Naslov
Rectifiable oscillations and singular behaviour of solutions of second-order linear differential equations
(Rectifiable oscillations and singular behaviour of solutions of second-order linear differential equatons)

Autori
Pašić, Mervan ; Raguž, Andrija

Izvornik
International Journal of Mathematical Analysis (1334-8671) 2 (2008); 477-490

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Linear equations ; oscillations ; graph ; rectifiability ; singular

Sažetak
Rectifiable oscillations on the finite interval $I=(0, 1)$ of all solutions of the equation $(P)$: $y''+f(x)y=0$, $x\in I$, has been recently introduced and studied in Pa\v{; ; ; s}; ; ; i\'{; ; ; c}; ; ; \cite{; ; ; Pasic07}; ; ; and Wong \cite{; ; ; Wong07}; ; ; , where $f(x)$ is of Euler type coefficient. It has been continued in \cite{; ; ; Pasic08}; ; ; , where $f(x)$ is of Riemann-Weber type and in \cite{; ; ; KPW}; ; ; , where $f(x)$ is of more general type, that is, $f(x)$ satisfies the so-called Hartman-Wintner asymptotic condition near $x=0$. These results show that the infiniteness of arclength of any solution curve $y$ of $(P)$ depends on asymptotic behavior of $f(x)$ near $x=0$. In this paper we do not suppose Hartman-Wintner asymptotic condition on $f$. Instead, we impose certain growth conditions on $f$ and we study connection of oscillatory solutions of $(P)$ and singular behavior of its solutions $y$ near $x=0$. In a sense, it generalizes results and methods presented in \cite{; ; ; Pasic07}; ; ; . Finally, we define some open questions.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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