Department of Mathematics, Masaryk University,
Janackovo nam. 2a, CZ-662 95 Brno, Czech Republic.
E-mail: dosly@math.muni.cz
janar@math.muni.cz
Abstract. We introduce the concept of the regular (nonoscillatory)
half-linear second order differential equation
$$ \left(r(t)\Phi(x')\right)'+c(t)\Phi(x)=0\,,\quad \Phi(x):=|x|^{p-2}x\,,\quad
p>1 \leqno{(*)}
$$ and we show that if (*) is regular, a solution $x$ of this equation
such that $x'(t)\ne 0$ for large $t$ is principal
if and only if $$ \int^\infty \frac{dt}{r(t)x^2(t)|x'(t)|^{p-2}}=\infty\,.
$$ Conditions on the functions $r,c$ are given
which guarantee that (*) is regular.
AMSclassification. 39C10.
Keywords. Regular half-linear equation, principal solution, Picone's identity, Riccati-type equation.