By Sarason D.
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T > Suppose Then there such that to + (2k-l)r ~ tk ~ to + 2kr, k = 1,2, ••• , and By the assumption on t ~ to. x(t)! < L for all Therefore, on the intervals ! x(t)! > 5/2, and hence By taking a large ! xt ! > 5/2. Therefore, L, if necessary, we can assume that these intervals do not overlap, and hence Let K(50 ,L) be the smallest integer ~ v(eo)/«5/L)w(5/2». then 52 If k> 1 + K(50 ,L), which is a contradiction. we have 2rK( BO,L) Therefore, at some e. \x t \ < 1 t l , such that to ~ tl ~ to + This proves the theorem.
3. For nonlinear equations, the equivalence classes Va may involve the consideration of trajectories which have a point in common after any preassigned times. r = 1, The following example was given by A. Hausrath. consider the scalar equation x(t) = ~[I xtl - x(t)]. For ~ > 0, For a given ~ in C = C([-l,O],R), this equation through If ~(O) In fact, since implies o x(t) which is continuous in (O,~) 2: 0, F 0, ~ 2: 0, => x(O) > 0 then 2: ~(O) x(t) > 0 and for for t 2: of (~,~,t). I xtl = x(t) for > 1.
1) at tl > to. 1) is stable at ~ for which ~ 3rrj2). x(t - ~ = exp(-3rr~j2), Since there is a ~O tl > is unstable for any >0 the function x(t) = satisfying this equation, 3rr. It seems to be very difficult to determine in general when stability at t O implies stability at t >t 1 - but the following result is very easy and indi- 0' cates the simplicity of autonomous and periodic systems. 1. If f(t,~) is either independent of bility (or asymptotic stability) of (or asymptotic stability) of Proof.
Blaschke products in B by Sarason D.