Abstract
In this paper, fixed point theorems of multiple values of self-mappings have been investigated in Banach space. In addition, a new modification of the same mentioned study has been considered in a metric space with at most three valued self-mappings. Moreover, we have extended and improved some existent results in a complete metric space. Finally, we present a slight improvement on some well-known lemmas that are related to nonexpansive mappings and nonincreasing distances mappings.
1. Introduction
In mathematics and related fields, the fixed point theory plays a significant role in several fields, see [1–4]. In this recent work, we continue studying some existent results by many researchers on fixed point theorems in Banach and metric spaces. In 2014, a general fixed point theorem for multivalued mappings that is not always a contraction was proposed by Abdul Latif[2]. In addition, he derived some modern studies on contraction mappings. In 2020, Singh and Anthony [5] studied and extended the concept of more than one fixed point into mappings on a cone Sb-metric space. Moreover, they provided some robust examples to validate their results.
The structure of our proposed paper is organized as follows: In Section 2, we present our first generalization of results by Ray [6], where we provide some well-known results consisting of multivalued mappings in a complete metric space . Moreover, we present the same study for the results provided by Taskovic [7]. In section 3, we provide some results about the fixed point theorem of three mappings , and in a complete metric space . Furthermore, a brief discussion of few common fixed point theorems is going to be presented. Also, we will give a study of the three mappings and that satisfy the following equation: where ; , , ; ; and are continuous function that satisfy the following condition: More studies have been done and considered by many authors regarding theorems that satisfy different functional inequalities and in terms of fixed point theorems in different spaces as well as several applications such as solving nonlinear and differential equations, see [8], [3, 9, 10], and [11–15]. In Section 4, we mainly extend the results that have been obtained by Kirk in [16] and in [1]. In fact, we work on modifying the given lemmas that are related to multivalued mappings in order to extend the results for the sake of generalization.
2. Fixed Point in Metric Space
Jleli and Bessem in [17] introduced a novel definition of generalized metric spaces, where they worked on extending several existing fixed-point findings along with the Banach contraction fundamental theorem [7]. In addition, Taskovic presented and proved a fixed-point theorem on a metric space along with a mapping , which is not necessarily continuous and satisfy a condition of the type,for all .
In the following theorem, we present the first result for two mappings and that generalizes the abovementioned result.
Theorem 1. Suppose that and are a couple self-functionals of a complete metric space with the following condition:for all such that the numbers and with and . Then and will contain a unique common fixed point.
Now, we introduce the below generalized theorem of Ray [6] for a couple of self-mapping on a metric space.
Theorem 2. Suppose that is a continuous functional in a complete metric space and suppose that is another mapping in the same space where the two mappings commute with each other, and satisfies the following condition:for each , where , and are monotonically decreasing functions from into with and . Then, the two mappings and have only one common fixed point in .
The proof of the above theorem is not going to be presented as it goes in a similar way as that in [18]. Therefore, the proof is omitted.
Remark 1. If happens to be an identity mapping, we get Theorem 2 of Ray [6] without the diagram of being closed. Further, if , we have the result of Rokotch [19].
Another generalization of Theorem 2 of Ray goes as follows:
Theorem 3. Suppose that and are two self-functionals in a complete metric space , where the two mappings satisfy the following condition:for every , such that , and are monotonically decreasing functions from into with , Then the two mappings and have only one common fixed point in .
3. Consequences
Let us assume that represent an upper semicontinuous function from the space to the space , where this function is monotonically increasing in every variable and for each number the following condition is satisfied:where such that .
In the following, we present the main theorem of this work.
Theorem 4. Suppose that , and are three self-functionals of a complete metric space , which satisfy the following conditions:for all . Further, let be continuous, then , and have only one common fixed point in . The proof of this theorem goes in a similar way to the one by Devi Prasad [9]. Therefore, we omit the proof.
The following result is a corollary to the previous theorem by taking , where represents the identity map.
Corollary 1. Let and be a couple of mappings from a complete metric space into itself, and they satisfy the following condition:for all , where is defined in (7), then and have only one common fixed point.
Now, we present the following three theorems, where the proof of each one follows from Theorem 4.
Theorem 5. Suppose that , and are three self-functionals of a complete metric space sin such a way that,
, then there is a non-negative and nonzero integers , and satisfying the following condition:for all , and if is continuous, then , and have only one common fixed point.
Theorem 6. Suppose that , and are sequences of self-functionals of a complete metric space such that , , and converge uniformly to self-mappings , and on with continuous. Suppose that for each , is a common fixed point of and , and is a common fixed point of and . Furthermore, let , and satisfy conditions (8) and (7). If is the common fixed point of , and , and and Then, and .
Theorem 7. Let , and be sequences of self-mappings of a complete metric space such that is continuous for each and , and satisfy conditions (8) and (7) for each . If , and are uniform limits of , and respectively, then, , and also satisfy conditions (8) and (7). Also, let be a sequence of the only one common fixed point of , and converges to the only one common fixed point of , and , whenever
4. Main Results
Suppose that is a metric space and let be a class of countably compact subsets of , and suppose that this class is stable under intersections, normal, and contains all the closed balls in . Also, suppose that , are subsets of . Moreover, we define as such that
We shall take this assumption in order to initiate our proposed theorems.
Theorem 8. If be a mapping such that,and for all .is a compact class of which each nonempty member is a compact subset of . For any satisfying , for all ,
Then, has a fixed point in .
Proof. A slight modification on the main lemma by kirk [1], we show that for each , there exists a nonempty set such that,for all and for which . For this, we take if . Otherwise, construct as follows:
Now, let. By the definition of , the set is nonempty, where represents the closed ball with the center and the radius . Letand let be a decreasing chain in . Let , then and . Furthermore, since is decreasing, it follows that for each , the family has a finite intersection property. So, by hypothesis (13),That is , thus every decreasing chain in has a lower bound. Therefore, by Zorn’s lemma, has a minimal element . Let , whereThen, So, . Also, for and So, . Hence, . Thus, .
Since is a minimal member of and , it follows that . LetThen, . Since , and let . Then, . Take . Let . Next take , then and by (13),Since and are nonempty compact sets and , there exists , such thatSo, , hence, . Thus, and . Since is a minimal element in , it follows that . So L, i.e., So, , which implies Now, let and for each , letThe rest of the proof is omitted as it goes in a similar way as the one by Kirk in [1].
Remark 2. The necessity of condition (14) of the theorem has been studied by Samanta in [20, 21].
In the following, we mention two theorems by Kirk in [1] and Samanta in [20] as we consider them corollaries of Theorem 4.
Corollary 2 (see [1]). Suppose that is a metric space, which is bounded and nonempty and let be a countably compact class of subsets in , which is normal and stable under random intersections. In addition, suppose that we have closed balls of that are contained in . Then, there exists a fixed point in each nonexpansive mapping of into itself.
Corollary 3 (see [20]). Let be bounded and a closed convex subset in a reflexive Banach space , and if is a mapping such that,for any closed convex subset of satisfying
Then, has a fixed point.
5. Conclusions
We have presented in this paper generalizations of various fixed-point theorems in a metric space and in a Banach space. A generalization with multiple values of self-mappings in those mentioned spaces have been intensively studied. We saw that mapping scheme of the fixed-point theorem is vast no doubt. Therefore, the existent result by Kirk was improved by extending the proof and working on the fixed-point theorem of multiself-mappings.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors have read and agreed to the published version of the manuscript.
Acknowledgments
The fifth author received financial support from Taif University Researches Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia. All authors contributed equally to this work.