To prove the lemma
we may assume deg(L [intersection] Z) = t + 1.
Choose [psi](t) = [t.sup.p], p > 1, and w(x) = 1 and let M be a ball B in Lemma
6; we find that the norms [[parallel] u [parallel].sub.p,B] and [[parallel]u - [u.sub.B] [parallel].sub.p,B] are comparable; that is,
The following Lemma
is due to Theorem 1(i) of Kocetova et al.
[E.sub.3)](1; F') = [E.sub.3)](1; G'), by Lemmas
2.2,2.3,2.4 and 2.7 we get f [equivalent to] g.
It is not difficult to see from (23) and Lemma
12 that, for sufficiently large k,
By the structure of G and Lemma
1, we can obtain that [b.sub.n] (G) > 5 when n is even, and [b.sub.n] (G) < -4 when n is odd.
By virtue of ([H.sub.3])-([H.sub.7]), we have firstly the following lemma
By the case for [alpha] = 1 and Lemma
2, [k.sub.w] is the product of a function in [F.sub.1] and a function in [F.sub.[alpha]-1].
We apply this fact to Lemma
4, so they are not adjacent vertices in the graph F.
It is sufficient to show that the assumption (iv) of Lemma
2 is satisfied.
2.3 (3) to [E.sub.11] -[E.sub.12] + [E.sub.21], then [[sigma].sub.l]([E.sub.11] - [E.sub.12] + [E.sub.21]) is an infinite set and its elements are of the same modulus and real part.
The following lemma
denotes the mathematical form of the ellipsoidal approximation of the central path H([mu]).