Analysis II | Part IB, 2005

Let F:[a,a]×[x0r,x0+r]RF:[-a, a] \times\left[x_{0}-r, x_{0}+r\right] \rightarrow \mathbf{R} be a continuous function. Let CC be the maximum value of F(t,x)|F(t, x)|. Suppose there is a constant KK such that

F(t,x)F(t,y)Kxy|F(t, x)-F(t, y)| \leqslant K|x-y|

for all t[a,a]t \in[-a, a] and x,y[x0r,x0+r]x, y \in\left[x_{0}-r, x_{0}+r\right]. Let b<min(a,r/C,1/K)b<\min (a, r / C, 1 / K). Show that there is a unique C1C^{1} function x:[b,b][x0r,x0+r]x:[-b, b] \rightarrow\left[x_{0}-r, x_{0}+r\right] such that



dxdt=F(t,x(t)).\frac{d x}{d t}=F(t, x(t)) .

[Hint: First show that the differential equation with its initial condition is equivalent to the integral equation

x(t)=x0+0tF(s,x(s))ds.]\left.x(t)=x_{0}+\int_{0}^{t} F(s, x(s)) d s .\right]

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