Completeness of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 23.1145 12.533" width="23.1145pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.017,0,0,-0.017,12.848,0)"><path id="g117-131" d="M548 311L516 323C497 274 452 240 400 240C372 240 342 260 316 278C277 305 235 330 188 330C116 330 66 268 40 199L72 187C91 236 136 270 188 270C216 270 246 250 272 232C311 205 353 180 400 180C472 180 522 242 548 311Z"/></g></svg>Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions

Ali, Basit; Abbas, Mujahid; Sen, Manuel de la

doi:https://doi.org/10.1155/2020/5183291

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Research Article | Open Access

Volume 2020 | Article ID 5183291 | https://doi.org/10.1155/2020/5183291

Completeness of Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions

Basit Ali,¹Mujahid Abbas,²and Manuel de la Sen³

Academic Editor: Xiaohua Ding

Received16 May 2020

Accepted15 Aug 2020

Published02 Sept 2020

Abstract

In this article, we present a completeness characterization of metric space via existence of fixed points of generalized multivalued quasicontractions. The purpose of this paper is twofold: (a) to establish the existence of fixed points of multivalued quasicontractions in the setup of metric spaces and (b) to establish completeness of a metric space which is a topological property in nature with existence of fixed points of generalized multivalued quasicontractions. Further, a comparison of our results with comparable results shows that the results obtained herein improve and unify the existing results in the literature applicable to the case where existing results fail.

1. Introduction and Preliminaries

In this paper, we consider an important problem in mathematics known as the “completeness problem.” Completeness problem is to know under what circumstances the underlying space is complete. Completeness of space guarantees the convergence of all Cauchy sequences. There are some other problems like fixed point problem and variational problem which are equivalent to the completeness problem and readers interested in equivalence of completeness and fixed point problem are referred to a survey article by Cobzaş [1]. Completeness problem in mathematics has correspondence with an important problem known as “the end problem” in behavioral sciences. The end problem is to determine where and when human dynamics defined as succession of positions that starts from an initial position and follows transitions ends somewhere. For details on the completeness problem and the end problem, we refer to [2] and references therein.

On the other hand, the notion of a distance between two objects of an abstract set plays a vital role in mathematics and other related disciplines such as routing theory, graph theory, matching problems, and decision-making processes. Due to its significance in solving problems in various disciples, the concept of a distance function (metric) has been generalized in many ways (see [3] and the reference therein). One of such generalizations is the notion of a bmetric introduced by Czerwik [4–6].

In Section 1, we fix the notations along with the provision of necessary tools to prove the main results in this paper. In Section 2, we solve fixed point problem in the context of metric spaces via more general multivalued quasicontractions. In Section 3, we obtain equivalence between completeness problem and fixed point problem in the context of metric spaces.

Throughout this article, let be a nonempty set and , , , and denote the sets of positive integers, nonnegative integers, reals, and nonnegative reals, respectively.

Czerwik [4–6] introduced metric spaces.

Definition 1. A bifunction is a b metric on if there exists a with such that for in , satisfies(i) if (ii)(iii)The pair is known as metric space (s) (shortly as MS (s)) with b-metric constant . Clearly, for , is a metric space, but there are metrics that are not metrics (see [4, 7, 8]).
Let be a MS and a sequence in . Then is a Cauchy sequence if for any given , there is a so that for all , or equivalently, is Cauchy iffor all . Further, is convergent if there is a in and for any , there exists so that for all , or equivalently, a sequence in converges to in ifand we write as . A subset is closed in if for every sequence in with as , implies and is bounded ifA MS is complete if every Cauchy sequence in converges. Some topological properties have been discussed in [9]. For instance, a b-metric is not continuous in both variables in general. However, if is continuous in one variable, then due to the symmetry property of the b-metric , it is continuous in the other variable as well. Moreover, a subsetin is not an open set (in general) but if is continuous in one variable, then is an open subset in . Moreover, suppose that the b-metric in this paper is continuous in one variable.
The following lemma provides an important tool to prove that a sequence that satisfies certain contractive condition is Cauchy in a MS.

