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
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1979 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.