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31 pages, 452 KB

Open AccessArticle

A Banach-Space Framework for Proposed (v,w)–s–Convex Response-Curve Certification in Machine Learning

by Ahad Hamoud Alotaibi, Muhammad Saeed Ahmad, Muhammad Waseem Asghar and Mujahid Abbas

Mathematics 2026, 14(12), 2209; https://doi.org/10.3390/math14122209 (registering DOI) - 19 Jun 2026

Viewed by 95

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s [...] Read more.

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s–convex response-curve certification. The response curve is treated as an element of the Banach space of continuous functions under the supremum norm, while derivative-based certificates are handled in a Lipschitz and Sobolev-type norm when required. Generalized convexity is represented through a bounded structural operator, whose order condition defines a closed convex acceptance set. The violation score is measured by the positive part of the operator residual, and the Hermite–Hadamard, Fejér, and Ostrowski quantities are interpreted as bounded certificate functionals. The auxiliary profiles are constructed from validation-curve residuals through a split-calibrated procedure and then tested on held-out triples. The framework certifies only scalar response-curve summaries under explicit structural and empirical assumptions; it does not certify a full learning system, guarantee generalization, or replace dense sampling when the structural gate fails. Full article

11 pages, 232 KB

Open AccessArticle

Fixed Point Results for Large Closed Four-Step Orbital Contractions in Metric Spaces

by Nawal Alharbi

Mathematics 2026, 14(10), 1680; https://doi.org/10.3390/math14101680 - 14 May 2026

Viewed by 231

Abstract

This paper introduces a higher-order orbital framework in fixed point theory based on a closed four-step orbital functional. Existing approaches, such as triangle-perimeter contractions, mainly rely on three-point configurations and first-order geometric interactions. In contrast, the proposed functional incorporates four successive iterates together [...] Read more.

This paper introduces a higher-order orbital framework in fixed point theory based on a closed four-step orbital functional. Existing approaches, such as triangle-perimeter contractions, mainly rely on three-point configurations and first-order geometric interactions. In contrast, the proposed functional incorporates four successive iterates together with a nonlocal comparison term involving second-order orbital displacements. Using this structure, we define a new class of large closed four-step orbital contractions and establish a corresponding fixed point theorem in complete metric spaces under a boundedness assumption on one orbit. The proof is based on a propagation mechanism that transfers contractive behavior along the orbit generated by the mapping. Several examples demonstrate that the proposed framework extends classical contraction settings such as Banach and triangle-perimeter contractions. Furthermore, an application to a nonlinear Volterra integral equation provides explicit analytical estimates showing how the four-step orbital contraction structure can be verified in functional settings. These results provide a higher-order orbital extension of existing contraction principles and may contribute to further developments in generalized metric spaces and nonlinear analysis. Full article

(This article belongs to the Topic Fixed Point Theory and Measure Theory)

33 pages, 458 KB

Open AccessArticle

Symmetric Analytic Functions on Banach Spaces Associated with the Cantor Set

by Iryna Chernega, Roman Dmytryshyn, Zoriana Novosad, Serhii Sharyn and Andriy Zagorodnyuk

Symmetry 2026, 18(5), 716; https://doi.org/10.3390/sym18050716 - 23 Apr 2026

Viewed by 304

Abstract

We consider Banach spaces

ℓ_{p} (C),

1 \leq p < \infty,

where the index set

C

is the classical Cantor set and study various groups of symmetries of

ℓ_{p} (C),

associated with the [...] Read more.

We consider Banach spaces

ℓ_{p} (C),

1 \leq p < \infty,

where the index set

C

is the classical Cantor set and study various groups of symmetries of

ℓ_{p} (C),

associated with the binary representation of

C .

The main purpose of the paper is the investigation of polynomials on

ℓ_{p} (C)

that are symmetric (i.e., invariant) with respect to the constructed groups

G .

We are interested in finding systems of generators of algebras of G-symmetric polynomials for different groups G and we discuss possible applications of G-symmetric polynomials to highly composite physical systems. The generators are useful for descriptions of spectra of algebras of G-symmetric analytic functions on

ℓ_{p} (C),

and for the construction of some nontrivial complex homomorphisms of these algebras. Finally, we establish the topological transitivity and hypercyclicity of some shift-like operators on

ℓ_{p} (C)

and its subspaces, and translation operators on algebras of symmetric analytic functions on

ℓ_{p} (C) .

Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

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25 pages, 447 KB

Open AccessArticle

Stability and Controllability of Coupled Neutral Impulsive ϱ-Fractional System with Mixed Delays

by F. Gassem, Mohammed Almalahi, Mohammed Rabih, Manal Y. A. Juma, Amira S. Awaad, Ali H. Tedjani and Khaled Aldwoah

Fractal Fract. 2026, 10(3), 192; https://doi.org/10.3390/fractalfract10030192 - 13 Mar 2026

Cited by 2 | Viewed by 842

Abstract

This study examines a comprehensive class of coupled nonlinear

ϱ

-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, [...] Read more.

This study examines a comprehensive class of coupled nonlinear

ϱ

-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, and impulsive disturbances, which have been studied separately by researchers. We utilize the Banach contraction principle and Krasnoselskii’s fixed-point theorem to provide suitable conditions for the existence and uniqueness of solutions within the product space of piecewise continuous weighted functions. In addition to existence, we examine Ulam–Hyers–Rassias (UHR) stability using a generalized Gronwall inequality, which guarantees the system’s robustness against functional perturbations. We also develop a controllability framework and a feedback control law that steer the system towards the desired terminal states. The theoretical results are supported by a numerical simulation using a complex kernel, implemented via a modified predictor-corrector algorithm, which validates the practical effectiveness of the proposed control and stability outcomes. Full article

(This article belongs to the Section Complexity)

24 pages, 913 KB

Open AccessArticle

A Semi-Analytical and Topological Study of Fractional Dynamical Systems in Banach Spaces Endowed with the Compact-Open Topology: Applications to Wave Propagation Phenomena

by Hasan N. Zaidi, Amin Saif, Muntasir Suhail, Neama Haron, Amira S. Awaad, Khaled Aldwoah and Ali H. Tedjani

Fractal Fract. 2026, 10(3), 181; https://doi.org/10.3390/fractalfract10030181 - 11 Mar 2026

Viewed by 511

Abstract

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer [...] Read more.

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case

0 < α < 1

. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems. Full article

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30 pages, 420 KB

Open AccessArticle

Subsymmetric Polynomials on Banach Spaces and Their Applications

by Vitalii Bihun, Daryna Dolishniak, Viktoriia Kravtsiv and Andriy Zagorodnuyk

Mathematics 2025, 13(22), 3693; https://doi.org/10.3390/math13223693 - 18 Nov 2025

Cited by 1 | Viewed by 537

Abstract

We investigate algebraic and topological properties of subsymmetric polynomials on finite- and infinite-dimensional spaces. In particular, we focus on the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, as well as possible extensions of subsymmetric polynomials and [...] Read more.

We investigate algebraic and topological properties of subsymmetric polynomials on finite- and infinite-dimensional spaces. In particular, we focus on the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, as well as possible extensions of subsymmetric polynomials and analytic functions to larger spaces. We consider algebras of subsymmetric analytic functions of bounded type and their spectra, and study linear subspaces in the zero-sets of subsymmetric polynomials, as well as subspaces where a subsymmetric polynomial is symmetric. In addition, we propose some possible applications of subsymmetric polynomials in cryptography and in operator theory. Full article

(This article belongs to the Section C: Mathematical Analysis)

17 pages, 325 KB

Open AccessArticle

Descriptions of Spectra of Algebras of Bounded-Type Block-Symmetric Analytic Functions

by Viktoriia Kravtsiv and Andriy Zagorodnyuk

Symmetry 2025, 17(11), 1974; https://doi.org/10.3390/sym17111974 - 15 Nov 2025

Cited by 1 | Viewed by 504

Abstract

This paper is devoted to the study of the algebra of bounded-type block-symmetric analytic functions on the Banach space

l_{1} (C^{s}) .

In particular, it presents a description of the spectrum of this algebra in terms of exceptional characters [...] Read more.

This paper is devoted to the study of the algebra of bounded-type block-symmetric analytic functions on the Banach space

l_{1} (C^{s}) .

