Radial pulsations of gravitationally decoupled anisotropic self-bound stars with a generalized linear equation of state

We construct anisotropic self-bound stellar configurations through gravitational decoupling in the minimal geometric deformation framework and investigate their stability against linear radial perturbations. Retaining the Tolman IV temporal potential, we impose the generalized linear equation of state $P_r=η(ρ-ρ_s)$ on the complete matter distribution. Central regularity and Schwarzschild matching determine the radial geometry and relate the surface density to the compactness and stellar radius. Closed-form solutions are obtained for $η=1,1/2,1/3,1/4$, including the MIT bag radial relation when $η=1/3$ and $ρ_s=4B$. Since the equilibrium anisotropy does not determine its dynamical response, we introduce two material prescriptions that preserve either the anisotropy of each fluid layer or its ratio to $ρ+P_r$. Including metric perturbations, we compute the fundamental mode and three overtones for twelve equilibrium configurations under both prescriptions. All 24 background response combinations have positive fundamental squared frequencies, establishing linear radial stability within the adopted closures. The fundamental eigenvalue decreases with compactness within each sampled family, while changing the anisotropic response produces shifts whose direction depends on the background. Tangential tensions in several configurations and the failed tangential profile diagnostic of the maximally stiff family limit their physical interpretation. These results provide analytically tractable self-bound backgrounds and demonstrate that identical equilibrium configurations can support distinct radial spectra depending on their anisotropic material response.