ICTS

In this talk, we will discuss several Lyapunov-type inequalities for second and third order linear, half-linear and fractional 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 Lyapunov-type inequalities. In the process, we also obtain sharper inequalities than many existing results in the literature. 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 Lyapunov-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.