12 Jack Palmer Sanders

Let 0 d min{d',d",d ,a,l-a}. Then 0 a-d a+d 1. Let

K = T T (Kn(xx[a-d,a+d]) ) x [a-d,a+d]. Let

1 X

L = [( U xx[t-d,t+d])n(Xxi)] u K , and let

(x,t)eKn(Xx[0,a])

L = [( u x*[t-d,t+d])n(Xxi)] u K . L and L are compact,

(x,t)eKn(Xx[a,l])

being closed subsets of T T (K) X I. We show that f e W(h(L ),U) and that

X 1

g £ W(k(L ) ,U) .

Suppose t h a t ( x , t ) £ L . Then e i t h e r t h e r e e x i s t s an

( x , t ' ) £ Kn(Xx[0,a]) such t h a t ( x , t ) £ x x [ t ' - d , t ' + d ] , o r t h e r e e x i s t s an

( x , t ' ) £ Kn(Xx[a-d,a+d]) such t h a t ( x , t ) £ x x [ a - d , a + d ] . In t h e f i r s t

c a s e , ( x , t ' ) £ B ( x , , t . ) fo r some i , 1 _ _ i _ n, and so

( x , t ' ) £ x x ( t . - d ( x . , t . ) , t . + d ( x , , t . ) ) . Then ( x , t ) £ (

x

x [ t ' - d , t ' + d ] ) n

(XXI)

c

1 1 1 1 1 1

[ x x ( t . - 2 d ( x . , t . ) , t . + 2 d ( x . , t . ) ) ] n (Xxi) c h~

1

(f~ (U)) . Hence f ( h ( x , t ) ) e U .

I i l l 1 1

In the second case, we have from above that (x,t) e xx[a-d,a+d] c v, or

p (f,g,a)(x,t) £ U. If t _ a, then f (x,t/a) •= f (h(x,t)) £ U, and if

t a, then (x,a) £ V, and p (f,g,a)(x,a) = f(x,l) = f(h(x,t)) £ U.

Thus f £ W(h(L ),U). That g £ W(k(L ),U) is proved similarly.

The function q(t) = (a+d-at)/(1+d-t) is increasing in the interval

(-°°,l+d) , and q(t) - a as t - -00. In particular,

2 2

aq(a) = (a+d-a )/(l+d-a), and (a+d-a )/(l+d-a) _ (a+d-at)/(1+d-t) for

a _ t _ 1. Let Z = (a/(l+d),a/(l-d) ) n (a-d,q(a)). It follows from the

facts that d a and d 1-a that Z c (0,1). Then

E = W(h(LJ,U) x w(k(L ),U) x z is an open set in

1 c 2 c

C(Xxi.Y) x C(Xxl,Y) x i, and (f,g,a) £ E. It is sufficient to show that

c c

P3(E) c W(K,U) .

Let (F,G,e) £ E, and let (x,t) £ K. Suppose first that t e.