Lyapunov-Type Inequalities for Third Order Nonlinear Equations

TL;DR

This paper establishes new Lyapunov-type inequalities for third order nonlinear differential equations involving generalized $ ext{ψ}$-Laplacian operators, providing insights into solution behavior, oscillation, and zero distribution.

Contribution

It introduces novel Lyapunov inequalities that incorporate $q_{+}$ and $q_{-}$, generalizing previous results and employing a different proof technique for third order nonlinear equations.

Findings

Derived inequalities constrain solution maxima.

Provided bounds on the number of zeros of solutions.

Analyzed properties of oscillatory solutions.

Abstract

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.