Oscillation criteria for impulsive dynamic equations on time scales.
2000 Mathematics Subject Classification: 34C10, 34C15.Some new criteria for the oscillation of all solutions of second order differential equations of the form (d/dt)(r(t)ψ(x)|dx/dt|α−2(dx/dt))+ p(t)φ(|x|α−2x,r(t) ψ(x)|dx/dt|α−2(dx/dt))+q(t)|x|α−2 x=0, and the more general equation (d/dt)(r(t)ψ(x)|dx/dt|α−2(dx/dt))+p(t)φ(g(x),r(t) ψ(x)|dx/dt|α−2 (dx/dt))+q(t)g(x)=0, are established. our results generalize and extend some known oscillation criterain in the literature.
Our aim in this paper is to present sufficient conditions for the oscillation of the second order neutral differential equation (x(t)-px(t-))"+q(t)x((t))=0.
Some results concerning oscillation of second order self-adjoint matrix differential equations are obtained. These may be regarded as a generalization of results for the corresponding scalar equations.
We state and prove two oscillation results which deal with bounded solutions of a forced higher order differential equation. One proof involves the use of a nonlinear functional.
We obtain sufficient conditions for every solution of the differential equation to oscillate or to tend to zero as approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when has sub-linear growth at infinity. Our results also apply to the neutral equation when has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.