Lemma 1 (see [10]). If a sequence in a MS satisfiesfor somethen it is Cauchy sequence in .

Recently, Suzuki [11] replaced condition (1) with and proved the following important lemma in the context of MSs. It is worth mentioning that the following lemma shows that the criteria to establish Cauchyness of a sequence in metric and ordinary metric spaces are the same.

Lemma 2 (see [11]). If a sequence in a MS satisfiesfor some then it is a Cauchy sequence in .

Let be a MS and () the set of nonempty, closed, bounded subsets (nonempty subsets) of . For , the mapping , defined asis called Hausdorff metric on induced by , where

The following lemma encompasses some important properties of MSs that we will use in proving the main results of this article.

Lemma 3 (see [4–6, 8]). For a MS , and , the following assertions hold: () is a MS. () For all . () For all in , . () For and , there is a so that . () For every and , there is a so that . () if and only if , where is the closure of in . () For any sequence in ,

Consider a self-mapping on , that is, and a multivalued mapping . The fixed point of or is a point in , if or , and we denote the fixed point set of or as or .

Regarding the existence of a unique fixed point of , one of the important theorems is Banach contraction principle (BCP) [12]. This principle states that if a self-mapping on (a complete metric space) satisfiesfor some , then is singleton. The mapping that satisfies (11) is called a Banach contraction. BCP has been generalized further in more than one directions. For example, Suzuki [13] presented a generalization of BCP that characterized the metric completeness. Kannan [14, 15] presented a fixed point result that is independent of BCP but characterizes the metric completeness (see [16]), whereas BCP does not characterize the completeness of underlying metric spaces (see [17, 18]). Among other generalizations of Banach contractions, Ciric [19] introduced a self-mapping that satisfiesfor some and termed it a quasicontraction. Further, they obtained fixed point results for quasicontractions in orbitally complete metric spaces.

Nadler [20] extended the BCP for multivalued mapping on as given below.

Theorem 1. Let be a complete metric space and such thatfor all and some ; then, is nonempty.

Amini-Harandi [21] generalized Theorem 1 by presenting the following result.

Theorem 2 (see [21]). Let be a complete metric space and . Iffor all and some , then is nonempty.

Kikkawa and Suzuki [22] generalized Theorem 1 using a function defined as

Theorem 3 (see [22]). Let be a complete metric space. If satisfiesfor all and for some , then is nonempty.

Jovanovic et al. [23] extended BCP in the framework of MSs.

Theorem 4. Let be a complete MS and be a self-mapping on satisfyingfor all and for some . Then, is singleton.

Dung and Hang [24] replaced the condition by in the above theorem. They obtained the following version of BCP which shows that BCP can be obtained in MSs without imposing any additional condition on .

Theorem 5. Let be a complete MS and be a self-mapping on satisfyingfor all and for some . Then, is singleton.

Note that there are fixed point theorems (see [7]) that cannot be transported from metric to bMSs as in the case of BCP.

Aydi et al. [25] obtained metric version of Theorem 2.

Theorem 6 (see [25]). Let be a complete MS and . If satisfiesfor all and for some with , then is nonempty.

Using a function defined as , Kutbi et al. [26] presented the next result in MSs.

Theorem 7 (see [26]). Let be a complete MS. If satisfiesfor all and for some with , then is nonempty.
Let be a MS, and and let be in , we use the notationswhere andwhere is the metric constant (see [27]).

Recently, Alolaiyan et al. [27] obtained the following theorem for quasicontractions which is a generalization of results in [7, 13, 21, 25, 28, 29].

Theorem 8 (see [27]). Let be a complete MS. If satisfiesfor all and for some with , where , then is nonempty.

Theorem 9. Further, in [27], by considering a class of metrics such that for any Cauchy sequence and any in , the sequence is Cauchy in . One typical example of such metric isfor , where is a metric on (see [27]). For the class of metrics, they have obtained the following result.

Theorem 10. Let be a MS such that . For with , assume that is a class of self-mappings of that satisfies the following.
(a) For ,where .

