Boletim da Sociedade Paranaense de Matemática

Iterated Bernstein-type $L_p$ Inequalities for Polynomials

2026-02-21T15:11:39+00:00

We develop several new Bernstein-type $L_p$ inequalities for complex polynomials by iterating the first-order differential operator $A_\alpha(P):=zP'(z)-\alpha P(z)$. Our results extend, unify, and sharpen $L_p$ inequalities of Zygmund, de Bruijn, and Jain as well as the recent $L_p$ extensions for $A_\alpha$ and its second-order companion shown in \cite{RatherBhatGulzar2024}. In particular, for any finite sequence $\alpha_1,\dots,\alpha_m$ with $\Rea(\alpha_j)\le n/2$ we obtain sharp bounds for $\norm{\prod_{j=1}^m A_{\alpha_j} P}_p$ in terms of $\norm{P}_p$ for all $0\le p\le\infty$, together with refined ``Erd\H{o}s--Lax''-type improvements when $P$ has no zeros in the open unit disc. As corollaries, we derive $L_p$-versions of higher order Bernstein inequalities for $z^k P^{(k)}$ and scale-invariant formulations on circles $\{|z|=r\}$.

Fixed Point Theorems for Four Mappings in a Complete Digital Metric Space

2026-02-21T15:18:41+00:00

We demonstrate a fixed-point theorem for digital photographs in this study. In particular, we prove a special digital fixed-point theorem for four self-mappings in an entire digital metric space. In the setting of digital metric space, our solution is a logical progression of the seminal work of Bhagwat and Singh [2].

Complex Valued Extended b-Metric Space and Its Fixed Points

2026-02-21T16:28:54+00:00

This work focuses on establishing a set of common fixed point theorems in complex valued
extended b-metric spaces, formulated under rational contraction conditions. The results obtained not only
extend the classical theorems of Azam et al. [1], Bhatt et al. [3], Bryant [4], and Rouzkard and Imdad [10],
but also provide a broader framework for their application. Furthermore, several corollaries are derived, and
illustrative examples are included to showcase the practical relevance of the theorems and the improvements
they offer over earlier results.

Solution of Non Linear Matrix Equation Using $\theta$-Hyperbolic Sine Distance Functions

2026-02-21T16:53:13+00:00

In this paper we introduce Ciric-type(I) $\theta$-hyperbolic $\mathcal{Z}$- contraction and discuss the existence and uniqueness of fixed point of a mapping satisfying Ciric-type(I) $\theta$-hyperbolic $\mathcal{Z}$- contraction in the setting of orbitally complete metric spaces. We furnish the result by suitable example. Further we apply the results to find solution of non liner matrix equation.

The The Vector $\overrightarrow{0}$ Algorithm for Solving Optimal Strategy Problems in Matrix Games and Experimental Computation in Matlab Environment

2026-02-21T19:03:13+00:00

From the Cone-Min Method presented in \cite{ref1}, we have built a Vector $\overrightarrow{0}$ algorithm to solve the primitive standard linear programming problem with an objective function with non-negative coefficients.
We have built a computer sample program for this algorithm to solve primal standard linear programming problems with size any on the Matlab environment. We present experimental results on the same problem with random data to solve some primal linear programming problems has form the problem of optimal strategy in matrix game, using Vector $\overrightarrow{0}$ algorithm compared with Simplex Method. The experimental calculation results with random data show that the number of iterations and calculation time according to this new algorithm compared to the Simplex algorithm is much less and noticeably faster. We hope that it will be one of the effective algorithms for solving medium and large size linear programming problems on any common computer today.

Approximation of Linear Positive Fuzzy Operators

2026-02-21T20:06:43+00:00

This study extends the main results of classical approximation theory to fuzzy theory. We begin by defining fuzzy valued functions, exploring their properties and then applying the fuzzy Korovkin theorem to approximate them. The study delves into the approximation by various linear positive fuzzy operators – Fuzzy Bernstein, fuzzy Szasz-Mirakyan and fuzzy Baskakov operators – utilizing them to approximate fuzzy-valued functions. Their Voronovskaya type asymptotic result is also proved using Taylor’s theorem.

