Some New Observations on Generalized Contractive Mappings and Related Results in <i>b</i>-Metric-Like Spaces

Došenović, Tatjana; De La Sen, Manuel; Paunović, Ljiljana; Rakić, Dušan; Radenović, Stojan

doi:https://doi.org/10.1155/2021/6634822

Journal of Mathematics

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Research Article | Open Access

Volume 2021 | Article ID 6634822 | https://doi.org/10.1155/2021/6634822

Some New Observations on Generalized Contractive Mappings and Related Results in b-Metric-Like Spaces

Tatjana Došenović,¹Manuel De La Sen,²Ljiljana Paunović,³Dušan Rakić,¹and Stojan Radenović⁴

Academic Editor: Jen-Chih Yao

Received10 Dec 2020

Revised11 Feb 2021

Accepted02 Mar 2021

Published20 Mar 2021

Abstract

In this paper, we consider, discuss, complement, improve, generalize, and enrich some fixed point results obtained for contractive conditions in ordered b-metric-like spaces. By using our new approach for the proof that one Picard’s sequence is Cauchy in the context of b-metric-like spaces, we get much shorter proofs than the ones mentioned in the recent papers. Also, by the use of our method, we complement and enrich some common fixed point results for contraction mappings. Our approach in this paper generalizes and modifies several comparable results in the existing literature.

1. Introduction

Fixed point theory is one of the most important areas of nonlinear analysis. At the beginning of the development, this part of analysis was related to the use of successive approximation in order to prove the existence and uniqueness of the solution of differential and integral equations. Later on, it is applied in various fields such as economics, physics, chemistry, differential and integral equations, partial differential equations, numerical analysis, and many others. Banach’s contraction principle in metric spaces [1] is one of the most important results in fixed point theory and nonlinear analysis in general. In 1922, when Stefan Banach formulated the concept of contraction and proved the famous theorem, scientists around the world started publishing new results that are related either to the generalization of the contractive mapping such as Kannan, Chatterjea, Hardy–Rogers, Ćirić, and many others or by generalizing space itself. By changing some axioms of ordinary metric space, new classes of so-called generalized metric spaces were obtained such as partial metric space, metric-like space, metric space, metric-like space, and others. For more details about fixed point theory in metric as well as generalized metric spaces, we encourage readers to see [2–9].

In each of them, Banach’s well-known theorem is true in -metric and metric-like spaces regardless of the magnitude of the coefficient in the triangle relation to each.

In [10], Matthews introduced the notion of a partial metric space where nonzero self-distance is considered, which has found great application in computer science. The second important generalization of metric spaces is so-called metric spaces. This concept was introduced by Bakhtin [11] and Czerwik [12] where the third axiom of metric spaces, referring to triangular inequality, weakened.

Furthermore, Amini Harandi [13] introduced the notion of metric-like space, as a generalization of a partial metric space, where all of the axioms of a metric is satisfied except that self-distance may be positive.

In [14], the concept of metric-like space which generalizes the notions of partial metric space, metric-like space, and b-metric space is introduced.

Relations of the metric spaces and mentioned generalizations are illustrated as follows [15, 16]: .

2. Preliminaries

Firstly, we present some definitions and basic notions of metric-like, b-metric-like spaces, admissible mappings, contractive mappings of type-I and type-II, altering distance functions, contraction mappings, and ordered metric-like, that is, ordered b-metric-like space.

Definition 1. (see [13]). Let be a nonempty set. A mapping is said to be metric-like if the following conditions hold for all :(i) implies (ii)(iii)In this case, the pair is called a metric-like space.

Definition 2. (see [14]). Let be a nonempty set and be a given real number. A metric-like space on a nonempty set is a function if the following conditions hold for all : (i) implying (ii)(iii)In this case, the pair is called a metric-like space with the coefficient constant .

For some examples, see [17–19] and [13, 20–25].

It is clear that each metric-like space is metric-like space, while the converse is not true. For more such examples and details see [9, 13, 14, 20, 24–27]. Also, for various metrics, but in the context of complex domain, see [28, 29].

Definition 3. A triple is said to be ordered b-metric-like space if is a partially ordered set and is a metric-like on .

Now, we give the definition of convergence of the sequences in b-metric-like space.

Definition 4. (see [14]). Let be a sequence in a metric-like space with the coefficient .(i)The sequence is said to be convergent to if (ii)The sequence is said to be Cauchy in if exists and is finite(iii)One says that metric-like space is a complete if for every Cauchy sequence in it there exists an , such that

Properties such as convergence, completeness, and Cauchyness are introduced in the same way for metric and b-metric spaces. And, in these two types of spaces, the limits of the sequence if exists, is unique, as well as the convergent sequence is a Cauchy. Otherwise, for the other 4 types of space (partial metric, partial b-metric, metric-like, and b-metric like), this is not the case.

