  This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for
linear systems of equations with Neumann boundary conditions and for any
constant $p \geq 1$. We consider ordinary and elliptic problems. The results
obtained in the linear case are combined with Schauder fixed point theorem to
provide new results about the existence and uniqueness of solutions for
resonant nonlinear problems. The proof uses in a fundamental way the nontrivial
relation between the best Lyapunov constants and the minimum value of some
especial minimization problems.
