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13 pages, 281 KiB

Open AccessArticle

m-Isometric Operators with Null Symbol and Elementary Operator Entries

by Bhagwati Prashad Duggal

Axioms 2025, 14(7), 503; https://doi.org/10.3390/axioms14070503 - 27 Jun 2025

Viewed by 181

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if [...] Read more.

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if

▵_{A, B}^{m} (X) = (\sum_{j = 0}^{m} (\begin{matrix} m \\ j \end{matrix}) L_{A}^{j} R_{B}^{j}) (X) = 0

and

▵_{A, B}^{m - 1} (X) \neq 0

, where

L_{A}

and

R_{B} \in B (B (X))

are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators

E_{A, B}

and

E_{A, B} (X)

by

E_{A, B} = L_{A} R_{B}

and

{(E_{A, B} (X))}^{n} = E_{A, B}^{n} (X)

for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating

(m, X)

-isometric properties of pairs

(A_{1}, A_{2})

and

(B_{1}, B_{2})

to pairs

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

and

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

is proved: if

A_{i}, B_{i}, S_{i}, X

are operators in

B (X)

,

1 \leq i \leq 2

and X a quasi-affinity, then the pair

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

(resp., the pair

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

) is strict

(m, X)

-isometric for all

X \in B (X)

if and only if there exist positive integers

m_{i} \leq m

,

1 \leq i \leq 2

and

m = m_{1} + m_{2} - 1

, and a non-zero scalar

β

such that

I - E_{β A_{1}, A_{2}} (S_{1})

is (strict)

m_{1}

-nilpotent and

I - E_{\frac{1}{β} B_{1}, B_{2}} (S_{2})

is (strict)

m_{2}

-nilpotent (resp.,

(β A_{1}, B_{1})

is strict

(m_{1}, I)

-isometric and

(\frac{1}{β} B_{2}, A_{2})

is strict

(m_{2}, I)

-isometric). Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

20 pages, 288 KiB

Open AccessArticle

On the Existence Theorems of Equilibrium and Quasi-Equilibrium Problems in Different Spaces

by Ali Farajzadeh, Mahmood Ghobadi and Jen-Chih Yao

Mathematics 2025, 13(4), 564; https://doi.org/10.3390/math13040564 - 8 Feb 2025

Viewed by 573

Abstract

In this article, the solvability of the equilibrium problem (EP), Minty equilibrium problem (MEP), and quasi-equilibrium problem (QEP) by using the notions of cyclically monotone and cyclically antimonotone in the setting of topological vector spaces and metric spaces is investigated. Also, the concepts [...] Read more.

In this article, the solvability of the equilibrium problem (EP), Minty equilibrium problem (MEP), and quasi-equilibrium problem (QEP) by using the notions of cyclically monotone and cyclically antimonotone in the setting of topological vector spaces and metric spaces is investigated. Also, the concepts transfer lower continuity, transfer weakly lower continuity, lower semicontinuity, from above, and sequentially weakly lower semicontinous which are weaker notions than the lower semicontinuity for establishing the existence results for EP and QEP, and the other forms of them are applied. Moreover, by using the famous results for the minimum points of a function, some existence theorems, by using the triangle property, of solutions for EP and QEP are given when the domains of bifunctions are compact and not compact. The results of this paper can be viewed as new versions of the corresponding published results with new and mild assumptions. Full article

(This article belongs to the Special Issue Variational Inequality, 2nd Edition)

17 pages, 304 KiB

Open AccessArticle

Quasi-Lower C² Functions and Their Application to Nonconvex Variational Problems

by Messaoud Bounkhel

Axioms 2024, 13(12), 870; https://doi.org/10.3390/axioms13120870 - 13 Dec 2024

Viewed by 670

Abstract

This study presents a novel category of nonconvex functions in Banach spaces, referred to as quasi-lower

C^{2}

functions on nonempty closed sets. We establish the existence of solutions for nonconvex variational problems involving quasi-lower

C^{2}

functions defined in Banach spaces. To [...] Read more.

