20

PATRICK FITZPATRICK AND JAC0B0 PEJSACHOWICZ

fixed u € X, a principal part of F at u differs from the above L(u)

by a compact operator. In order to utilize (1.4) and (1.5) to study (BVP)

and (BVP) , this is sufficient.

The smoothness conditions under which (1.8) is quasilinear Fredholm

are less stringent than those required to verify that (1.7) is quasilinear

Fredholm. We prove existence, bifurcation and multiplicity results for

(BVP) and (BVP) when A = I R or A = S , in the Holder context. The

2k+2 7

precise analytical conditions under which, if X = C (fi) and

Y =

C2,r(Q)x

n

C2+2k"mi,r(aQ),

F : X—Y is quasilinear Fredholm and

i=l

F : AxX—-»Y is a quasilinear Fredholm family, are prescribed in Section 10.

The general results from the earlier sections are then applied to (BVP) and

(BVP) to obtain very general existence results provided that there exist

a priori bounds for solutions of certain families, and to provide global

bifurcation and continuation theorems. As one example, when A = R and u = 0

is a solution of (BVP). for each X € IR, we prove that there is global

bifurcation from (X ,0) of nontrivial solutions of (BVP). provided that

the following two conditions hold: let £ = ^(0) and B =

(B ^(0) B .(0)) and write first order expansions at X = Xn.

1,A K,A U

2(X) = 2(AQ) + (X-XQ)T + R(X)

B(X) = B(XQ) + (X-XQ)S + R(X).

The first condition is that the dimension of the space of solutions of

2(Xn)v(x) =0, x € Q

U (1.12)

B(XQ)V(X) =0, x € an