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.
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