Jaroslav Jaroš

Generalized Picone Identity and Comparison of Half-Linear Differential Equations of Order 4m

abstract:
A Picone-type identity and the Sturm-type comparison theorems are established for ordinary differential equations of the form
$$ \big(p(t)\varphi(u^{(2m)})\big)^{(2m)} + q(t)\varphi(u) = 0 $$
and
$$ \big(P(t)\varphi(v^{(2m)})\big)^{(2m)} +Q(t) \varphi(v) = 0, $$
where $m\! \geq \!1$, $p,P \in C^{2m}([a,b],(0,\infty))$, $q,Q \in C([a,b],\mathbf{R})$, $\varphi(s):= |s|^\alpha\sgn s$ and $\alpha>0$.

Mathematics Subject Classification: 34C10

Key words and phrases: Picone's identity, comparison, half-linear differential operator