In particular, it presents a description of the spectrum of this algebra in terms of exceptional characters

ϕ_{α}

and characters that can be associated with exponential-type functions of several variables with plane zeros. Due to this representation, it is proven that every element of the spectrum is a convolution of an exceptional character with a point evaluation functional. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Applications, 2nd Edition)

21 pages, 395 KB

Open AccessArticle

An Efficient Iteration Method for Fixed-Point Approximation and Its Application to Fractional Volterra–Fredholm Integro–Differential Equations

by Ekta Sharma, Shubham Kumar Mittal, Sunil Panday and Lorentz Jäntschi

Axioms 2025, 14(11), 830; https://doi.org/10.3390/axioms14110830 - 11 Nov 2025

Viewed by 1256

Abstract

This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and [...] Read more.

This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and more reliably than established Picard-type methods. Its application to fractional models involving the Gamma function highlights the method’s efficiency and potential for broader use in nonlinear and fractional systems. Full article

(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

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26 pages, 382 KB

Open AccessArticle

Some Realisation of the Banach Space of All Continuous Linear Functionals on ℓ₁ Approximated by Weakly Symmetric Continuous Linear Functionals

by Mykhailo Varvariuk and Taras Vasylyshyn

Symmetry 2025, 17(11), 1896; https://doi.org/10.3390/sym17111896 - 6 Nov 2025

Viewed by 603

Abstract

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{1}

(i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation [...] Read more.

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{1}

(i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation is a periodic sequence. We consider the completion of the space of all such linear continuous functionals on

ℓ_{1}

with periods of Riesz representations equal to powers of 2. It is known that this completion is a Banach space with a Schauder basis. In this work, we construct a sequence Banach space with the standard Schauder basis

{e_{m} = (\underset{m - 1}{\underset{︸}{0, \dots, 0}}, 1, 0, \dots)}_{m = 1}^{\infty}

that is isometrically isomorphic to this completion. Results of the work can be used to describe spectra of topological algebras of analytic functions on

ℓ_{1}

that can be approximated by weakly symmetric functions. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis: 3rd Edition)

20 pages, 436 KB

Open AccessArticle

Numerical Solutions for Fractional Bagley–Torvik Equation with Integral Boundary Conditions

by Xueling Liu, Jing Huang, Junlin Li and Yufeng Zhang

Symmetry 2025, 17(10), 1755; https://doi.org/10.3390/sym17101755 - 17 Oct 2025

Cited by 2 | Viewed by 951

Abstract

The Bagley–Torvik equation (BTE) is an important model in mathematical physics and mechanics, but obtaining its analytical solution remains challenging. For its numerical treatment, the presence of composite functions in the generalized BTE poses additional difficulties, and efficient approaches for handling nonlinear terms [...] Read more.

The Bagley–Torvik equation (BTE) is an important model in mathematical physics and mechanics, but obtaining its analytical solution remains challenging. For its numerical treatment, the presence of composite functions in the generalized BTE poses additional difficulties, and efficient approaches for handling nonlinear terms are still lacking in the literature. This study proposes an improved numerical method for the fractional BTE with integral boundary conditions. By employing an integration technique, the original problem is transformed into a weakly singular Fredholm–Hammerstein (F–H) integral equation of the second kind. To address the nonlinear terms, an enhanced piecewise Taylor expansion scheme is developed to construct the discrete form, while the uniqueness of the solution is proven using the contraction mapping theorem in Banach spaces. The convergence and error analyses are rigorously carried out, and numerical experiments confirm the accuracy and efficiency of the proposed approach. Full article

(This article belongs to the Section Mathematics)

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11 pages, 452 KB

Open AccessArticle

A Banach Space Leap: Contraction Mapping Solutions for Stochastic Delay Systems

by Fatin Nabila Abd Latiff, Dawn A. Stoner, Kah Lun Wang and Kok Bin Wong

Mathematics 2025, 13(18), 3002; https://doi.org/10.3390/math13183002 - 17 Sep 2025

Viewed by 999

Abstract

We investigate the solvability and stability properties of a class of nonlinear stochastic delay differential equations (SDDEs) driven by Wiener noise and incorporating discrete time delays. The equations are formulated within a Banach space of continuous, adapted sample paths. Under standard Lipschitz and [...] Read more.

