length 2, i.e. and , joining two any points of X. Indeed is the unique fixed point of T but Theorem 2 is not applicable because X is not complete.
Remark 2. If 5) does not hold, Theorem 3 does not guarantee the uniqueness of the fixed point as proved in the following example:
Example 2. Let be endowed with metric for all . Define as follows: if for all such that or . Then X is a metric space with the partially defined binary relation . Define as if and if . Then property 1) holds because if . The property 2) holds because T is strictly increasing in both intervals and . The property 3) holds because it is enough to take strictly increasing sequences in . Property 4) holds also for . Property 5) fails because if and , for any finite -path of length k, , there exists at least certainly some such that and , hence . Note that T has two fixed points which are 0 and 1.
Remark 3. Theorem 2 is not applicable to Example 2 because X is not complete.
4. Relation Contractions and Continuous Selfmapss
In [19] the following theorem appears:
Theorem 4. Let be a nonempty partially ordered set and there exists a metric . Let T be a selfmap of X such that
1) there exists a point such that ;
2) T is continuous and non-decreasing;
3) if is a non-decreasing Cauchy sequence in X, then converges to a point ;
4) there exists such that for all with . Then T has a fixed point.
In the case T is assumed continuous, Theorem 2 becomes [1] :
Theorem 5. Let be a complete metric space, and T be a selfmap of X such that
1) there exists at least a point ;
2) is T-closed;
3) T is continuous;
4) there exists such that for all .
Then T has a fixed point.
Now we unify Theorems 4 and 5 with the following:
Theorem 6. Let be a metric space, and T be a selfmap of X. Suppose that
1) there exists in X a point such that ;
2a) is T-closed;
2b) T continuous;
3) if xn is a -preserving Cauchy sequence in X, then Txn converges to a point ;
4) there exists such that for all .
Then T has a fixed point in X.
Proof. As in the proof of Theorem 3, let , and for every . Because of properties (1) and (2.1), the sequence is -preserving. In virtue of property 4), we have that
and hence is a Cauchy sequence. By property 3), converges to a point z and therefore converges to because of property 2.2), thus because of the uniqueness of the limit.
Remark 4. Theorem 4 is generalized from Theorem 6 by defining the non-decreasing order “ ” as relation . Theorem 5 is generalized from Theorem 6 because if is a -preserving Cauchy sequence in X, the completeness of X and the continuity of T assure that converges to a point of X, i.e. the property 3) of Theorem 6 holds. The following example shows Theorem 5 is not applicable but Theorem 6 holds.
Example 3. Let with the metric for all . Define as follows: if for all . Define as for any . Obviously T is continuous in X and is T-closed. If is a -preserving (that is monotone non-decreasing) Cauchy sequence in X, then is a monotone non-decreasing bounded sequence and hence converging to a point , thus properties 1), 2), 3) hold. Too property 4) holds because it is enough to assume . is a metric space not complete, so Theorem 5 is not applicable while all the assumptions of Theorem 6 (or Theorem 4) are satisfied and 1 is the (unique) fixed point of T.
Remark 5. The uniqueness of the fixed point can be guaranteed from several additional properties of the relation (cf. [1] [3] [4] [7] [8] [10] [19] ) which we do not examine here. The following example, borrowed from [1] , shows that the continuity of T in Theorem 6 is necessary.
Example 4. Consider equipped with usual metric for all . is a complete metric space. Define as
and as , if , if . is T-closed but T is not continuous. Consider any -preserving sequence , then for all . Hence or for all . If is a -preserving Cauchy sequence in X, then we have definitively (resp., ), i.e. there exists some suitable integer m such that (resp., ) for every integer , which implies that (resp., ) for all . Further
where . Thus all the hypothesis of Theorem 6 hold except property 2.2 but T has no fixed points.
5. Conclusions
We have generalized fixed point theorems for theoretic-relation contractions about continuous selfmaps of metric spaces. Suitable examples prove the effective generalization of our results in metric spaces not necessarily complete.
Future studies shall be necessary for establishing extensions of the results here presented, essentially common fixed point theorems involving Cauchy sequences of Jungck type [6] [19] under a generalized condition of weak commutativity of two selfmaps such as, the weak compatibility (cf., e.g. [21] ).
Acknowledgements
Funds of the “Dipartimento di Architettura” (Università degli Studi di Napoli Federico II, Italy) of the second author cover this research.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Cite this paper
Diviccaro, M.L. and Sessa, S. (2019) Relation Contractive Selfmaps Involving Cauchy Sequences. Advances in Pure Mathematics, 9, 421-428. https://doi.org/10.4236/apm.2019.95020
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