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Volume 15
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Axioms, Volume 15, Issue 1 (January 2026) – 68 articles

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20 pages, 323 KB  
Article
On the Categories of ℒℱ-Ideals, ℒℱ-Grills, and ℒℱ-Topological Spaces
by Ahmed A. Ramadan and Anwar J. Fawakhreh
Axioms 2026, 15(1), 68; https://doi.org/10.3390/axioms15010068 (registering DOI) - 19 Jan 2026
Abstract
This paper is devoted to the study of the interrelationships among LF-grills, LF-ideals, LF-neighborhoods, LF-topologies, and LF-co-topologies. We establish a categorical framework that demonstrates the interconnections among these concepts. In addition, we investigate categorical connections from LF-ideal [...] Read more.
This paper is devoted to the study of the interrelationships among LF-grills, LF-ideals, LF-neighborhoods, LF-topologies, and LF-co-topologies. We establish a categorical framework that demonstrates the interconnections among these concepts. In addition, we investigate categorical connections from LF-ideal spaces to LF-topological spaces and from LF-grill spaces to LF-topological spaces using concrete functors, confirming the existence of Galois correspondences between these spaces. Finally, the practical relevance of the theoretical framework is illustrated through applications in information systems and medical diagnosis. Full article
9 pages, 505 KB  
Article
On the Qualitative Behaviors of Solutions of the Sunflower-Type Equation with Multiple Constant Delays
by Sultan Erdur
Axioms 2026, 15(1), 67; https://doi.org/10.3390/axioms15010067 - 18 Jan 2026
Abstract
This paper investigates several qualitative behaviors of the solutions for a class of second-order nonlinear delay differential equations (DDEs) characterized by multiple constant delays. Applying the Lyapunov–Krasovskii (LK) approach, together with LaSalle’s Invariance Principle, we derive new conditions for stability when [...] Read more.
This paper investigates several qualitative behaviors of the solutions for a class of second-order nonlinear delay differential equations (DDEs) characterized by multiple constant delays. Applying the Lyapunov–Krasovskii (LK) approach, together with LaSalle’s Invariance Principle, we derive new conditions for stability when p(t,x,y)≡0 and boundedness and integrability whenever p(t,x,y) is non-zero. The results obtained generalize some existing theorems in the literature to the case of multiple delay configurations, accommodating a wider class of sunflower-type equations. Full article
(This article belongs to the Section Mathematical Analysis)
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5 pages, 174 KB  
Editorial
Differential Geometry and Its Application, 3rd Edition
by Mića S. Stanković
Axioms 2026, 15(1), 66; https://doi.org/10.3390/axioms15010066 - 18 Jan 2026
Abstract
In this Editorial, we introduce the Special Issue of Axioms entitled “Differential Geometry and Its Application, 3rd Edition [...] Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)
16 pages, 351 KB  
Article
Iterative Integro-Differential Techniques Based on Green’s Function for Two-Point Boundary-Value Problems of Ordinary Differential Equations
by Juan I. Ramos
Axioms 2026, 15(1), 65; https://doi.org/10.3390/axioms15010065 - 17 Jan 2026
Viewed by 47
Abstract
Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed [...] Read more.
Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article
(This article belongs to the Section Mathematical Analysis)
12 pages, 382 KB  
Article
Numerical Solution of Fractional Third-Order Nonlinear Emden–Fowler Delay Differential Equations via Chebyshev Polynomials
by Mashael M. AlBaidani
Axioms 2026, 15(1), 64; https://doi.org/10.3390/axioms15010064 - 17 Jan 2026
Viewed by 48
Abstract
In the current study, we used Chebyshev’s Pseudospectral Method (CPM), a novel numerical technique, to solve nonlinear third-order Emden–Fowler delay differential (EF-DD) equations numerically. Fractional derivatives are defined by the Caputo operator. These kinds of equations are transformed to the linear or nonlinear [...] Read more.