A Study of Fractional Order SIS Model with Fear Effect and Beddington-De Angelis Incidence Rate

2026-02-21T20:12:26+00:00

Many studies have demonstrated that during epidemics, fear can significantly influence human behaviour, often leading to a decline in birth rates. In this work, we propose a fractional-order SIS compartmental model that incorporates the effects of fear and employs a Beddington-De Angelis type incidence rate. This incidence function captures the impact of preventive measures taken by both susceptible and infected individuals, thereby reflecting more realistic disease transmission dynamics. Following the formulation of the model, we establish fundamental properties such as positivity and boundedness of solutions. We then compute the basic reproduction number, R0, and demonstrate the existence of an endemic equilibrium when R0 > 1. Furthermore, we analyze the local stability of both the disease-free and endemic equilibria using the linearized system. To support our analytical results, we conduct numerical simulations using the Adams-Bashforth-Moulton Predictor-Corrector method.

Novel Iterative Approaches: Examining Stability, and Convergence

2026-02-21T20:16:47+00:00

In this paper, we propose a new iterative method for approximating fixed points of contraction
mappings. We prove a convergence theorem and evaluate its rate of convergence regarding the KF, AA,
Piri, and S∗ iterative algorithms. In addition, we present findings on stability. These findings contribute to
the ongoing development of iterative algorithms for nonlinear problems in various mathematical and applied
contexts.

Optimizing Production Strategies for Deteriorating Items in Two-level Manufacturing

2026-02-21T20:54:28+00:00

A modern business enterprise should develop a sound operational system of organization to meet the challenges of an ever-changing environment where conditions are somewhat too unpredictable. The need to increase production is determined by the demand for specific goods rising with time, which in many cases is done because of the growth in popularity and the value associated with the product. Thus, the organization faces pressure to increase its manufacturing rates to meet the increased demand. The proposed model optimizes a deteriorating goods inventory system with two different manufacturing rates and exponentially formulable demand. Production initially starts off at a constant rate, how is that production, demand, inherent deterioration and variable manufacturing rates combine to build up the inventory level progressively. The more the demand rate is increased, the more the production rate is increased by the factor known as a so that it is managed according to the market. The feedback system is important to ensure that supply and demand is balanced in a dynamic marketplace. The empirical results only further show that a 10% increase in production costs is equivalent to a 5% increase in total costs, which shows the non-linear relationship between cost input and the overall financial performance. To promote the reliability of the model, a sensitive analysis has been carefully conducted, and thus a robustness check of the model and predictive reliability of its operational results has been verified.

Mixed hemivariational inequality arising in a thermo-elastic frictional contact problem

2026-02-21T21:07:14+00:00

This paper presents a new mathematical model for the analysis of frictional contact between a thermoelastic body and a foundation. The contact interaction is described through a combination of unilateral frictional conditions, nonmonotone multivalued contact laws, and friction laws formulated via the Clarke subdifferential. A variational formulation of the problem is developed, and the existence and uniqueness of a weak solution are rigorously proved.

Analyzing the Limit Set of Rough Ideal $\lambda\gamma$-Statistical Convergence of Order $\varrho$ in Lattice-Valued Fuzzy Normed Spaces

2026-02-21T15:11:45+00:00

This study introduces the framework of rough $\mathcal{I}$-$\lambda\gamma

$-statistical convergence of order $\varrho$ within the setting of

$\mathcal{L}$-fuzzy normed spaces (lattice-valued fuzzy normed spaces). This

generalizes existing convergence notions by integrating ideal convergence

($\mathcal{I}$), generalized sequence transformations ($\lambda\gamma$), an

arbitrary order ($\varrho$), and the concept of roughness ($r$). A primary

focus is the characterization of the resulting rough limit set. We rigorously

establish that, contrary to classical convergence, the limit is inherently a

set. Furthermore, we prove that this limit set possesses key structural

properties, specifically closure and convexity, under the topology induced by

the $\mathcal{L}$-fuzzy norm. Finally, we define the corresponding notion of

$\mathcal{I}$-$\lambda\gamma$-statistical cluster points of order $\varrho$

and elucidate the relationship between this set of cluster points and the

rough limit set.

