In this paper, we employ the second method of Lyapunov to examine sufficient conditions for the uniform-ultimate boundedness of solutions and existence of at least one periodic solution to the following second order vector differential equation: Ẍ+ F(X, Ẋ ) Ẋ + H(X) = P(t, X, Ẋ ), when the non-linear term H(X) is: (i) differentiable, (ii) non-necessarily differentiable. The results contain in this paper are new and complement related ones in the literature.