Axioms

25 pages, 375 KB

Open AccessArticle

Jacobi–Sobolev Orthogonal Polynomials, Differential Properties and Structural Formulas

by Héctor Pijeira-Cabrera, Javier Quintero-Roba and Juan Toribio-Milane

Axioms 2026, 15(7), 525; https://doi.org/10.3390/axioms15070525 - 13 Jul 2026

In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass points and a positive semidefinite Sobolev product matrix. Using [...] Read more.

In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass points and a positive semidefinite Sobolev product matrix. Using Christoffel–Darboux kernels, we derive several structure and connection formulas involving two consecutive Jacobi and Jacobi–Sobolev polynomials. This representation leads to lowering and raising operators with rational coefficients; a second-order ordinary differential equation and a three-term recurrence relation, with polynomial coefficients. These coefficients depend on n. These results extend several classical structural properties of Jacobi polynomials to a general discrete Sobolev setting. Full article

(This article belongs to the Section Mathematical Analysis)

22 pages, 338 KB

Open AccessArticle

Model-Free Prediction of Multivariate Time Series

by Hanieh Saeidi and Adel Mohammadpour

Axioms 2026, 15(7), 524; https://doi.org/10.3390/axioms15070524 - 13 Jul 2026

Abstract

This paper extends a model-free prediction framework from univariate to multivariate time series. We show that, under a mild uniformly bounded first-moment condition, a multivariate time series admits a VARMA-type representation and an associated infinite-past linear solution. These results are obtained without assuming [...] Read more.

This paper extends a model-free prediction framework from univariate to multivariate time series. We show that, under a mild uniformly bounded first-moment condition, a multivariate time series admits a VARMA-type representation and an associated infinite-past linear solution. These results are obtained without assuming that the observed data are generated by a parametric VAR, VARMA, or other specifically postulated stochastic model. Motivated by this representation, we develop a practical forecasting method based on truncating the infinite-past solution and estimating the resulting coefficient matrices by multivariate least squares. The proposed procedure is simple to implement and is applicable to a broad class of stationary or nonstationary, linear or nonlinear multivariate time series when the required moment, identification, and empirical design conditions are plausible. We also describe recursive multi-step forecasting and give multivariate analogues of the main theoretical accuracy statements. The representation results require only a uniformly bounded first moment, whereas the forecast-accuracy results are stated under an explicit selected-solution identification assumption and additional empirical design conditions needed for multivariate least-squares estimation. Monte Carlo results, a real-data application, and additional sensitivity checks show that, in the lag-based implementation studied here, the method is empirically comparable to standard linear and regularized forecasting approaches, while not uniformly dominating them, and retains the conceptual advantage of a model-free formulation. Full article

27 pages, 2010 KB

Open AccessArticle

Geometric Algebra Quantum Gate Decomposition

by Youssef Amraoui and Zeno Toffano

Axioms 2026, 15(7), 523; https://doi.org/10.3390/axioms15070523 - 13 Jul 2026

Abstract

Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate [...] Read more.

Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of

π / 4

-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through

π / 8

-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications. Full article

(This article belongs to the Special Issue Research in Quantum Information Theory)

► Show Figures

Figure 1

15 pages, 545 KB

Open AccessArticle

IMEX–Crank–Nicolson Methods for the Merton Jump-Diffusion PIDE: Stability, Convergence, and Fast Jump Evaluation

by Mehran Paziresh, Karim Ivaz and Mariyan Milev

Axioms 2026, 15(7), 522; https://doi.org/10.3390/axioms15070522 - 13 Jul 2026

Abstract

This paper presents an efficient numerical framework for solving the Merton jump–diffusion partial integro-differential equation (PIDE) arising in European option pricing. To address the nonlocal integral term generated by asset price jumps, we employ an IMEX Crank–Nicolson time-stepping scheme that preserves the tri-diagonal [...] Read more.

This paper presents an efficient numerical framework for solving the Merton jump–diffusion partial integro-differential equation (PIDE) arising in European option pricing. To address the nonlocal integral term generated by asset price jumps, we employ an IMEX Crank–Nicolson time-stepping scheme that preserves the tri-diagonal structure of the resulting linear system. A nonuniform spatial grid and fast Gaussian quadrature with spline interpolation are incorporated to enhance accuracy and computational efficiency. We establish the unconditional stability and convergence of the IMEX–Crank–Nicolson scheme through a detailed theoretical analysis. Numerical experiments confirm the theoretical results and illustrate the effectiveness of the proposed method for representative jump–diffusion parameters. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 3rd Edition)

► Show Figures

Figure 1

34 pages, 484 KB

Open AccessArticle

Decomposition-Based Checking and Local Certification for Propositional Circumscription via Minimal Reducts

by Zhongtao Xie, Xin Zhou, Hongbo Hu and Xiang Du

Axioms 2026, 15(7), 521; https://doi.org/10.3390/axioms15070521 - 10 Jul 2026

Viewed by 80

Abstract

Propositional circumscription selects models that are minimal with respect to designated atoms while permitting another set of atoms to vary. For a clause theory

φ

, minimized atoms P, varied atoms Z, and a candidate interpretation M, the minimal-reduct characterization [...] Read more.

Propositional circumscription selects models that are minimal with respect to designated atoms while permitting another set of atoms to vary. For a clause theory

φ

, minimized atoms P, varied atoms Z, and a candidate interpretation M, the minimal-reduct characterization reduces candidate-model checking to the entailment

Red [φ; P; Z, M] ⊧ ⋀ (M \cap P)

. We establish a structural decomposition of this entailment using the collapsed negative dependency graph of the reduct. Each selected source component induces a scoped entailment, and the global entailment is equivalent to the finite sequence of local obligations generated by successive contraction. The proof combines graph-based source selection, constructive extension of scoped countermodels, and preservation under contraction. These results yield a sound and complete checker that contracts certified minimized atoms and records origin-preserving certificate fragments over the original clauses. We instantiate the framework by direct SAT-based entailment checking and MUS-based support extraction. Experiments on 5445 random 3CNF instances and 462 industrial CNF instances show complete agreement with the global reduct criterion, and every generated certificate is successfully replayed. MUS-based extraction reduces the mean accumulated support size by

97.4 %

and

99.85 %

on the two benchmark collections, respectively, while incurring additional running time. The results provide a formal basis for local, replayable certification of propositional circumscription models. Full article

(This article belongs to the Section Logic)

18 pages, 286 KB

Open AccessArticle

Umbral Methods, Function Factorisation and Mittag–Leffler Fourier-Type Integral Transform

by Giuseppe Dattoli, Roberto Ricci and Tommaso Severati

Axioms 2026, 15(7), 520; https://doi.org/10.3390/axioms15070520 - 10 Jul 2026

Viewed by 128

Abstract

We propose a systematic way to construct trigonometric-like functions beyond the classical sine–cosine pair by factorising rational expressions in the umbral operator and then evaluating their action on the vacuum. The guiding idea is simple: the usual trigonometric functions may be viewed as [...] Read more.