The Perturbed G-Metric Space and Fixed Point Results

2026-02-21T15:26:55+00:00

In this paper, we introduce and investigate fixed point results within the framework of perturbed G-metric space, a generalization of metric and perturbed metric spaces involving a generalized distance function defined on triples of points. We establish Banach amd Kannan fixed point theorems for mappings that contract with respect to a perturbed G-metric, proving existence and uniqueness of fixed points. Further, the results of Mustafa, Mustafa and Obiedat, Nutu and Pacurar, and Gaba are obtained as corollaries.

Thermal and mass transport in Maxwell-Boger nanofluid flow through a variable porous medium with endothermic and exothermic chemical reactions

2026-02-21T21:54:41+00:00

Riga surface with a variable porous medium plays a vital role in boundary layer flow and flow separation, and its major applications are found in submarines, biomedical, and aircraft. Therefore, the current work examines the impact of activation energy and a variable porous medium via a Riga surface in the existence of Maxwell-Boger nanofluid with endothermic/exothermic chemical reactions, active and passive control of nanoparticles. The similarity transformations are utilized for transforming partial differential equations into dimensionless ordinary differential equations. Additionally, the Runge-Kutta-Fehlberg 4^th 5^th order and shooting methods are used to resolve the reduced equations, and engineering factors were also examined. Graphical analysis is used to analyse the behaviour of dimensionless factors with their profiles. Notable findings reveal that an increment in activation energy, a drop in temperature in the exothermic, and a rise in the endothermic cases. For increasing values of the porous parameter, velocity profile drops. Streamline patterns and isothermal contours are also illustrated.

Solvability of Nonlinear Volterra-Hammerstein type Fractional Integral Equations in Orlicz Space

2026-02-21T16:33:29+00:00

In this paper, we are focused on analyzing new analytical properties of $g$-fractional type operators, such as continuity, boundedness and monotonicity in Orlicz spaces $L_\varphi $. Using these properties, along with Darbo’s Fixed-Point theorem and the measure of noncompactness, we investigate the existence and uniqueness of solutions to a nonlinear fractional integral equation in $L_\varphi $. The $ g$-fractional operators being investigated for the first time in the space $L_\varphi .$ Here we generalizes various fractional operators and encompassing and unifying the results of many specific cases of classical and quadratic fractional issues explored in the previous literature. Lastly, we provide some examples to illustrate our main results.

A mathematical modeling to determine the evolutionary relationships between Bat and Human corona virus

2026-02-21T17:01:47+00:00

In this article, we present novel correlation coefficients for measuring the relationship between two fuzzy sets. Here we put forward mathematical models to determine the evolutionary relationships between SARS-CoV \& SARS-CoV-2 and five other commonly known corona viruses in human which includes MERS-CoV, HCoV-OC43, HCoV-229E, HCoV-NL63 and HCoV-HKU1 and two bat corona viruses (RsSHC014 and RaTG13) with special emphasis on their correlation. The work can be considered as a pioneer in the field of application of fuzzy set theory in the study of this zoonsis.

Solvability Analysis of (k − ℓ)-Hilfer Fractional Differential Equations through Generalized Weak Wardowski Contractions

2026-02-21T18:43:54+00:00

In this work, we introduce the concept of generalized weak Wardowski contractions and establish the existence and uniqueness of fixed points for such mappings. Furthermore, we apply weak Wardowski contraction to investigate the existence of solutions for a novel (k − ℓ)-Hilfer fractional differential equation of order 2 < α ≤ 3 subject to specific boundary conditions. Finally, an example is provided to illustrate the applicability and effectiveness of the obtained theoretical results.

NONLOCAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS FOR MODELING ANOMALOUS DIFFUSION IN BIOLOGICAL TISSUES: A UNIFIED THEORETICAL FRAMEWORK

2026-02-21T18:47:37+00:00

Anomalous diffusion is a hallmark of complex biologically tissues where heterogeneity, microstructural barriers, and long-range correlations lead to significant deviations from classical Fickian behavior. These tissues include brain white matter, extracellular matrix networks, and tumor microenvironments where experimental data show nonlinear mean square displacement, memory effects, nonlocal interactions, and anisotropic transport patterns. Hence, traditional integer-order diffusion equations are insufficient to explain biological diffusion. This research builds a unified theoretical framework utilizing nonlocal fractional partial differential equations (FPDEs) that combine three essential aspects: generalized fractional temporal derivatives for modeling memory-driven subdiffusion, anisotropic fractional spatial operators for capturing the direction-dependent tissue microstructure, and kernel-based nonlocal interactions for explaining the long-range spatial coupling. The nonlocal boundary conditions integrated into the resultant FPDE model offer a mathematically rigorous description of the anomalous transport phenomena. Besides, the model analytical results such as existence, uniqueness, positivity preservation, energy estimates, and scaling laws, to name a few, provide evidence of the model’s physical consistency and well-posedness. To complete the picture, the authors have developed numerical schemes that approximate the full dynamics based on L1 time discretization, Fourier spectral methods, and nonlocal quadrature. Simulation results on drug transport, diffusion-weighted MRI, extracellular matrix diffusion, and tumor microenvironment modeling are representative of the framework's biological relevance and versatility. The current work serves as a solid basis for forthcoming multiscale, data-driven, and inverse modeling research on biological anomalous diffusion.

Adaptive Multi-Switching Chaos Synchronization of Lotka-Volterra Model to Replicate Complete and Anti-Behavior of Hindmarsh-Rose Neuron Model

2026-02-21T18:51:22+00:00

The generalized Lotka--Volterra ($\mathscr{L}$–$\mathscr{V}$) model, widely known for describing predator–prey population interactions, and the Hindmarsh–Rose ($\mathscr{H}$–$\mathscr{R}$) neuron model, a cornerstone in computational neuroscience, represent two distinct examples of nonlinear dynamical behavior in ecology and biology, respectively. In this work, an adaptive control framework is developed to achieve analytical results on multi-switching based hybrid projective synchronization between the $\mathscr{L}$–$\mathscr{V}$ and $\mathscr{H}$–$\mathscr{R}$ systems under eleven uncertain parameters. This approach relies minimally on exact parameter information, which enhances robustness and maintains efficient synchronization. Stability of synchronization errors is ensured through Lyapunov analysis. MATLAB simulations further verify the theoretical findings, demonstrating that synchronization remains successful despite the presence of multiple parameter uncertainties. Moreover, the framework encompasses several classical schemes, including complete synchronization, anti-synchronization, projective synchronization, and hybrid projective synchronization, as special cases. Alongside the synchronization results, a detailed dynamical analysis of the $\mathscr{L}$–$\mathscr{V}$ system is also conducted, providing additional insights into its complex nonlinear behavior.

A generalization of Darbo’s fixed point theorem and its applications to ψ Hilfer fractional hybrid differential equation

2026-02-21T18:59:58+00:00

In the present paper, we develop a new fixed point theorem based on a newly introduced
contraction operator. Our formulation is built upon essential ideas from the measure of noncompactness,
which serve as a foundation for our analysis. By using this framework, we study the existence results for the
solutions to the first order hybrid fractional differential equation involving the ψ-Hilfer fractional derivative.
The applicability of our results, we conclude with an example.

A nonhomogeneous Steklov problem with $(p,q)$-Laplace differential operator

2026-02-21T19:17:22+00:00

We consider a nonlinear Steklov problem driven by the $(p,q)$-Laplacian operator with a concave parametric term and an asymmetric perturbation. We prove a multiplicity theorem producing three non-trivial solutions all with sign information(two positive and one negative), when the parameter is sufficiently small. Under a oddness condition near the origin for the perturbation.

