Publication details

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

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ŘEHÁK Pavel TADDEI Valentina

Year of publication 2016
Type Article in Periodical
Magazine / Source Differential and Integral Equations
MU Faculty or unit

Faculty of Education

Field General mathematics
Keywords half-linear differential equation; positive solution; asymptotic formula; regular variation; class Gamma; class Pi
Description 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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