##
A singular version of Leighton's comparison theorem for forced

##
quasilinear second order differential equations

##
*Ondrej Dosly, Jaroslav Jaros*

**Address.**

Mathematical Institute, Czech Academy of Sciences

Zizkova 22, CZ-616 62 Brno, Czech Republic
Department of Mathematical Analysis,

Comenius University, Mlynska

Dolina, 842 15 Bratislava, Slovakia

**E-mail: **dosly@math.muni.cz**
**jaros@alpha.dcs.fmph.uniba.sk

**Abstract.**

We extend the classical Leighton comparison theorem

to a class of quasilinear forced second order differential equations

$$

(r(t)|x'|^{\alpha-2}x')'+c(t)|x|^{\beta-2}x=f(t)\,,\quad

1<\alpha\leq \beta,\ t\in I=(a,b)\,,
\tag{*}

$$

where the endpoints $a$, $b$ of the interval $I$ are allowed to be

singular. Some applications of this statement in

the oscillation theory of

(*) are suggested.

**AMSclassification.** 34C10

**Keywords. **Picone's identity, forced quasilinear equation,
principal solution.