Remark 1. In a metric-like space, the limit of a sequence need not be unique and a convergent sequence need not be a Cauchy sequence (see Example 7 in [20]). However, if the sequence is Cauchy such that in the complete metric-like space with coefficient , then the limit of such sequence is unique. Indeed, in such a case, if as , we get that . Now, if and , where , we obtain thatBy , it follows that . A contradiction.

Definition 5. Let be a nonempty set and suppose and are mappings. is called admissible if for all ,Also, we say that is admissible (or admissible) if for ,

The next definition and the corresponding proposition are important in the context of fixed point theory.

Definition 6. (see [30]). A function is called an altering distance function if it satisfies the following properties:(i) is continuous and nondecreasing(ii) iff

Definition 7. (see [31]). The self-mappings are weakly compatible if , whenever .

Proposition 1. (see [31]). Let and be weakly compatible self-maps of a nonempty set . If they have a unique point of coincidence , then is the unique common fixed point of and .

In this paper, we shall use the following result for the proof that some Picard’s sequence is Cauchy. The proof is completely identical with the corresponding in [32] (see also [33–35]).

Lemma 1. Let be a sequence in a metric-like space with the coefficient such thatfor some , and each .Then, is a Cauchy sequence in such that .

Remark 2. It is worth to notice that the previous lemma holds in the context of metric-like spaces for each . For more details, see [27].

Otherwise, many authors for the proof that some sequence in metric-like space is Cauchy use the next lemma.

Lemma 2. (see [26]). Let be a metric-like space with the coefficient and assume that and as . Then, we have

In the case that the coefficient , the given metric like space becomes the metric-like space . If is a given sequence in the metric-like space , then we have the following very useful result.

Lemma 3. (see [23, 36]). Let be a metric-like space and let be a sequence in it such that . If is not a Cauchy sequence in , then there exist and two sequences and of positive integers such that and the following four sequences tend to when :

3. Main Results

In [37], the authors introduced two new types of contractive mappings, namely, contractive mappings of type-I and of type-II in ordered metric-like spaces.

Definition 8. (see [37], Definition 1 and Definition 4). Let be a partially ordered metric-like space with the coefficient . A mapping is said to be contractive mapping of type-I (resp., of type II), if there exist two altering distance functions and such thatfor all comparable , whererespectively,

In [37], the authors proved the following results for contractive mapping of type-I (resp., of type II).

Theorem 1. (see [37], Theorem 2.2). Let be a -complete ordered metric-like space with the coefficient . Let be a contractive mapping of type-I. Assume that the following assertions hold:(1) is admissible and admissible (or -admissible)(2)There exists such that and (3) is continuous, non-decreasing with respect to , and if , then

Then, has a fixed point.

Theorem 2. (see [37], Theorem 2.3). Let be a -complete ordered metric-like space with the coefficient . Let be a contractive mapping of type-I. Suppose that the following conditions hold:(1) is admissible and admissible (or -admissible)(2)There exists such that and (3) is nondecreasing with respect to (4)If is a sequence in such that and for all , and , as , then and for all

Then, has a fixed point.

Theorem 3. (see [37], Theorem 2.5). Instead of the contractive mapping condition of type-I in Theorem 1, assume that contractive mapping condition of type-II is satisfied. Then, has a fixed point.

Similar to Theorem 2, the authors proved the following result.

Theorem 4. Let all the conditions of Theorem 2 are satisfied, apart from condition (8) which is replaced by (9). Then, has a fixed point.

Now, we give the shorter proofs of Theorems 1–4.

First, let . Now, by applying (5) and (6) from [37], we getwhich is according to (7) from. . It further follows that . Since , we get that , that is, the sequence is a Cauchy by to Remark 2 of Lemma 1. The proof is further the same as in [37].

If , then the given contractive conditions in all Theorems in [37] imply that the corresponding Picard’s sequence is Cauchy according to Lemma 1.

In [20], the authors introduced the so-called contraction mappings and obtained some common fixed point theorems for such contractions in the context of metric-like spaces.

