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

Haidong Liu and Cuiqin Ma

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: 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, ∞) 𝕋 : = [t0, ∞)∩𝕋, 𝕋 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] 𝕋1→ℝ is a strictly increasing right‐dense continuous function; p, q : [t0, ∞) 𝕋 → ℝ, k:[t0,∞) 𝕋×[a,b] 𝕋1→ℝ, δ : [t0, ∞) 𝕋 → [t0, ∞) 𝕋, and g:[t0,∞) 𝕋×[a,b] 𝕋1→[t0,∞) 𝕋 are right‐dense continuous functions; ξ:[a,b] 𝕋1→ℝ is strictly increasing. Our results extend and supplement some known results in the literature.

Date: 2014
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