(Φp(ρ(t)x (t))) + q(t)f(t, x(t), x(t)) = 0, t 6= ti, t ∈ J, ∆x(ti) = Ii(x(ti)), −∆Φp(ρ(ti)x (ti)) = Ji(x(ti)), i = 1, 2, . . . , m, x(0) = ax(ξ), lim t→+∞ ρ(t)x(t) = 0, here J = [0,+∞), Φpx := |x| x, p > 1, 0 = t0 < t1 < · · · < tm < ∞, a > 0, 0 ≤ ξ <∞, aξ < 1, ρ, Ii, Ji, q, f satisfy the following assumptions (H1) ρ ∈ C[0,+∞) ∩ C(0,+∞), ρ(t) > 0 is increasing on [0,+∞), ∫∞ 0 1 ρ(t) dt <∞; (H...