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].