This study presents a novel category of nonconvex functions in Banach spaces, referred to as quasi-lower

C^{2}

functions on nonempty closed sets. We establish the existence of solutions for nonconvex variational problems involving quasi-lower

C^{2}

functions defined in Banach spaces. To illustrate the applicability of our findings, an example is provided in

L^{p}

spaces. Full article

(This article belongs to the Special Issue New Developments in Analysis of Variational Inequalities and Related Fields)

25 pages, 350 KiB

Open AccessArticle

Fixed-Point Results with Applications in Generalized Neutrosophic Rectangular b-Metric Spaces

by Nawab Hussain, Nawal Alharbi and Ghada Basendwah

Axioms 2024, 13(12), 818; https://doi.org/10.3390/axioms13120818 - 24 Nov 2024

Cited by 1 | Viewed by 680

Abstract

In this paper, we introduce several new concepts: generalized neutrosophic rectangular b-metric-like spaces (GNRBMLSs), generalized intuitionistic rectangular b-metric-like spaces (GIRBMLSs), and generalized fuzzy rectangular b-metric-like spaces (GFRBMLSs). These innovative spaces can expand various topological spaces, including neutrosophic rectangular extended b [...] Read more.

In this paper, we introduce several new concepts: generalized neutrosophic rectangular b-metric-like spaces (GNRBMLSs), generalized intuitionistic rectangular b-metric-like spaces (GIRBMLSs), and generalized fuzzy rectangular b-metric-like spaces (GFRBMLSs). These innovative spaces can expand various topological spaces, including neutrosophic rectangular extended b-metric-like spaces, intuitionistic fuzzy rectangular extended b-metric-like spaces, and fuzzy rectangular extended b-metric-like spaces. Moreover, we establish Banach’s fixed point theorem and Ćirić’s quasi-contraction theorem with respect to these spaces, and we explore an application regarding the existence and uniqueness of solutions for fuzzy fractional delay integro-differential equations, as derived from our main results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications)

20 pages, 330 KiB

Open AccessArticle

Quasi-Compactness of Operators for General Markov Chains and Finitely Additive Measures

by Alexander Zhdanok

Mathematics 2024, 12(19), 3155; https://doi.org/10.3390/math12193155 - 9 Oct 2024

Viewed by 841

Abstract

We study Markov operators T, A, and

T^{*}

of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach [...] Read more.

We study Markov operators T, A, and

T^{*}

of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach space of all bounded countably additive measures. We construct an operator

T^{*}

, topologically conjugate to the operator T, acting in the space of all bounded finitely additive measures. We prove the main result of the paper that, in general, a Markov operator

T^{*}

is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator

T^{*}

is quasi-compact if and only if the Doeblin condition

(D)

is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T, A, and

T^{*}

are equivalent to each other. In addition, we obtain that, for an operator

T^{*}

to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator

T^{*}

in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin’s counterexample is provided. Full article

(This article belongs to the Section D1: Probability and Statistics)

16 pages, 276 KiB

Open AccessArticle

Monotonicities of Quasi-Normed Orlicz Spaces

by Dong Ji and Yunan Cui

Axioms 2024, 13(10), 696; https://doi.org/10.3390/axioms13100696 - 7 Oct 2024

Viewed by 848

Abstract

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces [...] Read more.

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces

L^{p} (p \geq 1)

and

L^{p} (0 < p < 1)

. The Orlicz spaces

L^{Φ}

are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given. Full article

15 pages, 291 KiB

Open AccessArticle

Orthogonal Stability and Solution of a Three-Variable Functional Equation in Extended Banach Spaces

by Jagjeet Jakhar, Shalu Sharma, Jyotsana Jakhar, Majeed A. Yousif, Pshtiwan Othman Mohammed, Alina Alb Lupas and Nejmeddine Chorfi

Mathematics 2024, 12(18), 2868; https://doi.org/10.3390/math12182868 - 14 Sep 2024

Cited by 2 | Viewed by 988

Abstract

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-

β

-Banach spaces and multi-Banach spaces. Additionally, the study [...] Read more.