We propose a systematic way to construct trigonometric-like functions beyond the classical sine–cosine pair by factorising rational expressions in the umbral operator and then evaluating their action on the vacuum. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an n-fold decomposition of rational umbral expressions. For each integer

n \geq 2

, the construction produces n functions which play the role of higher-order trigonometric-like components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case

n = 2

gives the Gaussian-trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case

n = 3

yields a three-component cyclic system and shows how the same idea extends beyond the usual even–odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin–Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations and local expansions for the functions obtained formally, while also fixing the branch and residue conventions needed for their analytic continuation. Finally, we indicate how the same cyclic kernels act as deformations of the Fourier transform. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

16 pages, 444 KB

Open AccessArticle

On the Structural Distortion Induced by the Inverse Box–Cox Transformation

by Rui Gonçalves

Axioms 2026, 15(7), 519; https://doi.org/10.3390/axioms15070519 - 10 Jul 2026

Viewed by 128

Abstract

The Box–Cox transformation is widely used to improve normality, stabilize variance, and enable Gaussian-based modelling in a transformed scale. After model fitting, conditional summaries are often mapped back to the original scale by applying the inverse transformation. This paper shows that this transform–fit–inverse [...] Read more.

The Box–Cox transformation is widely used to improve normality, stabilize variance, and enable Gaussian-based modelling in a transformed scale. After model fitting, conditional summaries are often mapped back to the original scale by applying the inverse transformation. This paper shows that this transform–fit–inverse procedure has a structural limitation: nonlinear inverse transformations do not, in general, preserve conditional expectations. Equivalently, conditional expectation and nonlinear inversion do not commute. Within the Box–Cox Gaussian framework, the admissible domain of the inverse transformation leads naturally to a truncated normal formulation in the transformed scale. Under this formulation, we derive a second-order decomposition showing that the original-scale conditional mean differs from the inverse-transformed truncated conditional mean by a curvature-driven correction term depending on the truncated conditional variance. The usual untruncated Gaussian expression is recovered as a local approximation when the inadmissible probability is negligible. A numerical sensitivity analysis, focused on

0 \leq λ_{Y} \leq 1

, illustrates how the distortion depends on the transformation parameter, correlation, and conditional dispersion. A real-data illustration using medical insurance charges further shows that the discrepancy can be visible in an applied regression setting and is not removed by changing the transformation of the explanatory variable. The results distinguish this structural invariance problem from classical retransformation bias and show that inverse-transformed fitted curves should be interpreted as transformation-induced structural curves, not automatically as conditional mean functions on the original scale. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

► Show Figures

Figure 1

8 pages, 270 KB

Open AccessArticle

Existence of Measurable Versions of Stochastic Processes

by Kazimierz Musiał

Axioms 2026, 15(7), 518; https://doi.org/10.3390/axioms15070518 - 10 Jul 2026

Viewed by 101

Abstract

Let

(X, A, P)

,

(Y, B, Q)

be two arbitrary probability spaces and

P : = {(A, P_{y}) : y \in Y}

be a regular conditional probability (rcp) [...] Read more.

Let

(X, A, P)

,

(Y, B, Q)

be two arbitrary probability spaces and

P : = {(A, P_{y}) : y \in Y}

be a regular conditional probability (rcp) on

A

with respect to Q. Denote by R the skew product of P and Q determined by

P

on the product

σ

-algebra

A \otimes B

and by

\hat{R}

its completion. I prove that if

(X, A, P)

is separable in the Fréchet–Nikodým pseudo-metric, then the stochastic process

{ξ_{y} : y \in Y}

has an equivalent measurable modification if and only if it is measurable with respect to a certain particular

σ

-algebra larger than

A \otimes B

. The theorem is a strong generalization of two earlier results of the author and coauthors, where it was only proved that a suitable class of liftings transfer a measurable process into a measurable process. It is known that not every process possesses an equivalent measurable modification. My approach is essentially different from the earlier trials. It reverts to an earlier paper of Talagrand, who proved the existence of an equivalent separable modification of a measurable process (in case of

R = P \times Q

), provided Y is endowed with a separable pseudo-metric. Full article

(This article belongs to the Special Issue Measure Theory and Related Topics)

13 pages, 274 KB

Open AccessArticle

Classification of Π-Manifolds Regarding the Pseudo-Riemannian Metric Associated Through the Structure

by Mancho Manev and Victoria Kuncheva

Axioms 2026, 15(7), 517; https://doi.org/10.3390/axioms15070517 - 9 Jul 2026

Viewed by 143

Abstract

The study of the so-called

Π

-manifolds has been continued. This is a short name for almost paracontact almost paracomplex manifolds with a pair of metrics—Riemannian and pseudo-Riemannian—that are mutually related and compatible with the structure of the manifold. The well-known classification of [...] Read more.

