Consider the Cauchy problem for the autonomous system : x' = f(x), r(to) = To , in which f : 0 -> R" is a Lipschitz function and $ is an open set of R". a) Show that if f is also bounded in ? then the problem has an unique so- lution define in some open set Ia(to) for any (to, To) E R" x Rn . b) Under the conditions of the previous item ,show that if ? = R" then Ia (to) = R for any (to, To) E R X R . c) Under the conditions of the item b, let y be the solution of our problem for some (to, To) fixed and arbitrary .Show that if p(t1 ) = To for some t > to then y is periodic with period T = t1 -to