Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 641 https://internationalpubls.com Menger Space and Some Contraction Mappings Ajay Kumar Chaudhary 1 , Chet Raj Bhatta 2 , and Dilip Kumar Sah 3* 1 Department of Mathematics, Tri-Chandra Multiple Campus, Tribhuvan University 2 Central Department of Mathematics, Kirtipur, Tribhuvan University 3 Department of Mathematics, Patan Multiple Campus, and PhD Scholar of Tribhuvan University, Kathmandu, Nepal * Corresponding e-mail: dilipofficial.121@gmail.com Article History: Received: 04-06-2024 Revised: 03-07-2024 Accepted: 30-07-2024 Abstract: Menger space is a probabilistic metric space introduced by K. Menger [15] in 1942 as one of the generalizations of metric space. The two different types of contraction mappings in probabilistic metric spaces are the creation of V.M. Sehgal [20-21], and T.L. Hicks [14]. Since then many researchers have introduced generalizations of these contraction mappings and established fixed point theorems in Menger space. In this paper, we discuss Menger space and contraction mapping in Menger space. Also, presents the interrelationship between some contraction mapping with examples. Keywords: Probabilistic metric space, Menger space, Sehgal, and Hicks contraction. 1. Introduction In the early 19 th century mathematicians were studying various spaces, mainly spaces of functions where they had various notions of convergence. For each space, its concepts of convergence were introduced and studied. To overcome this problem, in 1906 French mathematician M. Frechet [11] gave the axiomatic notions of metric space, this name metric space was given by F. Hausdorff in 1914. These metric spaces provide a deterministic framework for measuring distances between two points. Representing or giving the deterministic approach for the distance between two points in space by a single number is over-idealization. In such cases where we can’t predict the distance by a single number, give a probable answer. So, looking at the distance concept as a statistical/ probabilistic rather than a determinate one is appropriate. The Austrian mathematician Karl Menger in 1942 had given the idea of statistical metric space later called probabilistic metric space to overcome uncertainties in cases of the distance between points in spaces. The idea is to replace the distance function with a probability distance function. And ( )( ) ( ( ) ) That is the value of the distribution function in between and is equal to the probability that the distance between and is less than . So, Menger space is a space in which the concept of distance is considered to be probabilistic, rather than deterministic and the theory of Menger spaces is of fundamental importance in probabilistic functional analysis. Schweizer and Sklar [22] have investigated several of these structures. Particularly, a lot of work has been done on the existence of fixed points of mappings in such spaces. For more details on these spaces see references [2-4], [8-10], [23]- [28]. Fundamental works in probabilistic metric spaces for mathematics researchers were done by V. M. Sehgal [20], who introduced a natural probabilistic version of Banach contraction [1]. Using this contraction, V. M. Sehgal and A.T. Bharucha Reid [21] established the first fixed point theorem in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 642 https://internationalpubls.com Menger space in 1972. Another probabilistic contraction mapping was introduced by T. L. Hicks in 1983[14]. Then after many researchers introduced its extension and generalized forms see references [5-7], [[17-19]. 