Qualitative analysis on existence and stability of nonlinear fractional dynamic equations on time scales

This study explores the qualitative analysis of Dγh(ζ)=ℒ(ζ,h(ζ),Dγh(ζ)),\begin{array}{r}{{\mathcal{D}}}^{\gamma }{\mathfrak{h}}\left(\zeta )={\mathcal{ {\mathcal L} }}\left(\zeta ,{\mathfrak{h}}\left(\zeta ),{{\mathcal{D}}}^{\gamma }{\mathfrak{h}}\left(\zeta )),\end{array} where Dγ{{\mathcal{D}}}^{\gamma } is a Riemann–Liouville nabla (∇\nabla ) fractional nabla derivative, incorporating integral boundary conditions on arbitrary time scales. We rigorously investigate the existence, uniqueness, and stability of solutions to these equations by employing Krasnoselskii’s fixed point theorem and the Banach contraction principle. The incorporation of integral boundary conditions extends the applicability of the results to more general systems. To demonstrate the validity and practicality of the theoretical results, we present an illustrative example supported by MATLAB-generated graphical representations. This work contributes to the broader understanding of fractional dynamic systems and their stability across diverse time scales, offering insights with potential applications in engineering and applied sciences.