Forced Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

and Applied Analysis 3 exists, when σ t t here by s → t it is understood that s approaches t in the time scale and when x is continuous at t and σ t > t, xΔ t : x σ t − x t μ t . 1.7 Note that if T R, then the delta derivative is just the standard derivative, and when T Z the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales e.g., the time scale q0 : {1, q, q2, . . .}which is very important in quantum theory 8 . 2. Main Theorem Theorem 2.1. Assume that given any T ∈ a,∞ T , there exists points T ≤ s1 < t1 ≤ s2 < t2 such that f t ⎧ ⎨ ⎩ ≤ 0, t ∈ s1, t1 T, ≥ 0, t ∈ s2, t2 T. 2.1 Further assume that there exists a function u ∈ C1 rd such that for i 1, 2, one has