International Monthly Seminar on Time Scales Analysis

Delta Dynamic Equations on Time Scales

In this talk, we shall investigate the validity of the Wirtinger inequality within the framework of time scales, a unified approach to continuous and discrete analysis. By constructing explicit counterexamples, we demonstrate that the classical Wirtinger inequality does not hold universally across all time scales. Motivated by this finding, we propose a reformulation of the inequality by modifying its underlying conditions. Additionally, we establish several new improved Wirtinger-like inequalities, extending the theoretical foundation of the inequality on time scales.

1. INTRODUCTION
2. PRELIMINARIES
   3.PRODUCT OF TOEPLITZ MATRIX AND kTH-ORDER SLANT TOEPLITZ MATRIX.

In this work, the uniform stabilization of certain hyperbolic systems with Wentzell boundary conditions is considered, and a uniform energy decay rate for the problem is established, taking into account both internal localized damping and boundary feedback. The exponential stabilization is attained by constructing a new multiplier and using multiplier methods.

In this work, we investigate the existence and decay properties of global solutions for a class of second-order evolution equations incorporating memory effects, a nonlinear delay term, and a time-varying weight function. The model reflects realistic dynamics observed in viscoelastic and thermoelastic systems with hereditary characteristics and delayed feedback. Using appropriate energy methods and the construction of a Lyapunov functional, we establish the global existence of solutions under suitable assumptions on the kernel, delay, and nonlinearity. Furthermore, we derive general decay estimates for the energy, which unify and extend various known exponential and polynomial decay results. These findings contribute to the understanding of long-term behavior in complex dynamical systems with combined memory and delay effects.

In this work, we introduce a perturbed non-convex sweeping process with a class of subsmooth moving sets depending on the time and the state. In the first result we study the existence of solution and we present some topological properties of the attainable set, the perturbation considered here is an upper semi-continuous set-valued mapping with nonempty closed convex values unnecessarily bounded. In the second result we prove the existence to the minimal time problem and we give a description to the attainable set of control systems.

In this conversation, examines the long-term existence of solutions for a system of weakly coupled equations involving fractional evolution and various nonlinearities. The main focus is on analyzing the relationship between the regularity of initial data, memory terms, and the allowable range of exponents in a specific equation. Using L^p–L^q estimates for solutions of associated linear fractional $σ$-evolution equations with vanishing right-hand sides, along with a fixed-point method, the study establishes the existence of small-data solutions with in certain admissible exponent ranges.

In this talk we study the classes of
bounded linear operators $\Phi :\mathcal{L}\left( X,Y\right) \rightarrow \mathcal{L}\left( Z,W\right)$
such that $\left( T_{n}\right) \rightarrow \left( \Phi \left( T_{n}\right) \right) $ maps $l_{p}^{s}\left( X,Y\right) $ into $l_{p}\left( Z,W\right) $,
$l_{p}^{s}\left( X,Y\right) $ into $l_{p}^{s}\left( Z,W\right) $ and $% l_{p}^{w}\left( X,Y\right) $ into $l_{p}^{w}\left( Z,W\right) $. The Pietsch-type domination of $(l_{p}^{s},l_{p}) $-summing linear operators is also given .
\
\vspace{0.3cm}\
{\textbf {Keywords:}}$p-summing$ operator, Finite rank operator,
ideal property of $p-suming$ operators , Linear operator ideals,$(\ell^s_p,\ell_p)$-summing operators\
{\bf {2020 Mathematics Subject Classification:}} Primary 47A35, 60Fxx, 60G10.%-----------------------
\vspace{0.5cm}

