0 ) Pin the Convergence of Numerical Solutions to Ordinary Differential Equations

We will be concerned with the solution to the initial value problem $$ = gy, , YJxo) = L 7 \ where y is a point in the (real) Euclidean M-space RM and ;(y,) is a 4-. mapping of s onto itself satisfying the Lipschitz condition (2) for any pair of points y,z f s. L is a constant and iv/ for vC+s ~.. 'denotes a norm. Although the particular norm used is irrelevant for most purposes, a number of details in the results of this paper take a simpler frirm if' the norm used is defined by (3) \vJ =max I Y 3, v2,. .. , vM denoting the components of v. Accordingly, we adopt (3) as the definition of v. I I .,-It willbe necessary to consider sets of points v, v-2,. .. . v-4cqa and we shall regard such a set as corresponding to the point V = zl@[email protected]\. .. @~,y~c l$m-The norm of JJJ C k will be defined in a similar way to I (3) and a similar notation V I I will be used. Clearly ,- .