Generalized Statistical Convergence for Uncertain Double Sequences of Fuzzy Numbers Defined by an Orlicz Function

2026-02-21T19:20:50+00:00

In this study we propose the notion of I₂-statistical convergence for uncertain double sequences of fuzzy numbers defined by an Orlicz function. We investigate several associated convergence types such as convergence in measure, convergence in mean, convergence in distribution and uniformly almost sure convergence. Furthermore we present illustrative examples to clarify the connections and differences among these distinct convergence forms.

Quantum Difference Relative Uniform Convergence of Double Sequence Spaces of Sargent Type Functions

2026-02-21T20:01:21+00:00

This paper introduces two new double sequence spaces of Sargent type functions, denoted by ₂m(φ,ru,∇_{q}) and ₂n(φ,ru,∇_{q}), which are defined using the concept of relative uniform convergence in combination with the Jackson q-difference operator for double sequences. In this framework, we define bounded, p-absolutely summable, convergent, and null double sequences of functions based on the idea of quantum difference relative uniform convergence with respect to a scale function. These classes are represented by ℓ_{∞}(ru,∇_{q}), ℓ_{p}(ru,∇_{q}), ₂c(ru,∇_{q}) and ₂c₀(ru,∇_{q}), respectively. We also explore the inclusion relations and isomorphisms between these newly introduced spaces and other existing function spaces. Additionally, we investigate several algebraic and geometric properties, such as solidness and convexity.

Modular Space Stability for Cubic Functional Equations in Nonlinear Material Modeling

2026-02-21T20:58:47+00:00

This paper investigates the stability of a generalized cubic functional equation of the form
\begin{align*}
(m-n)\Big[(m+n)^3 f\Big(\tfrac{nu + mv}{\,n+m\,}\Big) + (n-m)^3 f\Big(\tfrac{nu - mv}{\,n-m\,}\Big)\Big]\\
+(m+n)\Big[(m+n)^3 f\Big(\tfrac{mu + nv}{\,m+n\,}\Big) + (m-n)^3 f\Big(\tfrac{mu - nv}{\,m-n\,}\Big)\Big]\\
=~mn(m^2+n^2)\big[f(u+v)+f(u-v)\big] + 2\,(m^4-n^4)f(u),
\end{align*}
within the setting of modular normed spaces. Using the direct method of Hyers--Ulam and a suitably defined control function, we establish explicit stability bounds for approximately cubic mappings. In addition, by employing an operator constructed in the modular space without $\Delta_2$-conditions and applying the fixed point alternative, we obtain existence, uniqueness, and generalized Hyers--Ulam stability of the exact cubic solution.

An application to nonlinear constitutive modeling in continuum mechanics is presented to illustrate the physical relevance of the cubic equation. The cubic stress--strain relation, widely used in modeling polymers, biological tissues, and metals under finite deformation, fits naturally into the functional framework developed in this study. The stability results guarantee that experimentally observed or numerically computed approximate constitutive laws admit a unique exact cubic model in their vicinity, enhancing robustness in material characterization and computational simulations. The findings demonstrate both the theoretical depth and practical significance of stability analysis for higher-order functional equations in modern applied mathematics.

A THREE-DIMENSIONAL MIXED FINITE ELEMENT SCHEME FOR NONLINEAR QUASI-STATIC EDDY CURRENT PROBLEM

2026-02-21T21:03:35+00:00

We present the first fully discrete mixed finite element formulation
for the stationary three-dimensional p-curl problem. The method relies on
N´ed´elec edge elements and Raviart–Thomas face elements, and yields a discrete
scheme that is consistent with the underlying continuous mixed formulation.
The core of the analysis is the proof of existence and uniqueness of the discrete
solution for p > 2. This work provides the first rigorous numerical framework
for the mixed finite element approximation of the three-dimensional p-curl
problem, and paves the way for future extensions, including non-homogeneous
boundary conditions, the fully time-dependent problem, and the implementation
of the method on representative test cases.