In the current study, we used Chebyshev’s Pseudospectral Method (CPM), a novel numerical technique, to solve nonlinear third-order Emden–Fowler delay differential (EF-DD) equations numerically. Fractional derivatives are defined by the Caputo operator. These kinds of equations are transformed to the linear or nonlinear algebraic equations by the proposed approach. The numerical outcomes demonstrate the precision and efficiency of the suggested approach. The error analysis shows that the current method is more accurate than any other numerical method currently available. The computational analysis fully confirms the compatibility of the suggested strategy, as demonstrated by a few numerical examples. We present the outcome of the offered method in tables form, which confirms the appropriateness at each point. Additionally, the outcomes of the offered method at various non-integer orders are investigated, demonstrating that the result approaches closer to the accurate solution as a value approaches from non-integer order to an integer order. Additionally, the current study proves some helpful theorems about the convergence and error analysis related to the aforementioned technique. A suggested algorithm can effectively be used to solve other physical issues. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
12 pages, 258 KB  
Article
The Sneddon R-Transform and Its Inverse over Lebesgue Spaces
by Hari Mohan Srivastava, Emilio R. Negrín and Jeetendrasingh Maan
Axioms 2026, 15(1), 63; https://doi.org/10.3390/axioms15010063 - 16 Jan 2026
Viewed by 82
Abstract
We study the Sneddon R-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel xcos(xt)+hsin(xt), the R-transform acts as a unifying operator [...] Read more.
We study the Sneddon R-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel xcos(xt)+hsin(xt), the R-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted Lp-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite R-transform. Full article
3 pages, 127 KB  
Editorial
Advances in Statistical Simulation and Computing
by Francisco Novoa-Muñoz and Bernardo M. Lagos-Álvarez
Axioms 2026, 15(1), 62; https://doi.org/10.3390/axioms15010062 - 16 Jan 2026
Viewed by 98
Abstract
In this Editorial, we are pleased to introduce the Special Issue of the journal Axioms entitled “Advances in Statistical Simulation and Computing” [...] Full article
(This article belongs to the Special Issue Advances in Statistical Simulation and Computing)
18 pages, 2195 KB  
Article
On the Expansion of Legendre Polynomials in Bicomplex Space and Coupling with Fractional Operators
by Ahmed Bakhet, Shahid Hussain, Mohra Zayed and Aya M. Mourad
Axioms 2026, 15(1), 61; https://doi.org/10.3390/axioms15010061 - 15 Jan 2026
Viewed by 74
Abstract
In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre [...] Read more.
In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre differential equation in bicomplex space. Additionally, we discuss both theoretical and practical contributions, especially in bicomplex Riemann Liouville fractional calculus. We numerically study the construction of bicomplex Legendre polynomials, orthogonality, spectral projection, coefficient decay, and spectral convergence in bicomplex space. The findings contribute to a deeper insight into bicomplex functions, paving the way for further developments in science and mathematical analysis, and providing a foundation for future research on special functions and fractional operators within the bicomplex setting. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
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11 pages, 269 KB  
Article
Shadows of Varieties Embedded in Projective Spaces
by Edoardo Ballico
Axioms 2026, 15(1), 60; https://doi.org/10.3390/axioms15010060 - 15 Jan 2026
Viewed by 86
Abstract
When two varieties X, X′ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one [...] Read more.
When two varieties X, X′ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one point of projection, there are many very different shadows with very different degrees. We give the geometric properties of some of them. These shadows are birational to the variety in which they are a shadow. We compute the minimum degree of all such shadows. For most smooth varieties X⊂Pr, r≥3, it is the integer deg(X)−1. Full article
15 pages, 851 KB  
Article
Partially Observed Two-Phase Point Processes
by Olivier Jacquet, Walguen Oscar and Jean Vaillant
Axioms 2026, 15(1), 59; https://doi.org/10.3390/axioms15010059 - 15 Jan 2026
Viewed by 143
Abstract
In this paper, a two-phase spatio-temporal point process (STPP) defined on a countable metric space and characterized by a conditional intensity function is introduced. In the first phase, the process is memoryless, generating completely random point patterns. In the second phase, the location [...] Read more.
