engleski

Rectifiable oscillations in second-order linear differential equations

We study the linear differential equation , on I=(0, 1), where the coefficient f(x) is strictly positive and continuous on I, and satisfies the Hartman– Wintner condition at x=0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P), characterized by the integrability of on I ; (ii) a stability result for rectifiable and unrectifiable oscillations of (P), in terms of a perturbation on f(x) ; (iii) the s-dimensional fractal oscillations (for which we assume also f(x)cx− α when x→ 0, α >2, and s=max{; ; ; ; ; ; 1, 3/2− 2/α }; ; ; ; ; ; ) ; and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman– Wintner condition on f(x). Explicit examples related to the above results are given.

linear equation; oscillation; graph; rectifiability; integral criterion; stability; co-existence; fractal dimension

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