2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

THE

S . L . Sobolev, Introduction to the T h e o r y of Cubature F o r m u l a s [in Russian], Nauka, Moscow (1974). V . I . Polovinkin," Cubature f o r m u l a s in I.~m)(~2), '' Dokh Akad. Nauk SSSR, 190, No. 1, 42-44 (1970). V . I . Polovinkin, " A s y m p t o t i c a l l y optimal sequences of cubature f o r m u l a s in L~n(~2),'' in: The T h e o r y of Cubature F o r m u l a s and Applications of Functional Analysis to Some P r o b l e m s in Mathematical P h y s i c s [in Russian], Nauka, Novosibirsk (1973), pp. 28-30. V . I . Polovinkin, "Asymptotic optimality:of sequences of cubature f o r m u l a s with a r e g u l a r boundary l a y e r for odd m," Sib. Mat. Zh., 1._66, No. 2, 328-335 (1975). V . I . Polovinkin, " A s y m p t o t i c a l l y best sequences of cubature f o r m u l a s , " Sib. Mat. Zh., 16, No. 6, 1255-1262 (1975). T . I . Khaitov, "Cubature f o r m u l a s with given derivatives," Dokh Akad. Nauk Tadzh. SSR, 1.~2, No. 10, 3-6 (1969). V . I . Polovinkin, "Composite cubature f o r m u l a s , " in: Topics in Computational and Applied Mathematics [in Russian], Voh 14, Inst. of Cybernetics and Computational Center, A c a d e m y of Sciences, Uzb. SSR, Tashkent (1972), pp. 53-62. V . I . Polovinkin, "Sequences of functionals with boundary layer," Sib. Mat. Zh., 16, No. 2, 328-335 (1975). V. L Polovinkin, "Some norm e s t i m a t e s for e r r o r functionals of cubature formulas," Mat. Zametki, 5, No. 3, 317-322 (1969). S . L . Sobolev, Applications of Functional Analysis in Mathematical Physics, A m e r . Math. Soc. (1969). V . I . Polovinkin and L. I. Didur, "On the o r d e r of convergence of cubature f o r m u l a s , " in: Topics in Computational and Applied Mathematics [in Russian], Voh 34, Inst. of Cybernetics and C o m p u t a t i o n a l Center, A c a d e m y of Sciences, Uzb. SSR (1975), pp. 3-14.

LYUSTERNIKS.

G.

SCHNIRELMAN

THEORY

Suvorov

UDC 519.3:517.944/947

Introduction In this article, the r e s u l t s of the papers [1-5] on the c r i t i c a l points of one functional (f) on the level s u r face of another (g) are generalized in the following directions: 1) less s m o o t h n e s s is required of the functionals f and g; 2) the functionals are defined in an a r b i t r a r y real Banach space E (for the results of Secs. 1-3 it is sufficient that E be a F r 6 c h e t space); 3) instead of assuming the functionals f and g are even, it is supposed that they are invariant under groups m o r e general than the group {-1, 1}; 4) condition C of P a l a i s - Smale [6] is replaced by two other conditions for the results of Secs. 1-3 which follow f r o m condition C in the case of Banach spaces. Section 1 introduces and studies two integer-valued set functions analogous to the genus of K r a s n o s e l ' s k i i [7, 8], viz., the inner and outer genus. Section 2 introduces (by analogy with the method of penalty functions [9]) an auxiliary functional ~N, and the main p r o p e r t y of its s t a t i o n a r y values is proved. In Sec. 3, the numb e r of s t a t i o n a r y points of this functional is estimated. F r o m the point of view of the functionals f and g, these e s t i m a t e s show the existence of a c e r t a i n number of solutions of the equation

/'(u) =~g'(u),

(1)

i . ~ T l ~ near a level surface of the functional g. Conditions are established in Sec. 4 for Banach spaces under which the solutions of Eq. (1) on ~g = ~} are limits of the s t a t i o n a r y points in Sec. 3, and c o r r e s p o n d i n g L y u s t e r n i k - Schnirelman t h e o r e m s are proved f o r this case. Finally, in Sec. 5 the case of variable levels of the functional g is considered.

T r a n s l a t e d f r o m Sibirskii Matematicheskii Zhurnal, Voh 19, No. 3, pp. 670-684, May-June, 1978. Original article submitted May 5, 1976.

