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Oscillation of solutions for third order functional dynamic equations. (English) Zbl 1203.34152

Summary: We study the oscillation of solutions to the third order nonlinear functional dynamic equation
\[ L_3(x(t))+\sum^n_{i=0}p_i(t)\Psi_k\alpha_ki(x(h_i(t)))=0, \]
on an arbitrary time scale \(\mathbb T\). Here
\[ L_0(x(t))=x(t),\quad L_k(x(t))=\left(\frac{[L_{k-1}x(t)]^\Delta}{a_k(t)}\right)^{\gamma_kk},\quad k=1,2,3 \]
with \(a_1,a_2\) positive rd-continuous functions on \(\mathbb T\) and \(a_3\equiv 1\); the functions \(p_i\) are nonnegative rd-continuous on \(\mathbb T\) and not all \(p_i(t)\) vanish in a neighborhood of infinity; \(\psi_kc(u)=|u|^{c-1}u\), \(c>0\). Our main results extend known results and are illustrated by examples.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
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