Periodic solutions of some classes of continuous second-order differential equations

We study the periodic solutions of the second-order differential equations of the form ẍ ± xn = μ f (t), or ẍ ± |x|n = μ f(t), where n = 4, 5,⋯, f(t) is a continuous T-periodic function such that ∫ 0T f(t)dt = 0, and μ is a positive small parameter. Note that the differential equations ẍ ± xn = μ f(t) are only continuous in t and smooth in x, and that the differential equations ẍ ± |x|n = μ f(t) are only continuous in t and locally-Lipschitz in x.