Solutions of half-linear differential equations in the classes Gamma and Pi
Solutions of half-linear differential equations in the classes Gamma and Pi
复制标题
Gamma 类和 Pi 类中半线性微分方程的解
DOI：
10.57262/die/1462298681
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发表时间：
2016
影响因子：
1.4
通讯作者：
V. Taddei
中科院分区：
文献类型：
--
作者：
P. Řehák;V. Taddei
本文利用正则变函数的Karamata理论和de Haan理论，研究了形式为$(r(t)|y'|^ {\alpha-1}\sgn y')'=p(t)|y|^{\alpha-1}\sgn y$的非振荡半线性微分方程的（全部）正解的渐近性，其中$\alpha\in(1,\infty)$和$r,p$是$[a,\infty)$上的连续正函数。我们展示了不断增加的响应。递减解属于de Haan类$\Gamma$。$\Gamma_-$在适当的假设下。进一步研究了建立渐近公式的慢变解的性质。我们的一些结果甚至在线性情况下也是新的$\alpha=2$。
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 $\alpha=2$.