One of interesting subjects in differential geometry is the biharmonic Lorentzian hypersurfaces of Lorentz 5-space form. A Lorentzian hypersurface ψ : M^4_1 → M^5_1(c) is said to be C-biharmonic if it satisfies the extended biharmonicity condition (C^2)ψ = 0. C is the well-known Cheng-Yau operator. We study weakly C-biharmonic Lorentzian hypersurfaces of M^5_1(c) with at most two distinct principal curvatures and constant mean curvature.
