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In this paper, first we give a notion for linear Weingarten spacelikehypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$.Furthermore, we study complete or compact linear Weingarten spacelikehypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying somecurvature conditions. By modifying Cheng-Yau's operator $\square$ given in{\cite{ChengYau77}}, we introduce a modified operator $L$ and give newestimates of $L(nH)$ and $\square(nH)$ of such spacelike hypersurfaces.Finally, we give partial generalizations of some conjectures in locallysymmetric Lorentz spaces $L_{1}^{n+1}$.

Partial generalizations of some Conjectures in locally symmetric Lorentz spaces