An autonomous system whose solutions are uniformly ultimately bounded but not uniformly bounded

Consider the ordinary differential equation (1) * = /(ί, x) for a continuous function f(t, x) on / x R n , I = [0, <*>).Then, the following properties are well established for (1) (cf.[2], [4]), where x(t) denotes an arbitrary solution of (1): ( i ) If there exists a continuous function L(t, s, r) on I 3 such that