This paper investigates the complex dynamical behavior of a discrete-time Cournot triopoly game in which firms pursue relative profit maximization under bounded rationality. Unlike the standard absolute-profit framework, relative profit objectives introduce strategic aggressiveness and stronger competitive interactions, significantly enriching the market dynamics. Starting from a nonlinear three-dimensional adjustment process, we derive the Cournot–Nash equilibrium and analyze its local stability through linearization and Jacobian-based conditions. Explicit analytical thresholds for flip (period-doubling) and Neimark–Sacker bifurcations are obtained, revealing the mechanisms through which stability is lost as key parameters, such as the adjustment speed and relative profit weight, vary.

Beyond local analysis, we explore the global dynamics of the triopoly system using numerical simulations, including bifurcation diagrams, phase portraits, and basins of attraction. The results uncover a wide range of behaviors, from stable equilibria and periodic cycles to quasi-periodic motions and chaos. To quantitatively characterize the complexity of these dynamics, spectral entropy analysis is employed, providing a robust measure that complements traditional bifurcation techniques and clearly distinguishes between regular and chaotic regimes.

Furthermore, an effective chaos control strategy based on parameter feedback is proposed to stabilize unstable dynamics and restore market equilibrium. The controlled system exhibits reduced complexity, confirmed by a significant decrease in spectral entropy. The findings highlight how relative profit considerations can destabilize oligopolistic markets, while appropriate control mechanisms can enhance stability. Overall, this study offers new analytical and numerical insights into triopoly competition, contributing to the understanding of nonlinear phenomena in economic systems.

In this study, we develop a two-variable fractional differential equation with respect to the temporal variable and the spatial variable , where the derivative is fractional in time using the Caputo derivative and classical in space. The equation is derived from the differential equation of orthogonal Chebyshev polynomials, with a perturbation introduced via involution, representing a generalization of standard fractional models.

The equation is solved using the separation of variables method, representing the solution as an infinite series with Chebyshev polynomials forming a basis in the weighted Hilbert space . This leads to a spectral problem in the spatial variable , from which the eigenvalues and eigenfunctions are determined. The main problem is then addressed through the series expansion, resulting in a linear fractional equation previously studied in the literature, yielding an explicit analytical solution suitable for theoretical analysis.

To ensure stability and differentiability, temporal boundary conditions are imposed, which are verified through the convergence of the solution series. Finally, the uniqueness of the solution is established based on the completeness of the Hilbert basis and the initial condition at t=0 .

This methodology can be generalized to other polynomial bases, such as Legendre or Hermite polynomials, and extended to construct other fractional equations using alternative derivatives, including Riemann–Liouville or Letnikov derivatives, offering a flexible framework for modeling complex time-space fractional systems.

In this talk, we present a mathematical model describing the evolution of tumor density under treatment while accounting for a delayed effect. After a medical motivation, we introduce a conceptual scheme linking the observed biological mechanisms to the choice of model variables. The resulting system is formulated as a fractional differential inclusion with Caputo derivative and two maximal monotone operators, representing respectively the intrinsic tumor dynamics and the treatment-induced response. This modeling approach provides a rigorous framework to incorporate complex biological phenomena and delayed therapeutic effects.

The main analytical challenges arise from the multivalued nonlinear structure and the memory property of the fractional term. We employ advanced tools from functional analysis, in particular monotone operator theory and regularization techniques, to establish an existence result for the solution. This solution u(t) exhibits a medically consistent temporal behavior and reflects key biological insights observed in clinical studies. The proposed framework allows us to interpret the “delayed” effect observed in certain therapies and may be used to predict tumor evolution, adjust dosing frequency, optimize treatment strategies, or analyze the effectiveness of novel therapeutic interventions.

We conclude with several perspectives for future work: detailed stability analysis, fractional optimal control, extensive numerical simulations, and systematic comparison with clinical data, highlighting the potential practical impact of the model in guiding personalized treatment.

Introduction:
First-order nonlinear partial differential equations tend to lose smoothness quickly, so classical solutions only describe the evolution for a short time. After that, one has to rely on variational ideas and weak-solution frameworks to make sense of the equation. This work looks at Hamilton–Jacobi equations and scalar conservation laws and tries to bring out the shared analytical structure that appears in both settings once characteristics begin to break down.

