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Regular half-linear second order differential equations

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*O. Dosly, J. Reznickova*

**Address.**

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.

**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.