In this paper, a two-phase spatio-temporal point process (STPP) defined on a countable metric space and characterized by a conditional intensity function is introduced. In the first phase, the process is memoryless, generating completely random point patterns. In the second phase, the location and occurrence time of each event depend on the spatial configuration of previous events, thereby inducing spatio-temporal correlation. Theoretical results that characterize the distributional properties of the process are established, enabling both efficient numerical simulation and Bayesian inference. A statistical inference framework is developed, for the setting in which the STPP is observed at discrete calendar dates while the spatial locations of events are recorded, their exact occurrence times are unobserved, i.e., interval-censored. This partial observation scheme commonly arises in ecological and epidemiological applications, such as the monitoring of plant disease or insect pest spread across a spatial grid over time. The methodology is illustrated through an analysis of the spatio-temporal spread of sugarcane yellow leaf virus (SCYLV) in an initially disease-free sugarcane plot in Guadeloupe, FrenchWest Indies. Full article
(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)
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23 pages, 350 KB  
Article
Application of Stochastic Elements in the Universality of the Periodic Zeta-Function: The Case of Short Intervals
by Marius Grigaliūnas, Antanas Laurinčikas and Darius Šiaučiūnas
Axioms 2026, 15(1), 58; https://doi.org/10.3390/axioms15010058 - 14 Jan 2026
Viewed by 86
Abstract
Let a={am:m∈N} be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip [...] Read more.
Let a={am:m∈N} be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip {s=σ+it:1/2<σ<1} by shifts ζ(s+iτ;a) of the zeta-function defined, for σ>1, by ζ(s;a)=∑m=1∞amm−s and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval [T,T+V] with T23/70⩽V⩽T1/2 as T→∞. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions. Full article
(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)
16 pages, 415 KB  
Article
Investigations of Compactness-Type Attributes in Interval Metric Spaces
by Rukhsar Khatun, Maryam G. Alshehri, Md Sadikur Rahman and Asoke Kumar Bhunia
Axioms 2026, 15(1), 57; https://doi.org/10.3390/axioms15010057 - 13 Jan 2026
Viewed by 93
Abstract
Discovering the compactness properties in generalized-type metric spaces opens up a fascinating area of research. The present study tries to develop a theoretical framework for compactness with key properties in the recently developed interval metric space. This work begins with explaining the covers [...] Read more.
Discovering the compactness properties in generalized-type metric spaces opens up a fascinating area of research. The present study tries to develop a theoretical framework for compactness with key properties in the recently developed interval metric space. This work begins with explaining the covers and open covers to define compact interval metric spaces and their main features. Next, a similar definition of compactness using the finite intersection property is introduced. Then, the famous Heine–Borel theorem for compactness is extended in the case of interval metric spaces. Also, the concepts of sequential-type compactness and Bolzano–Weierstrass (BW)-type compactness for interval metric spaces are introduced with their equivalency relationship. Finally, the notion of total boundedness in interval metric spaces and its connection with compactness is introduced, providing new insights into these mathematical concepts. Full article
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31 pages, 13946 KB  
Article
The XLindley Survival Model Under Generalized Progressively Censored Data: Theory, Inference, and Applications
by Ahmed Elshahhat and Refah Alotaibi
Axioms 2026, 15(1), 56; https://doi.org/10.3390/axioms15010056 - 13 Jan 2026
Viewed by 79
Abstract
This paper introduces a novel extension of the classical Lindley distribution, termed the X-Lindley model, obtained by a specific mixture of exponential and Lindley distributions, thereby substantially enriching the distributional flexibility. To enhance its inferential scope, a comprehensive reliability analysis is developed under [...] Read more.