472

0037-4466/78/1903-0472507.50

9 1979 Plenum Publishing Corporation

We now i n t r o d u c e the m a i n n o t a t i o n u s e d in t h i s a r t i c l e and w r i t e out the c o n d i t i o n s which the f u n c t i o n a l s satisfy. Notation: If u~E; v'~E', then
~g,~----{u:u~E, g(u)-----a}, J [ [ ~ , . ~ . d = { u : u ~ E , al~g(u)~a~}; herea c o n d i t i o n s wi[l be u n d e r s t o o d in what f o l l o w s without f u r t h e r mention;

s0,

0

if ~0 is a f u n c t i o n a l in E, t h e n {q0~c} ~ {u : u~E, rg(u) -> c}, and {~0 > c} is d e f i n e d a n a l o g o u s l y , e t c . ; Im ~0 is the r a n g e of the f u n c t i o n a l ~o; if A ~ E i s a n y s e t and H is a s e t of m a p p i n g s f r o m E into E, then HA = U hu. A s e t with the p r o p e r t y A = HA is c a l l e d H - i n v a r i a n t . u~A,R~H V a r i o u s c o n s t a n t s w i l l be d e n o t e d b y the l e t t e r s c, e, e t c . , without u s i n g i n d i c e s to d i f f e r e n t i a t e a m o n g t h e m w h e r e t h i s d o e s not l e a d to c o n f u s i o n . Conditions

Satisfied

by

the

Funetionals

a) f and g a r e d e f i n e d in a F r 6 c h e t s p a c e E (in S e e s . 4, 5 and in Sec. 3, T h e o r e m 4 , g is c o n t i n u o u s in E \ 0 , f i s c o n t i n u o u s in E;

E is a B a n a c h s p a c e ) ;

b) f and g a r e i n v a r l a n t u n d e r a f i n i t e g r o u p H of a u t o m o r p h i s m s of E which c o n s i s t s of m o r e t h a n a s i n g l e e l e m e n t ; the z e r o e l e m e n t 0 of E is the unique f i x e d point of e a c h n o n i d e n t i t y a u t o m o r p h i s m h~H; the m e t r i c p of E is a s s u m e d to be i n v a r i a n t u n d e r t r a n s l a t i o n s and u n d e r a l l the hEH, w h i c h d o e s not involve a l o s s of g e n e r a l i t y [8, 10]; c) f and g p o s s e s s GGteaux g r a d i e n t s e v e r y w h e r e , f r o m E \ 0 into the weak dual (E', a);

with the p o s s i b l e e x c e p t i o n of 0, which a r e c o n t i n u o u s

d) the o p e r a t o r s P , g' t a k e the s e t s Jt[~,,~.] into s e t s which a r e b o u n d e d in the s t r o n g d u a l (E ~, r c o n d i t i o n is u s e d in S e e s . 4, 5, i . e . , (E', r is a n o r m e d dual s p a c e ] ;

[this

e) t h e r e e x i s t s an e v e r y w h e r e c o n t i n u o u s f u n c t i o n c l(t) p o s i t i v e f o r t g 0 such t h a t c 1 (g(u)) -< ( g ' (u), u ) (in S e e s . 4, 5); f) f(0) = 0; g) g(u) > 0 f o r u r

g(tu) - - 0 a s t - - 0; g ( t u ) - - r

ast--~o,ur

h) t h e r e e x i s t s an e v e r y w h e r e c o n t i n u o u s f u n c t i o n c2(t) s u c h t h a t If(u)l -< c2(g(u)). T h e two c o n d i t i o n s r e p l a c i n g the P a l a i s - S m a l e c o n d i t i o n will be s t a t e d in Sec. 3. T h e y will be d e r i v e d f r o m t h e P a l a i s - S m a l e c o n d i t i o n in the B a n a c h c a s e and v e r i f i e d in c e r t a i n o t h e r c a s e s in Sec. 3. Since the f o r m u l a t i o n of the P a l a i s - S m a l e c o n d i t i o n g i v e n in Sec. 3 is not s t a n d a r d [it is not s t a t e d f o r the p a i r (f, g) but f o r the a u x i l i a r y function ~0N], p r o p e r t i e s of the f u n c t i o n a l s f and g will a l s o be p o i n t e d out w h i c h c a n be u s e d to v e r i f y the P a l a i s - S m a l e c o n d i t i o n . w

Inner

and

Outer

Genus

of

Sets

T h e l e t t e r J/f will d e n o t e one of the s e t s : E \ O , Jr..e, J[[=,~,]. The r e s u l t s a r e t r u e f o r a n y of t h e m . D e f i n i t i o n 1. The i n n e r g e n u s of a c o m p a c t s e t K ~ J [ r e l a t i v e to the g r o u p H is d e n o t e d b y H r , K and d e f i n e d a s f o l l o w s : H r , K = 0 tf and o n l y if K = r in the c a s e w h e n K # | if HE c o n t a i n s no H - i n v a r i a n t c o n n e c t e d s u b s e t ; H r . K = n if K c a n n o t be c o v e r e d b y l e s s t h a n n c o m p a c t s e t s of g e n u s one. D e f i n i t i o n 2. L e t A ~ f be an a r b i t r a r y s e t . T h e n i t s i n n e r g e n u s H r , A is d e f i n e d a s s u p H r , K , w h e r e the s u p r e m u m is t a k e n o v e r a l l c o m p a c t s e t s K ~ H A . We w r i t e H r , A = ~ if HA c o n t a i n s c o m p a c t s e t s of a r b i trarily large genus. D e f i n i t i o n 3. The n u m b e r Hr*A = i n f H r , G , w h e r e the i n f i m u m is t a k e n o v e r a l l s e t s G ~ ( open in J[ and c o n t a i n HA, is c a l l e d the o u t e r g e n u s of the s e t A.

which a r e

We r e s e r v e the n o t a t i o n HrA f o r t h o s e s e t s A f o r which H r , A = Hr*A (cf. p r o p e r t y V below). We s t a t e the m a i n p r o p e r t i e s of the f u n c t i o n s H r , and H r * . T h o s e p r o p e r t i e s which a r e o b v i o u s o r o b t a i n e d s i m p l y b y c a r r y i n g o v e r the r e s u l t s of [7, 8, 111 to the c a s e of JZ and H a r e not p r o v e d .

