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Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions
Georgian Mathematical Journal ( IF 0.7 ) Pub Date : 2020-08-06 , DOI: 10.1515/gmj-2020-2070
Kusano Takaŝi 1 , Jelena V. Manojlović 2
Affiliation  

We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

中文翻译:

半线性微分方程和广义卡拉马塔函数解的渐近行为

我们研究二阶半线性微分方程(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime}的最终正解的渐近行为)^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,其中 q 是一个连续函数,它可以在无穷大的任何邻域中取正值和负值并且 p 是满足条件之一的正连续函数\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or} \quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty. 使用 Karamata 理论建立广义规则变化解的渐近公式规律的变化。
更新日期:2020-08-06
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