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Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales. (English) Zbl 1303.34073

The authors consider an even order advanced type dynamic equation with mixed nonlinearities as \[ \bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t)\bigr)\bigr]^\Delta+p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr)\bigr)+\sum\limits_{i=1}^kp_i(t)\varPhi_{\alpha_i}\bigl(x\bigl(\delta(t)\bigr)\bigr)=0 \quad (1) \] on an arbitrary time scale \(\mathbb T\) with \(\sup T=\infty\). Moreover, they assume that the following properties are satisfied:
i) \(n\geq 2\) is even, \(\alpha_{1}>\dots>\alpha_{m}>\alpha>\alpha_{m+1}>\dots>\alpha_{k}>0\),
ii) \(r,p,p_{i}\in C_{rd}([t_0,\infty)_{T},(0,\infty)), i=1,\dots,k\) and \(\int_{t0}^{\infty}r^{-1/\alpha}(t)\Delta t=\infty\),
iii) \(\varPhi_{\ast}(u)=|u|^{\ast-1}u, \delta\in C_{rd}(T,T)\) and \(\delta(t)\geq t, t\in [t_0,\infty)_{T}\).
Using Hardy-Littlewood-Polya inequality, arithmetic-geometric mean inequality, introducing parameter functions and developing a generalized Riccati technique, the authors obtain some oscillation criteria for (1).

MSC:

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

References:

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