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

Author

Listed:
  • Haidong Liu
  • Cuiqin Ma

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.

Suggested Citation

  • Haidong Liu & Cuiqin Ma, 2014. "Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann‐Stieltjes Integrals," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:395381
    DOI: 10.1155/2014/395381
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    References listed on IDEAS

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    1. Yuangong Sun, 2011. "Interval Oscillation Criteria for Second-Order Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-14, October.
    2. Yuangong Sun, 2011. "Interval Oscillation Criteria for Second‐Order Dynamic Equations with Nonlinearities Given by Riemann‐Stieltjes Integrals," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    3. Martin Bohner & Allan Peterson, 2001. "Dynamic Equations on Time Scales," Springer Books, Springer, number 978-1-4612-0201-1, March.
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