Interval Oscillation Criteria for Second‐Order Dynamic Equations with Nonlinearities Given by Riemann‐Stieltjes Integrals

Yuangong Sun

Abstract and Applied Analysis, 2011, vol. 2011, issue 1

Abstract: By using a generalized arithmetic‐geometric mean inequality on time scales, we study the forced oscillation of second‐order dynamic equations with nonlinearities given by Riemann‐Stieltjes integrals of the form [p(t)ϕα(xΔ(t))] Δ+q(t)ϕα(x(τ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))Δξ(s)=e(t), where t ∈ [t0, ∞) 𝕋 = [t0, ∞) ⋂ 𝕋, 𝕋 is a time scale which is unbounded from above; ϕ*(u) = |u|*sgn u; γ:[a,b] 𝕋1→ℝ is a strictly increasing right‐dense continuous function; p, q, e : [t0, ∞) 𝕋 → ℝ, r:[t0,∞) 𝕋×[a,b] 𝕋1→ℝ, τ : [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. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:719628

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