2. Preliminaries notes: Definition 2.1 [5]: A function F: + is said to be a distribution function if a function is non-decreasing function, left continuous with inf * ( ) + , and sup* ( ) + We denote L for the set of all distribution functions, and H stands for the heavy side function, which is defined as: ( ) { For reference, we also record the definition of metric space. Definition 2.2 [8]: A metric space is an ordered pair ( ) where is an abstract set and is a mapping of , satisfying the following axioms: M1: ( ) if and only if ( Identity); M2: ( ) ( Positivity); M3: ( ) ( ) ( Symmetry); M4: ( ) ( ) ( ) ( Triangle inequality). Definition 2.2 [7]: Let (set of all distribution functions) be a distribution function i.e., associates a distribution function ( ) with every pair ( ) of points in a non-empty set . Then, a pair ( ) is said to be a Probabilistic metric space (abbreviated as Pm-space) if the distribution function ( ) also denoted by satisfies the following conditions: (I) ( ) for every if and only if , (II) ( ) for every , (III) ( ) ( ) for every and (IV) ( ) if and only if ( ) and ( ) Here, ( ) represents the value of Example 2.1: Let ( ) be metric space where , -with usual metric ( ) | | and distribution function defined as: ( ) { ( ) for all Then, ( ) be PM space. Definition 2.3 [13]: A function , - , - , - is referred to as a Triangular norm Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 643 https://internationalpubls.com (shortly T-norm) if it satisfies the following conditions: T1: ( ) T2: ( ) for all , -, T3: ( ) ( ) for all , -, T4: then ( ) ( ) and T5: ( ( ) ) ( ( )), where , - Example 2.2 of t-norms ( ) *( ) + and ( ) * + The four basic standard t-norms are: (i) The minimum t-norm, , is defined by ( ) * +, (ii) The product t-norm, , is defined by ( ) , (iii) The Lukasiewicz t-norm, , is defined by ( ) * +, (iv) The weakest t-norm, the drastic product, , is defined by ( ) { ( ) ( ) Concerning the point-wise ordering, we have the following inequalities . Definition 2.4: [13] A Menger space is a triplet ( ) where is a non-empty set, is a function defined on to the set of distribution functions and is a -norm such that the following are satisfied: (I) ( ) for every if and only if , (II) ( ) for every , (III) ( ) ( ) for every and (IV) ( ) ( ( ) ( )), for every Definition 2.5: [6] Let ( ) be a Menger Space and be a continuous t-norm (i) A sequence * + in is said to converge to a point in (written ) if for every and ( ), there exists positive integer such that ( ) for all . (ii) A sequence * + in is said to be a Cauchy if, for every and ( ), there exists positive integer such that ( ) for all . (iii) A Menger space ( ) is said to be complete if every Cauchy sequence is convergent in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 644 https://internationalpubls.com (iv) A mapping is said to be continuous in Menger space ( ) at point if for each ( ) there exists a real number ( ) satisfying the condition: ( ) ( ) for each and Definition 2.6: [8] Fixed point of a self-mapping function is an element such that ( ) . Example 2.3: ( ) have two fixed points but ( ) have no fixed point. Definition 2.7: [8] Common fixed point of self-mapping functions is an element such that ( ) ( ) Example 2.4: Let be functions such that ( ) and ( ) , then is a common fixed point of and Definition 2.8: [1] Let ( ) be a metric space. Then, a mapping is said to be a contraction mapping if there exists a fixed constant , ) such that ( ( ) ( )) ( ) Example 2.5: Let function , - , - be defined by ( ) , - ( - Then, is a contraction but is not a contraction. (why?) 3. Contraction Mapping in Menger Space: In 1966, V. M. Sehgal [20] first defined probabilistic contraction in his PhD dissertation at Wayne State University as: Definition 3.1: Let ( ) be a probabilistic metric space. A mapping is a probabilistic contraction or Sehgal contraction if there exists ( ) such that ( ) ( ) for all and . The interpretation of Sehgal Contraction is as follows: The probability that the distance between the image points and is less than is at least equal to the probability that the distance between that is less than . In 1983, T.L. Hicks [14] defined another contraction mapping in probabilistic metric space as: Definition 3.2: A mapping in probabilistic metric space ( ) is said to be Hicks contraction or C-contraction if there exists ( ) such that for every , and every ( ) ( ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 645 https://internationalpubls.com In 2005, D. Mihet [16] introduced a weaker form of Hicks contraction and defined it as: Definition 3.3: A mapping is said to be weak - Hicks contraction (w-H contraction) if there exists ( ) such that, for all . ( ) ( ) ( ) ( ) Example: Let , ) and ( ) ( ) ( ) Then, ( ) be a complete Menger space under triangular norm . It can be seen that the mapping , ( ) * is a contraction for ( ) 3.1 Generalized form of probabilistic contraction: A probabilistic (m, k) contraction is a generalization of Sehgal contraction, where and ( ) and is defined as: Definition 3.1.1: [13] If ( ) is a PM - space, and ( ), a function is called probabilistic (m, k)-contraction if for any there is an with such that for every , ( ) ( ) If and ( ) then a probabilistic ( )-Sehgal contraction, is a probabilistic Sehgal contraction. As a generalization of C-contraction, we have Definition 3.1.2: [13] If ( ) is a PM - space, and ( ), a function is called a (m, k)-C-contraction if for any there is an with such that for every . ( ) ( ) If and ( ) then a probabilistic ( )-C-contraction is a probabilistic C-contraction. g-contraction mapping is another generalization of Hick’s C-contraction in probabilistic metric space which is defined as: Definition 3.1.3: [16] Let be two mappings defined on a Menger space ( ) with values into itself, and let us suppose that is bijective. The mapping is called a probabilistic g- contraction with a constant ( ) if and ( ) ( )( ) impies ( ) ( )( ) 4. Results and Conclusions: 4.1 Interrelationship: 4.1.1 Every metric space is a probabilistic metric space: Every metric space can be shown as a probabilistic metric space if we set ( ) ( ( )) for every pair of points ( ) in the metric space. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 646 https://internationalpubls.com This can be illustrated as follows: ( ) ( ( )) ( ) ( ) as ( )is a distribution function. This shows that ( ) for every if and only if For proving ( ) , consider ( ) ( ( )) ( ) For proving, ( ) ( ) consider ( ) ( ( )) ( ( )) as ( ) ( ) ( ) Finally, we consider ( ) ( ( )) ( ) ( ) ( ) ( ( )) ( ) ( ) Therefore, ( ) ( ( )) ( ), for ( )) , for every Thus, ( ) if and only if ( ) and ( ) Theorem 4.1.1: Let ( ) be a complete metric space and be a mapping satisfying the following condition: there exists a constant ( ) such that ( ( ) ( )) ( ) Then, has a fixed point in , and for any Proof: Defining mapping by ( ) ( ( )) We know that ( ) is a complete Menger space. Since, for each we have ( ) ( ( )) ( ( )) ( ( ) ( ) This implies that is contraction mapping in Hence, by theorem, every contraction mapping in complete Menger space has a fixed point in ,-, we have a fixed point in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 647 https://internationalpubls.com 4.1.2: Contraction condition and their relationship:  Sehgal contraction: ( ) ( );  Generalized (m, k)- Sehgal contraction: ( ) ( );  Hicks C-contraction: ( ) ( )  w-H contraction: ( ) ( )  g-contraction: ( ) ( )( ) impies ( ) ( )( ) ;  (m, k)-C-contraction: ( ) ( ) 4.1.3: Interrelationship:  (m, k) contraction ⟹ C-contraction ⟹ Sehgal contraction ⟹ Banach contraction  g-contraction ⟹ C-contraction  C-contraction ⟹ w-H contraction. Conclusions: Menger solves the uncertainty cases of the distances between two points in spaces by introducing the probabilistic distance function in metric space. The structure of the probabilistic metric space allows probabilistic generalizations of the contraction mapping principle in some inequivalent ways. Probabilistic contraction mappings extend the study and research in probabilistic metric space which helps not only in mathematical cases but also in the geometric study of quantum mechanics. Lastly, this paper helps mathematicians and researchers in the study of contraction mapping, its generalization, and its interrelationship in probabilistic metric space. Acknowledgement: Authors are grateful to editors and reviewers for their kind suggestions for making betterment of this article. The corresponding author also remembers and thanks to University Grants Commission, Nepal for considering financial support to PhD scholars for their research activities. References [1] Banach S. (1922), Sur les operations dans les ensembles abstraits et leur applications aux equations integral. Fund. Math. 3, 133-181. [2] Bharucha-Reid, A. T. 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