Abstract
We establish several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation. Two illustrative examples are presented to show the significance of the results obtained.
Highlights
We are concerned with the oscillation of a class of third-order damped dynamic equations of neutral type (r (t) φγ (zΔΔ (t)))Δ + d (t) φγ (zΔΔ (t))
Throughout, we suppose that the following conditions are satisfied: (C1) γ ≥ 1 is a constant
It follows from the proof of Lemma 4 and z(h(t)) > b that
Summary
We are concerned with the oscillation of a class of third-order damped dynamic equations of neutral type (r (t) φγ (zΔΔ (t)))Δ + d (t) φγ (zΔΔ (t)). A great deal of interest in oscillation of solutions to different classes of dynamic equations on time scales has been shown; we refer the reader to [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Whereas Qiu and Wang [20] considered a second-order damped dynamic equation (r (t) φγ (xΔ (t)))Δ + p (t) φγ (xΔ (t)) + f (t, x (g (t))) (6). Equation (1) is said to be almost oscillatory if all its solutions either are oscillatory or converge to zero asymptotically
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