Abstract
In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results.
Keywords:
metric space; presymmetric w-distance; basic p-contraction of Suzuki-type; p-contractive self map; fixed point MSC:
54H25; 54E50; 47H10
1. Introduction
It has been widely recognized for many years that the so-called w-distances, introduced and discussed by Kada et al. in [], provide an impactful and very accurate kind of generalized metric. This structure allows us to refine, enhance, and extend classical and important results as Caristi’s fixed point theorem and its “equivalent” Ekeland’s Variational Principle, Nadler’s fixed point theorem, and many others (see, e.g., [,,,,,,,,]), as well as to characterize complete metric spaces and complete fuzzy metric spaces by means of fixed point results [,]. The recent monograph by Rakočević [] and the references therein provide a valuable and updated source to the study of w-distances and their applications to the fixed point theory.
In an article that is already a classic [], Suzuki presented an elegant generalization of Banach’s contraction principle that he used to characterize complete metric spaces. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts (see, e.g., [,,,,,,,,,,,,]), we here show by means of an easy example that its full extension to the framework of w-distances presents suggestive difficulties that we think deserve attention.
Motivated by this scenario, we introduce and analyze the concept of presymmetric w-distance on metric spaces. In particular, we give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between this theorem and the w-distance full generalization of the Banach contraction principle, obtained by Suzuki and Takahashi in [] (Theorem 1) and which they used to characterize complete metric spaces (see [] (Theorem 4)). In this context, we introduce and examine an alternative notion of basic contraction of Suzuki-type, by attaining a new fixed point theorem which we compare with our main result through some examples. The last part of the paper is devoted to obtaining necessary and sufficient conditions for a metric space to be complete, which is made by combining our fixed point results with both Suzuki’s characterization and Suzuki–Takahashi’s characterization.
At this point, we wish to remark the following: There are many publications, mainly about the fixed point theory, that deal (separately) with w-distances and contractions of Suzuki-type. For this reason, and in order not to saturate the amount of bibliographical references cited in this paper, we have stuck to those given in the first two paragraphs of this section because we think they provide a sufficient background to help the reader.
2. Preliminaries
As far as possible, our notation and terminology will be standard. In what follows, we design by the set of natural numbers.
Next, we establish the following special case of Suzuki’s theorem [] (Theorem 2), which will be sufficient for our targets here.
Theorem 1
([]). Let be a complete metric space and T be a self map of X. If there exists a constant such that, for any , the next contraction condition holds:
then, T has a unique fixed point in X.
In line with [], a self map T of a metric space satisfying the contraction condition (1) will be called a basic contraction of Suzuki-type (on ).
If is a sequence in a metric space that converges to an , we simply will write or .
Remind ([]) that a w-distance on a metric space is a function that verifies the following conditions:
- (w1)
- for all ;
- (w2)
- for each the function is lower semicontinuous;
- (w3)
- for each there exists such that and imply
Given a w-distance p on a metric space , we shall denote by the function defined on as for all . Notice that clearly satisfies conditions (w1) and (w3) above, but not condition (w2) in general, as the following example shows.
Example 1.
Let and d be restriction of the usual metric to . It is routine to check that the function defined as and otherwise, is a w-distance on However, we obtain and for all so is not lower semicontinuous. Therefore, condition (w2) does not hold, and, thus, is not a w-distance on
According to [] (p. 3118), we say that a w-distance p on a metric space is symmetric if for all . Obviously, whenever p is symmetric.
Several interesting examples of w-distances on metric spaces are given, among others, in [,,]. In fact, every metric d on a set X is a symmetric w-distance on the metric space . Another very interesting example of a symmetric w-distance can be found in [] (Example 7) and in [] (Lemma 2).
We conclude this section by recalling a couple of representative instances of w-distances, which will be useful later on.
Example 2
(compare [] (Example 4)). Let X be a (non-empty) subset of . Then, the function defined as for all , is a w-distance for the metric space , where by d we denote the restriction of the usual metric to X.
