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.

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.

In this paper we study some topological density properties of certain functional spaces on the time scales and its relationships to Lebesgue spaces in the sense of ∇-integrals on time scales. Our results are provided with applications

In this paper, we investigate the structural $\nabla$-derivative on arbitrary time scales. We begin by recalling the classical $\nabla$-derivative and then extend it to the structural setting by incorporating a structural function $p(\cdot)$ and a positive parameter $\alpha$. Several illustrative examples are provided to emphasize the distinction between the standard $\nabla$ derivative and its structural counterpart. Fundamental properties are established, including monotonicity results and a chain rule for function compositions. These results not only unify existing approaches-such as the classical $\nabla$, weighted, and fractional-type operators-but also open new perspectives for applications in dynamic equations, stability analysis, and variational problems within the time scale framework.

We present a new concept of Riemann-Liouville Fractional Sobolev Spaces in the context of time scale calculus in this paper. This novel method unifies continuous and discrete analysis by extending conventional fractional Sobolev space theory to dynamic domains. Our study’s primary contribution is the first description of L_{Δ}^{p}-representability in relation to time scales, which lays the groundwork for future advancements in fractional dynamic equations. Our findings offer novel ideas regarding fractional.

In this paper, we introduce and develop the concept of tempered distributions on time scales. This work is based on the framework of Lebesgue and Sobolev spaces $L^{p}_{\Delta}(\mathbb{T})$ and $W^{1,p}_{\Delta}(\mathbb{T})$, which provide the appropriate setting for extending classical distribution theory to the delta calculus. Our approach establishes a unified formulation that remains valid for both continuous and discrete cases, thereby contributing to the functional and harmonic analysis on time scales.