Abstract
In this manuscript, we obtain common fixed point theorems in the neutrosophic cone metric space. Also, notion of -weak contraction is defined in the neutrosophic cone metric space by using the idea of altering distance function. Finally, we review many examples of cone metric spaces to verify some properties.
1. Introduction
The concept of fuzzy sets was introduced by Zadeh [1]. The study of fuzzy topological spaces was initiated by Chang [2]. The notion of intuitionistic fuzzy sets was introduced by Atanassov [3]. The notion of intuitionistic -topological spaces was introduced by Atanassov and Stoeva [4] by extending -topology to intuitionistic -fuzzy setting. The notion of the intuitionistic fuzzy topological space was introduced by Çoker [5]. The concept of generalized fuzzy closed set was presented by Balasubramanian and Sundaram [6]. Smarandache extended the intuitionistic fuzzy sets to neutrosophic sets [7]. After the introduction of the neutrosophic set concept [8, 9] in 2019 by Smarandache and Shumrani on the nonstandard analysis, the nonstandard neutrosophic topology was developed. In recent years, neutrosophic algebraic structures have been investigated. Neutrosophy has laid the foundation for a whole family of new mathematical theories, generalizing both their classical and fuzzy counterparts, such as a neutrosophic theory in any field, see [10, 11]. Recently, there were introduced neutrosophic mapping and neutrosophic connectedness. The concept of the neutrosophic metric space presented by [12] Al-Omeri et al. is a generalization of the intuitionistic fuzzy metric space due to Veeramani and George [13]. In 2019 and 2020, Smarandache generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) whose operations and axioms are partially true, partially indeterminate, and partially false as extensions of Partial Algebra and to AntiAlgebraic Structures (or AntiAlgebras) whose operations and axioms are totally false. And in general, he extended any classical structure, in no matter what field of knowledge, to a NeutroStructure and an AntiStructure, see [14, 15]. In 2007, Huang and Zhang [16] introduced the concept of cone metric space and proved some fixed point theorems for contractive mappings. Recently, Öner et al. [17] introduced the concept of the fuzzy cone metric space that generalized the corresponding notions of the fuzzy metric space by George and Veeramani [13] and proved the fuzzy cone Banach contraction theorem. In 2010, Vetro et al. [18] extended the notion of -weak contraction to fuzzy metric spaces and proved some common fixed point theorems for four mappings in fuzzy metric spaces by using the idea of an altering distance function. Gupta et al. and Wasfi et al. [19, 20] introduced the notions of E. A and common E. A on the modified intuitionistic generalized fuzzy metric space. They extended the notions of the common limit range property and E. A property for coupled maps on modified intuitionistic fuzzy metric spaces. This paper is devoted to the study of extending and generalizing the -weak contraction to the neutrosophic cone metric space and prove some results. In Section 2, we will recall some materials which will be used throughout this paper. In Section 3, we give definitions and present the cone neutrosophic metric space and explain a number of properties. In Section 4, the results obtained from theorems and theoretical application of the neutrosophic fixed point are also presented. The last section contains the conclusions of the paper.
2. Preliminaries
Definition 1 (see [21]). Let be a non-empty fixed set. A neutrosophic set (briefly, NS) is an object having the form , where , and which represent the degree of membership function (namely, ), the degree of indeterminacy (namely, ), and the degree of nonmembership (namely, ), respectively, of each element to the set .
A neutrosophic set can be identified to an ordered triple in on .
Remark 1 (see [21]). By using symbol for the NS, .
Definition 2 (see [13]). Let be a NS on . The complement of may be defined as three kinds of complements:(1)(2)(3)We have the following NSs (see [21]), which will be used in the sequel:(1) or(2),(3),(4).(1) or(2),(3),(4).
Definition 3 (see [21]). Let be an arbitrary family of NSs in . Then,(1) may be defined as follows:(i)(ii)(2) may be defined as follows:(i)(ii)
Definition 4 (see [21]). For any , let neutrosophic sets and be in the form and . The two possible definitions of are as follows:(1)(2)
Definition 5 (see [22]). A neutrosophic topology (NT for short) and a nonempty set is a family of neutrosophic subsets in satisfying the following axioms:(1)(2) for any (3), The elements of are called open neutrosophic sets. The pair is called a neutrosophic topological space, and any neutrosophic set in is known as the neutrosophic open set (NOS) in . A neutrosophic set is closed if its complement is neutrosophic-open, denoted by . Throughout this paper, we suppose that all cone metrics have nonempty interior.
For any in [23], we have and .
Definition 6. A subset of is said to be a cone in the following cases:(1)If both and , then (2)If , , and , then (3) is closed, nonempty, and For a given cone, partial ordering () on via is defined by iff . will stand for and , while will stand for .
If a constant such that for all , implies , and the least positive number satisfying this property is called the normal constant of , where is the normal.
Definition 7. Let be a nonempty set and be a given real number. A function is said to be a cone metric on if the following conditions hold:(1) for all (2) for all (3)(4) iff The pair is called a cone metric space (shortly, ).
Definition 8. A t-norm is continuous for any binary operation if verifies the following statements:(1) is continuous(2) is commutative and associative(3) whenever and for all (4) for all
Definition 9. Let be a CMS. Then, for any and , , , and such that and .
Example 1. and .
Example 2. and .
