OSCILLATION OF SOLUTION TO SECOND-ORDER HALF-LINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

This article concerns the oscillation of solutions to second-order half-linear dynamic equations with a variable delay. By using integral averaging techniques and generalized Riccati transformations, new oscillation criteria are obtained. Our results extend Kamenev-type, Philos-type and Li-type oscillation criteria. Several examples are given to illustrate our results

Oscillation of second order neutral dynamic equations with distributed delay

In this paper, we establish new oscillation criteria for second order neutral dynamic equations with distributed delay by employing the generalized Riccati transformation. The obtained theorems essentially improve the oscillation results in the literature. And two examples are provided to illustrate to the versatility of our main results

Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: (p(t)xΔn-1(t)α-1xΔn-1(t))Δ+f(t,x(δ(t))) + ∫aσ(b)k(t,s)x(g(t,s))θ(s)sgn(x(g(t,s)))Δξ(s)=0, where t∈[t0,∞):=[t0,∞)∩, a time scale which is unbounded above, n⩾2 is even, f(t,u)⩾q(t)uα, α>0 is a constant, and θ:[a,b]1→ℝ is a strictly increasing right-dense continuous function; p,q:[t0,∞)→ℝ, k:[t0,∞)×[a,b]1→ℝ, δ:[t0,∞)→[t0,∞), and g:[t0,∞)×[a,b]1→[t0,∞) are right-dense continuous functions; ξ:[a,b]1→ℝ is strictly increasing. Our results extend and supplement some known results in the literature