Example 3
(compare [] (Example 3)). Let X be a (non-empty) subset of and let be constants such that . Then, the function defined as for all , is a w-distance for the metric space , where by d we denote the restriction of the usual metric to X. Clearly, w is symmetric whenever .
3. Presymmetric -Distances and a Generalization of Suzuki’s Theorem
We begin by generalizing the notion of a basic contraction of Suzuki-type to the framework of w-distances.
Definition 1.
Let p be a w-distance on a metric space . We say that a self map T of X is a basic p-contraction of Suzuki-type (on ) if there exists a constant such that, for any , the next contraction condition holds:
In light of Theorem 1, the following question naturally arises:
Let p be a w-distance on a complete metric space and T be a basic p-contraction of Suzuki-type (on ). Under these conditions, has T a fixed point in X?
The next example shows that, unfortunately, this question has a negative answer, in general.
Example 4.
Let and let d be the restriction of the usual metric to Consider the self map T of X defined as and for all
Obviously, T has no fixed points. We shall show that, nevertheless, it is a basic p-contraction of Suzuki-type, where and p is the w-distance on given in Example 2, i.e., for all .
Indeed,
- If we obtain
- If and we obtain
- If , we obtain
With the aim of lessening the difficulties evidenced by Example 2, we propose the following notion.
Definition 2.
A w-distance p on a metric space is called presymmetric provided that it fulfills the next property:
Whenever is a sequence in X such that and for some then there is a subsequence of satisfying for all
Proposition 1.
Every symmetric w-distance is presymmetric.
Proof.
Let p be a symmetric w-distance on a metric space If is a sequence in X such that and for some , there is a subsequence of such that for all By the symmetry of p, we conclude that for all □
Example 5.
Let p be a w-distance on a metric space for which there is a constant such that for all . Then, it is clear that p is a presymmetric w-distance on .
Example 6.
Let X be a subset of be such that , let d be the restriction of the usual metric to X, and let p be the w-distance on given by for all , where a and b are real constants such that and (compare Examples 2 and 3).
- If , the w-distance p is not presymmetric: Indeed, take the sequence Then, and for all so Since , there exists an such that for all Therefore, for all
- If the w-distance p is symmetric, and, hence, it is presymmetric by Proposition 1.
- If , the w-distance p is presymmetric: Indeed, suppose that is a sequence in X fulfilling and for some Then, and, hence, for allNote that, in this case, p is not symmetric.
Notice that in both Proposition 1 and Examples 5 and 6, the convergence of the sequence to x is not essential. We now give an example, which may be compared with Examples 3 and 6, where such convergence plays a decisive role.
Example 7.
Let , Y be a non-empty subset of , and d be the discrete metric on Define as for all with constants such that . Then, p is a non-symmetric presymmetric w-distance on
We first show that p satisfies conditions (w1), (w2), and (w3):
- (w1)
- It is trivially satisfied.
- (w2)
- Let and be a sequence in X such that Since d is the discrete metric on , eventually. Therefore, , eventually.
- (w3)
- Given , put Let and Then, and ; therefore, in particular, and , which implies that .
We have shown that p is a w-distance on . Clearly, it is not symmetric.
Now let be a sequence in X such that and for some . Since d is the discrete metric on X we infer that , eventually, which obviously implies that p is presymmetric.
Next we state and show the main result of this paper. In the first part of its proof, we will adapt the methods customary applied when both contractions of Suzuki-type and w-distances are involved.
Theorem 2.
Let p be a presymmetric w-distance on a complete metric space and let T be a basic p-contraction of Suzuki-type. Then, T has a unique fixed point . Furthermore,
Proof.
By assumption, there exists a constant for which the contraction condition (2) holds.
Fix Put for all By standard arguments, we first show that is a Cauchy sequence in
As , it follows from (2) that so, for all Therefore, by the triangle inequality (w1),
for all and
Given let for which condition (w3) holds. Take an such that
Then, it follows from (3) that and whenever which implies, by (w3), that whenever . Since is arbitrary, we conclude that is a Cauchy sequence in the complete metric space
Let be such that
For the given , , and above, choose any with By (w2), we can find such that
Next we prove that is a fixed point of
To this end, suppose that there is an verifying that for all . Then, we obtain
for all Thus, for all As we have reached a contradiction because, by assumption, p is presymmetric.
