Lyapunov-Type Inequalities for Third Order Nonlinear Equations

We derive Lyapunov-type inequalities for general third order nonlinear equations involving multiple $\psi$-Laplacian operators of the form \begin{equation*} (\psi_{2}((\psi_{1}(u'))'))' + q(x)f(u) = 0, \end{equation*} where $\psi_{2}$ and $\psi_{1}$ are odd, increasing functions, $\psi_{2}$ is super-multiplicative, $\psi_{1}$ is sub-multiplicative, and $\frac{1}{\psi_{1}}$ is convex, and $f$ is a continuous function which satisfies a sign condition. Our results utilize $q_{+}$ and $q_{-}$, as opposed to $|q|$ which appears in most results in the literature. Additionally, these new inequalities generalize previously obtained results, and the proofs utilize a different technique than most other works in the literature. Furthermore, using the obtained inequalities, we obtain a constraint on the location of the maximum of a solution, properties of oscillatory solutions, and an upper bound for the number of zeroes.