NONLINEAR BOUNDAR Y VALU E PROBLEM S 9 (/, c) e ke r QT an d moreover Q T i s a projector and is continuous, as easily verified. Con - sequently, using (1.3), we get &p s .Qite C W = Sl f ®ds + (M + ^)s l 3 r f * N i o AO* ) (1.4) = Hmds + ( M + N)slTA(f, c) , f G/ , having in this way an explicit expression of the projectors Ps, Q T an d of th e corresponding generalized inverse Kp Q i n terms of projectors in Rn ont o ker(M + iV) and Im(Af + TV) and of the related generalized inverse of M + N. When c = 0 , i.e. in the case of homogeneous boundary conditions, there is another natural way o f associatin g a n abstrac t operato r t o th e differentia l problem let u s tak e X = {x e OJ, RnY Mx(0) + TVx(l) = 0} , Z = L l (I, R n )9 do m L = { x G £: x i s absoultely con- tinuous on /} an d define L b y Lx = x' s o that ker L = {x G do m L: x i s a constant mapping whose value belongs to ker(M + TV)}. Now, Problem (1.1) with c = 0 being clearly equivalent t o x(f) = x(0) + ff(s)ds, Mx(p) + Nx(\) = 0 , we obtain at once tha t Im L = J/GZ : Nf^f(s)ds G Im(M + 7V)| = ^(ImCM + TV)) if 5: Z —• i?n is the linear continuous mapping defined by Bf = Nf l Q f(s)ds. Thus, dim ker L = di m ker(A/ + TV ) i s finite, an d Im L i s closed moreover, at the mapping / *~ / o/ fr° m % *nt0 Rn i s onto , it is not difficult t o see that codim Im L = dim(I m N/Jm N n Im(A f + TV)), so that in this formulation L i s not necessarily o f index zero however, if det( M + N) = £ 0, then ker L = {0 } an d codim I m L = dim(I m W/Im A0 = 0 so that L i s invertible it can also be shown that if rank(Af, N) = n L is of index zero. This last condition is in particular satisfied fo r the case of periodic boundary conditions x(0) - x(l ) = 0 for which dim ker L = n = codi m Im L . Now if we define P s o n Xt wit h 5 a projector in Rn suc h that Im S = ker(Tl f + TV) , b y Psx = S(x(0)) (where the right-hand membe r denotes the element o f X having the constant value £(x(0))), we clearly obtai n a continuous projector such that
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