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

• Main Authors: Haidong Liu, Cuiqin Ma
• Format: Article
• Language: English
• Published: Hindawi Limited 2014-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2014/395381

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.