This paper introduces a novel extension of the classical Lindley distribution, termed the X-Lindley model, obtained by a specific mixture of exponential and Lindley distributions, thereby substantially enriching the distributional flexibility. To enhance its inferential scope, a comprehensive reliability analysis is developed under a generalized progressive hybrid censoring scheme, which unifies and extends several traditional censoring mechanisms and allows practitioners to accommodate stringent experimental and cost constraints commonly encountered in reliability and life-testing studies. Within this unified censoring framework, likelihood-based estimation procedures for the model parameters and key reliability characteristics are derived. Fisher information is obtained, enabling the establishment of asymptotic properties of the frequentist estimators, including consistency and normality. A Bayesian inferential paradigm using Markov chain Monte Carlo techniques is proposed by assigning a conjugate gamma prior to the model parameter under the squared error loss, yielding point estimates, highest posterior density credible intervals, and posterior reliability summaries with enhanced interpretability. Extensive Monte Carlo simulations, conducted under a broad range of censoring configurations and assessed using four precision-based performance criteria, demonstrate the stability and efficiency of the proposed estimators. The results reveal low bias, reduced mean squared error, and shorter interval lengths for the XLindley parameter estimates, while maintaining accurate coverage probabilities. The practical relevance of the proposed methodology is further illustrated through two real-life data applications from engineering and physical sciences, where the XLindley model provides a markedly improved fit and more realistic reliability assessment. By integrating an innovative lifetime model with a highly flexible censoring strategy and a dual frequentist–Bayesian inferential framework, this study offers a substantive contribution to modern survival theory. Full article
(This article belongs to the Special Issue Recent Applications of Statistical and Mathematical Models)
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16 pages, 327 KB  
Article
Left-Symmetric Algebras Arising from Modified DNA Insertion Operations
by Chen Yuan, Zhixiang Wu and Jing Wang
Axioms 2026, 15(1), 55; https://doi.org/10.3390/axioms15010055 - 12 Jan 2026
Viewed by 96
Abstract
DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct left-symmetric algebras arising from DNA insertion operations. That is, we define a modified insertion operation by weighting the simplified insertion. It generalizes [...] Read more.
DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct left-symmetric algebras arising from DNA insertion operations. That is, we define a modified insertion operation by weighting the simplified insertion. It generalizes the left-symmetric algebra constructed from the simplified DNA insertion operation. We prove that the algebra F(R) (over a field F of characteristic 0, with R being an infinite free semigroup generated by DNA nucleotides {A,G,C,T}) forms a left-symmetric algebra if and only if the function f satisfies a certain multiplicative condition for all positive integers m, n, and p. A key example of such a function is f(m,n)=exp{g(m,n)}, where g(m,n)=k·mn, and k is a fixed positive number, which effectively models length-dependent DNA insertion dynamics. This work contributes an algebraic framework that may be useful for quantitative modeling of DNA recombination processes. Full article
(This article belongs to the Section Algebra and Number Theory)
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13 pages, 275 KB  
Article
On the Structure and Homological Regularity of the q-Heisenberg Algebra
by Yabiao Wang and Gulshadam Yunus
Axioms 2026, 15(1), 54; https://doi.org/10.3390/axioms15010054 - 12 Jan 2026
Viewed by 106
Abstract
The q-Heisenberg algebra hn(q) is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and [...] Read more.
The q-Heisenberg algebra hn(q) is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincare´-Birkhoff-Witt) basis techniques to systematically investigate hn(q). Our main results establish that: hn(q) possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of hn(q), while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over hn(q), opening new avenues for further research in deformation quantization and quantum algebra. Full article
12 pages, 272 KB  
Article
Upper Semicontinuous Representations of Semiorders as Interval Orders
by Gianni Bosi, Gabriele Sbaiz and Magalì Zuanon
Axioms 2026, 15(1), 53; https://doi.org/10.3390/axioms15010053 - 10 Jan 2026
Viewed by 138
Abstract
We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair (u,v) of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the [...] Read more.
