ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 9
For certain classes M of quasilinear Fredholm mappings it is
possible to choose an orientation so that, for homotopies whose sections
are mappings in M, the corresponding degree is homotopy invariant. For
instance, if G is a subset of $n(X,Y) which is closed under compact
perturbations and which has the property that the parity of each admissible
path in G depends only on the end-points, then there is an orientation of
GL(X,Y) which induces a homotopy invariant degree for quasilinear Fredholm
mappings whose principal part lies in G. One example of such a choice,
which is important in application to elliptic boundary value problems, is
the following: Let X be compactly embedded in Y, and G be a convex
subset of $ (X,Y) having the property that for each pair of isomorphisms
T and T in X there exists Xm 0 such that tT + (l-t)T + XI is
an isomorphism if t [0,1] and A Xm; here, I denotes the inclusion
of X into Y. There is an orientation e of GL(X,Y) having the
property that if T is an isomorphism in G, then e(T) = (-1) , where m
is the sum of the algebraic multiplicities of the negative eigenvalues of
T. Moreover, for a C quasilinear Fredholm mapping whose derivatives at
each point lie in G, (1.5) is sharpened to the classic regular value
formula
deg(f,0,O) = Y (-l)n(x), (1.6)
x€0nf-1(O)
where n(x) is the number, counted with algebraic multiplicity, of negative
eigenvalues of f'(x). For quasilinear Fredholm homotopies having a
representation whose principal part, when restricted to the origin, is an
admissible path in G, the degree is homotopy invariant.
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