Definition 10. A t-conorm of a binary operation is continuous if verifies the following statements:(1) is continuous(2) is associative and commutative(3) whenever and for all (4) for all
Definition 11 (see [12]). is said to be a neutrosophic cone metric space if is NCMS of , is an arbitrary set, is a N-continuous t-conorm, is a N-continuous t-norm, and are neutrosophic sets on , which satisfy the following statements: and (that is, and ):(1)(2) iff (3), where is permutation(4)(5) ⌋0−, 1+⌊ is neutrosophic-continuous
Definition 12 (see [12]). Let be a . For , the open ball with center and radius is defined by .
Example 3. Let . Then, is a normal cone, and is a normal constant. Let , , and , defined by and .
Definition 13 (see [12]). An neutrosophic cone metric is called complete neutrosophic if any sequence which is Cauchy in is convergent.
Definition 14 (see [12]). is said to be a neutrosophic if is a neutrosophic cone metric (shortly, NCMS) of , where is an arbitrary set, is a neutrosophic continuous t-norm, is a neutrosophic continuous t-conorm, and are neutrosophic sets on , which satisfy the following statements: and (that is, and ):(1) iff (2)(3), where is permutation(4)(5) ⌋0−, 1+⌊ is neutrosophic-continuous(6)(7) ⌋0−, 1+⌊ is neutrosophic-continuous(8)(9) if and only if (10)(11), where is permutationThen, is called a neutrosophic cone metric on .
The functions and are defined by the degree of non-nearness between and with respect to , respectively.
Definition 15 (see [12]). Let be a , , and be a sequence in . Then, is said to be convergent to if for all and all , there exists such that for any . We defined that or as .
Definition 16. A function is an altering distance if is monotone increasing and continuous, and iff .
Definition 17. Let be a metric space and let . Defined and for any , and let and be fuzzy sets on represented by and .
3. Main Result
Definition 18. Let be a neutrosophic cone metric space (CMS) and be two mappings. Mapping is said to be neutrosophic -weak contraction if there exists a function with and for and an alternating distance function such thathold for all and each . If is the identity map, then is called a neutrosophic -weak contraction mapping.
Definition 19. Let be a neutrosophic cone metric space and be two mappings. Point is said to be a coincidence point in of and if .
Definition 20. Let and be two finite families of self-mappings on . They are called pairwise commuting if(1), where (2), where (3), where and
Theorem 1. Let be a neutrosophic cone metric space and be a neutrosophic -weak contraction with respect to . If and or is a complete subset of , then and have a unique common fixed point in provided that is a continuous function.
Proof. Let be an arbitrary point. Let point such that . This can be done since . Continuing this process, we obtain a sequence such that . We assume that for all ; otherwise, and have a coincidence point. Now, we getwhich suppose that mapping is nondecreasing; hence, . Hence, is an increasing sequence of positive real numbers in . Let . We prove that . If not, there exists such that . Then, from the above inequality on taking , we obtainwhich is a contradiction. Therefore, as . Now, for each , by Definition 18, we getIt follows that . At the same time, we havein which considering that the mapping is nondecreasing, then . Thus, is a decreasing sequence of positive real numbers in . Let . We show that for all . If this is not the case, there exists such that . Then, it follows from (5), by taking , that , which is a contraction. Therefore, as . Now, for each , by Definition 14 (9), we haveIt follows that . Hence, is a Cauchy sequence. If is complete, then there exists such that as . The same holds if is complete with . Let and . Now, we shall show that is a coincidence point of and . In fact, we have takenfor every , in which by letting ,Therefore, . Now, we shall prove that . If it is not so, then we havewhich is a contradiction. By inequalities (4) and (5) we prove the uniqueness. The desired equality is obtained.
Example 4. Let be a complete neutrosophic cone metric space, , be a maximum norm, and be a minimum norm. Let be defined byAlso, define by and , for all , , and . Obviously, , and is a continuous function. Then, we haveFrom the above inequality and the fact that , we conclude that the conditions (1) and (2) in Definition 2.18 are satisfied. Thus, is a neutrosophic -weak contraction with respect to .
Corollary 1. Let be a neutrosophic cone metric space and be a neutrosophic -weak contraction. If is continuous, then has a unique fixed point.
Corollary 2. Let be a neutrosophic cone metric space. Then, is a mapping satisfyingfor each , , and .
Theorem 2. Let be a neutrosophic cone metric space and be two finite self-mappings on with and such that and . Suppose be a generalized neutrosophic -weak contraction which is given with respect to . If and or is a complete subset of , then have a common fixed point in which is unique, provided a description of is a continuous function and the families commute pairwise.
Proof. By Theorem 1, we obtain that and have a common fixed point that is unique, say . In order to prove that remains as a fixed point of all self-mappings, letSince the other conditions are similarly proved, we can show that , , and , which imply that , , and are other fixed points of mapping . For the uniqueness of and of self-mappings, we get , which shows that is a common fixed point of and , .
Example 5. Let be a complete neutrosophic cone metric space, , and . Define by , for all and two families of self mappings and where byThen, we haveFrom the above and the idea of , we get that statements (i) and (ii) hold. All statements of Theorem 2 hold; therefore, and have uniqueness.
4. Conclusion
In this paper, the definition of the neutrosophic cone metric space is introduced and studied. Based on this definition, we also stated and proved some fixed point theorems on the neutrosophic CMS. We provided a description of the example and investigated some properties in Section 3. We established and extended the definition of the -weak contraction in the intuitionistic generalized fuzzy cone metric space.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding this manuscript.