6
AN OVERVIEW
matters and we will confine our attention to an example from partial differentia l
equations in Chapter 10.
Our fina l topic , covere d i n Chapte r 11 , concerns application s o f minima x
methods to bifurcation problems . Suc h problems are of interest since bifurcation
phenomena occu r i n a wid e variet y o f setting s i n nature . Conside r th e ma p
F: RxE-+E., wher e
(1.10) F(A,tt ) = L u + Jr(t4)--Att,
E i s a real Banach space, A R, u E, L is a continuous linear map of E int o
E, an d H C
X(E,E)
wit h H{u) = o(||t*|| ) a s u - 0 . Not e that F(A,0 ) = 0
for all A R. W e call these zeros of F trivial Solutions of F(A, u) = 0 . A point
(p, 0) R x E i s called a bifurcation point for F i f every neighborhood of (#, 0)
contains nontrivial Solutions of F(A, u) = 0. It is well known—see Chapter 11—
that a necessary condition for (ß,0) t o be a bifurcation poin t i s that ß r(L),
the spectrum of L. Simpl e counterexamples show this necessary condition is not
sufficient. Howeve r in Chapte r 11 , it wil l be shown tha t i f (1.10 ) correspond s
to an equation of the form J'(u ) = 0 , ß £ a(L) i s also a sufficient conditio n fo r
(/i, 0) to be a bifurcation point . Othe r sharper results will give more Information
about th e nontrivia l Solution s o f F(A , u) = 0 for (A , u) nea r (# , 0) bot h a s a
function o f ||i*|| and as a function o f A.
Lastly there are two appendices. Th e first, Appendix A, is mainly concerned
with an important tool called the Deformation Theorem. I t is used to help prove
all of our abstract critica l point theorems. Appendi x B contains some technical
results which ar e usefu l i n verifyin g abstrac t condition s lik e I C
1
(£?,R) o r
(PS) in a partial differential equation s setting.
Some other sources of material on minimax methods and critical point theory
in general axe [BW, Bg, Ch2, CC, K, LLS , Mi, N2, P2, R2, S2, Va].
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