We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair (u,v) of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the representing functions is a one-way utility for the characteristic weak order ≺0 associated with the semiorder. Such a description generalizes the upper semicontinuous threshold representation. To this end, we introduce a suitable upper semicontinuity condition concerning a semiorder, namely strict upper semicontinuity. We further characterize the mere existence of an upper semicontinuous one-way utility for this characteristic weak order, with a view to the identification of maximal elements on compact metric spaces. Full article
(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)
23 pages, 475 KB  
Article
Matrix-Theoretic Investigation of Generalized Geometric Frank Matrices: Spread Estimates and Arithmetic Insights
by Bahar Kuloğlu
Axioms 2026, 15(1), 52; https://doi.org/10.3390/axioms15010052 - 9 Jan 2026
Viewed by 177
Abstract
This study introduces a novel generalization: the generalized geometric Frank matrix, which extends the classical Frank matrix and its known variants. We systematically examine its algebraic structure, providing detailed analyses of its factorizations, determinant, inverse, and various norm computations. Furthermore, we investigate the [...] Read more.
This study introduces a novel generalization: the generalized geometric Frank matrix, which extends the classical Frank matrix and its known variants. We systematically examine its algebraic structure, providing detailed analyses of its factorizations, determinant, inverse, and various norm computations. Furthermore, we investigate the reciprocal form of the reciprocal generalized geometric Frank matrix and reveal a variety of its intriguing algebraic properties. To illustrate the applicability of our theoretical results, we present a compelling example using Fibonacci number entries within the Frank matrix framework. Additionally, we analyze how the spread’s upper bounds are influenced by variations in the parameter r and the matrix dimension. Also, to formally assess the computational implications of these structural choices, we use Big O notation to describe how the computational cost scales with the matrix size n and the iteration count k(r). Our findings demonstrate that selecting r<1 and utilizing lower-dimensional generalized geometric Frank matrices can yield tighter bounds and significantly reduce computational complexity. These results highlight the potential of the proposed matrix class for optimization problems where efficiency is critical. Full article
(This article belongs to the Section Algebra and Number Theory)
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18 pages, 576 KB  
Article
A Gravity Tensor and Gauge Equations for Newtonian Dynamics
by Jing Tang Xing
Axioms 2026, 15(1), 51; https://doi.org/10.3390/axioms15010051 - 9 Jan 2026
Viewed by 124
Abstract
It is revealed that the material derivative of a variable in gravity field is its directional derivative, from which and energy/complementary-energy conservations with exterior derivatives, two sets of gauge equations of Newton’s dynamic gravity field are derived, which has same mathematical structure with [...] Read more.
It is revealed that the material derivative of a variable in gravity field is its directional derivative, from which and energy/complementary-energy conservations with exterior derivatives, two sets of gauge equations of Newton’s dynamic gravity field are derived, which has same mathematical structure with the gauge ones for the Maxwell equations in electromagnetic fields, revealing that gravity force and curl momentum in Newton’s gravity field, respectively, play the roles like the electric E  and the magnetic B of the Maxwell equations in the electromagnetic field. The gravity tensor of Newton’s gravitational field is constructed, and an example is given to validate it. This finding allows Newton’s gravity to be governed by a gauge theory, addressing the historic issue that “Newton’s gravitation is an exception to the Yang–Mills gauge theory”. Full article
(This article belongs to the Section Mathematical Physics)
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14 pages, 20048 KB  
Article
A Physical Model of the Reeb Foliation
by Gianluca Bande and Gregorio Franzoni
Axioms 2026, 15(1), 50; https://doi.org/10.3390/axioms15010050 - 9 Jan 2026
Viewed by 111
Abstract
In this paper, after explaining some basic aspects of the modern theory of foliations with the aim of describing the celebrated Reeb foliation, we propose the first construction of a comprehensive physical model of it. The construction of the model is achieved through [...] Read more.