The paper deals with the following fractional Hardy-Sobolev equation with nonhomogeneous term
\begin{equation}
%\label{eq1}
\begin{cases}
{(-\Delta)}^{s}u-\mu \frac{u}{|x|^{2s}}=|u|^{2_{s}^{}-2}u+\lambda \frac{u}{|x|^{2s-\alpha}}+f(x),&x\in \Omega,\
u=0&x\in \partial\Omega,
\end{cases}
\end{equation}
being $02s)$ containing the origin $0$ in its interior, $0\leq \mu <\overline{\mu_{s}}:=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$, $\lambda$ is a positive parameter, $0<\alpha<2s$, $2_{s}^{*}=\frac{2N}{N-2s}$ is the fractional critical Hardy-Sobolev exponent. The fractional Laplacian $(-\Delta)^{s}$ is defined by
\begin{equation}
-2(-\Delta)^{s}u(x)=C_{N,s}\underset{\mathbb{R}^{N}}{\int}\dfrac{u(x+y)+u(x-y)-2u(x)}{|x-y|^{N+2s}}dy
\end{equation*}
where
$$C_{N,s}=\dfrac{4^{s}\Gamma(N\setminus2+s)}{\pi^{N\setminus2}|\Gamma(-s)|}.$$ $\Gamma$ is the Gamma function, $f$ is a given bounded measurable function. by virtue of Ekeland’s Variational Principle and the Mountain Pass Lemma and for which we consider the following hypothesis \begin{equation*} \inf\left\lbrace \gamma_{N,s}(T(u))^{\frac{N+2s}{4s}}-\underset{\Omega}{\int}f u dx:\;u\in X,\underset{\Omega}{\int}|u|^{2_{s}^{*}} dx=1\right\rbrace>0.\;\;(\mathcal{F}) %\label{ast} \end{equation*} Where $X$ is a Hilbert space defined as $$X=\lbrace u\in H^{2s}(\mathbb{R}^{N}):u=0\;\text{in}\;\mathbb{R}^{N}\setminus\Omega\rbrace,$$ where $H^{2s}(\mathbb{R}^{N})$ the usual fractional Sobolev space, $$\gamma_{N,s}=\frac{4s}{N-2s}(\frac{N-2s}{N+2s})^{\frac{N+2s}{4s}}$$ and $$T(u)=C_{N,s}\underset{\mathbb{R}^{N}}{\int}\underset{\mathbb{R}^{N}}{\int}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy-\mu \underset{\Omega}{\int}\frac{u^{2}}{|x|^{2s}}dx-\lambda\underset{\Omega}{\int}\frac{u^{2}}{|x|^{2s-\alpha}}dx.$$\\ Moreover, the following eigenvalue problem with Hardy potentials and singular coefficient \begin{equation*} \begin{cases} {(-\Delta)}^{s}u-\mu \frac{u}{|x|^{2s}}=\lambda \frac{u}{|x|^{2s-\alpha}}& x\in\Omega, \\ u=0 & x\in\partial \Omega, \end{cases} \end{equation*} where $0 < \alpha <2s$, $\lambda \in \mathbb{R}$, has the first eigenvalue $\lambda_{1}$ given by: \begin{equation*} \lambda_{1}= \underset{u\in X\setminus\lbrace0\rbrace}{\inf}\dfrac{C_{N,s}\underset{\mathbb{R}^{N}}{\int}\underset{\mathbb{R}^{N}}{\int}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy-\mu \underset{\Omega}{\int}\frac{u^{2}}{|x|^{2s}}dx}{\underset{\Omega}{\int}\frac{u^{2}}{|x|^{2s-\alpha}}dx}. \end{equation*} We get the following results: \\ Let $0<\mu<\overline{\mu_{s}}$, $0<\lambda<\lambda_{1}$ and $f$ is a bounded measurable function satisfying the condition $(\mathcal{F})$, then the problem has at least two nontrivial solutions, if $0<\alpha<2\beta^{+}(\mu)+2s-N.$ \\ Where $\beta^{+}(\mu)$ comes through the processes and techniques of calculations.
%\label{th}

