Regular half-linear second order differential equations

O. Dosly, J. Reznickova

Mathematical Institute, Academy of Sciences of the Czech Republic,
Zizkova 22, CZ-616 62 Brno, Czech Republic.

Department of Mathematics, Masaryk University,
Janackovo nam. 2a, CZ-662 95 Brno, Czech Republic.


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.