Perturbing Uniform Ultimate Bounded Differential Systems

We obtain results on the eventual uniform boundedness and eventual uniform ultimate boundedness of solutions of the differential equation \[ \dot x = f(t,x) + g(t,x)\] given that solutions of the equation $\dot x = f(t,x)$ are uniformly bounded and uniformly ultimately bounded. By assuming various regularity conditions on f we obtain admissible classes of g such that the boundedness properties are preserved. By use of various examples the admissible classes of g are shown to be “maximal.”