In this work, we investigate the approximation of spatially nonstationary spatio-temporal GARCH (ST-GARCH) processes by spatially stationary counterparts at fixed locations. This approach enables a localized analysis of complex spatio-temporal volatility structures. Building upon the model's recursive formulation, we establish that the ST-GARCH process can be represented as a sum of random matrix products, allowing us to derive conditions under which the process admits a Lipschitz continuous approximation. We prove that, under mild regularity and continuity assumptions, the nonstationary process (X^2_t(s)) can be closely approximated by a spatially stationary process (X^2_{t,s_0}(s)) at a fixed point (s_0), with a convergence rate governed by the spatial distance (|s - s_0|_\infty). Furthermore, using a Taylor expansion and a derivative-based construction, we refine this approximation by including the first-order spatial derivative, yielding an improved representation as a linear combination of two spatially stationary processes. Our theoretical findings lay the groundwork for practical localized modeling and inference in real-world applications involving heterogeneous spatio-temporal data.

In extreme value theory (EVT), estimating the tail index of heavy-tailed distributions is crucial for understanding rare and extreme events. Traditional estimators such as the Hill and Maximum Likelihood Estimators (MLE) perform well with large samples but struggle
with small sample sizes due to increased bias and variance. In this paper, we introduce a novel estimation technique the Maximum Lq-Likelihood Estimator (MLqE), which incorporates a distortion parameter q, making it more robust to extreme observations and more accurate in small-sample scenarios. We demonstrate that the MLqE is consistent and asymptotically normal, outperforming the classical MLE in terms of mean squared error in
moderate and small sample sizes. Moreover, we present simulation results that highlight the superior performance of the MLqE, particularly when comparing it to the MLE in tail
index estimation. This method not only offers a significant improvement in the accuracy of heavy-tailed distribution parameter estimation but also provides a versatile tool for various
real-world applications, including finance, hydrology, and risk management.

Cohen has introduced the notion of strongly $p$-summing and $p$%
-nuclear for linear operators. Many authors have considered these notions by
generalizing in several directions, namely the multilinear, sublinear and
Lipschitz cases. In the same circle of ideas, we will make an extension of
these notions in order to produce the class of Cohen M-strictly Lipschitz $p$-nuclear operators.\newline

\\
\vspace{0.3cm}\\
{\textbf {Keywords:}}
Lipschitz $p$-summing operators,  MS-Lipschitz $p$-summing operators, MS-Cohen Lipschitz $p$-summing\\
{\bf {2020 Mathematics Subject Classification:}} Primary 47A35, 60Fxx, 60G10.%-----------------------

The present presentation investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and a general source term, where we assume that the initial data are of Yudovich type.

This talk investigates the conditional Hyers–Ulam stability of first-order nonlinear logistic models, both continuous and discrete. Identifying bounds on both the relative size of the perturbation and the initial population size is an important issue for nonlinear Hyers–Ulam stability analysis. Utilizing a novel approach, for h-difference equations we derive explicit expressions for the optimal lower bound of the initial value region and the upper bound of the perturbation amplitude, surpassing the precision of previous research. Furthermore, we obtain a sharper Hyers–Ulam stability constant, which quantifies the error between true and approximate solutions, thereby demonstrating enhanced stability. The Hyers–Ulam stability constant is proven to be in terms of the step-size h and the growth rate, but independent of the carrying capacity. Detailed examples are provided illustrating the applicability and sharpness of our results on conditional stability.

Autoregressive (AR) models are fundamental tools in time series analysis, capturing temporal dependencies through lagged observations. While traditional approaches often focus on long-term dynamics, many real-world phenomena—such as high-frequency financial data, climate fluctuations, and energy demand—exhibit behaviors that are best understood at shorter time scales and are often influenced by seasonal effects. In this work, we introduce and study the Short Time Scale Autoregressive (STAR) process, designed to model short-range temporal correlations with particular attention to rapidly evolving structures in the data while explicitly incorporating seasonal components.