We introduce new properties of Riemann-Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.

The stability analysis of Timoshenko beam systems that incorporate delays and fractional boundary controls is a complex area of study in the field of viscoelasticity. Our study aims to balance the symmetric influence of internal viscoelastic damping and boundary fractional damping in a structured way. The goal is to establish a system where both effects contribute symmetrically to the overall stability and dynamics. In this paper, we study the stability of certain hyperbolic evolution problems, in particular, a Timoshenko system in viscoelasticity with fractional time delay and fractional boundary controls. We prove, under assumptions on the data, the lack of exponential stability decay rate when $\eta \ge 0$ and polynomial stability decay rate when $\eta >0$ using energy methods.

We present here more general concepts of Hausdorff derivatives (structural Nabla derivatives) on a timescale. We examine structural Nabla integration on temporal scales. Using the fixed-point theorem, we establish adequate criteria for the question of existence and uniqueness of the solution to an initial value problem characterized by structural Nabla derivatives on timescales. Furthermore, some features of the new operator are proven and illustrated by using concrete examples.

In this paper, we propose a new functional space that unifies $L_{\Delta }^{p}\left( \mathbb{T},\mathbb{R}\right) $ and $L_{\nabla }^{p}\left( \mathbb{T},\mathbb{R}\right) $, motivated by the need for a more comprehensive framework to study dynamic equations involving both delta and nabla operators. This unified space is designed to better capture the analytical properties of such equations on time scales, providing a more suitable setting for future theoretical developments and applications.

This paper discusses the asymptotic controllability of fractional-order Sobolev-type perturbed stochastic control systems with Brownian motion and nonlocal fractional-order Sobolev stochastic conditions. A new set of sufficient conditions is established using the theory of semigroups together with iterative methods, with some advancements on the Brownian motion properties. Our main results are obtained assuming that the associated linear system is controllable. A most essential example is brought to illustrate the analysis obtained.

The strong stability of the Timoshenko beam in viscoelastic systems with a fractional time delay and fractional boundary controls is a complex yet fascinating area of study. This work aims to investigate a mathematical model wherein nontrivial methods are used to solve for both quantitative and qualitative properties. This paper considers a particular Timoshenko system in viscoelastic systems with a fractional time delay and fractional boundary controls. Under assumptions about the data, we establish the well‐posedness of the solution and the strong stability. The paper addresses the question of how the constraints used in the boundary in the fractional type affect the stability of our model in the presence of a fractional delay. Proving strong stability in such systems requires a careful analysis of energy decay, taking into account the interaction between fractional delays and boundary feedback. The semigroup approach will be used to show the well‐posedness, while energy methods, along with the fractional Sobolev embedding, play a critical role in the proof.