Hence, there is a subsequence of such that for all By condition (2), we obtain
for all Since we deduce that
Again, given let for which condition (w3) holds. Since there is such that for all
Choose be such that and Therefore, and, by condition (w3), we obtain . Since is arbitrary, we conclude that
Now we check that Indeed, we have , so, by condition (2), , which implies that
Finally, we see that is the unique fixed point of Suppose that satisfies As , we obtain From condition (2) it follows that Hence, Thus, by condition (w3). □
Corollary 1.
Let p be a symmetric w-distance on a complete metric space and let T be a basic p-contraction of Suzuki-type. Then, T has a unique fixed point . Furthermore,
The following example shows that Corollary 1, and, hence, Theorem 2, is a real generalization of Theorem 1.
Example 8.
Let and let d be the restriction of the usual metric to Consider the self map T of X defined as and for all
We first show that we cannot apply Theorem 1 to this case. Indeed, for and , we obtain , but .
Next we show that we can apply Corollary 1 and, thus, Theorem 2 for the symmetric w-distance on given by for all (see Example 6). Indeed,
- If , we obtain
- If , we obtain
- If and , we obtain
4. On the Relationship between -Contractive Self Maps, Basic -Contractions of Suzuki-Type, and the Corresponding Fixed Point Theorems
In accordance with Suzuki and Takahashi [], given a w-distance p on a metric space a self map T of X is called p-contractive (or weakly contractive) if there is a constant such that for all Then, they proved the following w-distance generalization of Banach’s contraction principle.
Theorem 3
([]). Let p be a w-distance on a complete metric space Then, each p-contractive self map T of X has a unique fixed point Furthermore,
It is clear that the Banach contraction principle is a direct consequence of both Theorems 1 and 3. In contrast to this fact, we have that Theorem 3 is not a direct consequence of Theorem 2 as the following easy example shows.
Example 9.
Let d be the restriction of the usual metric to p be the w-distance on given in Example 2, and T be the self map of X defined as for all Since for each we deduce that T is p-contractive. Hence, all conditions of Theorem 3 are satisfied. However, we cannot apply Theorem 2 because p is not presymmetric.
Remark 1.
Evidently, every p-contractive self map is a basic p-contraction of Suzuki-type. Example 4 furnishes an instance of a basic p-contraction of Suzuki-type T (on a complete metric space ) without fixed point, so, by Theorem 3, it is not p-contractive. In fact, we have for all . In addition, and as an immediate consequence of Theorem 2, we conclude that the involved w-distance p is not presymmetric.
Remark 2.
In [] (Example 1), Suzuki presented an instance of a complete metric space and a self map T of X that is a basic p-contraction of Suzuki-type but not p-contractive, for .
Despite Example 9, in Remark 3 below we shall see that it is still possible to deduce Theorem 3 from Theorem 2 by applying the following consequence of a result from Shioji et al. [] (Theorem 1).
Theorem 4
([]). Let p be a w-distance on a metric space and let T be a p-contractive self map of Then, there exists a symmetric w-distance q on such that T is q-contractive.
Remark 3.
Let p be a w-distance on a complete metric space and T be a p-contractive self map of X (with contraction constant ). By Theorem 4, there exists a symmetric w-distance q on such that T is q-contractive. Therefore, T is a basic q-contraction of Suzuki-type, so, by Corollary 1, it has a unique fixed point and We also have because .
Shioji et al. also explored in [] a class of contractions which we here define as follows: Given a w-distance p on a metric space , a self map T of X is -contractive provided that there is a constant such that for all Then, they proved the following.
Theorem 5
([]). Let p be a w-distance on a metric space and let T be a p-contractive self map of Then, there exists a w-distance q on such that T is -contractive.
