Hypersurfaces satisfying $\mathrm{△}\vec{H}=\lambda \vec{H}$ in ${\mathbb{E}}_{\text{lowercase}s}^5$

In this paper, we study hypersurfaces $M_r^4$ $(r=0,1,2,3,4)$ satisfying $\mathrm{△}\vec{H}=\lambda \vec{H}$ ($\lambda$ a constant) in the pseudo-Euclidean space ${\mathbb{E}}_s^5$ $(s=0,1,2,3,4,5)$. We obtain that every such hypersurface in ${\mathbb{E}}_s^5$ with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in ${\mathbb{E}}_s^5$ with diagonal shape operator must be minimal.