Sturmian Comparison Theory for Half-Linear and Nonlinear Differential Equations via Picone Identity

Abstract

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form
\begin{equation}\label{e2c.}
(m(t)\Phi_\beta(y'))'+\sum_{i=1}^nq_i(t)\Phi_{\alpha_i}(y)=0
\end{equation}
and the second is the half-linear differential equations
\begin{equation}\label{e2.}
(k(t)\Phi_\beta(x'))'+p(t)\Phi_\beta(x)=0
\end{equation}
where $\Phi_*(s)=|s|^{*-1}s$ and $\alpha_1>\ldots>\alpha_m>\beta>\alpha_{m+1}>\ldots>\alpha_n>0$. Under the assumption that the solution of Eq.~\eqref{e2.} has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq.~\eqref{e2c.} by employing the new nonlinear version of Picone's formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq.~\eqref{e2c.}. Examples are given to illustrate the relevance of the results.