λ-Biharmonic hypersurfaces in the product space L^{m}\times $\mathbb{R}$

In this paper, we study λ-biharmonic hypersurfaces in the product space L^{m}\times$\mathbb{R}$, where L^{m} is an Einstein space and $\mathbb{R}$ is a real line. We prove that λ-biharmonic hypersurfaces with constant mean curvature in L^{m}\times$\mathbb{R}$ are either minimal or vertical cylinders, and obtain some classification results for λ$-biharmonic hypersurfaces under various constraints. Furthermore, we investigate λ-biharmonic hypersurfaces in the product space L^{m}©\times$\mathbb{R}$, where L^{m}© is a space form with constant sectional curvature c, and categorize hypersurfaces that are either totally umbilical or semi-parallel.