10 J. MAWHI N hnPs = kerZ,. On the other hand, let V be a projector in Rn suc h that Im V = N~l(lm(M + N)) and define th e operator Q v i n Z by where again the right-hand membe r denotes the corresponding constant element in Z then Qv i s a continuous projector i n Z such that ker Qv = {/€Z: f*f(s)ds = K / J /(*)& } = {/ Z : f* f(s)ds €AT HMJI/ + N))\ - J / 6 Z : NJ l f(s)ds e Im( M + JV)} = Im I . Now, ifiS is the projector in Rn define d above and if/€ Z , it is not difficult t o check that ** x fM = -W + N)s l Nvflf(s)ds + /„(/(*) - (/ - v) ) o f(u)du)ds, t e i. In the special case of periodic boundary value conditions, one has M = -N = /, s o that we can take S = I, V 0, which implies p I x=x(o), Q 0 r= Urn*, and Let us also note that when /€ C(/ , Rn) on e can respectively replac e Z and Z b y C(/, /?") x £n an d C(I, R n ) an d replace the requirement of absolute continuity in the domains of the linear part by continuous differentiability withou t altering the conclusions and the formulae above. 3. Boundar y value problems for functional differential equations. Th e above considera- tions can be extended to boundary value problems for functional differentia l equations . Le t r 0, C = C([-r, 0] , Rn) and , if x 6 C([-r 9 1 ], Rn) an d t e / , let us denote by xt th e element of C defined by x t (s) = x(t + s), se[-r, 0} .
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