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Asymptotic properties of solutions of certain third-order dynamic equations. (English) Zbl 1239.34112

Summary: The well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r 2 (t)((r 1 (t)x Δ (t)) Δ ) γ ) Δ +q(t)f(x(t))=0

on time scale 𝕋, where γ1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when 𝕋= and 𝕋=. Two examples of dynamic equations on different time scales are given to show the applications of our main results.


MSC:
34N05Dynamic equations on time scales or measure chains
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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