£y" + f(x,y,y',6) = O, 0 0 is a small parameter and "prime" denotes differentiation with respect to x. We shall formulate conditions under which the problem (i.i), (1.2) has a unique solution y(x, e) existing on the entire interval [0,b], for e sufficiently small. We shall also obtain explicit bounds on y(x,E) and y' (x,e) as g + 0. In particular, we find conditions under which y(x,£) is uniformly bounded as g + 0. The conditions we place on f(x,y,z,£) are quite different from the standard uniform Lipschitz condi~f tion; other than smoothness, our essential requirements are that f3(x,y,z,e) ~)z ~f be strictly positive and bounded away from zero and that f2(x,y,z,£) ~ be bounded above o_nl_~ for z = 0. The precise conditions are stated below. We shall indicate a couple of applications of our results at the end. More serious applications will hopefully appear elsewhere. The spirit of this work is very close to that of [i] and [2].