Lyapunov-type inequalities for higher order half-linear differential equations

The Lyapunov inequality for second order linear differential equations has been extended to higher order linear differential equations in various forms. It has also been extended to second and third order half-linear equations. However, it has not yet been developed for higher order half-linear differential equations due to the nonlinear feature of the equation and the complexity caused by the multiple p-Laplacian operators in the equation. In this paper, by subtle applications of forward and backward inductions, we establish several Lyapunov-type inequalities for even and odd order half-linear differential equations. These inequalities are applied to determine the nonexistence of nontrivial solutions of higher order boundary value problems and to estimate eigenvalues for higher order eigenvalue problems. Our results cover many results in the literature when the equations become linear.