The homotopy dependence of the degree can be formulated as follows:
Let H : [0,1] x X —» Y be a quasilinear Fredholm homotopy of the form
H(t,x) = L(t,x)x + C(t,x), let a(t) = L(t,0) and assume that a(0), a(l)
are invertible. If 0 is an open bounded set such that H(t,0 does not
vanish on the boundary of 0 for any t, then
deg (H(0,0,0,0) = e(a(a))r(a,[0,l])e(a(b)) deg (H(1,0,0,0). (1.4)
The index of an isolated, non-degenerate solution of f(x) =0 is not
given directly by the degree of the derivative at that point, as it is in
the case of a compact vector field, but it can be easily computed from the
parity. More generally, we prove that if f: 0 Y is C and has 0
as a regular value, then
deg (f,0,O) = e(f(0)) £
where e (f ,x) = r(f'(tx), [0,1]) if f'(x) GL(X,Y).
From (1.5) it follows that if f(0) = 0 and f'(0) is invertible,
then deg(f,U,0) = e(f'(0)) if f"1(0) n U = {0}. From this and (1.4) it
follows that if H : [0,1] x X Y is as in the above paragraph and, in
addition, t i— D H(t,0), 0tl, defines a path with invertible
end-points, then
deg(H(0,0,0,0) = 8 deg(H(l,0,0,0),
8 = deg |DxH(t,0)|r|DxH(t,0),[0,1]|deg|D H(1,0)|.
This is precisely what is needed in order to extend the Rabinowitz Global
Bifurcation Theorem [Ra] to quasilinear Fredholm mappings.
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