In this paper, after explaining some basic aspects of the modern theory of foliations with the aim of describing the celebrated Reeb foliation, we propose the first construction of a comprehensive physical model of it. The construction of the model is achieved through an implementation of geometric methods for 3D printing. Full article
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14 pages, 264 KB  
Article
Relations Established Between Hypergeometric Functions and Some Special Number Sequences
by Sukran Uygun, Berna Aksu and Hulya Aytar
Axioms 2026, 15(1), 49; https://doi.org/10.3390/axioms15010049 - 9 Jan 2026
Viewed by 125
Abstract
In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By [...] Read more.
In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By employing Binet-type formulas, recurrence relations, Chebyshev polynomial connections, and classical transformation properties of Gauss hypergeometric functions, we derive several explicit and alternative representations for the Pell and Jacobsthal numbers. These representations unify known identities, yield new closed-form expressions, and reveal deeper structural parallels between hypergeometric functions and linear recurrence sequences. The results demonstrate that hypergeometric functions provide a systematic and versatile analytical tool for studying special number sequences beyond the Fibonacci case, and they suggest potential extensions to broader families such as Horadam-type sequences and their generalizations. Full article
(This article belongs to the Section Algebra and Number Theory)
17 pages, 333 KB  
Article
Multivalued Fixed Point Results in Rectangular m-Metric Spaces
by Safeer Hussain Khan, Muhammad Zahid and Ali Raza
Axioms 2026, 15(1), 48; https://doi.org/10.3390/axioms15010048 - 8 Jan 2026
Viewed by 165
Abstract
In this paper, we initiate the study of multivalued fixed point results in the framework of rectangular m-metric spaces. We establish fixed point theorems for Reich–Rus–Ćirić-type contractions and analyze two distinct cases based on the sum of the interpolative exponents: when the [...] Read more.
In this paper, we initiate the study of multivalued fixed point results in the framework of rectangular m-metric spaces. We establish fixed point theorems for Reich–Rus–Ćirić-type contractions and analyze two distinct cases based on the sum of the interpolative exponents: when the sum is less than one and when it is greater than one. Furthermore, by introducing the Hausdorff metric structure induced by rectangular m-metrics, our results generalize and extend various existing results in the literature. Illustrative examples are also provided to support and validate the obtained results. Full article
21 pages, 311 KB  
Article
Geometrical Analysis on Submanifolds in Riemannian Manifolds Attached with Silver Structure
by Shadab Ahmad Khan, Fatemah Mofarreh, Toukeer Khan, Mohd Danish Siddiqi and Anis Ahmad
Axioms 2026, 15(1), 47; https://doi.org/10.3390/axioms15010047 - 8 Jan 2026
Viewed by 209
Abstract
In this paper, we analyze a silver Riemannian structure on a Riemannian manifold. We compute some fundamental properties of the induced structure on submanifolds immersed in a silver Riemannian manifold and also obtain some results for induced structures on submanifolds of codimension 2. [...] Read more.
In this paper, we analyze a silver Riemannian structure on a Riemannian manifold. We compute some fundamental properties of the induced structure on submanifolds immersed in a silver Riemannian manifold and also obtain some results for induced structures on submanifolds of codimension 2. Moreover, we explore the conditions for totally geodesic and minimal submanifolds in a silver Riemannian manifold. Finally, we also give an example of submanifolds in a silver Riemannian manifold. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)
12 pages, 265 KB  
Article
Advanced Generalizations of Weighted Opial-Type Inequalities in the Framework of Time Scale Calculus
by Nadiah Zafer Al-Shehri, Mohammed M. A. El-Sheikh, Mohammed Zakarya, Hegagi M. Ali, Haytham M. Rezk and Fatma M. Khamis
Axioms 2026, 15(1), 46; https://doi.org/10.3390/axioms15010046 - 8 Jan 2026
Viewed by 162
Abstract
This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with Hölder’s inequality and Keller’s chain rule. These results extend the classical Opial-type inequalities by embedding them [...] Read more.
