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

Abstract EN

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.