Let be the set of mappings on that satisfies (a).

(b) is denumerable and every subset is closed.

Then, statements (i)–(iii) are equivalent.(i) is complete.(ii) is nonempty for every for all where .(iii) is nonempty for every for some that satisfies .

Motivated by the work in [27], we provide the following definition.

Definition 2. Let be a MS. A mapping is a generalized multivalued Ciric–Suzuki-type (shortly type) quasicontraction if there exists with such thatfor all , with , where . If is replaced with a self-mapping , then is generalized type quasicontraction.
In this article, we provide with some new fixed point theorems for generalized multivalued type quasicontractions in MSs. It is worth noting that the existence of fixed point sets of type quasicontractions further establishes the completeness of underlying MSs. The authors in [27] obtained a completeness characterization of a specific class of MSs. We extend their result and present a completeness characterization of any MS.

2. Fixed Points of Generalized Multivalued Type Quasicontractions

The following is the first main result about the fixed points of generalized multivalued type quasicontractions of MSs.

Theorem 11. Let be a complete MS and be a generalized multivalued type quasicontraction. Then, is nonempty.

Proof. Note that for any in with , we have . As , we can choose a positive real number satisfyingIf then and . Let be any point in and . If , . So, assume that .So, by the given assumption on , we get thatAs , by in Lemma 3, we have that satisfiesFrom (29)–(30), we obtainThat is,Continuing like this, we obtain a sequence in that satisfiesfor all . Set ; then, from (33), we havewhere . As and , we haveThat is, . By Lemma 2, is a Cauchy sequence in , and hencefor some . Now, we show thatfor all . From (36), we choose so thatfor all and for all . Hence,for all . This further implies thatfor all . By applying the limit as , we obtainIf , then we geta contradiction. Consequently, (37) holds for all .
Now, assume that for all . As , . Since is nonempty for all in , forthere exists such thatHence, we obtainSincewe obtainIfthenwhich implies that either , that is , or , a contradiction. Consequently, we haveIfthen (37)–(50) imply thatAgain if , then either or , soAlso, from (37)–(53), we obtain thatFrom (45), (53), and (54), we havea contradiction. Hence, .

Remark 1. Asfor all , , we obtain Theorem 8 as a particular case of Theorem 11. Further, Theorem 8 is a generalization of Theorems 1–3, 6, and 7.

Corollary 1. Let be a complete MS and . Iffor all , such that , and for some , then is nonempty.

Corollary 2. Let be a complete MS and . Iffor all and for some , then is nonempty.

The next two corollaries are the extensions of Theorem 6 in MSs as (56) holds for .

Corollary 3. Let be a complete MS and . Iffor all , such that , and for some , then is nonempty.

Corollary 4. Let be a complete MS and . Iffor all , such that , and for some , then is nonempty.

Remark 2. If in Corollary 4, we get Theorem 2.

Example 1. Let and be defined asAs is not a metric on . For , is a metric on . Let , . Then,Define a mapping byFor all ,Note thatFurtherMoreover,Hence, for and ,Consequently, for and , the implicationholds true. For , , we getHence, all the assumptions in Theorem 11 are satisfied.
Now based on the above calculations, we present a comparison among Theorem 11 and some results from the literature.(1)If , and , then but for any , so Theorem 8 (a main result in [27]) cannot be applied in this example. Hence, Theorem 8 is a generalization of Theorem 8.(2)Now, andOn the other hand,for any . Hence, Theorem 6 is not applicable in this example. So, Theorem 11 is a proper generalization of Theorem 6 which itself is a generalization of Theorems 1 and 2 and Theorem 1 of [7].
We give another example to show the fact that Theorem 11 is applicable to certain cases where several existing results are not.

Example 2. Let . Suppose that bifunction is given byAs , is not a metric on . For , is a metric on . Consider defined asIf we consider and , thenFor , we haveHence, for and ,That is,Hence,for all , . That is, all the assumptions of Theorem 11 are met. If , , thenBut for any . Hence, Theorem 7 cannot be applied in this situation. So, Theorem 11 is a proper generalization of Theorem 7 which generalizes Theorems 1 and 3.

