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Asymptotic behavior of solutions of higher-order dynamic equations on time scales. (English) Zbl 1219.34120

Summary: We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation
\[ x^{\Delta^n}(t)+f(t,x(t),x^\Delta(t),\dots,x^{\Delta^{n-1}}(t))=0, \]
on an arbitrary time scale \(\mathbb T\), where the function \(f\) is defined on \(\mathbb T\times \mathbb R^n\). We give sufficient conditions under which every solution \(x\) of this equation satisfies one of the following conditions: (1) \(\lim_{t\to\infty}x^{\Delta^{n-1}}(t)=0\); (2) there exist constants \(a_i\) \((0\leq i\leq n-1)\) with \(a_0\neq 0\), such that \(\lim_{t\to\infty} x(t)/\sum^{n-1}_{i=0}a_ih_{n-i-1}(t,t_0)=1\), where \(h_i (t,t_0)\) \((0\leq i\leq n-1)\) are suitable functions iteratively defined.

MSC:

34N05 Dynamic equations on time scales or measure chains
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:

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