473

I.

H r , A --< Hr*A;

if A = B , t h e n H r , A -< H r , B and Hr*A -< Hr*B.

II. Let T : M - - M be a c o n t i n u o u s m a p p i n g which c o m m u t e s with the a c t i o n of H. and Hr*A -< H r * T A . III.

T h e n H r , A --< H r , T A

If A is c o m p a c t , t h e n H r , A < 0%

IV. If A and B a r e c l o s e d , t h e n Hr, (A (J B) ~ Hr,A ~ Hr,B and as a c o n s e q u e n c e Hr, ( A \ , B ) > H r , A -- Hr,B, if H r , ] 3 < 0% V.

If A is c o m p a c t , t h e n H r , A = Hr*A.

T h i s is a l s o t r u e for s e t s open in J/.

F o r open s e t s , t h i s f o l l o w s d i r e c t l y f r o m the d e f i n i t i o n . If. on the o t h e r hand, A is c o m p a c t , the proof follows b y c a r r y i n g o v e r the r e s u l t s of [11] in a n o b v i o u s way to the c a s e of the g r o u p H. F o r c o m p a c t s e t s , p r o p e r t y V m e a n s t h a t t h e r e e x i s t s a n e > 0 such that H r , A = H r , S ( A , e) = H r , S ( A , e), w h e r e S(A, e) is a n e - n e i g h b o r h o o d of the s e t A. VI.

If A ~1 B is open in B and H r , A < ,o, t h e n H r , ( B \ A ) ~ H r , B - - H r , A .

Indeed, let K be a n a r b i t r a r y c o m p a c t s e t in ]3. The s e t K \ . A is c o m p a c t by h y p o t h e s i s ; thus t h e r e e x i s t s a n e > 0 s u c h that Hr, ( K \ A ) ~ Hr,S (K \ A . e). The s e t C = K \ S (K \ A , e) is c o n t a i n e d in A and c o n s e q u e n t l y H r , C -< H r , A . C l e a r l y , K \ C ~ S(K \ A,e). It follows f r o m what h a s b e e n s a i d that Hr,(B \ A ) ~ H r , ( K ~ A ) = Hr,S(K\A,e)~.~Hr,(K\C)~Hr,K--Hr,C~Hr,K--Hr,A. If H r , ] 3 < r then t a k i n g a c o m p a c t s e t K ~ B such t h a t H r K = H r , B , we o b t a i n Hr, (B \ A) >~Hr, B - - Hr,A. If Hr,]3 = r then c h o o s i n g a c o m p a c t s e t K = B of a r b i t r a r i l y l a r g e g e n u s , we get ( b e a r i n g in m i n d the c o n d i t i o n H r , A < ,o) that H r , ( B \ A ) = ~o. In this c a s e , both s i d e s of the i n e q u a l i t y H r , ( B \ . A ) ~ H r , B - - H r , A a r e i n f i n i t e , and this is the s e n s e in which we u n d e r s t a n d the inequality. If A and B a r e a n y two s e t s in ~ ' , the i n e q u a l i t y Hr* (AtJ B ) ~ H r * A - k Hr*B holds and h e n c e Hr*(A\B)~Hr*A--Hr*B if Hr*]3 < ~o. VII.

P r o o f . Let GI and G2 be open s e t s in ~ ' with A=G1, B~G2, Hr*A~HrG~, Hr*B----HrG2. By p r o p e r t y I Hr, [(Gl ( j G~) \ G~] ~ HrG1. By p r o p e r t y V I , Hr, [(G1 U G~) \ G~] ~ H r (GI (A G~) -- HrG~, i. e., Hr (G, U G~) Hr G1 -}- Hr G~. But t h e n Hr* (A (J B ) ~ Hr (G1 [J G~) ~ HrG 1 -~ HrG 2 " Hr*A -}-Hr*B. Since A C (A \ B) IJ B, we have Hr*A .~ Hr* ( A \ B ) -}- Hr*B, i. e., H r * ( A \ B) ~ H r * A -- Hr*B. T h i s p r o v e s the p r o p e r t y . F o r the i n n e r g e n u s t h i s p r o p e r t y c a n be shown to be f a l s e by c o u n t e r e x a m p l e s . VIII.

Let {Gi} be a s e q u e n c e of open s e t s , G~=G~+,, G = kJ Gi, HrG=n.

