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

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), wheret∈[t0,∞)T=[t0,∞)  ⋂  T,Tis a time scale which is unbounded from above;ϕ*(u)=|u|*sgn u;γ:[a,b]T1→ℝis 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, andg:[t0,∞)T×[a,b]T1→[t0,∞)Tare right-dense continuous functions;ξ:[a,b]T1→ℝis 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.