The study of the so-called

Π

-manifolds has been continued. This is a short name for almost paracontact almost paracomplex manifolds with a pair of metrics—Riemannian and pseudo-Riemannian—that are mutually related and compatible with the structure of the manifold. The well-known classification of these manifolds with respect to the Riemannian metric has been used to derive an alternative classification of the same manifolds with respect to the pseudo-Riemannian metric. The interrelations between the two classifications and the corresponding classification tensors are investigated. Finally, a three-dimensional explicit example on a Lie group is given, which illustrates the most interesting case of the transition from one classification to the other. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 3rd Edition)

20 pages, 316 KB

Open AccessArticle

Sign-Changing Solutions for the Fractional Choquard Equation

by Ying-Xin Cui and Qiaoyan Li

Axioms 2026, 15(7), 516; https://doi.org/10.3390/axioms15070516 - 9 Jul 2026

Viewed by 73

Abstract

We study a class of fractional Choquard equation with continuous potential. This equation is a doubly nonlocal problem which has two nonlocal term: fractional Laplacian operator and convolution term. Using variaotional metheod and some estimates, we give a sign-changing solution with minimal energy [...] Read more.

We study a class of fractional Choquard equation with continuous potential. This equation is a doubly nonlocal problem which has two nonlocal term: fractional Laplacian operator and convolution term. Using variaotional metheod and some estimates, we give a sign-changing solution with minimal energy for this equation under suitalbe condition. The results extend previous papers on fractional Choquard equation with constant potential and provide a method for general fractional Choquard equations with non-compact potential. Full article

25 pages, 387 KB

Open AccessArticle

Geometric Singular Perturbation Analysis of Poisson– Nernst–Planck Models with Large Permanent Charges and Multi-Cation Transport

by Jianing Chen

Axioms 2026, 15(7), 515; https://doi.org/10.3390/axioms15070515 - 9 Jul 2026

Viewed by 213

Abstract

Assuming the permanent charge density is much larger than the ion concentrations on two boundaries of an open ion channel, this paper studies the impacts of large permanent charges which are positioned along the channel wall and play a significant role in channel [...] Read more.

Assuming the permanent charge density is much larger than the ion concentrations on two boundaries of an open ion channel, this paper studies the impacts of large permanent charges which are positioned along the channel wall and play a significant role in channel functioning. Through this work, the quasi-one-dimensional Poisson–Nernst–Planck (PNP) system is utilized to analyze the electrodiffusion properties of ionic flows through an ion channel. The geometric singular perturbation theory is applied to derive the existence and local uniqueness of solutions to the corresponding PNP system containing large permanent charges and two monovalent cations. The matching asymptotic expansions are conducted to generate the expansions of state variables in terms of the permanent charge density, from which the permanent charge effects on individual fluxes can be discussed by further using the regular perturbation analysis. The mechanisms of the inhibited role played by large permanent charge in the co-ions’ flow are also elucidated from the perspective of internal dynamics. We hope this work could provide more comprehensive information on large permanent charge effects on ion channels and display more realistic interactions between related physical parameters such as channel geometry, diffusion coefficients, boundary potential and boundary ion concentrations. Full article

(This article belongs to the Special Issue Recent Advances in Nonlinear Mathematical Physics and Complex Systems)

► Show Figures

Figure 1

42 pages, 2790 KB

Open AccessArticle

Functionally Constructed Semi Linear Copulas from Fuzzy Implication Operators

by Panagiotis G. Mangenakis and Basil K. Papadopoulos

Axioms 2026, 15(7), 514; https://doi.org/10.3390/axioms15070514 - 8 Jul 2026

Viewed by 111

Abstract

This paper proposes a framework for generating symmetric two-branched copulas directly from fuzzy implications via monotone function composition. The proposed approach is based on a two-branched composition principle induced by fuzzy implications and yields four explicit construction schemes for broad classes of genuine [...] Read more.

This paper proposes a framework for generating symmetric two-branched copulas directly from fuzzy implications via monotone function composition. The proposed approach is based on a two-branched composition principle induced by fuzzy implications and yields four explicit construction schemes for broad classes of genuine copulas. Under suitable analytical assumptions, the resulting copulas satisfy the boundary conditions, symmetry, and global 2-increasingness. The framework also establishes a structural relation between fuzzy implications and the corresponding copulas. In contrast to related approaches that often remain at the level of quasi-copulas or semicopulas, the present method produces fully valid copulas while allowing distinct monotone behavior in each branch. The resulting families inherit analytical properties from the underlying fuzzy implications and may exhibit strong dependence, as reflected by high values of Spearman’s ρ and Kendall’s τ. Full article

(This article belongs to the Special Issue Theory and Applications in Functional Analysis)

22 pages, 446 KB

Open AccessArticle

Time Series Forecasting with New Type-2 Fuzzy T-Norms and T-Conorms

by Pablo Hernández-Varela, Pedro Huidobro, Francisco Javier Talavera, Carmen Torres-Blanc, Susana Cubillo and Jorge Elorza

Axioms 2026, 15(7), 513; https://doi.org/10.3390/axioms15070513 - 8 Jul 2026

Viewed by 120

Abstract

This paper presents new families of triangular norms (t-norms) and conorms (t-conorms) specifically constructed for type-2 fuzzy sets. Our approach considers several partially ordered sets of membership functions defined on the unit interval, each characterized by different structural properties and suited to representing [...] Read more.

This paper presents new families of triangular norms (t-norms) and conorms (t-conorms) specifically constructed for type-2 fuzzy sets. Our approach considers several partially ordered sets of membership functions defined on the unit interval, each characterized by different structural properties and suited to representing particular kinds of uncertainty. For these settings, we define operators that consistently represent fuzzy intersection and union, filling gaps where no such type-2 operators were previously available. It is the first time that t-norms and t-conorms are obtained with respect to both usual partial orders in the set of normal membership functions. Building on this framework, we integrate the proposed operators into a type-2 fuzzy time series. A large-scale evaluation on multiple benchmark time series shows that the new operators consistently achieve the best predictive accuracy, in terms of MAPE, in comparison with classical type-2 and type-1 fuzzy time series baselines. These results demonstrate that the algebraic design of type-2 operators has a direct impact on forecasting performance and can substantially improve the modeling of uncertainty in time-dependent data. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic and Fuzzy Implications)

25 pages, 436 KB

Open AccessArticle

Exact-Penalty Prox-Linear Methods for Bilevel Optimization with ℓ₁ Lower-Level Gradient Penalty

by Yutong Zheng, Jiani Li and Qingna Li

Axioms 2026, 15(7), 512; https://doi.org/10.3390/axioms15070512 - 8 Jul 2026

Viewed by 102

Abstract

Bilevel optimization is a fundamental framework for hierarchical decision-making, but its solution is challenging due to the implicit and typically set-valued nature of the lower-level optimality condition. In this paper, we study bilevel optimization problems through an exact-penalty reformulation based on the [...] Read more.

