Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities

Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form (r(t)Φα(xΔ(t)))Δ+p0(t)Φα(x(τ0(t)))+∑i=1npi(t)Φβi(x(τi(t)))=e(t),t∈[t0,∞)T where TT is a time scale, t0∈Tt0∈T a fixed number; [t0,∞)T[t0,∞)T is a time scale interval; Φ∗(u)=|u|∗−1uΦ∗(u)=|u|∗−1u; the functions r,pi,e:[t0,∞)T→Rr,pi,e:[t0,∞)T→R are right-dense continuous with r>0r>0 nondecreasing; τk:T→Tτk:T→T are nondecreasing right-dense continuous with τk(t)≤tτk(t)≤t, limt→∞τk(t)=∞limt→∞τk(t)=∞; and the exponents satisfy β1>⋯>βm>α>βm+1>⋯βn>0.β1>⋯>βm>α>βm+1>⋯βn>0. All results are new even for T=RT=R and T=ZT=Z.Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives. Two examples are provided to illustrate one of the theorems.