Corollary 5. Let be a complete MS and be a generalized type quasicontraction. Then, is singleton.

Proof. From Theorem 11, there exists . To prove the uniqueness, assume with . Aswe havea contradiction. Hence, is singleton.

Corollary 6. Let be a complete MS and . Iffor all and for some , then is singleton.

Corollary 7. Let be a complete MS and . If there exists and with such thatfor all , then is singleton.

Proof. It follows from Corollary 5.

Corollary 8. Let be a complete MS and . If there exists with such thatfor all , where , then is singleton.

Proof. ConsiderHence, . If and , thenHence, the result follows from Corollary 7.

3. Completeness Characterization of b Metric Spaces

In this section, we discuss the completeness of MSs via fixed point results.

In 1959, Connel [17] studied some important topological properties of fixed point sets of mappings of metric spaces and presented an example of a separable and locally contractible incomplete metric space such that every Banach contraction on has a fixed point. This shows that the converse of BCP does not hold true (see [18]). For the sake of completeness and convenience, we present the example initially given in ([17], Example 3).

Example 3. Let be any Banach contraction on , whereWe prove that is nonempty. Assume that is empty. Let be the restriction of on . DefineSince is a continuous mapping on , there is .
Claim: for some integer . If for all integers , thenMoreover,Otherwise, if for some , thenfor some . Consequently, we getBut implies that , a contradiction, as for all integers . Hence, (94) holds. Thus, andFrom (93)–(97), we havea contradiction as has no fixed point. Hence, for some integer . As is empty,for some . SetIf , there is a sequence in such that as and there is such thatWe claim that .

3.1. Case (I)

If , then there exists such that and eventually. Aseventually. This implies

But eventually, a contradiction to .

3.2. Case (II)

If , then there is such that and eventually. Sinceeventually and

But (eventually). Hence, there exists an integer such that

But . This implies , a contradiction as . Hence, , and

That is, is nonempty but is not complete as it is not closed.

Remark 3. As every metric is a metric, Theorem 5 and Example 3 show that BCP in MSs does not characterize the completeness of underlying MS.
Park and Rhoads [30] commented on completeness characterization of metric spaces and Alolaiyan et al. [27] presented those comments in the context of MSs as follows. “Let be a MS and be a class of mappings of such that if any map in has a fixed point, then is complete. Let be a class of mappings of containing such that completeness of implies the existence of fixed point of any map in .”

Theorem 12 (see [27, 30]). If is a MS, then

Now, we provide a characterization of completeness of MS.

Theorem 13. Let be a MS and , such that(a)For , where and .(b)For , where and .(c) is denumerable.(d)Every is closed.Let be a class of mappings of that satisfies (a), be a class of mappings of that satisfies (b), and be a set of mappings of that satisfies (b), (c), and (d). Then, statements (1)–(4) are equivalent:(1) MS is complete.(2) is nonempty for every for all with .(3) is nonempty for every for all with .(4) is nonempty for every for some with .

Proof. By Corollary 8, (6) implies (11).
As , . Hence, (6) implies (29). Since , (29) implies (30). (30) implies (6), and assume on contrary that (30) holds but is incomplete. That is, there is a sequence which is Cauchy but does not converge. Define asfor . As is Cauchy, for , we can choose so that for all ,Hence, for all , we getwhich implies that the sequence is bounded in for every . This further implies that the function is well defined. Further, for all in . For , there exists such that for all ,for all . Hence, we get for all ,for all . That is,From (118), for every , there exists such thatIf , thenfor all . From (120), we have ; hence, for all . That is, is empty. As , (c) holds. Note that (d) holds as well. Further, satisfiesNow fix such thatWe need to show that (111) holds. Observe thatWe have two cases. Case (1) Suppose ; then, Case (2) Whenever , from (123),Hence,for all . From (30), mapping is nonempty, a contradiction. Hence, is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2020 Basit Ali 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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