Naslov
Two-point oscillations in second-order linear differential equations

Autori
Pašić, Mervan ; Wong, James S.W.

Izvornik
Differential equations & applications (1847-120X) 1 (2009), 1; 85-122

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

Ključne riječi
oscillations ; Dirichlet problem ; linear ; rectifiability

Sažetak
A second-order linear differential equation $(P)$: $y''+f(x)y=0$, $x\in I$, where $I=(0, 1)$ and $f\in C(I)$, is said to be two-point oscillatory on $I$, if all its nontrivial solutions $y\in C(\bar{; ; I}; ; )\cap C^{; ; 2}; ; (I)$, oscillate both at $x=0$ and $x=1$, i.e. having sequences of infinite zeros converging to $x=0$ and $x=1$. It necessarily implies that all solutions $y(x)$ of $(P)$ must satisfy the Dirichlet boundary conditions and that $f(x)$ must be singular at both end points of $\bar{; ; I}; ; $. We first describe a class of two-point oscillatory equations of $(P)$. Secondly, we prove that $(P)$ is two-point oscillatory if $f(x)$ satisfies certain Hartman-Wintner type asymptotic conditions. Furthermore, we study the arclength of the graph $G(y)$ of solutions curve $y(x)$ on $I$. Two-point oscillatory equation $(P)$ is said to be two-point rectifiable (unrectifiable) oscillatory if the arclengths of all solutions are finite (infinite). We give conditions on $f(x)$ which imply $(P)$ is two-point rectifiable (unrectifiable) oscillatory. When $(P)$ is two-point unrectifiable oscillatory, we determine the fractal dimension of its solution curves for a special class of $f(x)$ similar to the Euler type equations when $f(x)$ is only singular at one end point of $I$. Finally, the preceding results motivate a study on two-sided oscillations of $(P)$ at an interior point of $\bar{; ; I}; ; $.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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