Oscillation of second-order nonlinear dynamic equations with positive and negative coef“cients

on an arbitrary time scale T with supT =∞, subject to the following conditions: (C) t ∈ T and [t,∞)T := {t ∈ T : t ≥ t} is a time scale interval in T; (C) r ∈ Crd( [t,∞)T , (,∞)) and ∫ ∞ t  r(t) t =∞; (C) p,q ∈ Crd( [t,∞)T , [,∞)); (C) ξ , δ ∈ Crd(T,T), limt→∞ ξ (t) = limt→∞ δ(t) = ∞, δ has the inverse function δ– ∈ Crd(T,T), v := δ– ◦ ξ ∈ Crd(T,T), ξ , v ∈ Crd( [t,∞)T , (,∞)), ξ (t), v(t)≤ t for t ∈ [t,∞)T, ξ ( [t,∞)T) = [ξ (t),∞)T and v( [t,∞)T) = [v(t),∞)T, where ξ ( [t,∞)T) := {ξ (t) : t ∈ [t,∞)T} and v( [t,∞)T) := {v(t) : t ∈ [t,∞)T}; (C) f ,h ∈ C(R,R), there exist positive constants L, L and M such that f (u)/u ≥ L,  < h(u)/u≤ L and |h(u)| ≤M for u = , and Lp(t) – Lq(v(t))v (t) >  for t ∈ [t,∞)T; (C) ∫ ∞ t [  r(s) ∫ s v(s) q(u) u] s <∞ for every sufficiently large t ∈ T. Recall that a solution of () is a nontrivial real function x such that x ∈ C rd( [tx,∞)T ,R) and rx ∈ C rd( [tx,∞)T ,R) for a certain tx ≥ t and satisfying () for t ≥ tx. Our attention is