From Theorem 5, it follows that the relevant Theorem 3 is a consequence of the next result. We emphasize that, to our surprise, we have not found that result in the literature. In any case, we formulate and prove it for the sake of completeness.
Theorem 6.
Let p be a w-distance on a complete metric space Then, each -contractive self map T of X has a unique fixed point Furthermore,
Proof.
Let T be a -contractive self map of Then, there exists a constant such that
for all
Now, fix and put for all
By the contraction condition (5), we obtain
Similarly,
so,
Continuing this process, we obtain
if n is odd, and if n is even.
Hence,
for all Hence, exactly as in the proof of Theorem 2, we deduce the existence of a point such that and
We shall prove that is the unique fixed point of and also that
From condition (5) we deduce that and, thus, by the triangle inequality (w1).
Since we obtain So, by condition (w3), for all i.e.,
Consequently,
which implies that Therefore, by the triangle inequality (w1), and , so, , by condition (w3).
Finally, suppose that satisfies Then,
Therefore, Since we obtain by condition (w3). □
In [] (Proposition 1), it was presented an example of a w-distance p on a complete metric space and a -contractive self map of X that has a unique fixed point, but that is not a q-contractive self map for any w-distance q on . Hence, this example shows that Theorem 6 is a real generalization of Theorem 3.
Since Theorem 2 can be interpreted as an extension of Theorem 3, it seems natural to ask whether it is possible to obtain an analogous extension of Theorem 6. We shall prove that this question has an affirmative answer.
Definition 3.
Let p be a w-distance on a metric space . We say that a self map T of X is a basic -contraction of Suzuki-type (on ) if there exists a constant such that, for any , the next contraction condition holds:
Theorem 7.
Let p be a presymmetric w-distance on a complete metric space and let T be a basic -contraction of Suzuki-type. Then, T has a unique fixed point Furthermore,
Proof.
By assumption, there exists a constant for which the contraction condition (6) holds.
Fix and put for all
Since we have Hence, so, by condition (6),
Again, condition (6) implies that Therefore,
Following this process, we deduce that for all
Then, and as in the proof of Theorem 2, there exists such that and
By the presymmetry of p, there exists a subsequence of such that for all This implies that
Since we can repeat the argument given in the proof of Theorem 2 to deduce the existence of a subsequence of satisfying for all . Hence, condition (6) implies that
for all Since , we deduce that Taking into account that we obtain by condition (w3). Moreover, by condition (6).
Finally, suppose that satisfies Once again, condition (6) implies that
so, By condition (w3), we conclude that □
We finish this section by exemplifying that Theorems 2 and 7 are of a different value from each other.
Example 10.
Let d be the discrete metric on and be defined as for all
Then, p is a presymmetric w-distance on as Example 7 shows.
Now let T be the self map of X defined as if and if
We prove that T is p-contractive with contraction constant and, hence, it is a basic p-contraction of Suzuki-type.
- If we obtain
- If and we obtain
- If and , we obtain
- If we obtain
Therefore, all conditions of Theorem 2 (and of Theorem 3) are satisfied.
However, we have but Thus, T is not a basic -contraction of Suzuki-type, so, we cannot apply Theorem 7 to these p and T.
Example 11.
Let let d be the discrete metric on be a constant such that and be defined as and for all for all for all and for all
It is routine to check that p is a presymmetric w-distance on the complete metric space (for instance, (w2) follows immediately from the fact that d is the discrete metric, and to verify (w3), one must choose for any
Now let T be self map of X defined as for all and
We first show that T is a basic p-contraction of Suzuki-type (with contraction constant and, unlike Example 10, it is not p-contractive.
Indeed,
- If we obtain
- If and we obtain
- If we obtain
- If and we obtain
- If and we obtain (recall that
We conclude that T is a basic p-contraction of Suzuki-type and, hence, all conditions of Theorem 2 are satisfied.
However, T is not p-contractive because
Finally, note that but which implies that T is not a basic -contraction of Suzuki-type, so we cannot apply Theorem 6 to these p and
Example 12.
