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Generalized Picone's formula and forced oscillations in quasilinear differential
equations of the second order

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*J. Jaros, T. Kusano and N. Yoshida*

**Address.** J. Jaros, Department of Mathematical Analysis, Faculty
of Mathematics and Physics, Comenius University, 842 48 Bratislava, SLOVAK
REPUBLIC
T. Kusano, Department of Applied Mathematics, Faculty of Science, Fukuoka
University, Fukuoka, 814-0180, JAPAN

N. Yoshida, Department of Mathematics, Faculty of Science, Toyama University,
Toyama, 930-8555, JAPAN

**E-mail:** jaros@fmph.uniba.sk

tkusano@cis.fukuoka-u.ac.jp

nori@sci.toyama-u.ac.jp

**Abstract.** In the paper a comparison theory of Sturm-Picone type
is developed for the pair of nonlinear second-order ordinary differential
equations first of which is the quasilinear differential equation with
an oscillatory forcing term and the second is the so-called half-linear
differential equation. Use is made of a new nonlinear version of the Picone's
formula.

**AMSclassification.** 34C10 .

**Keywords.** Forced quasilinear differential equation of the second
order, nonlinear Picone's identity, Sturmian comparison thorems