We investigate the solvability and stability properties of a class of nonlinear stochastic delay differential equations (SDDEs) driven by Wiener noise and incorporating discrete time delays. The equations are formulated within a Banach space of continuous, adapted sample paths. Under standard Lipschitz and linear growth conditions, we construct a solution operator and prove the existence and uniqueness of strong solutions using a fixed-point argument. Furthermore, we derive exponential mean-square stability via Lyapunov-type techniques and delay-dependent inequalities. This framework provides a unified and flexible approach to SDDE analysis that departs from traditional Hilbert space or semigroup-based methods. We explore a Banach space fixed-point approach to SDDEs with multiplicative noise and discrete delays, providing a novel functional-analytic framework for examining solvability and stability. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications: 3rd Edition)

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14 pages, 280 KB

Open AccessArticle

Essential Norm of Products of Volterra-Type Operators and Composition Operators on Iterated Banach-Type Spaces

by Rabab Alyusof, Shams Alyusof and Nacir Hmidouch

Axioms 2025, 14(5), 352; https://doi.org/10.3390/axioms14050352 - 4 May 2025

Viewed by 939

Abstract

Let

H (D)

be the set of analytic functions on the open unit disk

D

. For

n \in N_{0} : = N \cup {0}

, define the iterated weighted-type Banach space [...] Read more.

Let

H (D)

be the set of analytic functions on the open unit disk

D

. For

n \in N_{0} : = N \cup {0}

, define the iterated weighted-type Banach space

V_{n} : = \{f \in H (D) : {sup}_{z \in D} {(1 - | z |}^{2}) | f^{(n)} (z) | < \infty\} .

In this work, we study the boundedness and the essential norm of products of Volterra-type operators and composition operators on iterated weighted-type Banach spaces. Full article

25 pages, 368 KB

Open AccessArticle

LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

by G. F. Helminck and J. A. Weenink

Geometry 2025, 2(2), 4; https://doi.org/10.3390/geometry2020004 - 15 Apr 2025

Viewed by 1484

Abstract

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of [...] Read more.

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of the deformations of the powers

{S^{i}, i ⩾ 1}

. They are called the

k [S]

-hierarchy and the strict

k [S]

-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in

GL (B)

can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup

G_{c p t} (B)

of

GL (B)

such that the procedure with LU factorizations works for each

g \in G_{c p t} (B)

. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

39 pages, 391 KB

Open AccessArticle

Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations

by Chenkuan Li and Wenyuan Liao

Fractal Fract. 2025, 9(4), 200; https://doi.org/10.3390/fractalfract9040200 - 25 Mar 2025

Cited by 4 | Viewed by 1037

Abstract

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a [...] Read more.

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a generalized Mittag-Leffler function, as well as a few popular fixed-point theorems. These studies have good applications in general since uniqueness, existence and stability are key and important topics in many fields. Several examples are presented to demonstrate applications of results obtained by computing approximate values of the generalized Mittag-Leffler functions. (ii) We use the inverse operator method and newly established spaces to find analytic solutions to a number of notable partial differential equations, such as a multi-term time-fractional convection problem and a generalized time-fractional diffusion-wave equation in

R^{n}

with initial conditions only, which have never been previously considered according to the best of our knowledge. In particular, we deduce the uniform solution to the non-homogeneous wave equation in n dimensions for all

n \geq 1

, which coincides with classical results such as d’Alembert and Kirchoff’s formulas but is much easier in the computation of finding solutions without any complicated integrals on balls or spheres. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

19 pages, 4920 KB

Open AccessArticle

Analytical and Computational Investigations of Stochastic Functional Integral Equations: Solution Existence and Euler–Karhunen–Loève Simulation

by Manochehr Kazemi, AliReza Yaghoobnia, Behrouz Parsa Moghaddam and Alexandra Galhano

Mathematics 2025, 13(3), 427; https://doi.org/10.3390/math13030427 - 27 Jan 2025

Viewed by 1534

Abstract

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, [...] Read more.

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics. Full article

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

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