Methods:
The approach uses convex analysis, weak convergence, and tools from Sobolev spaces and geometric measure theory. For Hamilton–Jacobi equations with convex Hamiltonians, the Legendre transform and the Hopf–Lax formula are used to build viscosity solutions and to understand why these variational representations remain meaningful after the loss of classical regularity. For scalar conservation laws, the analysis centers on entropy admissibility, the Rankine–Hugoniot jump condition, and the way shocks form. The Lax–Oleinik formula is examined in detail because it ties the two equations together and shows that both rest on similar minimization principles.

Results:
The study shows that viscosity and entropy solutions naturally emerge from the same underlying variational structure. The Hopf–Lax and Lax–Oleinik formulas give explicit solution representations that stay valid beyond the classical regime. Convexity ensures stability, and Sobolev/GMT techniques help describe the limiting behavior and the formation of singularities.

Conclusions:
The work provides a unified theoretical viewpoint on first-order nonlinear PDEs, where variational principles, convex duality, and weak-solution ideas fit together in a consistent and natural way. Many qualitative properties of these equations can be understood directly from analysis without relying on computational or numerical methods.

This work develops a comprehensive theory of what we call (p, q)-compact Bloch mappings, extending to the Bloch setting the concept of (p, q)-compact linear operators introduced by Ain, Lillemets, and Oja. We introduce and deeply study holomorphic mappings from the open unit disc in the complex plane in a complex Banach space, whose measures of the size of (p, q)-compactness associated with the Bloch range are finite. The collection of all zero-preserving (p, q)-compact Bloch mappings, endowed with a suitable norm, is shown to constitute a surjective s-Banach ideal of normalized Bloch mappings, where the exponent s is given by s = pq/(p + q). This ideal becomes regular for the collection of all reflexive complex Banach spaces.

This paper establishes several structural properties of these mappings. We prove invariance under Möbius transformations and a linearization theorem that identifies a Bloch mapping as (p, q)-compact precisely when its associated continuous linear operator on the Bloch-free space is (p, q)-compact. This correspondence allows us to extend many classical results on operator ideals to the nonlinear Bloch framework. We also obtain factorization theorems for (p, q)-compact Bloch mappings through compositions involving compact Bloch mappings and (p, q)-compact linear operators. Furthermore, we introduce and analyze the subclass of (t, u, v)-nuclear Bloch mappings, providing characterizations and their ideal structure parallel to those known in operator theory.

Hence, this study unifies and generalizes previous results on compact and p-compact Bloch mappings, establishing a deep interaction between operator ideals and the geometry of holomorphic mappings on the open complex unit disc, and demonstrating the robustness of the (p, q)-compact setting under linearization, factorization, and Möbius invariance.

The development of various generalizations of metric spaces has significantly enriched the study of fixed point theory, enabling the extension of classical results to broader and more flexible frameworks. In this talk, we examine several of these generalized metrics, with particular attention to the bipolar metric, suprametric, and perturbed metric. We investigate their topological properties, including completeness, compactness, and continuity structures, and discuss how these properties interact with fixed point principles.

A key aspect of our study is the introduction and analysis of remetrization techniques—methods of redefining distance functions while preserving completeness of the underlying space. Approaches can provide deeper insights into the behavior of iterative schemes and broaden the applicability of fixed point theorems. The necessary and sufficient assumptions of fixed point theorems will be compared in generalized metric space and induced metric space.

We also explore the stability properties of fixed point problems within generalized and remetrized spaces, highlighting differences and similarities in the convergence behavior of Picard iterative sequences. Comparative analysis reveals how these generalizations influence not only the existence and uniqueness of fixed points but also the efficiency and robustness of numerical methods used to approximate them. The presented results offer a unified perspective both on fixed point results and the scope of their applications, in that way combining both pure and applied mathematical contexts.

Hepatitis B remains a major public health challenge, especially due to its potential to progress into chronic infection and severe liver disease. This study develops a mathematical model to understand the role of Interferon therapy, particularly comparing Standard IFN-α and Pegylated IFN-α (Peg-IFN) in controlling HBV infection. The model tracks four key groups: susceptible individuals, those with acute infection, chronic cases, and recovered individuals.

We determine the basic reproduction number ( R0R_0R0​ ) and analyze the conditions under which the disease can be eradicated or persist. Our results show that when R0<1R_0 < 1R0​<1, the infection dies out, leading to a stable disease-free equilibrium (DFE). However, when R0>1R_0 > 1R0​>1, the disease persists at an endemic equilibrium (EE). Through numerical simulations, we explore how Interferon therapy influences HBV progression. The findings suggest that Peg-IFN is more effective than standard interferon due to its longer-lasting effects and stronger suppression of the virus, reducing the number of infected individuals and increasing recovery rates.