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-

β

-Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

17 pages, 312 KiB

Open AccessArticle

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

by María A. Navascués

Axioms 2024, 13(7), 474; https://doi.org/10.3390/axioms13070474 - 13 Jul 2024

Cited by 2 | Viewed by 955

Abstract

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations [...] Read more.

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

22 pages, 315 KiB

Open AccessArticle

Fixed Point Dynamics in a New Type of Contraction in b-Metric Spaces

by María A. Navascués and Ram N. Mohapatra

Symmetry 2024, 16(4), 506; https://doi.org/10.3390/sym16040506 - 22 Apr 2024

Cited by 8 | Viewed by 1784

Abstract

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity [...] Read more.

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

19 pages, 346 KiB

Open AccessArticle

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

by Aeshah Abdullah Muhammad Al-Dosari

Fractal Fract. 2024, 8(4), 235; https://doi.org/10.3390/fractalfract8040235 - 17 Apr 2024

Cited by 2 | Viewed by 1448

Abstract

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area [...] Read more.

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (

E_{α, β}

), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

16 pages, 306 KiB

Open AccessFeature PaperArticle

On Protected Quasi-Metrics

by Salvador Romaguera

Axioms 2024, 13(3), 158; https://doi.org/10.3390/axioms13030158 - 28 Feb 2024

Cited by 1 | Viewed by 1865

Abstract

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, [...] Read more.

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations. Full article

(This article belongs to the Section Geometry and Topology)

18 pages, 315 KiB

Open AccessArticle

On Generalized Sehgal–Guseman-Like Contractions and Their Fixed-Point Results with Applications to Nonlinear Fractional Differential Equations and Boundary Value Problems for Homogeneous Transverse Bars

by Muhammad Din, Umar Ishtiaq, Muzammil Mukhtar, Salvatore Sessa and Hassan Ali Ghazwani

Mathematics 2024, 12(4), 541; https://doi.org/10.3390/math12040541 - 8 Feb 2024

Cited by 3 | Viewed by 1278

Abstract

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in

S

-metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, [...] Read more.

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in

S

-metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, quasi-contractions, Ćirić–Reich–Rus-type contractions, Hardy–Rogers-type contractions, Reich-type contractions, interpolative Kannan contractions, interpolative Chatterjee contractions, among others, with their generalizations in

S

-metric spaces. We offer significant examples to substantiate the reliability of our results. This study also establishes consequential fixed-point results and applies them to nonlinear fractional differential equations and the boundary value problem for homogeneous transverse bars. At the end of the manuscript, we present an important open problem. Full article

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 - 1 Feb 2024

Viewed by 1153

Abstract

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)

20 pages, 325 KiB

Open AccessArticle

On Normed Algebras and the Generalized Maligranda–Orlicz Lemma

by Mieczysław Cichoń and Kinga Cichoń

Symmetry 2024, 16(1), 56; https://doi.org/10.3390/sym16010056 - 31 Dec 2023

Cited by 2 | Viewed by 1461

Abstract

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent [...] Read more.

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

17 pages, 333 KiB

Open AccessArticle

Stochastic Quasi-Geostrophic Equation with Jump Noise in L_p Spaces

by Jiahui Zhu, Xinyun Wang and Heling Su

Mathematics 2023, 11(22), 4608; https://doi.org/10.3390/math11224608 - 10 Nov 2023

Viewed by 827

Abstract

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild

L^{p} (D)

-solutions for the dissipative quasi-geostrophic equation with a full range [...] Read more.

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild

L^{p} (D)

-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers

α \in (\frac{1}{2}, 1]

by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in

L^{\infty} (0, T; L^{p} (D))

space for all

p > \frac{2}{2 α - 1}

, which is consistent with the deterministic case. Full article

(This article belongs to the Special Issue Stochastic Models with Applications, 2nd Edition)

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