By Jack K. Hale (auth.)
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9. AUTONOIDUS SYSTEMS Due to the remarks of Section 8, it is impossible to develop a theory of autonomous functional differential equations which is as comprehensive as the one for ordinary differential equations. However, some of the meaningful concepts are given in this section. f: C ~Rn Suppose bounded sets of Rn. If is continuous, takes closed bounded sets of C into x(cp) is the solution of the autonomous equation x(t) through (O,cp), we also suppose that the solution is defined on unique, and therefore tion x(cp) of x(cp)(t) is continuous in function of xt(cp).
We now show that T is a contraction on ~(a,'i3). 5) I (T~) ( t) I ::: I (T~)(t) for all t E [-a,O], ~,e contraction. w(a,~). Q1(a,~) theorem. 1. R n D, and the solution is unique. then T(t,cr) Proof. 1) through any T(t,cr): C -? C, t ? cr, is defined by ~t (cr,~) " ~t (cr,~), ~t(cr,~) 1 1 ~(cr,~)(t), y(t) " ~(cr,*)(t), then x(t) in the domain of definition of Theorem for (cr,~) T(t,cr)~" ~t(cr,~), is one-to-one. 1 implies there are an a [tl-r-a,tl-r] through ses, it follows that f ~ in f ~t(cr,~), cr ::: t < t l • = f(t'~t)' y(t) Since ~.
Or periodic in at to t, then sta- implies stability t l ? to. to But, stability at implies stability at to + kT for which the solutions are defined on implies sta[tl,tO+kT]. Continuous dependence of solutions on initial values implies there is a such that for any defined on ~ in [tl,tO+kT]. and is The lemma is proved. For a scalar equation, the following result is true. 2. For x b = b(kT) a scalar, consider the equation 48 L(t,~) where in e =I x(t) ( 10. 2 ) o (t,~) is continuous for for each fixed x(t+e)d~(t,e) -r E R+ X C.