Research Article | Open Access
Basit Ali, Mujahid Abbas, Manuel de la Sen, "Completeness of Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 5183291, 13 pages, 2020. https://doi.org/10.1155/2020/5183291
Completeness of Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions
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 . 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  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  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.
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 . 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 ). If a sequence in a MS satisfiesfor somethen it is Cauchy sequence in .
Recently, Suzuki  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 ). 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) . 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  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 ), whereas BCP does not characterize the completeness of underlying metric spaces (see [17, 18]). Among other generalizations of Banach contractions, Ciric  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  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.
Theorem 2 (see ). Let be a complete metric space and . Iffor all and some , then is nonempty.
Theorem 3 (see ). Let be a complete metric space. If satisfiesfor all and for some , then is nonempty.
Jovanovic et al.  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  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 ) that cannot be transported from metric to bMSs as in the case of BCP.
Theorem 6 (see ). Let be a complete MS and . If satisfiesfor all and for some with , then is nonempty.
Using a function defined as , Kutbi et al.  presented the next result in MSs.
Theorem 7 (see ). 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 ).
Theorem 8 (see ). Let be a complete MS. If satisfiesfor all and for some with , where , then is nonempty.
Theorem 9. Further, in , 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 ). 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 , 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  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, .
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.
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.
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 get