Bilevel optimization is a fundamental framework for hierarchical decision-making, but its solution is challenging due to the implicit and typically set-valued nature of the lower-level optimality condition. In this paper, we study bilevel optimization problems through an exact-penalty reformulation based on the

ℓ_{1}

-norm of the lower-level gradient. Under suitable regularity assumptions, we show that this penalty defines a distance-bound function and yields an exact penalty property for sufficiently large penalty parameters. To solve each fixed-penalty problem, we apply a prox-linear procedure that keeps the nonsmooth

ℓ_{1}

penalty in its original form and linearizes the smooth mappings. We prove a stationarity-oriented convergence guarantee for the fixed-penalty prox-linear loop. For the unconstrained simple bilevel setting, the prox-linear subproblem admits explicit dual reformulation as a box-constrained quadratic program. This dual structure enables the use of a nonmonotone spectral projected gradient method together with a closed-form primal recovery formula. Numerical experiments on the Minimum Norm Solution Problem show that the proposed method consistently achieves lower-level feasibility and upper-level accuracy, and attains a higher success rate than several existing methods on the tested instances. A Lipschitz least-squares variant is further included to provide a numerical illustration of the global exactness theorem. Full article

(This article belongs to the Special Issue Recent Advances in Mathematical Optimization and Its Applications)

14 pages, 333 KB

Open AccessArticle

Towards Effective Recognition of Black Box Rings

by Alexandre Borovik and Şükrü Yalçınkaya

Axioms 2026, 15(7), 511; https://doi.org/10.3390/axioms15070511 - 8 Jul 2026

Viewed by 96

Abstract

Black box algebra is a part of computational algebra focused on probabilistic methods of solving problems in very large finite algebraic objects where deterministic approaches simply do not work. The present paper provides a construction of structural proxies for black box rings encrypting [...] Read more.

Black box algebra is a part of computational algebra focused on probabilistic methods of solving problems in very large finite algebraic objects where deterministic approaches simply do not work. The present paper provides a construction of structural proxies for black box rings encrypting rings of

2 \times 2

matrices over finite fields. Full article

(This article belongs to the Section Algebra and Number Theory)

14 pages, 417 KB

Open AccessArticle

Monotonicity and Composition Analyses for Fractional Differences Involving Generalized Mittag-Leffler Kernels

by Mera Arab, Alina Alb Lupas, Christopher S. Goodrich and Pshtiwan Othman Mohammed

Axioms 2026, 15(7), 510; https://doi.org/10.3390/axioms15070510 - 6 Jul 2026

Viewed by 209

Abstract

This study establishes a rigorous monotonicity and composition analysis for fractional difference operators defined via the generalized Mittag-Leffler kernel. A fundamental criterion linking the sign of the discrete Atangana–Baleanu–Riemann (ABR) fractional difference to a new generalized form of monotonicity ( [...] Read more.

This study establishes a rigorous monotonicity and composition analysis for fractional difference operators defined via the generalized Mittag-Leffler kernel. A fundamental criterion linking the sign of the discrete Atangana–Baleanu–Riemann (ABR) fractional difference to a new generalized form of monotonicity (

w (1 - w)

−monotonicity) of the function is derived at first. Subsequently, a crucial composition rule for the ABR difference and its associated sum is proved, providing an explicit formula that elegantly incorporates initial conditions. The use of our theoretical findings is demonstrated practically by its application in solving linear discrete initial value problems, transforming them into explicit solutions by employing the composition theorem. The outcomes reveal the underlying dynamics of discrete models, significantly enhancing the theoretical frameworks and applications. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)

► Show Figures

Figure 1

19 pages, 312 KB

Open AccessArticle

Generalized Convexity via Clarke Subdifferential of Interval Mappings and Its Optimization Applications

by Shexiang Hai and Yanmei Zhang

Axioms 2026, 15(7), 509; https://doi.org/10.3390/axioms15070509 - 6 Jul 2026

Viewed by 188

Abstract

Clarke subdifferential and directional derivative theories are well developed for real functions. However, the nonsmooth analysis framework for interval mappings is still incomplete. Few works reveal the inherent relationship between Clarke directional derivatives of interval mappings and their endpoint functions. This paper addresses [...] Read more.

Clarke subdifferential and directional derivative theories are well developed for real functions. However, the nonsmooth analysis framework for interval mappings is still incomplete. Few works reveal the inherent relationship between Clarke directional derivatives of interval mappings and their endpoint functions. This paper addresses these gaps. We first explore the relationships between Clarke directional derivatives of interval mappings and their endpoint functions. The Clarke subdifferential of interval mappings is defined by means of directional derivatives and interval order relations, with its fundamental properties established. A new generalized convexity concept for interval mappings is further introduced based on the proposed Clarke subdifferential. Finally, sufficient conditions for efficient solutions to interval nonsmooth optimization are derived under the new generalized convexity framework. Full article

(This article belongs to the Special Issue Advances and Applications in Mathematical Modeling and Optimization)

49 pages, 632 KB

Open AccessArticle

EPiC: A Four-Valued Evidential Constraint Calculus for First-Order Reasoning

by José Oscar Olmedo-Aguirre, Isaac Machorro-Cano, Giner Alor-Hernández, Lisbeth Rodríguez-Mazahua, José Luis Sánchez-Cervantes and Aura Lucina Kantún-Montiel

Axioms 2026, 15(7), 508; https://doi.org/10.3390/axioms15070508 - 6 Jul 2026

Viewed by 185

Abstract

This article introduces the Evidence Propagation Calculus (EPiC), an operational framework for first-order reasoning built on a simple but productive observation: familiar inference patterns such as Modus Ponens and Modus Tollens behave like the movement of evidential markers across a structured graph. Positive [...] Read more.

