Lower bounds for generalized eigenvalues of the quasilinear systems

In this paper, we establish several new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which generalize or improve all related existing ones. Then we use the Lyapunov-type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one-dimensional quasilinear elliptic system with the Dirichlet boundary conditions.