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 Title: On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE. (English) Author: Vampolová, Jana Language: English Journal: Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica ISSN: 0231-9721 Volume: 52 Issue: 1 Year: 2013 Pages: 135-152 Summary lang: English . Category: math . Summary: We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^{\prime }(t) \right)^{\prime } + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^{\prime }(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb {R}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace$. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions. (English) Keyword: singular ordinary differential equation of the second order Keyword: time singularities Keyword: unbounded domain Keyword: asymptotic properties Keyword: Kneser solutions Keyword: damped solutions Keyword: non-oscillatory solutions MSC: 34A12 MSC: 34D05 idZBL: Zbl 06285760 idMR: MR3202755 . Date available: 2013-08-02T08:05:21Z Last updated: 2014-07-30 Stable URL: http://hdl.handle.net/10338.dmlcz/143397 . Reference: [1] Abraham, F. F.: Homogeneous Nucleation Theory. Acad. Press, New York, 1974. Reference: [2] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: On oscillatory solutions of quasilinear differential equations. J. Math. Anal. Appl. 320 (2006), 108–120. 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