This article introduces the Evidence Propagation Calculus (EPiC), an operational framework for first-order reasoning built on a simple but productive observation: familiar inference patterns such as Modus Ponens and Modus Tollens behave like the movement of evidential markers across a structured graph. Positive evidence at an antecedent propagates forward to the consequent; negative evidence at a consequent propagates backward. When both markers coexist at a node, the system is locally inconsistent but not operationally broken. To make this observation precise, EPiC grounds reasoning in a four-valued evidential domain

V = {N, T, F, B}

, where N denotes absence of evidence, T positive evidence, F negative evidence, and B their coexistence. Each logical connective is assigned a local evidential table, and inference is treated uniformly as the progressive restriction of admissible configurations under an evidential order: inadmissible values are eliminated, minimal surviving values are selected as the next effective evidential states, and the resulting restrictions propagate across shared variables. Compound formulas are decomposed into families of local unary and binary constraints through auxiliary variables, making the propagation process explicit and structurally uniform. Within this setting, Modus Ponens, Modus Tollens, and polarity-switching negation are not postulated as primitive rules. They emerge as derived consequences of the same local table calculus. The framework distinguishes different operational routes of justification. In some cases, positive support reaches the target formula directly through successive local restrictions. In others, propagation first stabilizes the relevant components and the target occurrence is then fixed by the corresponding connective table. Consistency is not a second basic notion of justification but a distinguished property of certain justified outcomes. The article establishes local and global soundness, conservativity over the classical fragment, and a conditional adequacy result. It further develops a translation between decomposed formulas and informational graphs, with a reverse reconstruction theorem for well-formed graphs. The result is a unified operational account of first-order reasoning situated between model-theoretic and proof-theoretic approaches, in which semantics, propagation, and graphical structure are mutually supporting rather than independently layered. Full article

(This article belongs to the Special Issue 15th Anniversary of Axioms: Logic)

► Show Figures

Figure 1

28 pages, 369 KB

Open AccessArticle

Stability Conditions in Multiple-Input Multiple-Output Systems

by Macarena Boix and Begoña Cantó

Axioms 2026, 15(7), 507; https://doi.org/10.3390/axioms15070507 - 6 Jul 2026

Viewed by 160

Abstract

This paper investigates the stabilization of unstable third-order Multiple-Input Multiple-Output (MIMO) systems whose interaction structure is described by a doubly stochastic combined matrix, also known as the Relative Gain Array (RGA). Starting from systems with negative Niederlinski index, we derive necessary and sufficient [...] Read more.

This paper investigates the stabilization of unstable third-order Multiple-Input Multiple-Output (MIMO) systems whose interaction structure is described by a doubly stochastic combined matrix, also known as the Relative Gain Array (RGA). Starting from systems with negative Niederlinski index, we derive necessary and sufficient conditions under which stability can be recovered through diagonal perturbations while preserving the doubly stochastic structure of the combined matrix. By exploiting the canonical representation of matrices associated with a prescribed combined matrix and the invariance properties under diagonal equivalence, the problem is reduced to a structured parametric form that allows a complete algebraic characterization. Special attention is given to perturbations involving the (1, 1) entry and one additional diagonal entry, leading to explicit bounds on the perturbation parameters that guarantee stabilization. The results extend previous papers on diagonal perturbations of combined matrices and provide a constructive method for stabilizing MIMO systems without altering their interaction pattern. Numerical examples illustrate the applicability of the proposed approach. Full article

(This article belongs to the Section Mathematical Analysis)

► Show Figures

Figure 1

30 pages, 17844 KB

Open AccessArticle

Hysteresis and Optimal Pricing of Subscriptions with Cancellation Cost

by Dmitrii Rachinskii

Axioms 2026, 15(7), 506; https://doi.org/10.3390/axioms15070506 - 5 Jul 2026

Viewed by 138

Abstract

We develop a stochastic Stackelberg model of a subscription market with cancellation costs. A representative consumer chooses when to subscribe to and cancel a service as the utility derived from the subscription evolves according to a diffusion process, while the firm selects the [...] Read more.

We develop a stochastic Stackelberg model of a subscription market with cancellation costs. A representative consumer chooses when to subscribe to and cancel a service as the utility derived from the subscription evolves according to a diffusion process, while the firm selects the subscription fee and cancellation cost to maximize its expected payoff. The consumer’s problem is equivalent to the classical real-options model of entry and exit under uncertainty with adjustment costs and exhibits a two-threshold policy with an inaction band and hysteresis. Unlike the standard formulation, in which the optimal thresholds are characterized implicitly through a system of nonlinear equations, we derive an explicit parametric solution in closed form. This solution reduces the firm’s optimization problem to a two-dimensional unconstrained problem and yields a detailed characterization of the optimal pricing policy. We show that the firm’s strategy exhibits three qualitatively distinct regimes depending on the initial utility level. For small utility levels, the optimal cancellation cost is zero. In an intermediate regime, the firm’s optimal policy induces the consumer to set the entry threshold equal to the initial utility level, resulting in immediate subscription. For sufficiently large utility levels, the firm induces permanent lock-in by setting a high cancellation cost and a low subscription fee: the consumer subscribes immediately and never subsequently unsubscribes. The transition between the latter two regimes is discontinuous and results from competition between two local maxima of the firm’s payoff function. We then extend the model to a heterogeneous population of consumers. The superposition of individual two-threshold subscription strategies generates a Preisach hysteresis operator describing the aggregate dependence of the firm’s revenue on the utility dynamics. The discontinuous regime transition persists under heterogeneity, demonstrating the robustness of the underlying mechanism. The Preisach representation predicts complex history dependence and long-term effects of temporary utility shocks. For a gamma distribution of consumer preferences, the firm’s expected payoff is obtained in closed form in terms of incomplete gamma functions. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications, 2nd Edition)

► Show Figures

Figure 1

28 pages, 396 KB

Open AccessArticle

A Foundational Analysis of Local Kernel-Based Calculus

by Pierros Ntelis

Axioms 2026, 15(7), 505; https://doi.org/10.3390/axioms15070505 - 5 Jul 2026

Viewed by 154

Abstract

We introduce the local kernel-based calculus, a unifying framework for local differential and integral operators based on an arbitrary positive continuous kernel function. This framework encompasses conformable, non-conformable, and our newly introduced local Euler-kernel derivatives as special cases. The parameter of the kernel [...] Read more.

