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Solutions of half-linear differential equations in the classes Gamma and Pi

Autoři

ŘEHÁK Pavel TADDEI Valentina

Druh Článek v odborném periodiku
Časopis / Zdroj Differential and Integral Equations
Fakulta / Pracoviště MU

Pedagogická fakulta

Citace
Obor Obecná matematika
Klíčová slova half-linear differential equation; positive solution; asymptotic formula; regular variation; class Gamma; class Pi
Popis We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $(r(t)|y'|^ {\alpha-1}\sgn y')'=p(t)|y|^{\alpha-1}\sgn y$, where $\alpha\in(1,\infty)$ and $r,p$ are positive continuous functions on $[a,\infty)$, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class $\Gamma$ resp. $\Gamma_-$ under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case $\alpha=2$.
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