Abstract and Applied Analysis
Volume 2011 (2011), Article ID 719628, 14 pages
Research Article

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

School of Mathematics, University of Jinan, Jinan, Shandong 250022, China

Received 18 May 2011; Accepted 30 July 2011

Academic Editor: Paul Eloe

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.


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 is a strictly increasing right-dense continuous function; p,q,e:[t0,)T, r:[t0,)T×[a,b]T1, τ:[t0,)T[t0,)T, and g:[t0,)T×[a,b]T1[t0,)T are right-dense continuous functions; ξ:[a,b]T1 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.