We introduce the local kernel-based calculus, a unifying framework for local differential and integral operators based on an arbitrary positive continuous kernel function. This framework encompasses conformable, non-conformable, and our newly introduced local Euler-kernel derivatives as special cases. The parameter of the kernel is unrestricted and may take negative values, reflecting its role as a genuine parameter rather than an order of fractional differentiation. Within this general setting, we rigorously prove a complete set of foundational theorems: linearity, the product rule, continuity, Rolle’s theorem, the mean value theorem, and the fundamental theorem of calculus via the associated integral operator. We also derive a new formulation of the chain rule that expresses the chain rule entirely in terms of the kernel-based derivatives. While algebraically equivalent to the classical form, this representation preserves the intuitive structure of the chain rule without reference to the classical derivative. We further establish the Fundamental Theorem of Local Euler Calculus and its generalization, the Fundamental Theorem of Local Kernel-Based Calculus, confirming that the derivative and integral operators are genuine inverses, with the classical fundamental theorem recovered as special cases when the kernel reduces to unity. As an important illustration, we develop the local Euler calculus with the exponential kernel in full detail, providing explicit derivative and integral formulas for elementary functions. This special case demonstrates the simplicity and power of the functional approach. Overall, the local kernel-based calculus provides a solid, self-contained foundation that unifies a wide class of local operators and extends far beyond the traditional setting. Full article

(This article belongs to the Section Mathematical Analysis)

33 pages, 1338 KB

Open AccessArticle

Disquisition of a Retrial Queueing System with Batch Markovian Arrival Process, Nonidentical Service Devices and Phase-Type Distribution of Service Times

by Mei Liu and Alexander N. Dudin

Axioms 2026, 15(7), 504; https://doi.org/10.3390/axioms15070504 - 3 Jul 2026

Viewed by 161

Abstract

We study a retrial queueing system with N ranked heterogeneous service devices where processing times at each device follow a phase-type (

P H

) distribution with device-dependent parameters. Requests arrive according to a Batch Markovian Arrival Process (

B M A P

[...] Read more.

We study a retrial queueing system with N ranked heterogeneous service devices where processing times at each device follow a phase-type (

P H

) distribution with device-dependent parameters. Requests arrive according to a Batch Markovian Arrival Process (

B M A P

). The system uses a preemptive priority rule: idle devices with smaller serial numbers are preferred, and when a lower-numbered device completes service, the request being processed at the highest-numbered busy device is moved there and its service restarts. Requests that cannot be served immediately join an orbit of infinite capacity and retry after random time intervals. We describe the system dynamics by a multidimensional continuous-time Markov chain with a block upper-Hessenberg generator. A sufficient ergodicity condition for this Markov chain is derived. We present formulas for the key performance measures, including the mean orbit length, device utilizations, and the probability of immediate service. Numerical experiments show how the arrival rate and the coefficient of variation of processing times affect system performance. In particular, higher processing-time variability (hyperexponential case) in the considered example widens the stability region, while lower variability (Erlang case) narrows it, compared with exponential service. A supplementary study shows that higher arrival correlation under the chosen set of the system parameters amplifies orbit congestion and shifts utilization from the fastest server to the slower ones. Full article

(This article belongs to the Special Issue Recent Stochastic and Statistical Approaches for Modeling Complex Systems and Dependent Variables)

► Show Figures

Figure 1

43 pages, 1553 KB

Open AccessArticle

Adaptive Phase-Field Fracture Modeling Using C¹ PHT-Splines: A Consistent High-Order Isogeometric Formulation

by Abdel Ahad El Mahmi, Ahmed El Khalfi, Abdeslam El Akkad, Maria Luminița Scutaru and Sorin Vlase

Axioms 2026, 15(7), 503; https://doi.org/10.3390/axioms15070503 - 3 Jul 2026

Viewed by 263

Abstract

This work develops a locally adaptive isogeometric phase-field framework for two-dimensional quasi-static brittle fracture using cubic

C^{1}

polynomial splines over hierarchical T-meshes (PHT-splines). The aim is not to introduce a new crack-density functional or a new degradation law, but to provide a [...] Read more.

This work develops a locally adaptive isogeometric phase-field framework for two-dimensional quasi-static brittle fracture using cubic

C^{1}

polynomial splines over hierarchical T-meshes (PHT-splines). The aim is not to introduce a new crack-density functional or a new degradation law, but to provide a consistent variational-to-discrete setting in which second- and fourth-order phase-field regularizations can be treated within the same locally refined spline framework. Starting from the energy functional, the formulation is carried through admissible weak forms to the corresponding discrete residual equations. The second-order formulation is posed in an

H^{1} (Ω)

setting, whereas the fourth-order model is treated directly through a Laplacian-based

H^{2} (Ω)

-compatible approximation without auxiliary phase-field variables. The formulation combines history-field irreversibility, the tension–compression split of the elastic energy, and an adopted cubic degradation law with

s = 10^{- 4}

, whose nonlinear tangent contribution is handled by a Taylor-stabilized staggered Newton scheme. Numerical tests on a single-edge notched tensile benchmark and a notched perforated beam under asymmetric bending show that local refinement captures the fracture zone while maintaining critical-load deviations of about

0.8 %

and

0.3 %

, respectively, relative to the reference critical loads used for the two benchmark problems. The contribution therefore lies in the coherent coupling of higher-order regularity, admissible weak forms, local PHT-spline adaptivity, and stabilized nonlinear degradation treatment within a spline-based phase-field fracture implementation. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics, 2nd Edition)

► Show Figures

Figure 1

21 pages, 354 KB

Open AccessArticle

Explicit Runge–Kutta–Nyström-Type Schemes for Third-Order Systems y‴ = f(x, y, y′)

by Rubayyi T. Alqahtani, Theodore E. Simos and Charalampos Tsitouras

Axioms 2026, 15(7), 502; https://doi.org/10.3390/axioms15070502 - 3 Jul 2026

Viewed by 156

Abstract

Initial value problems of the third order featuring explicit dependence on velocity, denoted as

y^{‴} = f (x, y, y^{'})

, emerge regularly across applications such as electromechanical networks, structural mechanics, and robotic trajectory control. Despite their [...] Read more.

