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14 pages, 262 KB

Open AccessArticle

On Weak Probabilistic φ-Contractions in Menger Probabilistic Metric Spaces

by Dingwei Zheng and Qingming He

Axioms 2025, 14(9), 658; https://doi.org/10.3390/axioms14090658 - 27 Aug 2025

Viewed by 394

Two new concepts are introduced in this paper: ZH-set and weak probabilistic

φ

-contraction. Firstly, a fixed-point theorem for weak probabilistic

φ

-contraction is given in Menger probabilistic metric spaces, which generalizes and unifies several results in the existing literature. Then, a best [...] Read more.

Two new concepts are introduced in this paper: ZH-set and weak probabilistic

φ

-contraction. Firstly, a fixed-point theorem for weak probabilistic

φ

-contraction is given in Menger probabilistic metric spaces, which generalizes and unifies several results in the existing literature. Then, a best proximity point theorem is obtained. Finally, two examples are provided to prove the effectiveness of our results. Full article

12 pages, 254 KB

Open AccessArticle

Solving Fredholm Integral Equations Using Probabilistic F-Contractions

by Ismail Tahiri, Youssef Achtoun, Mohammed Lamarti Sefian and Stojan Radenović

Axioms 2025, 14(2), 119; https://doi.org/10.3390/axioms14020119 - 5 Feb 2025

Viewed by 1381

Abstract

Fixed-point theory plays a pivotal role in addressing equations that model various real-life applications, offering robust methods to find solutions that remain stable under different conditions and dynamics. The main objective of this paper is to introduce and study the novel concept of [...] Read more.

Fixed-point theory plays a pivotal role in addressing equations that model various real-life applications, offering robust methods to find solutions that remain stable under different conditions and dynamics. The main objective of this paper is to introduce and study the novel concept of probabilistic

F

-contraction within the framework of Menger spaces. This innovative approach extends classical fixed-point results to probabilistic settings, leveraging the probabilistic structure of Menger spaces to handle uncertainty and variability in the modeling process. By establishing the existence and uniqueness of fixed points in this versatile class of spaces, the study highlights the broader applicability and deeper significance of the probabilistic

F

-contraction. We explore the intricate interrelations among these fixed points, shedding light on their implications across different contexts and presenting insights into various theorem versions that enhance our understanding of their utility. Additionally, we propose a straightforward and effective approach for solving a system of Fredholm integral equations using fixed-point techniques specifically tailored for Menger spaces, illustrating their practical utility in tackling the complex mathematical models encountered in diverse fields. Full article

(This article belongs to the Section Mathematical Analysis)

13 pages, 305 KB

Open AccessArticle

On Prešić-Type Mappings: Survey

by Youssef Achtoun, Milanka Gardasević-Filipović, Slobodanka Mitrović and Stojan Radenović

Symmetry 2024, 16(4), 415; https://doi.org/10.3390/sym16040415 - 2 Apr 2024

Cited by 2 | Viewed by 1362

Abstract

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known [...] Read more.

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić’s method, which generalizes Banach’s theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić’s approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić’s works while providing valuable insights into this specialized domain. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

11 pages, 276 KB

Open AccessArticle

The Hausdorff–Pompeiu Distance in Gn-Menger Fractal Spaces

by Donal O’Regan, Reza Saadati, Chenkuan Li and Fahd Jarad

Mathematics 2022, 10(16), 2958; https://doi.org/10.3390/math10162958 - 16 Aug 2022

Viewed by 1240

Abstract

This paper introduces a complete

G n

-Menger space and defines the Hausdorff–Pompeiu distance in the space. Furthermore, we show a novel fixed-point theorem for

G n

-Menger-

θ

-contractions in fractal spaces. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

11 pages, 291 KB

Open AccessArticle

Common Fixed Points Theorems for Self-Mappings in Menger PM-Spaces

by Rale M. Nikolić, Rajandra P. Pant, Vladimir T. Ristić and Aleksandar Šebeković

Mathematics 2022, 10(14), 2449; https://doi.org/10.3390/math10142449 - 13 Jul 2022

Cited by 3 | Viewed by 1687

Abstract

The purpose of this paper is to prove that orbital continuity for a pair of self-mappings is a necessary and sufficient condition for the existence and uniqueness of a common fixed point for these mappings defined on Menger PM-spaces with a nonlinear contractive [...] Read more.

The purpose of this paper is to prove that orbital continuity for a pair of self-mappings is a necessary and sufficient condition for the existence and uniqueness of a common fixed point for these mappings defined on Menger PM-spaces with a nonlinear contractive condition. The main results are obtained using the notion of R-weakly commutativity of type

A_{f}

(or type

A_{g}

). These results generalize some known results. Full article

13 pages, 541 KB

Open AccessArticle

An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

by Aurora Espinoza-Valdez, Jesús Leaños, Christophe Ndjatchi and Luis Manuel Ríos-Castro

Symmetry 2021, 13(6), 1050; https://doi.org/10.3390/sym13061050 - 10 Jun 2021

Cited by 3 | Viewed by 2459

Abstract

Let P be a set of

n \geq 3

points in general position in the plane. The edge disjointness graph

D (P)

of P is the graph whose vertices are the

(\binom{n}{2})

closed straight line segments with endpoints in P [...] Read more.

Let P be a set of

n \geq 3

points in general position in the plane. The edge disjointness graph

D (P)

of P is the graph whose vertices are the

(\binom{n}{2})

closed straight line segments with endpoints in P, two of which are adjacent in

D (P)

if and only if they are disjoint. In this paper we show that the connectivity of

D (P)

is at most

\frac{7 n^{2}}{18} + Θ (n)

, and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of

D (Q_{n})

, where

Q_{n}

denotes an n-point set that is almost 3-symmetric. Full article

(This article belongs to the Special Issue Recent Studies on Topological and Geometrical Properties of Discrete Structures)

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13 pages, 288 KB

Open AccessArticle

Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

by Thodsaporn Kumduang and Sorasak Leeratanavalee

Symmetry 2021, 13(4), 558; https://doi.org/10.3390/sym13040558 - 27 Mar 2021

Cited by 20 | Viewed by 2882

Abstract

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools [...] Read more.

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups. Full article

(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)

14 pages, 292 KB

Open AccessArticle

Ternary Menger Algebras: A Generalization of Ternary Semigroups

by Anak Nongmanee and Sorasak Leeratanavalee

Mathematics 2021, 9(5), 553; https://doi.org/10.3390/math9050553 - 5 Mar 2021

Cited by 11 | Viewed by 2771

Abstract

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups [...] Read more.

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

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