8 J. MAWHI N (1.2) (M + N)x(0) = c-NJ l f(s)ds and therefore Im L = {(/ , c) e Z: c-N^fE lm(M + N)\ = A ~l(}m(M + N)) if we define th e linear mapping A: Z * R n b y ^ac) = c-^j o V(s)ds. With the usual norms on Z,1^, R n ) an d /*w an d the correspondin g product norm on Z, A i s clearly continuous, which implies that Im L i s closed in Z. Now , ker A i s closed and -4 is onto Rn, s o that codi m ker A = « an d Z = ke r ^4 © ( 7 (topologically) with dim U = w thus, if i4 ^ i s -4 restricted to {/ , the n ^_1(Ini(M + N)) = ke r ^ eA^} l (lm(M + JV)) = ke r ,4 © K with dim V = di m Im(Af4 - TV) consequently codim I m L = n - di m Im(M + -/V) = di m ker(M + N) = di m kerZ,, so thatZ, w a Fredholm mapping of index zero. W e shall show now the construction o f projectors P and Q such that Im P = ke r Lf I m Z, = ke r Q and of the correspondin g gener- alized inverse. Let S: Rn— * /? " b e a projector suc h that Im 5 = ker( M + TV ) an d let (M + A0 5 b e the restriction o f M + N to ker 5 then (M + N) s i s a bijection fro m ker S onto Im(Af 4-TV), and if (/ , c) E ImZ,, equation (1.2) is equivalent t o *(0) = S(x(0) ) + ( M + N)s l (c-Nf*Mb) which gives, for the boundary value problem, the solutio n x(t) = S(x(0)) + J J / W * + ( M + TV)^ 1 (c -Nf l Q f(s)ds). If we defin e therefor e th e projector Ps o n A" by Ps(x) = 5(^(0)) , with the right-hand side understood a s the constant mapping in X o f value S(x(0)), it follows fro m the above rela- tion that, for any (/, c) I m L, we have {*PS(f. cW) = f f o f(s)ds + (M + N)s l (c-N j l o f(s)ds) (1.3) ^ V J = f f Q f(s)ds + (M + tf)j U(/ f c ) ( r G Z). Let now T be a projector in Rn suc h that Im T = Im(A f + TV ) an d define Q T o n Z by Qr(f c ) = = (0» (/ " T)Mf c ))- % th e above discussion , (/, c ) G Im Z if and only if
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