Copyright © 2011 Yuangong Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.