Additionally, sensitivity analysis highlights key factors—such as treatment rates, immune response, and disease progression—that influence infection dynamics. Our results reinforce the importance of early intervention and suggest that combining pegylated interferon with antiviral therapies may be the best approach for HBV control.

In this work, we consider the following fractional reaction system:

, in

, in

or for all , on

for all , in

where u = (u₁, . . . , u_m) , m ≥ 2, Ω is a bounded and regular domain of R^N with boundary ∂Ω, N ≥ 2, u_i = u_i (t, x), 1 ≤ i ≤ m for (t, x) ∈ Q_T = (0, T ) × Ω and ƒ_i are real functions, the presence of the non-local operator , 0<<1 for all 1 ≤ i ≤ m, which accounts for the anomalous diffusion, meaning that the sub-populations face some obstacles that slow their movement, and the constants of diffusion d_i are assumed to be non-negative. ƒ_i : R^m →R^m are regular enough and are non-negative functions in L¹ (Ω) for all cases where 1 ≤ i ≤ m.

The local existence in time of the solution is classical. The positivity of the solution stems from the positivity of , which is assumed to be continuous for all cases where 1 ≤ i ≤ m.

Material extrusion 3D printing is one of the most versatile and accessible additive manufacturing techniques used by researchers and engineers across the globe. The technique is limited by manufacturing parameter-induced defects. These defects develop during the fabrication process, such as warping, poor layer adhesion, and dimensional inaccuracy. The defects can be controlled by optimizing process parameters, such as nozzle temperature, print speed, layer height, and cooling rate. Traditionally, parameter selection relies on iterative trial-and-error or single-objective optimization. These approaches often fail to capture the inherent trade-offs between conflicting quality characteristics. In this research, we used a systematic multi-objective optimization framework (MOO) to minimize manufacturing defects in the fabrication processing conditions.

We produced the specimens using the material extrusion 3D printing (ME3DP) method. We used polylactic acid (PLA) for fabricating the specimens in the ME3DP technique. PLA has good manufacturability and is a biodegradable material. We produced manufacturing defects by controlling the temperature and the basic components of material design used in the fabrication process, such as nozzle temperature, heated bed temperature, layer thickness, and infill density. We applied one of the mathematical and multi-attribute decision-making techniques for the MOO framework, used as ‘Multi-Objective Optimization on the Basis of Ratio Analysis (MOORA)’. We used the (MOORA) technique to find the most significant configuration for evaluating manufacturing defect formation during the fabrication process. The MOORA technique is prevalently used in process engineering and optimization in industrial applications. The method simultaneously integrates both beneficial and non-beneficial attributes by using a ratio-based system. It then converts a decision matrix into a ranking system-based sample configuration. The ranking system determines the best parameters to control to reduce the manufacturing defects during the fabrication process. Our investigation shows that the MOORA method successfully determines the best manufacturing processing parameters for the reduction in defects.

This research investigates an abstract Cauchy problem associated with a coupled reaction–diffusion system of two equations, subject to homogeneous Dirichlet or Neumann boundary conditions. The primary objective is to establish the global existence of a weak solution for this nonlinear system.

The authors simplify the analysis by applying a linear transformation (v̄ = v − (c/(a − d))u) to the original system. This transformation effectively decouples the principal parabolic operators, resulting in a reformulated system where each equation features a single diffusion term with constant coefficients.

The central theoretical achievement (Theorem 2.1) is the proof of existence for a positive global weak solution (u, v̄) to the transformed system. The solution is constructed within the functional framework C([0, T]; L¹(Ω)) ∩ L¹(0, T; ) and is represented using variation-of-constants formulas involving two distinct contraction semigroups, {Sₐ(t)} and {}, generated by the operators aΔ and dΔ in L¹(Ω), respectively.

Supporting lemmas establish crucial a priori estimates. Lemma 2.1 confirms the positivity of the solution, while Lemma 2.2 provides a uniform L¹-bound on the sum u + v̄, dependent on initial data and domain parameters, which is vital for proving global existence.

The overall methodology hinges on the theory of compact semigroups and a priori estimates to overcome the nonlinearities represented by the reaction terms f̃ and g̃. This work contributes to the broader mathematical theory of nonlinear parabolic systems, with potential implications for modeling phenomena in physics, biology, and ecology where such coupled reaction–diffusion equations are prevalent.