The method of upper and lower solutions and the generalized quasilinearization technique for second-order nonlinear m-point dynamic equations on time scales of the type xΔΔ(t)=f(t,xσ), t∈[0,1]T=[0,1]∩T, x(0)=0, x(σ2(1))=∑i=1m-1αix(ηi), ηi∈(0,1)T, ∑i=1m-1αi≤1, are developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.