A telescoping principle for oscillation of second order differential equations on a time scale. (English) Zbl 1156.34021
The authors study the self adjoint second order scalar equation
\[ (p(t) x^\Delta(t))^\Delta + q(t) x(\sigma(t)) = 0 \]
on time scales. The \(\Delta\) symbol denotes the generalized differentiation operator on a time scale (arbitrary nonempty closed subset of the reals). The telescoping principle is an “interval shrinking” transformation from the given time scale to another more appropriate time scale, that preserves the essential information about oscillatory behavior of the above equation. The second part of the paper contains many applications and examples for the telescoping technique.
\[ (p(t) x^\Delta(t))^\Delta + q(t) x(\sigma(t)) = 0 \]
on time scales. The \(\Delta\) symbol denotes the generalized differentiation operator on a time scale (arbitrary nonempty closed subset of the reals). The telescoping principle is an “interval shrinking” transformation from the given time scale to another more appropriate time scale, that preserves the essential information about oscillatory behavior of the above equation. The second part of the paper contains many applications and examples for the telescoping technique.
Reviewer: Stefan Hilger (Eichstätt)
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 39A10 | Additive difference equations |
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