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Axioms
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New Perspectives in Operator Theory and Functional Analysis
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New Perspectives in Operator Theory and Functional Analysis

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

Deadline for manuscript submissions: 30 April 2026 | Viewed by 4035

Special Issue Editor

Prof. Dr. Chung-Chuan Chen

E-Mail Website
Guest Editor
Department of Mathematics Education, National Taichung University of Education, Taichung 403, Taiwan
Interests: operator theory; functional analysis; abstract harmonic analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Operator theory and functional analysis are important and significant topics in mathematics with various applications in other research areas. Indeed, operator theory includes the general theory of linear operators; special classes of linear operators; individual linear operators as elements of algebraic systems; integral, integrodifferential, and pseudodifferential operators; equations and inequalities involving nonlinear operators and linear spaces; algebras of operators; and so on. Also, functional analysis covers topological linear spaces and related structures for function spaces, normed linear spaces and Banach spaces, inner product spaces and their generalizations, linear function spaces and their duals, topological algebras, normed rings and algebras, Banach algebras for group algebras, and self-adjoint operator algebras.

Through this Special Issue, we expect to collect research papers on the most recent progress, concerns, and questions in this direction. Therefore, we invite researchers to contribute their original and significant research papers which will inspire advances in operator theory and functional analysis.

Prof. Dr. Chung-Chuan Chen
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

  • operator theory
  • functional analysis
  • linear operator
  • nonlinear operator
  • operator algebras

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Published Papers (7 papers)