Initial value problems of the third order featuring explicit dependence on velocity, denoted as

y^{‴} = f (x, y, y^{'})

, emerge regularly across applications such as electromechanical networks, structural mechanics, and robotic trajectory control. Despite their practical prevalence, these differential equations remain insufficiently addressed by standard numerical integration techniques. Orthodox Runge–Kutta–Nyström (RKN) schemes are fundamentally formulated for differential equations lacking the first derivative, specifically

y^{‴} = f (x, y)

. Due to this algorithmic constraint, researchers frequently resort to computationally demanding first-order system reductions or rely upon standard Runge–Kutta methods. The present study resolves this methodological gap by defining an explicit s-stage integration architecture that natively incorporates the first derivative within the internal stage evaluations. Such structural modifications require the deployment of a supplementary coefficient matrix, denoted as D, to formulate the corresponding order theory. The complete set of algebraic order conditions is systematically established up to the seventh order, accompanied by a generic mathematical framework for generating schemes of arbitrary order. Based on this analytical foundation, an embedded

6 (4)

method is constructed. This specific pair achieves strict error tolerances utilizing merely six function evaluations per integration step, representing a substantial operational reduction compared to the eight computations strictly required by equivalent Runge–Kutta pairs. Direct numerical integration of the native third-order system prevents the dimensionality increase from reducing to first-order systems. Performance validation of the numerical solver involves two representative physical benchmarks: a coupled robotic appendage subjected to platform excitation and an electromechanical actuator array regulated by transient control inputs. Both dynamical systems exhibit severe velocity-dependent dissipation mechanisms and nonlinear external forcing. Quantitative numerical evaluations confirm that the constructed 6(4) pair yields higher precision and demands less computational expenditure than prevailing RK and RKN integrators. The analytical and empirical findings establish that derivative-capable Nyström integration algorithms furnish mathematically rigorous and computationally efficient numerical solutions for velocity-coupled third-order dynamics. Full article

29 pages, 448 KB

Open AccessFeature PaperArticle

Efficient Numerical Methods for Fractional- and Integer-Order Ordinary Differential Equations

by Marian Milev, Radan Miryanov and Yuri Dimitrov

Axioms 2026, 15(7), 501; https://doi.org/10.3390/axioms15070501 - 2 Jul 2026

Viewed by 353

Abstract

This paper proposes numerical methods for solving ordinary differential equations and fractional ordinary differential equations. The proposed methods are based on discretizations of first- and second-order derivatives, the L1 approximation of the Caputo fractional derivative, and a shifted L1-based approximation on a uniform [...] Read more.

This paper proposes numerical methods for solving ordinary differential equations and fractional ordinary differential equations. The proposed methods are based on discretizations of first- and second-order derivatives, the L1 approximation of the Caputo fractional derivative, and a shifted L1-based approximation on a uniform mesh. The discretizations of the integer-order derivatives depend on a free parameter, which enables the construction of numerical schemes with any prescribed order of accuracy in the interval

(0, 2]

and supports the development of efficient, fast algorithms for computation of the solution. The discretizations of fractional derivatives employ the weights of the L1 approximation together with values of the Riemann zeta function. The convergence and accuracy of the numerical methods are analyzed theoretically. Numerical experiments confirm the theoretical results and demonstrate the improvement of the proposed methods over L1 schemes for the numerical solution of ordinary fractional differential equations. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis and Its Applications)

► Show Figures

Figure 1

20 pages, 337 KB

Open AccessArticle

A Chemotaxis-Based Model for the Aggregation Behavior of Students

by Jieqiong Shen

Axioms 2026, 15(7), 500; https://doi.org/10.3390/axioms15070500 - 2 Jul 2026

Viewed by 255

Abstract

Understanding the aggregation behavior of high-achieving student groups is critical for optimizing educational resource allocation and upgrading institutional talent development systems. To address the prevailing gap in dynamic modeling for this specific phenomenon, this paper develops a two-equation chemotaxis-based framework to investigate the [...] Read more.

Understanding the aggregation behavior of high-achieving student groups is critical for optimizing educational resource allocation and upgrading institutional talent development systems. To address the prevailing gap in dynamic modeling for this specific phenomenon, this paper develops a two-equation chemotaxis-based framework to investigate the emergence and evolution of such aggregations. The first equation captures the dynamics of an attractiveness field shaped by peer learning attraction, knowledge gravity, and individual behavioral tendencies, while the second delineates the spatiotemporal evolution of the density of high-achieving students. Applying this framework, we first identify the parameter regimes that can generate hotspots of high-achieving students. Subsequently, well-posedness and stability analyses reveal a key insight: insufficient institutional management and incentive policies can lead to the gradual decline or even complete disappearance of these populations. This work thus contributes a theoretical model for understanding student group dynamics, while simultaneously providing educational institutions with a robust foundation for formulating inclusive and sustainable talent development strategies. Full article

(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)

16 pages, 1664 KB

Open AccessArticle

Solving the Klein–Gordon–Fock Equation Using Separation of Variables in the Light-Front Coordinates

by Gislan Silveira Santos, Jorge Henrique de Oliveira Sales and Cássio Almeida Lima

Axioms 2026, 15(7), 499; https://doi.org/10.3390/axioms15070499 - 2 Jul 2026

Viewed by 169

Abstract

In this article, we present a methodological and systematic approach to solving the Klein–Gordon–Fock equation using the separation of variables method, with particular emphasis on its formulation in light-front coordinates. Although the plane-wave solution is well known in relativistic quantum mechanics, the explicit [...] Read more.