T h e n t h e r e e x i s t s an i0 such that

HrGi0 = n. I n d e e d , l e t K=G be a c o m p a c t s e t s u c h that H r K = HrG. F r o m the c o v e r i n g of K c o n s i s t i n g of the s e t s G i we c h o o s e a f i n i t e s u b c o v e r i n g {Gik} and l e t i 0 = m a x {ik}. T h e n c l e a r l y H r G i 0 = H r K = n. w

The

Main

Property

of

Stationary

Values

In a n a l o g y with the p e a a l t y f u n c t i o n m e t h o d [9], we i n t r o d u c e a new f u n c t i o n a l gON(U) = f(u) - Nc3(g(u) - a ) . H e r e c3(t) is a n e v e n , e v e r y w h e r e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n which i n c r e a s e s s t r i c t l y f o r t > 0 and is s u c h t h a t c3(0) = 0 and limc~(t)/ca(t) = 0 , w h e r e c2(t) is the f u n c t i o n of c o n d i t i o n (h) in the i n t r o d u c t i o n and N and a are positive real numbers.

The n u m b e r a is not i n d i c a t e d in the n o t a t i o n f o r the f u n c t i o n a l s i n c e u n t i l Sec.

5, a = c o a s t .

It is n e c e s s a r y to m a k e a r e m a r k h e r e . In what f o l l o w s it is a s s u m e d that {f > 0} ~ !3 and all c o n s t r u c t i o n s with the f u n c t i o n a l (PN a r e t a k e n on the d o m a i n {q~N -> 0}. If, on the o t h e r hand, {f > 0} = ~ , it is n e c e s s a r y to take a n o t h e r f u n c t i o n a l ~N(U) = f(u) + Nc3(g{u) - a ) , and all f u r t h e r c o n s t r u c t i o n s with a n i n c r e a s e in CN m u s t be c a r r i e d o v e r s y m m e t r i c a l l y to the c a s e of a d e c r e a s e of ~ N in the d o m a i n {~N -< 0}. T h i s will be e x p l a i n e d in m o r e d e t a i l in Sec. 3. D e f i n i t i o n 4. A point aGE is c a l l e d N - s t a t i o n a r y if ~ ( u ) = 0. p o i n t 0 is N - s t a t i o n a r y (O will d e n o t e the z e r o e l e m e n t in E and E ' ) .

F o r c o n v e n i e n c e we a l s o s a y that the

We l e t A N d e n o t e the s e t of a l l N - s t a t i o n a r y p o i n t s . L e t AN .c = AN f~ {q)N = C}, A*N,c = AN n {q)N~c},

474

AN.[..... ] -- AN N {Cl~_(~rr

We now c o n s t r u c t a " s m a l l " continuous m a p p i n g of E into E c o m m u t i n g with the action of H and l e a v i n g A N fixed, and which s t r i c t l y i n c r e a s e s the functional at e v e r y n o a s t a t i o n a r y point. Let ur and r~, ~ S (0,t) be a v e c t o r such that <~0~l(u), r~u) > 0. By condition c) of the introduction, t h e r e e x i s t s a n u m b e r e(u) > 0 such that f o r all v ~ S ( u , e(u))

(~',, (~,), ,\we c h o o s e a l o c a l l y finite s u b c o v e r i n g {S(u T, ~3,)}, w h e r e e T = e(u?), f r o m the c o v e r i n g of E \ A N by the n e i g h b o r h o o d s S(u, e(u)). This c a n be done by the p a r a c o m p a c t n e s s of E [12, p. 50]. We e n l a r g e the c o v e r i n g {S}uT, e T ) } t o a n H - i n v a r i a n t c o v e r i n g , k e e p i n g the e a r l i e r notation {S(u T, aT) }, and c o r r e s p o n d i n g l y the set {ru} is e n l a r g e d to U U ]lr,~, with the s a m e notation {r~u}. (It is c l e a r that E \ A x is H - i n v a r i a n t just as AN is.) "; h ~ H

In each set HuT we c h o o s e an a r b i t r a r y point v and c o r r e s p o n d i n g v e c t o r rlv in {r~u} (any v e c t o r if t h e r e a r e s e v e r a l ) . We put r~ = r v f o r the points chosen. F o r the points u = hv we take r~u = hriv o B e a r i n g in mind the l i n e a r i t y of all the h and the i n v a r i a n c e of ~0N and the m e t r i c u n d e r H, it is e a s y to v e r i f y that the r e s u l t i n g set of v e c t o r s has the p r o p e r t i e s that r~xu = hr~u, and p r o p e r t y (2) is r e t a i n e d f o r the v e c t o r s r~u in pIace of r~u f o r the s a m e e(u). We now c o n s t r u c t a v e c t o r field r3u on all of E',.A.n- as follows:

w h e r e the partition of unity is given by

O.

~ ~ S (nv, ~),

d (., ~?) = : 0 (rt, u~.) ~

(~'' -- p ( ' ' "v')) p (., %,)

!

if

l,

u

=

(41 u v.

In (4) the s u m m a t i o n is c a r r i e d out only o v e r t h o s e X'~ {~} f o r which u ~ S (uv., ew). T h e r e a r e f i n i t e l y m a n y such T' (E is p a r a c o m p a c t ) , and t h e r e f o r e the s u m (4) e x i s t s . F o r the s a m e r e a s o n , the n u m b e r s d{u, "/) a r e n o n z e r o f o r all but a finite n u m b e r of indices T, so that the s u m in (3) a l s o e x i s t s . The field r3u is continuous in E \ . A x , and if hu~/ = uT0, then d(u, T) = d(hu, To). Thus, the p r o p e r t y hr~3~ = r~u holds and f o r any t ~ [ 0 , 1] the m a p p i n g u ~ u + trau is a m a p p i n g of the set E A x into E which c o m m u t e s with H. o\ over, f o r all z ~ E'\Ax we have (gci~-(u), r,~ 3. = E J ( . , v)(~'~-('.), ~,;., / > o. V

\

More~

*

f Let t u denote the largest number such that f o r all t ~ [ 0 , t~], (q~N(U + tr3u), r 3 ) > 0. Take an a r b i t r a r y point u o ~ E \A~- and a neighborhood S(u0, fl) of u o which intersects f i n i t e l y m a n y o f the neighborhoods S(UTk, eTk). The field r 3 is finite dimensional in S(u 0, /~) and t h e r e f o r e f o r u,v~S(u0, ~), the function (~}q(u), r~v ) of the two v a r i a b l e s u and v is continuous. Thus, there exists 5(u 0) > 0 such that f o r all u, v~S(uo, 26(uo) we have 3\ 3 (q21~-(u), r ~ / ~ 1/e <~i~"(no), r~,o), If we take a n u m b e r c(u) f o r u~S(uo, 25) such that u - - tr~ ~ S(u 0, 26) f o r all 3 t ~ [ 0 , c(u)], then tu -> c(u). Since the set U r~ is bounded, t h e r e e x i s t s a unique S(u 0) > 0 such that f o r

u~Stuo,6)

any u~S(u0, 6) we have c(u) -> S(u0). We get the r e s u l t that f o r e v e r y u o ~ E ' \ A~. t h e r e e x i s t 5(u 0) > 0, S(u0) > 0 such that f o r all v~S(uo, 5(uo)) the inequality tv -> S(u0) holds.

Repeating the a r g u m e n t s in the c o n s t r u c t i o n of r3u, i . e . , c h o o s i n g an H - i n v a r i a n t l o c a l l y finite s u b e o v e r ing {S(ufl, 6fl)} of {S(n, 5(u))} and f o r m i n g a p a r t i t i o n of unity d(u, fl) by taking t~ = ~ d ( u , ~)S(ue), we get an H - i n v a r i a n t continuous functional t ~ such that f o r all u ~ E \ , A ~ . , t ~ [0, t c) we have (~o~l(u + tr3u), r3u } > 0. If we c o n s i d e r the field r4u = f 0u r fi, 3 then in place of the p r e c e d i n g inequality we m a y w r i t e f o r all u ~ E\AN~ t ~ [0, ~) r~> > 0.

(5)

F i n a l l y , we a l t e r the field one last t i m e : in place of r4u we c o n s i d e r r u = pl(u, AN)r~, w h e r e pl(u, v) = p ( u , v ) f o r p(u, v) -< l and pl(u, v) = l f o r p ( u , v ) > 1 . We take ru = 0 f o r u~A~-. It is c l e a r that f o r a n y ~:~[0, t] the m a p p i n g ~ ( u , t ) = u + t r ~ is the d e s i r e d one. In the sequel we will u s e the notation ~ ( u ) = ~

(u, 1/2).

475

L E M M A 1.

L e t c~Im%~..

T h e n H r * A N , c -> Hr*{q~ N -> c} - Hr{~0 N > c}.

P r o o f . 1. T h e p o i n t c is i n t e r i o r f o r Imq~N. L e t A~.o~G, w h e r e G is an open s e t and H r G = Hr*AN, c, w h i l e Gl -~{u : ~ ( u ) ~ { q D ~ - > c } . B y p r o p e r t y II of the g e n u s , Hr*G1 -< Hr{q~ N > c}. A t the s a m e t i m e , it is c l e a r t h a t the s e t G1 is open and c o n t a i n s { ~ N ~ c}\G. T h u s , G~ U G ~ { r c}, w h e n c e H r * { ~ N -> c} - H r G 1 + H r G -<

Hr{cPN > c} + Hr*AN, e. 2. The point c is an upper boundary point for lineN. Hr{~ N > c} + Hr*A by property VII of the genus.

Then { ~ c }

----AN,~U{~.~>c}, i.e., Hr*{~ N -> c} -<

3. The point c is a lower boundary point for Im~PN. Then {~N =c} =AN, c , and if G~A.,.~, then Hence, Hr*{~0N >- c} -< Hr{(PN > c} + HrG = Hr{r N > c} + Hr*AN, c for a suitable choice

{~N>c}[JG~{~n~c}. of G.

T h e l e m m a in i t s g e n e r a l f o r m is p r o v e d without u s i n g a n y p r o p e r t i e s of the f u n c t i o n a l CN a p a r t f r o m its c o n t i n u i t y and the p o s s i b i l i t y of c o n s t r u c t i n g t h e m a p p i n g ~ ( u ) . In p a r t i c u l a r , f o r a c o n c r e t e CN(U) = f(u) Nc3(g(u) - s ) , c a s e 3) in the l e m m a n e e d not be c o n s i d e r e d s i n c e Im ~0N h a s no l o w e r b o u n d a r y p o i n t s . The

w

Critical

D e f i n i t i o n 5. D e f i n i t i o n 6. I m ~ N if

Points

of

the

Functional

T h e n u m b e r c is c a l l e d a s t a t i o n a r y v a l u e of the f u n c t i o n a l r

if AN, c # 0 .

We w i l l s a y t h a t t h e f u n c t i o n a l CN h a s the g e n u s s e m i c o n t i n u i t y p r o p e r t y a t the point c ~ lira Hr { ~ c ~ c--e} =Hr* { q ~ c } .

e~-}-0

It is c l e a r t h a t t h i s p r o p e r t y m a y b e w r i t t e n a s : limHr{~.~.>c--e}=Hr*{~.,. >~ c}, o r a s :

t h e r e e x i s t s an

~-~-{-0

a>0

s u c h t h a t Hr{(p N > c - e} = Hr*{~ N >- c}. We c a n now s t a t e the r e m a i n i n g two c o n d i t i o n s c o n c e r n i n g the f u n c t i o n a l s f and g. (i) f o r a n y N > 0, c > 0, c ~ I m ~N, the f u n c t i o n a l q~N h a s the g e n u s s e m i c o n t i n u i t y p r o p e r t y ; (J) CN a t t a i n s a m a x i m u m in E.

In S e c s . 4, 5 it i s a s s u m e d t h a t E i s a B a n a c h s p a c e , and i n s t e a d of (i), (]) we a s s u m e t h a t the P a l a i s S m a l e c o n d i t i o n h o l d s (cf. D e f i n i t i o n 7 b e l o w ) . If we c o n s i d e r the f u n c t i o n a l (PN, t h e n c o n d i t i o n s (i), (]) t a k e the f o r m : (i') f o r a n y N > 0 , c < 0 , c ~ I m ~ lira Hr. {r~.~-~ c-i-e} -----Hr* { ~ ~ c} ; ~-]-0

(J') ~ N h a s a m a x i m u m in E. We now s t a t e the P a l a i s - S m a l e c o n d i t i o n [6]. F o r the p u r p o s e s of Sec. 5, we m u s t s t a t e it in a m a x i m a l l y g e n e r a l f o r m , a s s u m i n g t h a t the n u m b e r s N and s in the f u n c t i o n a l ~N(U) = f(u) - Nc3(g(u) - a ) c a n v a r y . F o r t h i s r e a s o n we d e n o t e t h i s f u n c t i o n a l b y ~N, s (u) ( h e r e and in Sec. 5). E is a B a n a c h s p a c e . D e f i n i t i o n 7.

T h e f u n c t i o n a l CN, s s a t i s f i e s the P a l a i s -

(1) the s e q u e n c e

S m a l e c o n d i t i o n if f o r a n y c > 0 the c o n d i t i o n s

{ui} H { , I N ~ , ~ c } ,

(2) ll(P~i' ai(ui) lIE, -* 0, (3) Ni >- No > 0, 0 < a -< s i < b < ~ imply that a converging subsequence {uij} can be extracted from {ui}. It f o l l o w s a t o n c e f r o m c o n d i t i o n (c) of t h e I n t r o d u c t i o n t h a t if Ni - - N < r &i "" s0, t h e n l i m u i j is an N - s t a t i o n a r y p o i n t f o r CN, s 0" It is c l e a r how to r e f o r m u l a t e D e f i n i t i o n 7 f o r t h e f u n c t i o n a l ~N, s L e t Ni - N > 0, s i = s > 0, with the f u n c t i o n a l d e n o t e d b y ~0N a s a b o v e . L E M M A 2.

If the P a l a i s -

S m a l e c o n d i t i o n h o l d s in a B a n a c h s p a c e , the f u n c t i o n a l ~ N h a s the g e n u s

s e m i c o n t i n u i t y p r o p e r t y a t e a c h p o i n t c ~ I m q0~, c ~ 0 . Proof.

476

~ 1. In the p r o o f we r e q u i r e the f o l l o w i n g o b j e c t s f r o m Sec. 2: r ~, ~ r~,

e(u), A v.~, A .~ + ~, ~ ( u ) , t~, 0 r~.

L e t the open s e t G contain {~PN -> c} and have the genus H r G = Hr*{r N -- c}. By the c o m p a c t n e s s of A ~ , a f o r all d > 0 (this follows f r o m the P a l a i s - S m a l e condition) t h e r e e x i s t e > 0, b > 0 such that S(A+,~, 3 e ) ~ G, Ax. , r ~ S ( A+N,~, e). Let i~e denote the s e t { ~ N ~ c --6}\S(A+N,~, e); 9.2s and fZae a r e defined a n a l o g o u s l y . By the P a l a i s - Smale condition inf [1~ (u)iI -~ ~ > O. We c o n s t r u c t a m a p p i n g ~ (u) by c h o o s i n g f o r all u~O~ v e c t o r s rlu such that <(p~-(u), r ~ ) ~ 1/2~. We c h o o s e rlu a r b i t r a r i l y outside ~e, e x c e p t that we r e q u i r e ( ~ ( u ) , r ul ) > 0 . We add the condition e ( u ) < a f o r e(u). F o r the points of gt2e we have (TN(U), r~> ~ x/4~ and min ( ~ v (u-~ try), ru/[lrul[ ) - c o a s t > 0. Thus the m a p p i n g h a s the p r o p e r t y 0-~t~l/2

~ (~) (u) ) - ~

(u) >1 da (u)

(6)

at the polnts of ~2e, w h e r e d = c o a s t and a(u) is the shift of the point u u n d e r the m a p p i n g .~(u). L e t ul~-.q)(u),

u 2 ~ ) ( u l ) , etc.

If the length of the t r a j e c t o r y ~ , u 1, u 2. . . . }, i . e . , ~ll~t~--uh_!lt, is finite,

t h e r e e x i s t s a l i m i t point u ! = l i m u k. Using this point, we c o n s t r u c t a t r a j e c t o r y {u, u~, u 1,2 . . . } etc. Using Z o r n ' s 1emma we p r o v e the e x i s t e n c e of a m a x i m a l " c o m p o s i t e " t r a j e c t o r y which we denote by Tr(u). If u~Q:o, then the length of Tr(u) is at l e a s t a. T a k i n g this into account we denote by Tre(u) the initial s e g m e n t of length e of the t r a j e c t o r y Tr(u) (for u ~ 2 ~ ) . F o r points outside ~22e we w r i t e T r s ( u ) f o r the t r a j e c t o r y {u, ~ ( u ) } ~ {u, ul}. [Of c o u r s e , the endpoints of the t r a j e c t o r i e s Tre(u) do not give a continuous i m a g e of the initial points.] We r e m a r k that if Tre (u)N S (A~,r 2 ~ ) = ~ , then the functional ~N i n c r e a s e s along Tre(u) by not l e s s than s o m e q u a n t i t y T > 0 [by Eq. (6)]. We d i s p l a y this r e s u l t f o r f u t u r e r e f e r e n c e . (.) t h e r e e x i s t s a T > 0 such that f o r all u ~ { ~ > c - - ~ } e i t h e r Trc (~) n {%~ > c} =/=~ or Tr~ (~) ~ S (A~,c, 2e) :#= Z . H e r e the n u m b e r T is taken f r o m the beginning of the proof of the l e m m a and does not e x c e e d 5. 2. We now c o n s t r u c t an open s e t G containing {goN -> c}, having genus H r G = Hr*~p N >- c}, and p o s s e s s i n g the p r o p e r t y that ~ ( G ) ~ . The l e t t e r G below d e n o t e s the open s e t i n t r o d u c e d in the beginning of p a r t 1 above. L e t G~ = {u : z~~ {q3N > C - - '~} ~ G, ~ (U) ~ {(pN>C}}, G2 =tr[(SA+,~, 2e) ,q {%~ > c -- 7}], w h e r e

tru = r ~5)~ (u), h=0

= G~ U G~. T h e set G is open (this is obvious) and contained in G. Indeed, G ~ G , and as f o r G~, a n y t r a j e c t o r y in S (A+,r 2e) f/{%-~ > c -- ~} e i t h e r r e m a i n s in S (A~,:, 3e) N {q~.v> c -- ?} ~ G o r e l s e g o e s outside S (A+r 3a), but then e n t e r s ~c~>c}~G by the definition of ~. Thus, H r G --< HrG. But since G c o n t a i n s {gvN ~ c}, we have H r G -> H r {~p~ - c~ = H r G , i.e., the genus of G is as r e q u i r e d . If u~G, w e h a v e the following p o s s ibilities: 1) u ~[S(A-~.r 2e) ~ {q~v > c - ~,}], and then ~ (u) ~ tr~ ~ G: ~ U, 2) ~ ~ iS(A+ :, 2s) ~ {~r~ > c -- ~}]: and then ~(~)~{~->c}~G. Thus G is the d e s i r e d set. 3. Now let G~ = G ( J ~ - ~ ( G ) (here ~ - ~ is the i n v e r s e of ~ ) . The genus H r G i = H r G . Indeed, ~ ( G ~ ) = ~ G @ ~ - - ~ ] (G)]== ~ ( G ) i A G ~ G , i . e . , HrG~ -< HrG, and the o p p o s i t e inequality is obvious, it is a l s o c l e a r that G~ = G~-~ U ~ ) - ; (G~-~) has the s a m e genus as G, i . e . , the genus is equal to Hr*{~ N ~ c}. B y p r o p e r t y VIII of the genus, Go = U G~ has genus H r G 0 = Hr*{cpN -> c}. h