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Research

13 pages, 295 KB  
Article
On Corresponding Cauchy–Riemann Equations Applied to Laplace-Type Operators over Generalized Quaternions, with an Application
by Ji Eun Kim
Axioms 2025, 14(9), 700; https://doi.org/10.3390/axioms14090700 (registering DOI) - 16 Sep 2025
Abstract
In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra Hα,β, with α,β∈R∖{0}. Starting from left/right difference quotients, we [...] Read more.
In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra Hα,β, with α,β∈R∖{0}. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm N vanishes, these first-order systems are necessary and, under C1 regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization Δα,β=D¯D=DD¯ shows that each real component of a differentiable mapping is Δα,β-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case (α,β)=(−1,−1), we present a Poisson-type representation solving a model Dirichlet problem on the unit ball B⊂R4, recovering mean-value and maximum principles. For computation and symbolic verification, real 4×4 matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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28 pages, 604 KB  
Article
A Study of Global Dynamics and Oscillatory Behavior of Rational-Type Nonlinear Fuzzy Difference Equations with Exponential Decay
by Sara Saud, Carlo Cattani, Muhammad Tanveer, Muhammad Usman and Asifa Tassaddiq
Axioms 2025, 14(8), 637; https://doi.org/10.3390/axioms14080637 - 15 Aug 2025
Viewed by 463
Abstract
The concept of fuzzy modeling and fuzzy system design has opened new horizons of research in functional analysis, having a significant impact on major fields such as data science, machine learning, and so on. In this research, we use fuzzy set theory to [...] Read more.
The concept of fuzzy modeling and fuzzy system design has opened new horizons of research in functional analysis, having a significant impact on major fields such as data science, machine learning, and so on. In this research, we use fuzzy set theory to analyze the global dynamics and oscillatory behavior of nonlinear fuzzy difference equations with exponential decay. We discuss the stability, oscillatory patterns, and convergence of solutions under different initial conditions. The exponential structure simplifies the analysis while providing a clear understanding of the system’s behavior over time. The study reveals how fuzzy parameters influence growth or decay trends, emphasizing the method’s effectiveness in handling uncertainty. Our findings advance the understanding of higher-order fuzzy difference equations and their potential applications in modeling systems with imprecise data. Using the characterization theorem, we convert a fuzzy difference equation into two crisp difference equations. The g-division technique was used to investigate local and global stability and boundedness in dynamics. We validate our theoretical results using numerical simulations. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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14 pages, 286 KB  
Article
Element-Based Construction Methods for Uninorms on Bounded Lattices
by Ümit Ertuğrul, Merve Yeşilyurt and Radko Mesiar
Axioms 2025, 14(8), 552; https://doi.org/10.3390/axioms14080552 - 22 Jul 2025
Viewed by 221
Abstract
Uninorms are aggregation operators that generalize the t-norms (t-conorms), which are extensions of the logical connectives ∧(∨) to the fuzzy set theory. The methods of constructing uninorms on more general algebraic structures (such as bounded posets, lattices, etc.) are an [...] Read more.
Uninorms are aggregation operators that generalize the t-norms (t-conorms), which are extensions of the logical connectives ∧(∨) to the fuzzy set theory. The methods of constructing uninorms on more general algebraic structures (such as bounded posets, lattices, etc.) are an important subject of study, including an extensive work concerning these operations on the unit real interval [0, 1]. The construction of uninorms on bounded lattices has been extensively studied using various aggregation functions, such as t-norms, t-conorms, and t-subnorms. In this paper, we present construction methods for uninorms, based on the elements of a lattice, without using the existence of the mentioned operators. We determine the necessary and sufficient conditions for the introduced construction methods to result in the uninorms. Then, we show the differences between our methods and several methods known from the literature, including some illustrative examples. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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17 pages, 288 KB  
Article
Positive Solutions for Discrete Robin Problem of Kirchhoff Type Involving p-Laplacian
by Zhi Chen and Zhan Zhou
Axioms 2025, 14(4), 285; https://doi.org/10.3390/axioms14040285 - 10 Apr 2025
Viewed by 365
Abstract
The aim of this paper is to investigate the existence of positive solutions for a discrete Robin problem of the Kirchhoff type involving the p-Laplacian by the means of critical point theory. Our results demonstrate that the problem admits at least three [...] Read more.
The aim of this paper is to investigate the existence of positive solutions for a discrete Robin problem of the Kirchhoff type involving the p-Laplacian by the means of critical point theory. Our results demonstrate that the problem admits at least three solutions, or at least two solutions under different conditions on the nonlinear term f. We establish a strong maximum principle for the problem and obtain the existence and multiplicity of positive solutions. Finally, we give three examples to verify our results. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
17 pages, 307 KB  
Article
Hammerstein Nonlinear Integral Equations and Iterative Methods for the Computation of Common Fixed Points
by María A. Navascués
Axioms 2025, 14(3), 214; https://doi.org/10.3390/axioms14030214 - 15 Mar 2025
Cited by 1 | Viewed by 855
Abstract
In the first part of this article, a special type of Hammerstein nonlinear integral equation is studied. A theorem of the existence of solutions is given in the framework of L2-spaces. Afterwards, an iterative method for the resolution of this kind [...] Read more.
In the first part of this article, a special type of Hammerstein nonlinear integral equation is studied. A theorem of the existence of solutions is given in the framework of L2-spaces. Afterwards, an iterative method for the resolution of this kind of equations is considered, and the convergence of this algorithm towards a solution of the equation is proved. The rest of the paper considers two modifications of the algorithm. The first one is devoted to the sought of common fixed points of a family of nearly asymptotically nonexpansive mappings. The second variant focuses on the search of common fixed points of a finite number of nonexpansive operators. The characteristics of convergence of these methods are studied in the context of uniformly convex Banach spaces. The iterative scheme is applied to approach the common solution of three nonlinear integral equations of Hammerstein type. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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15 pages, 560 KB  
Article
Characterization Results of Extremization Models with Interval Values
by Savin Treanţă and Omar Mutab Alsalami
Axioms 2025, 14(3), 151; https://doi.org/10.3390/axioms14030151 - 20 Feb 2025
Viewed by 384
Abstract
The present paper investigates new connections and characterization results on interval-valued minimization models. Specifically, we describe the solution set of the considered control problem with mixed constraints by employing the solution set associated with a class of controlled split variational inequalities. These equivalence [...] Read more.
The present paper investigates new connections and characterization results on interval-valued minimization models. Specifically, we describe the solution set of the considered control problem with mixed constraints by employing the solution set associated with a class of controlled split variational inequalities. These equivalence results are also accompanied by suitable numerical experiments. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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19 pages, 798 KB  
Article
Multifunctional Expectile Regression Estimation in Volterra Time Series: Application to Financial Risk Management
by Somayah Hussain Alkhaldi, Fatimah Alshahrani, Mohammed Kbiri Alaoui, Ali Laksaci and Mustapha Rachdi
Axioms 2025, 14(2), 147; https://doi.org/10.3390/axioms14020147 - 19 Feb 2025
Cited by 1 | Viewed by 927
Abstract
We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of [...] Read more.
We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of the long memory function of this type of model, we analyze the reproduced movements using recent algorithms in the field of functional data analysis (FDA). In fact, we develop, in particular, new risk tools based on the asymmetric least squares loss function. We build an estimator using the multifunctional kernel (MK) method and then establish its asymptotic properties. The multidimensionality of the Volterra process is explored through the dispersion component of the convergence rate, while the nonparametric path of the risk tool affects the bias component. An empirical analysis is conducted to demonstrate the ease of implementation of our proposed approach. Additionally, an application on real data is presented to compare the effectiveness of expectile-based measures with Value at Risk (VaR) in financial risk management for multiple assets. Full article
(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)
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