  In this paper, the generalized eigenvalue complementarity problem for tensors
(GEiCP-T) is addressed, which arises from the stability analysis of finite
dimensional mechanical systems and find applications in differential dynamical
systems. The general properties of the (GEiCP-T) have been studied. We
establish its relationship with the generalized tensor eigenvalue problem. It
follows that if exist, the number of $\lambda$-solutions can be bounded. We
also give some sufficient conditions for the existence of the solution. In
particular, there exists a unique solution of the (EiCP-T) (i.e., $J=[n]$) for
irreducible nonnegative tensors. For the symmetric case, we derive a sufficient
and necessary condition for the solvability of the (GEiCP-T) by reformulating
it as a nonlinear program. It has also been proved that deciding the
solvability of the (EiCP-T) is NP-hard in general. Moreover, a shifted
projected power method is proposed to solve the symmetric (GEiCP-T). The
monotonic convergence is also established. The numerical experiments
demonstrate convergence behavior of our method and show that the algorithm
presented is promising.