In this article, we present a methodological and systematic approach to solving the Klein–Gordon–Fock equation using the separation of variables method, with particular emphasis on its formulation in light-front coordinates. Although the plane-wave solution is well known in relativistic quantum mechanics, the explicit procedure leading to this solution is not always developed in detail, especially when the equation is written in light-front variables. We first revisit the Klein–Gordon–Fock equation for a free particle in Minkowski spacetime, showing how the usual separation between temporal and spatial variables leads to the expected plane-wave form. This treatment is used as a reference for the corresponding analysis in light-front coordinates. We then rewrite the equation in light-front coordinates, adopting

α_{L F} = \sqrt{2}

, and apply the separation of variables method to the coordinates

x^{+}

,

x^{-}

, and

x^{⊥}

. In this formulation,

x^{+}

and

x^{-}

appear coupled through the mixed derivative term

\partial_{+} \partial_{-}

, with the separation process requiring an additional decoupling step involving an inverse relation and a nonzero constant

λ

. We show that an appropriate choice of this constant, together with a suitable choice of the superposition coefficients, allows the separated solution to recover the plane-wave structure obtained from the covariant transformation of the scalar product

p_{μ} x^{μ}

. Thus, the results clarify the consistency between the direct coordinate-transformation approach and the explicit solution of the differential equation in light-front coordinates, while also highlighting the usefulness of separation of variables as a methodological tool in the study of relativistic wave equations. Full article

(This article belongs to the Special Issue Mathematical Foundations for Physical Sciences)

► Show Figures

Figure 1

21 pages, 498 KB

Open AccessArticle

Method for Automated Decomposition of Monolithic Software Systems Based on Graph Neural Networks

by Yaroslav Kornaga, Oleksandr Hubariev, Serhii Yevseiev, Petro Yablonskyi and Tetiana Pyrohovska

Axioms 2026, 15(7), 498; https://doi.org/10.3390/axioms15070498 - 2 Jul 2026

Viewed by 181

Abstract

The purpose of this study is to improve the architectural quality of software systems with microservice architecture by investigating and improving methods for automated transition from monolithic architecture to microservice architecture. The study was performed using methods of static source code analysis, graph [...] Read more.

The purpose of this study is to improve the architectural quality of software systems with microservice architecture by investigating and improving methods for automated transition from monolithic architecture to microservice architecture. The study was performed using methods of static source code analysis, graph theory, algorithms for transforming graph structures (detecting strongly connected components, redundant and cyclic dependencies), cluster analysis algorithms, graph neural networks, as well as multi-criteria assessment of internal cluster consistency and inter-cluster connectivity. The results presented in the study are a comprehensive solution to the scientific problem of ensuring the automated transformation of monolithic software systems into a microservice architecture with controlled inter-service connectivity and a high level of internal consistency of architectural components. Full article

(This article belongs to the Special Issue Modern Trends and Application of Decision-Making Theory, Stability and Control, 2nd Edition)

► Show Figures

Figure 1

16 pages, 1122 KB

Open AccessArticle

Analytical Study of Complex Heat Transfer During Steady-State Natural Convection near a Vertical Surface

by Andriy A. Avramenko, Igor V. Shevchuk, Nataliia P. Dmitrenko, Vladimir G. Demchenko, Andrii I. Tyrinov, Kyryl O. Fedortsev and Andrii S. Kobzar

Axioms 2026, 15(7), 497; https://doi.org/10.3390/axioms15070497 - 2 Jul 2026

Viewed by 189

Abstract

In this study we derived an analytical solution to the problem of radiation heat transfer under free convection near a vertical plate with a slip condition on its surface. While solving the problem, new equations for temperature and velocity profiles, boundary layer thickness, [...] Read more.

In this study we derived an analytical solution to the problem of radiation heat transfer under free convection near a vertical plate with a slip condition on its surface. While solving the problem, new equations for temperature and velocity profiles, boundary layer thickness, and Nusselt number were obtained. The obtained expressions make it possible to estimate the influence of slip and radiation effects on free convection, and to identify effects that favor heat transfer enhancement. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics, 2nd Edition)

► Show Figures

Figure 1

18 pages, 307 KB

Open AccessArticle

A Caratheodory Approximation Approach to Fixed Points of Measurable-Selection-Valued Correspondences Arising in Game Theory

by Jing Fu and Frank Page

Axioms 2026, 15(7), 496; https://doi.org/10.3390/axioms15070496 - 1 Jul 2026

Viewed by 170

Abstract

We establish a new fixed point result for measurable-selection-valued correspondences with nonconvex and possibly disconnected values arising from the composition of Caratheodory functions with an upper Caratheodory (

u C

) correspondence. Using Caratheodory approximation methods, we show that for any such upper [...] Read more.

We establish a new fixed point result for measurable-selection-valued correspondences with nonconvex and possibly disconnected values arising from the composition of Caratheodory functions with an upper Caratheodory (

u C

) correspondence. Using Caratheodory approximation methods, we show that for any such upper Caratheodory composition correspondence, if in each state, the upper semicontinuous part of the underlying upper Caratheodory correspondence contains an upper semicontinuous sub-correspondence taking contractible values, then the underlying upper Caratheodory correspondence is Caratheodory approximable, further implying that the induced measurable-selection-valued correspondence has fixed points—all accomplished without the induced selection correspondence being convex-valued or upper semicontinuous in the appropriate topologies (in the case the weak star topologies). An excellent example of such a composition correspondence is provided by discounted stochastic games (

D S G

). In particular, the Nash payoff selection correspondence of the parameterized collection of state-contingent one-shot games underlying a discounted stochastic game is gotten by composing players’ parameterized collection of state-contingent Caratheodory payoff functions with the upper Caratheodory Nash equilibrium correspondence (i.e., the

u C

Nash correspondence). We are able to conclude via our fixed point result that if the

u C

Nash correspondence has an upper semicontinuous part containing a contractibly valued upper semicontinuous sub-correspondence, implying that the

u C

Nash correspondence is Caratheodory approximable, then the Nash payoff selection correspondence induced by the

u C

Nash correspondence has fixed points. It then follows from Blackwell’s Theorem (1965–extended to games) that the

D S G

to which the selection correspondence belongs has stationary Markov perfect equilibria. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

Journal Menu

Journal Browser

Axioms, Volume 15, Issue 7 (July 2026) – 57 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI