281
Structural Hypotheses.
(H"4) A1(J.t,W) > 0 for all (J.t,w) E ~\ (0,0);
(H"5) Ao(J.t,W) = 0 for only finitely many (J.t,w) E ~
and obtain:
Theorem 3. (cf. [Hetzer/Schmidt (1995)]) Let (H"O) - (H"5) be satisfied. Then
a) ~ is the trace of a Jordan curve in R+ x C+(M): there exists a homeomorphism
'Y : R+ --+ ~ (onto), which is C 1 from (0,00) onto ~ \ {(O, O)} with 'Y'(p) =1= (0,0) for
all p E (0,00).
b) ~ is "S-shaped": P1'l 0 'Y(p) --+ 00, inf P1'2 0 'Y(p) --+ 00 as p --+ 00, and P1'l O'Y has
an even number of strict local extrema.
c) Given (J.t, w) E ~, (s, x) 1-+ W is an asymptotically stable (unstable) restpoint of U I-'
if"'('("'(-l(J.t, w)) > 0 « 0).
P1'oof, Note that a) and b) follow from Theorem 1. and that a principle of linearized
stability is available (cf. [Parrott (1989)]). Therefore it remains to show that the
principal eigenvalue Xo(J.t, w) of
-div( k grad 1jJ)(x) + g'(w(x))1jJ(x) + J.tQ(x)(82a)(x,w(x),w(x))1jJ(x)
= AC(X)1jJ(x) - J.tQ(x)(83 a))(x, w(x), w(x))1jJ(x) lOT b(s)e-ASds
has the same sign as ).,o(fl, w). Observe the nonlinear dependence on the eigenparameter,
which is characteristic for functional differential equations. In order to overcome this
difficulty, one introduces a second parameter K, E R and writes the last equation as
-div( k grad 1jJ)(x) + g'(w(x))1jJ(x) + J.tQ(x)(82a)(x,w(x),w(x))1jJ(x)
(21,,)
= AC(X)1jJ(x) - J.tQ(x)(83 a)(x,w(x),w(x))1jJ(x) lOT b(s)e-KSds.
Now, denoting by ~(K,; fl, w) the smallest real eigenvalue of (21,,) for a given K, one needs
to show that ~(O; J.t, w) and the only fixed point Xo(J.t, w) of ~(.; J.t, w) have the same
sign. This follows from a result in [Kerscher/Nagel (1984)], which uses a semigroup
approach. To this end one observes that
B : I' J 1-+ ~ (div ( k grad I'J) - (g' 0 w)I'J - J.tQ (82a )(', w, w)I'J)
C