Denote by the set of all finite sequences whose terms are natural numbers. Thus, if , we put where for all The number is called the length of x as is denoted by
Let where by ω we denote the infinite sequence that we represent as follows:
Set is even
Now, let d be the discrete metric on X and be the function defined as if , if and and otherwise.
It is straightforward to check that p is a presymmetric w-distance on the complete metric space
Define a self map T of X as if and if where is the unique element of obtained by repeating once the terms of i.e., if then . Thus,
It is routine to check that for all (in particular, if and we obtain
Hence, T is -contractive, so it is a basic -contraction of Suzuki-type. Thus, all conditions of Theorem 7 (and of Theorem 6) are fulfilled.
However, for and we obtain but This shows that T is not a basic p-contraction of Suzuki-type, so, we cannot apply Theorem 2 to these p and T.
5. Characterizing Complete Metric Spaces
Suzuki and Takahashi showed in [] (Theorem 4) that Theorem 3 characterizes complete metric spaces. More precisely, they proved the following result.
Theorem 8
([]). For a metric space the following statements are equivalent:
- (1)
- is complete.
- (2)
- For any w-distance p on , every p-contractive self map of X has a (unique) fixed point.
Observe that by combining Theorems 5, 6, and 8, we obtain the following variant of Theorem 8.
Theorem 9.
For a metric space the following statements are equivalent:
- (1)
- is complete.
- (2)
- For any w-distance p on , every -contractive self map of X has a (unique) fixed point.
On the other hand, Theorem 1 also characterizes metric completeness (compare [] (Corollary 1)). More precisely, we have the following result.
Theorem 10
([]). For a metric space the following statements are equivalent:
- (1)
- is complete.
- (2)
- Every basic contraction of Suzuki-type has a (unique) fixed point.
By using Theorem 2, Corollary 1 and the preceding characterization, we deduce the next result.
Theorem 11.
For a metric space the following statements are equivalent:
- (1)
- is complete.
- (2)
- For any presymmetric w-distance p on , every basic p-contraction of Suzuki-type has a (unique) fixed point.
- (3)
- For any presymmetric w-distance p on , every basic -contraction of Suzuki-type has a (unique) fixed point.
- (4)
- For any symmetric w-distance p on , every basic p-contraction of Suzuki-type has a (unique) fixed point.
Proof.
Apply Theorem 2.
Apply Theorem 7.
and are obvious.
Let T be a basic contraction of Suzuki-type on Then, it is a basic p-contraction of Suzuki-type for Hence, by assumption, T has a unique fixed point. Therefore, is complete by Theorem 10. □
Remark 4.
, in Theorem 11, can be also deduced from Theorem 8. Indeed, suppose that is not complete. Thus, there exists a Cauchy sequence in that does not converge. Then, following the proof of [] (Theorem 4), we can construct a symmetric w-distance p on and a p-contractive self map T of X that has no fixed point. Since T is a basic p-contraction of Suzuki-type, we have reached a contradiction.
6. Conclusions
We have discussed the question of extending the classical Suzuki fixed point theorem to the framework of w-distances on metric spaces. We have presented a simple example showing that a possible natural extension does not hold. Then, we introduced the notion of a presymmetric w-distance. We gave some examples and proved a real generalization of Suzuki’s theorem for this class of of w-distances. We have examined the relationship between the so-called p-contractive self maps of Suzuki and Takahashi, our basic p-contractions of Suzuki-type, and the corresponding fixed point results derived from them. We continued this study by introducing the notion of a basic -contraction of Suzuki-type, which allows us to obtain a new fixed point theorem which we compared with the obtained via basic p-contractions of Suzuki-type through some examples. Finally, a characterization of the metric completeness in terms of our fixed point results was given.
Author Contributions
S.R. and P.T. contributed equally in writing this article. All authors read and approved the final manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data is contained within the article.
Acknowledgments
The authors thank the reviewers for several useful comments and suggestions.
Conflicts of Interest
The authors declare no conflicts of interest.
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