We will establish a new interval oscillation criterion for second-order half-linear dynamic equation(r(t)[xΔ(t)]α)Δ+p(t)xα(σ(t))=f(t)on a time scaleTwhich is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second-order half-linear differential equation([x′(t)]α)′+csintxα(t)=cost, whereα=p/q,p,qare odd positive integers.
In this paper, we obtain an extended Halanay inequality with unbounded coefficient functions on time scales, which extends an earlier result in Wen et al. (J. Math. Anal. Appl. 347:169-178, 2008). Two illustrative examples are also given.
This paper mainly studies oscillatory of all solutions for a class higher order linear functional equations of the form
x(g(t))=P(t)x(t)+∑^m_i=1 Q_i(t)x(g^(k+1)(t))
Where P, Q, g:[t_0,∞] → R^+ =[0,∞] are given real valued functions and g(t) ≠t, lim_(t→∞) g(t)=∞.
Some sufficient conditions are obtained. Our results generalize or improve some results in some literature given. An example is also given to illustrate the results.
We present some new oscillation criteria for second-order neutral partial functional differential equations of the form(∂/∂t){p(t)(∂/∂t)[u(x,t)+∑i=1lλi(t)u(x,t-τi)]}=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)),(x,t)∈Ω×R+≡G, whereΩis a bounded domain in the EuclideanN-spaceRNwith a piecewise smooth boundary∂ΩandΔis the Laplacian inRN. Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.
The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time scales r t x Δ t Δ p t |x σ t | γ sgnx σ t 0, γ > 1, is discussed under the condition that P t lim τ → ∞ τ t p s Δs exists and P t ≥ 0 for large t.
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