10 1. Differential Equations and Their Solutions 1.1.4. Proposition. A continuous map x : I Rn is a solution of the initial value problem x (t) = V (x(t),t), x(t0) = x0 if and only if x is a fixed point of F V,x0 . Now if you have had some experience with fixed-point theorems, that should make your ears perk up a little. Not only are there some very general and powerful results for proving existence and uniqueness of fixed points of maps, but even better, there are nice algorithms for finding fixed points. One such algorithm is the so-called Method of Successive Approximations. (If you are familiar with Newton’s Method for finding roots of equations, you will recognize that as a special case of successive approximations.) If we have a set X and a self-mapping f : X X, then to apply successive approximations, choose some “initial approximation” x0 in X and then inductively define a sequence xn+1 = f(xn) of “successive approximations”. Exercise 1–3. Suppose that X is a metric space, f is continuous, and that the sequence xn of “successive approximations” converges to a limit p. Show that p is a fixed point of f. But is there really any hope that we can use successive approxi- mations to find solutions of ODE initial value problems? Let us try a very simple example. Consider the (time-independent) vector field V on Rn defined by V (x, t) = x. It is easy to check that the unique solution with x(0) = x0 is given by x(t) = etx0. Let’s try using suc- cessive approximations to find a fixed point of F V,x0 . For our initial approximation we choose the constant function x0(t) = x0, and fol- lowing the general successive approximation prescription, we define xn inductively by xn+1 = F V,x0 (xn), i.e., xn+1(t) = x0 + t 0 xn(s) ds. Exercise 1–4. Show by induction that xn(t) = Pn(t)x0, where Pn(t) is the polynomial of degree n obtained by truncating the power series for et (i.e., ∑n j=0 1 j! tj). That is certainly a hopeful sign, and while one swallow may not make a spring, it should give us hope that a careful analysis of successive approximations might lead to a proof of the existence and uniqueness theorems for an arbitrary vector field V . This is in fact the case, but
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