Differential and Integral Equations

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

Valentina Taddei and Pavel Řehák

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Abstract

We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $$ (r(t)|y'|^ {\alpha-1}\text{sgn}\, y')'=p(t)|y|^{\alpha-1}\text{sgn}\, y , $$ 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 $\alpha=2$.

Article information

Source
Differential Integral Equations Volume 29, Number 7/8 (2016), 683-714.

Dates
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1462298681

Mathematical Reviews number (MathSciNet)
MR3498873

Zentralblatt MATH identifier
06604491

Subjects
Primary: 34C11: Growth, boundedness 34C41: Equivalence, asymptotic equivalence 34E05: Asymptotic expansions 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]

Citation

Řehák, Pavel; Taddei, Valentina. Solutions of half-linear differential equations in the classes Gamma and Pi. Differential Integral Equations 29 (2016), no. 7/8, 683--714. https://projecteuclid.org/euclid.die/1462298681.


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