Mathematics

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13 pages, 277 KiB

Open AccessArticle

The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces

by Ştefan Cobzaş

Mathematics 2024, 12(3), 471; https://doi.org/10.3390/math12030471 - 01 Feb 2024

Viewed by 489

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. [...] Read more.

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results). Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

16 pages, 313 KiB

Open AccessArticle

Riemann Integral on Fractal Structures

by José Fulgencio Gálvez-Rodríguez, Cristina Martín-Aguado and Miguel Ángel Sánchez-Granero

Mathematics 2024, 12(2), 310; https://doi.org/10.3390/math12020310 - 17 Jan 2024

Viewed by 480

Abstract

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with [...] Read more.

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel

σ

-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each

μ

-measurable function is Riemann-integrable with respect to

μ

. Moreover, if

μ

is the Lebesgue measure, then the Lebesgue integral on a bounded set of

R^{n}

meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

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11 pages, 277 KiB

Open AccessArticle

Presymmetric w-Distances on Metric Spaces

by Salvador Romaguera and Pedro Tirado

Mathematics 2024, 12(2), 305; https://doi.org/10.3390/math12020305 - 17 Jan 2024

Viewed by 552

Abstract

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show [...] Read more.

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

7 pages, 241 KiB

Open AccessArticle

On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces

by Valentín Gregori, Juan-José Miñana, Bernardino Roig and Almanzor Sapena

Mathematics 2024, 12(2), 287; https://doi.org/10.3390/math12020287 - 16 Jan 2024

Cited by 1 | Viewed by 617

Abstract

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it [...] Read more.

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t-strong, and we establish an alternative version of Miheţ’s theorem for p-complete t-strong fuzzy metrics (Theorem 2). In addition, an example of t-strong fuzzy metric spaces that are not non-Archimedean is provided. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

13 pages, 283 KiB

Open AccessArticle

A New Notion of Fuzzy Function Ideal Convergence

by Dimitrios Georgiou and Georgios Prinos

Mathematics 2024, 12(2), 260; https://doi.org/10.3390/math12020260 - 12 Jan 2024

Viewed by 489

Abstract

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of [...] Read more.

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class,

L

, on a non-empty set, X, consisting of triads

(f, e, I)

, where f is a function from a non-empty set, D, to the set

FP (X)

of fuzzy points in X, which we call fuzzy function,

e \in FP (X)

, and

I

is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology,

δ

, on X, such that

(f, e, I) \in L

iff f

I

-converges to e, relative to the fuzzy topology

δ

. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

11 pages, 287 KiB

Open AccessArticle

Applications to Solving Variational Inequality Problems via MR-Kannan Type Interpolative Contractions

by Rizwan Anjum, Andreea Fulga and Muhammad Waqar Akram

Mathematics 2023, 11(22), 4694; https://doi.org/10.3390/math11224694 - 19 Nov 2023

Viewed by 631

Abstract

The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique [...] Read more.

The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique fixed point for this class of operators. Thirdly, we study Ulam-Hyers stability, well-posedness, and periodic point properties. Finally, an application of the main results to the variational inequality problem is given. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

16 pages, 311 KiB

Open AccessArticle

Existence of Fuzzy Fixed Points and Common Fuzzy Fixed Points for

FG

-Contractions with Applications

by Dur-e-Shehwar Sagheer, Zainab Rahman, Samina Batul, Ahmad Aloqaily and Nabil Mlaiki

Mathematics 2023, 11(18), 3981; https://doi.org/10.3390/math11183981 - 19 Sep 2023

Viewed by 560

Abstract

This article contains results of the existence of fuzzy fixed points of fuzzy mappings that satisfy certain contraction conditions using the platform of partial b-metric spaces. Some non-trivial examples are provided to authenticate the main results. The constructed results in this work [...] Read more.

This article contains results of the existence of fuzzy fixed points of fuzzy mappings that satisfy certain contraction conditions using the platform of partial b-metric spaces. Some non-trivial examples are provided to authenticate the main results. The constructed results in this work will likely stimulate new research directions in fuzzy fixed-point theory and related hybrid models. Eventually, some fixed-point results on multivalued mappings are established. These theorems provide an excellent application of main theorems on fuzzy mappings. The results of this article are extensions of many already existing results in the literature. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

13 pages, 295 KiB

Open AccessArticle

Fuzzy Metrics in Terms of Fuzzy Relations

by Olga Grigorenko and Alexander Šostak

Mathematics 2023, 11(16), 3528; https://doi.org/10.3390/math11163528 - 15 Aug 2023

Cited by 1 | Viewed by 752

Abstract

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we [...] Read more.

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression

d (x, y) = t

. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value

d (x, y)

and the other is a number within the range

[0, + \infty)

. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

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14 pages, 321 KiB

Open AccessArticle

Ulam–Hyers Stability and Well-Posedness of Fixed Point Problems in C*-Algebra Valued Bipolar b-Metric Spaces

by Manoj Kumar, Pankaj Kumar, Ali Mutlu, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby and Stojan Radenović

Mathematics 2023, 11(10), 2323; https://doi.org/10.3390/math11102323 - 16 May 2023

Cited by 1 | Viewed by 871

Abstract

Here, we shall introduce the new notion of

C^{*}

-algebra valued bipolar b-metric spaces as a generalization of usual metric spaces,

C^{*}

-algebra valued metric space, b-metric spaces. In the above-mentioned spaces, we shall define [...] Read more.

Here, we shall introduce the new notion of

C^{*}

-algebra valued bipolar b-metric spaces as a generalization of usual metric spaces,

C^{*}

-algebra valued metric space, b-metric spaces. In the above-mentioned spaces, we shall define

(α_{A} - ψ_{A})

contractions and prove some fixed point theorems for these contractions. Some existing results from the literature are also proved by using our main results. As an application Ulam–Hyers stability and well-posedness of fixed point problems are also discussed. Some examples are also given to illustrate our results. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

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