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Axioms
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Research on Functional Analysis and Its Applications
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Research on Functional Analysis and Its Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 25 December 2024 | Viewed by 4440

Special Issue Editor

Dr. Ji Eun Kim

E-Mail Website
Guest Editor
Department of Mathematics, Dongguk University, Gyeongju-si 38066, Republic of Korea
Interests: calculus; quaternion analysis; Clifford analysis; mathematical physics; computational methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue of Axioms, entitled “Research on Functional Analysis and Its Application”, will present original research papers on all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists from a variety of interdisciplinary areas will be published, with an emphasis on the field of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. This Special Issue also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.

This publication will include carefully selected original research papers on nonlinear functional analysis and applications. Potential topics may ordinary differential equations, all kinds of partial differential equations, functional differential equations, integrodifferential equations, control theory, approximation theory, optimal control, optimization theory, numerical analysis, variational inequalities, asymptotic behavior of solutions, fixed-point theory, dynamic systems, and complementarity problems.

Dr. Ji Eun Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • quaternion analysis
  • mathematical physics
  • computational methods
  • partial differential equations
  • several complex analysis

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (6 papers)

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Research

14 pages, 296 KiB  
Article
I-Convergence Sequence Paranormed Spaces of Order (α, β)
by Lian-Ta Su, Ravi Kumar, Sunil K. Sharma, Ajay K. Sharma and Qing-Bo Cai
Axioms 2024, 13(9), 626; https://doi.org/10.3390/axioms13090626 - 12 Sep 2024
Viewed by 484
Abstract
In this paper, we introduce and rigorously define a novel class of difference sequence spaces, denoted by wI(M,∆vu,r)αβ, [...] Read more.
In this paper, we introduce and rigorously define a novel class of difference sequence spaces, denoted by wI(M,∆vu,r)αβ, w0I(M,∆vu,r)αβ, w∞I(M,∆vu,r)αβ, and w∞(M,∆vu,r)αβ. These spaces are constructed through the application of the concept of I-convergence of sequences, combined with a Musielak–Orlicz function of order (α, β). The primary focus of our work is to thoroughly investigate the algebraic and topological properties of these defined sequence spaces. We explore their linearity, examine their structure within the framework of paranormed spaces, and analyze various other algebraic characteristics pertinent to these spaces. In addition, we examine the topological nature of these sequence spaces, identifying the conditions under which they exhibit specific topological properties. A significant part of our study is dedicated to examining the inclusion relationships between these sequence spaces, thereby providing a comprehensive understanding of how these spaces are interrelated. Our analysis contributes to the broader field of functional analysis and sequence space theory, offering new insights and potential applications of these advanced mathematical constructs. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
28 pages, 436 KiB  
Article
Locally Convex Spaces with Sequential Dunford–Pettis Type Properties
by Saak Gabriyelyan
Axioms 2024, 13(7), 491; https://doi.org/10.3390/axioms13070491 - 22 Jul 2024
Viewed by 754
Abstract
Let p,q,q′∈[1,∞], q′≤q. Several new characterizations of locally convex spaces with the sequential Dunford–Pettis property of order (p,q) are given. We introduce and [...] Read more.
Let p,q,q′∈[1,∞], q′≤q. Several new characterizations of locally convex spaces with the sequential Dunford–Pettis property of order (p,q) are given. We introduce and thoroughly study the sequential Dunford–Pettis* property of order (p,q) of locally convex spaces (in the realm of Banach spaces, the sequential DP(p,∞)* property coincides with the well-known DPp* property). Being motivated by the coarse p-DP* property and the p-Dunford–Pettis relatively compact property for Banach spaces, we define and study the coarse sequential DP(p,q)* property, the coarse DPp* property and the p-Dunford–Pettis sequentially compact property of order (q′,q) in the class of all locally convex spaces. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
14 pages, 402 KiB  
Article
Some Statistical and Direct Approximation Properties for a New Form of the Generalization of q-Bernstein Operators with the Parameter λ
by Lian-Ta Su, Esma Kangal, Ülkü Dinlemez Kantar and Qing-Bo Cai
Axioms 2024, 13(7), 485; https://doi.org/10.3390/axioms13070485 - 18 Jul 2024
Viewed by 545
Abstract
In this study, a different generalization of q-Bernstein operators with the parameter λ∈[−1,1] is created. The moments and central moments of these operators are calculated, a statistical approximation result for this new type of [...] Read more.
In this study, a different generalization of q-Bernstein operators with the parameter λ∈[−1,1] is created. The moments and central moments of these operators are calculated, a statistical approximation result for this new type of (λ,q)-Bernstein operators is obtained, and the convergence properties are analyzed using the Peetre K-functional and the modulus of continuity for this new operator. Finally, a numerical example is given to illustrate the convergence behavior of the newly defined operators. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
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43 pages, 436 KiB  
Article
A Theory for Interpolation of Metric Spaces
by Robledo Mak’s Miranda Sette, Dicesar Lass Fernandez and Eduardo Brandani da Silva
Axioms 2024, 13(7), 439; https://doi.org/10.3390/axioms13070439 - 28 Jun 2024
Viewed by 573
Abstract
In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without [...] Read more.
In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
24 pages, 374 KiB  
Article
An Introduction to Extended Gevrey Regularity
by Nenad Teofanov, Filip Tomić and Milica Žigić
Axioms 2024, 13(6), 352; https://doi.org/10.3390/axioms13060352 - 24 May 2024
Viewed by 844
Abstract
Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in [...] Read more.
Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
13 pages, 266 KiB  
Article
Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field
by Ji-Eun Kim
Axioms 2024, 13(5), 291; https://doi.org/10.3390/axioms13050291 - 25 Apr 2024
Viewed by 786
Abstract
In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as [...] Read more.
In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure. Full article
(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)
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