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Rectifiable oscillations and singular behaviour of solutions of second-order linear differential equatons (CROSBI ID 137255)

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Pašić, Mervan ; Raguž, Andrija Rectifiable oscillations and singular behaviour of solutions of second-order linear differential equatons // International journal of mathematical analysis, 2 (2008), 10; 477-490. doi: 10.12988/ijma

Podaci o odgovornosti

Pašić, Mervan ; Raguž, Andrija

engleski

Rectifiable oscillations and singular behaviour of solutions of second-order linear differential equatons

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.

Linear equations ; oscillations ; graph ; rectifiability ; singular

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Podaci o izdanju

2 (10)

2008.

477-490

objavljeno

1312-8876

1314-7579

10.12988/ijma

Povezanost rada

Matematika

Poveznice