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...

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...

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...

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...

We discuss existence results for solutions to some classes of nonlinear impulsive dynamic equations on time scales. By applying modern techniques in fixed point theory, specifically expansive mappings, -set contractions, and cone methods, we derive criteria for the existence of at least one and multiple solutions. Our methodology removes the standard boundedness assumptions often required for...

This talk provides a definition of an α-integrated semi-group
on a time scale, and we prove some properties associated with this concept. It also examines the relationship between the infinitesimal generator of the α-integrated semi-group on a time scale.

Using tools from generalized quantum calculus, we derive new delay-dependent versions of classical integral inequalities of Chebyshev, Poincaré, Sobolev, Opial, and Ostrowski type. These results extend and refine existing inequalities by incorporating delayed arguments, leading to improved bounds. The proposed inequalities are well suited for applications in the analysis of delayed dynamic...

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...

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...

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...

communication networks, computer systems, and service industries. Traditional models often assume complete vacations, but more realistic scenarios include working vacations, where the server operates at a reduced service rate. At the same time, modern applications must also account for impatient customer behaviors such as balking and reneging, as well as feedback mechanisms where unsatisfied...

This paper explores the existence of solutions and the stability of first-order nonlinear nabla dynamic equations on time scales. The findings on the existence of solutions are obtained by employing Schauder’s fixed point theorem and the technique of lower and upper solutions. Further, the results concerning uniform stability and the Ulam stability of the nabla dynamic initial value problems...

We discuss existence results for solutions to some classes of nonlinear impulsive dynamic equations on time scales. By applying modern techniques in fixed point theory, specifically expansive mappings, $k$-set contractions, and cone methods, we derive criteria for the existence of at least one and multiple solutions. Our methodology removes the standard boundedness assumptions often required...

Through the framework of concavity, we derive novel reversed nabla Pachpatte-type dynamic inequalities. These formulations not only encompass their corresponding delta analogues but also subsume both discrete and continuous versions as particular instances. Moreover, in the absence of concavity, they naturally lead to generalized forms of delta and nabla Bennett–Leindler-type dynamic...

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...

Fractional differential equations are an effective mathematical tool
to study anomalous diffusion phenomena, which are structurally
presented in real-life models such as oil pollution, tumor growth,
oxygen transport through capillaries to tissues, heat conduction in
living tissues. The feature of these anomalies in
diffusion/transport processes is that the mean square displacement
of the...

This talk is on several versions of the chain rule for the nabla derivative on measure chains. After building the theory required to establish these chain rules, we discuss why the chain rules from the known calculus do not carry on to nabla derivatives. We also attempt to generalize some versions of the chain rules to normed linear spaces. As a consequence of these chain rules, the nabla...

This paper provides a definition of an α-integrated semi-group
on a time scale, and we prove some properties associated with this concept. It also examines the relationship between the infinitesimal generator of the α-integrated semi-group on a time scale.