A contraction shrinks distances between Points As our first

A contraction shrinks distances between Points As our first application, we study the existence of solutions of differential equations. We give a specific context, although the technique is We shall generalize it in dollar 7.5. Consider a continuous function f(t, x) defined in a neighborhood of (t_0, x_0 R^2. Assume that the following Lipschitz condition holds: |f(t,x_1)-f(t,x_2)| lessthanorequalto K|x_1 -x_2| for all (t,x_1) and (t,x_2) in a neighborhood of (f_0,x_o). If f is differentiable in x and (partial differential f/partial differential x)(t,x) is continuous, then the condition is automatic (by the mean value theorem) Under the above assumptions, there is a partial differential > 0 such that the equation dx/dt = f(t,x), x(t_0) = x_0 has a unique C^1 solution x + phi (t), with phi (t_0) = x_0, for t_0 - delta

Solution

Equation (1) has no k(x,y) or a in it!

As it stands the equation (1) can be solved by assuming a powerseries for x(t) (which exists because of the contractibility of the mapping x->x on [0,1]).

would help if correct equation with notation is posted