This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with Hölder’s inequality and Keller’s chain rule. These results extend the classical Opial-type inequalities by embedding them into the time scale setting, which unifies both continuous and discrete analyses. Consequently, various integral and discrete inequalities emerge as particular cases of our main results, thereby broadening the applicability of Opial-type inequalities to dynamic systems and discrete models. Full article
(This article belongs to the Section Mathematical Analysis)
20 pages, 324 KB  
Article
Coupled Fixed Point Theory over Quantale-Valued Quasi-Metric Spaces (QVQMS) with Applications in Generalized Metric Structures
by Irem Eroğlu
Axioms 2026, 15(1), 45; https://doi.org/10.3390/axioms15010045 - 8 Jan 2026
Viewed by 127
Abstract
In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of θs-completeness [...] Read more.
In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of θs-completeness and, as an application of the main theorems, we derive some results in both quantale-valued partial metric spaces and probabilistic metric spaces. Full article
(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)
17 pages, 299 KB  
Article
Existence and Uniqueness of Solutions to Abstract Discrete-Time Cauchy Problems in Vector-Valued Weighted Spaces
by Jagan Mohan Jonnalagadda and Carlos Lizama
Axioms 2026, 15(1), 44; https://doi.org/10.3390/axioms15010044 - 8 Jan 2026
Viewed by 119
Abstract
This article studies the abstract discrete-time Cauchy problem involving the Riemann–Liouville type difference operator. Sufficient conditions for the existence of unique solution to the semilinear Cauchy problem in Lebesgue and weighted Lebesgue vector-valued spaces are shown. Finally, some examples are presented to illustrate [...] Read more.
This article studies the abstract discrete-time Cauchy problem involving the Riemann–Liouville type difference operator. Sufficient conditions for the existence of unique solution to the semilinear Cauchy problem in Lebesgue and weighted Lebesgue vector-valued spaces are shown. Finally, some examples are presented to illustrate the main results. Full article
(This article belongs to the Section Mathematical Analysis)
32 pages, 1118 KB  
Article
On the Invariant and Geometric Structure of the Holomorphic Unified Field Theory
by John W. Moffat and Ethan James Thompson
Axioms 2026, 15(1), 43; https://doi.org/10.3390/axioms15010043 - 8 Jan 2026
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Abstract
We present the invariant structure of a Holomorphic Unified Field Theory in which gravity and gauge interactions arise from a single geometric framework. The theory is formulated using a product principal bundle, with one connection, and curvature equipped with a Hermitian field on [...] Read more.
We present the invariant structure of a Holomorphic Unified Field Theory in which gravity and gauge interactions arise from a single geometric framework. The theory is formulated using a product principal bundle, with one connection, and curvature equipped with a Hermitian field on a complexification of spacetime. From a single Diff(M)×G-invariant action, variation yields the Einstein and Yang–Mills equations together with their paired Bianchi identities. A compatibility condition is implemented either definitionally or through an auxiliary penalty functional. It enforces that the antisymmetric part of our Hermitian field is the gauge field’s exact curvature on the real slice. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)
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25 pages, 328 KB  
Article
Solutions to the Sine-Gordon Equation: From Darboux Transformations to Wronskian Representations of the k-Negaton-l-Positon-n-Soliton Solutions
by Pierre Gaillard
Axioms 2026, 15(1), 42; https://doi.org/10.3390/axioms15010042 - 7 Jan 2026
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Abstract
With a specific Darboux transformation, we construct solutions to the sine-Gordon equation. We use both the simple Darboux transformation as well as the multiple Darboux transformation, which enables the obtainment of compact solutions of this equation. We give a complete description of the [...] Read more.