First, let denote the class of functions , respectively, satisfying the following conditions:(i) is nondecreasing, continuous function, and , if and only if (ii) is lower semicontinuous and , if and only if

Definition 9. (see [20], Definition 5). Let be a metric-like space with the coefficient . Let the constant and . The nonlinear self-mappings are called contraction mappings if for all .where , and

Theorem 5. (see [20], Theorem 1). Let be a complete metric-like space with the coefficient , and be mapping satisfying the following conditions:(i)(ii)The pair is a contraction(iii)Then, and have a point of coincidence in (iv)Moreover, if and are weakly compatible, then and have a unique common fixed point in

Remark 3. Some important remarks regarding the previous definition and theorem, namely, it is worth to notice that, for the set of real numbers from (11), we haveIndeed, this follows from the known estimation:
Therefore, instead of the set in [20], we introduce the next set:

Now, we give the new formulation and the proof of Theorem 1 from [20] taking the set instead of .

Theorem 6. Let be a complete metric-like space with the coefficient , and be a self-mapping satisfy the following conditions:(i) and at least one of , is closed subset in the (ii)the pair is an contraction(iii)Then and have a unique point of coincidence in (iv)Moreover, if and are weakly compatible, then and have a unique common fixed point in

Proof.

Step 1. Uniqueness of point of coincidence for pair . First, suppose that the pair has at least one point of coincidence . If it has other point of coincidence for example , this means that there are two points from such that and . According to (11) where is given with (14), we havewhereNow, (15) becomesthat is, which is a contradiction because .

Step 2. Existence of the point of coincidence.
Let be an arbitrary point in . Since , there exists such that . By continuing this process inductively, we get two sequences and in such thatIf for some , then is a unique point of coincidence for the pair . Furthermore, let , for all . Now, we shall prove thatfor some . Indeed, according to (11) where instead of , we take , if , we havewhereFrom (20) and (21), we getIt is clear that . Putting , we obtain (19). Then, by Lemma 1, the sequence is a Cauchy sequence in the complete metric-like space such that .
Let, for example, the subset be closed. The proof if is closed is similar. Then, in the first case, there is a unique point such that converges to . Since , there exists such that . We shall show that . For this proof, we firstly haveLetting the limit in (23) as , we obtain(25) is possible only if . Hence, is a unique point of coincidence. The result further follows by Proposition 1 from [31]. This completes the proof of Theorem 6.

As corollaries of our Theorem 6, we obtain the next results:

Corollary 1. Let be a complete metric-like space with the coefficient , and be a weakly compatible mappings satisfying the following conditions:(i) and at least one of is a closed subset in the space (ii)Assume that and such that the conditionholds for all

Then, and have a unique common fixed point in .

Proof.
By (26), it follows thatfor all . The result further follows from Theorem 6.

Corollary 2. Let be a complete metric-like space with the coefficient , and be a weakly compatible mappings satisfying the following conditions:(i) and at least one of is a closed subset in (ii)Assume that and such that the conditionholds for all

Then, and have a unique common fixed point in .

Proof.
Condition (28) implies thatthat is,for all . Hence, the result follows by the previous corollary.

Corollary 3. Let be a complete metric-like space with the coefficient , and be a weakly compatible mappings satisfying the following conditions:(i) and at least one of is a closed subset in (ii)Assume that and such that the conditionholds for all

Then, and have a unique common fixed point in .

Proof. It is clear that condition (31) implies (30). The proof then follows from Corollary 2.

In particular, by taking and in Theorem 6, we have the following immediate result.

Corollary 4. Let be a complete metric-like space with the coefficient , and be a given self-mapping that satisfiesfor all , where is a closed subset in .

Then, has a unique fixed point.

Putting and in Theorem 6, we can get the next result.

Corollary 5. Let be a complete metric-like space with the coefficient , and be a weakly compatible mappings satisfying the following conditions:(i) and at least one of is a closed subset in the space (ii)Assume that and such that the conditionholds for all

Then, and have a unique common fixed point in .

Proof. It is an immediate consequence of Theorem 6.

Remark 4. By the known relation for all real numbers , authors in [20] obtained the next result as an immediate consequence of theirs (Theorem 1).

Corollary 6. Let be a complete metric-like space with the coefficient . If be self-mappings, there exist and real constants with such thatfor all .

Then, and have a unique common fixed point, provided that the pair is a weakly compatible.

Remark 5. Lemmas 1 (for ) and 3 (for ) give the answer to a common question: does the given contractive condition imply that Picards’s sequence initiated by any point is Cauchy?

Now, we have the following interesting question: does the weakly compatible self-mappings which satisfy condition (34) have a unique common fixed point in

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first, third, and fourth authors are supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (nos. 451-03-68/2020-14/200134 and 174002). The second author thanks the Basque Government for its support of this work through Grant IT1207-19.

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Copyright © 2021 Tatjana Došenović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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