II. Comments. (a) The theorem shows that starting with an arbitrary function yo(.x) C(I) and calculating the sequence of successive approxima. tions" given by Yk+10.7) = 1 + f(t. yu(t)) dt (k = 0,1,2, ...), (6) one obtains a sequence that converges in the norm, and hence uniformly in J, to the solution y(x) of the initial value problem. This iteration procedure can also be used to determine a numerical approximation to the solution. In numerical approximations, it is a good idea to start with a function yo(x) that is as close as possible to the solution. However, if nothing is known about the solution. then yo(.r) = 1 is not a bad choice. (b) The following is a sufficient condition for the Lipschitz condition (2) to hold: f is differentiable with respect to y, and \fy(r. y) SL (the proof uses the mean value theorem). (c) Existence and Uniqueness Theorem to the Left of the Initial Value. Let J. = {{ a, ) (a > 0). If f is continuous in the strip S_ := J_ R and the Lipschitz condition (2) holds in S_, then the initial value problem y = f(x,y) for -a Sx 0). If f is continuous in the strip S_ := J_ R and the Lipschitz condition (2) holds in S_, then the initial value problem y = f(x,y) for -a Sx