With a specific Darboux transformation, we construct solutions to the sine-Gordon equation. We use both the simple Darboux transformation as well as the multiple Darboux transformation, which enables the obtainment of compact solutions of this equation. We give a complete description of the method and the corresponding proofs. We explicitly construct some solutions for the first orders. Using particular generating functions, we give Wronskian representations of the solutions to the sine-Gordon equation. In this case, we give different solutions to this equation. We deduce generalized Wronskian representations of the solutions to the sine-Gordon equation. As an application, we give the general expression of the k-negaton-l-positon-n-soliton solutions of the sine-Gordon equation and we construct some explicit examples of these solutions as well as m complexitons. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
24 pages, 1444 KB  
Article
Extended Parametric Family of Two-Step Methods with Applications for Solving Nonlinear Equations or Systems
by Ioannis K. Argyros, Stepan Shakhno and Mykhailo Shakhov
Axioms 2026, 15(1), 41; https://doi.org/10.3390/axioms15010041 - 6 Jan 2026
Viewed by 124
Abstract
The parametric family of two-step methods, with its special cases, has been introduced in various papers. However, in most cases, the local convergence analysis relies on the existence of derivatives of orders that the method does not require. Moreover, the more challenging semi-local [...] Read more.
The parametric family of two-step methods, with its special cases, has been introduced in various papers. However, in most cases, the local convergence analysis relies on the existence of derivatives of orders that the method does not require. Moreover, the more challenging semi-local convergence analysis was not introduced for this class of methods. These drawbacks are considered in this paper. We determine the radius of convergence and the uniqueness of the solution based on generalized continuity conditions. We also present the semi-local convergence analysis for this family of methods, which has not been studied before, using majorizing sequences. Numerical experiments and basins of attraction are included to validate the theoretical conditions and demonstrate the stability of the methods. Full article
(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)
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27 pages, 424 KB  
Article
Constraint Qualifications and Optimality Criteria for Nonsmooth Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds
by B. B. Upadhyay, Arnav Ghosh, I. M. Stancu-Minasian and Andreea Mădălina Rusu-Stancu
Axioms 2026, 15(1), 40; https://doi.org/10.3390/axioms15010040 - 6 Jan 2026
Viewed by 133
Abstract
Nonsmooth multiobjective mathematical programming problems with equilibrium constraints (NMMPEC) are studied in this article in the Hadamard manifold setting. In the context of (NMMPEC), the generalized Guignard constraint qualification (GGCQ) is formulated within the framework of Hadamard manifolds. Moreover, Karush–Kuhn–Tucker [...] Read more.
Nonsmooth multiobjective mathematical programming problems with equilibrium constraints (NMMPEC) are studied in this article in the Hadamard manifold setting. In the context of (NMMPEC), the generalized Guignard constraint qualification (GGCQ) is formulated within the framework of Hadamard manifolds. Moreover, Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions are derived for (NMMPEC). Thereafter, we explore constraint qualifications (CQ) tailored to (NMMPEC) in the Hadamard manifold setting. Interrelations between these constraint qualifications are subsequently derived. It is further demonstrated that the proposed constraint qualifications, when satisfied, ensure that GGCQ holds. It is noteworthy that constraint qualifications and optimality conditions for (NMMPEC) have not been investigated in the Hadamard manifold setting. Full article
(This article belongs to the Section Mathematical Analysis)
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12 pages, 293 KB  
Article
Multidimensional Widder–Arendt Theorem in Locally Convex Spaces
by Marko Kostić
Axioms 2026, 15(1), 39; https://doi.org/10.3390/axioms15010039 - 5 Jan 2026
Viewed by 206
Abstract
In this research article, we formulate and prove the multidimensional Widder–Arendt theorem and the integrated form of the multidimensional Widder–Arendt theorem for functions with values in sequentially complete locally convex spaces. Established results seem to be new even for scalar-valued functions. Full article
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