Liapunov-type inequalities for third-order half-linear equations and applications to boundary value problems

Abstract

We derive Liapunov-type inequalities for third-order half-linear differential equations. These inequalities utilize integrals of both q+(t) and q−(t) rather than those of |q(t)| as in most papers in the literature for higher-order Liapunov-type inequalities. Furthermore, by combining these inequalities with the “uniqueness implies existence” theorems by several authors, we establish the uniqueness and hence existence–uniqueness for several classes of boundary value problems for third-order linear equations. We believe that this is the first time for Liapunov-type inequalities to be used to deal with boundary value problems and expect that this approach can be further applied to study general higher-order boundary value problems.