In this paper we examine the asymptotic periodicity of the solutions of some second-order differential inclusions on [0,∞)associated with maximal monotone operators in a Hilbert space H, whose forcing terms are periodic functions perturbed by functions from L^1(0,∞; H; tdt). It is worth pointing out that strong solutions do not exist in general, so we need to consider weak solutions for this class of evolution inclusions. Similar second-order difference inclusions are also addressed. Our main results on asymptotic periodicity represent significant extensions of the previous theorems proved by R.E. Bruck (1980) and B. Djafari Rouhani (2012).