M o r e o v e r , G, U.~ -~ (Go) = G~,

since ( u ~ G o ) ~ k ( u ~ ) = ~ ( ~ ) - ~ ( u ) ~ G ~ + ~ ) O (~-~(u)~Go). We p r o v e that G o c o n t a i n s the s e t {~PN > c - TJ~. A s s u m e that this is f a l s e and t h e r e e x i s t s u0 ~ {~N>C--?}~\Go. B y the c o n s t r u c t i o n of G 0, the endpoint of the s e g m e n t Tre(U 0) lies in Go [cf. {*)], i . e . , t h e r e e x i s t s a point u~ ~ Go on T r e ( u 0) such that ~)(u~)~G0. Thus, ~ - i (Go).\Go=/: ~ , which is i m p o s s i b l e . T h u s Hr{~a N > c - T} -< HrG0 = Hr*{~PN -> c). l e m m a is proved.

At the s a m e t i m e Hr{r N > c - T) -> Hr*{~N ~ c}, and the

A s f o r the P a l a i s - S m a l e condition, it is v e r i f i e d in o u r c a s e u n d e r the usual a s s u m p t i o n s , viz., that f' is c o m p a c t , g' is u n i f o r m l y monotone and bounded (or condition S in [4] in place of the u n i f o r m monotonicity), and that we have (llu~[[-~)=~ (g(u d --.-co) and (l[/'(u~) [I -*- 0) =*- (](u,) --~0): Indeed, on the s e q u e n c e {ui} in Definition 7, we have t N i , ai(ui) -> c > 0, i . e . , f i r s t of all, f(u i) - c > 0 o r llf'(ui)[i -> c 1 > 0, and secondly, the g(u i) a r e bounded, i . e . , the n o r m s Ilui[I a r e bounded, so that the n o r m s IIg' (ui)ll a r e a l s o bounded. Hence, s i n c e II~p~li, ai(ui)ll tends to z e r o [i. e., [If'(ui) - Nic~(g(ui) - a)g'(ui)[I tends to z e r o ] , it follows that s t a r t i n g with s o m e i 0 we have ]Nic~(g(ui) - a i ) l _> c 2 > 0. Consequently, if i, j -~ ~, then

/ N~c"3 ( g ( u'i ) -- % ) "+Ni(zi

'

I I Nic"3 ( g ( u i ) -- ai)

(ttl) - - zVjc"3 (g (ul) -- ~zj) " (PNi'al (uj) =

Njc~ (g (uj)" __ ~])

l

gr (lti) -~ gf (U]) --> O.

477

The s e t

(J f' (ul)/Nic"3 (g ( u l ) - a~) is p r e c o m p a c t , i . e . , t h e r e e x i s t s a s u b s e q u e n c e {Uik} such that if u,, um ~ {ui h}

and n, m --* ~o, then N , c ' , (/'g ((un) u,)--a)

1' (urn) ~m) --~ 0. Nm/3(g(u,,)--

Thus, Ilg' (u n) - g' (um)il ~ 0 and it follows f r o m

the p r o p e r t i e s of g [3, 4] that a c o n v e r g i n g subs e q u e n c e can be e x t r a c t e d f r o m {Uik}. LEMMA 3.

The functional CN has a m a x i m u m in E.

P r o o f . The functional ~PN is bounded f r o m above, as follows f r o m condition (h) of the introduction and the p r o p e r t i e s of the function c s. A s in L e m m a 2, we c o n s t r u c t t r a j e c t o r i e s Tr(u) f o r each point u ~ {~0N~ 0} ~ A , . M o r e o v e r , in the p r e s e n t c a s e we do not w o r r y about the continuity of ~)(u), but c o n s t r u c t the t r a j e c t o r l e s indef p e n d e n t l y f o r e a c h point but in such a w a y that r - CN(Uk_ i) - cllu k - Uk_lJl JJg0N(Uk_l)li, w h e r e c = c o n s t > 9 ! 0 (as is e a s i l y seen, this is posmble). It follows f r o m the boundedness of CN that ]]r ~ 0 along Tr(u), i . e . , the l i m i t points (u) A of the t r a j e c t o r y Tr(u) lies in A~, ~pN(u)" Thus, if {u[} is a m a x i m i z i n g sequence, t h e r e a l s o e x i s t s a m a x i m i z i n g s e q u e n c e in U (U~)A contained in the c o m p a c t (by the P a l a i s - Smale condition) i

+

s e t AN, CN(U0), which p r o v e s the l e m m a . We state a n o t h e r l e m m a which c a n be helpful in v e r i f y i n g the g e n u s s e m i c o n t i n u i t y p r o p e r t y . LEMMA 4. Let the F r ~ c h e t s p a c e E i be d e n s e l y imbedded (by the identity o p e r a t o r I) in the F r ~ e h e t s p a c e E 2. A s s u m e that the functional rp is continuous in E i and in E2; that the s a m e g r o u p H a c t s on both E i and E2; and that the functional r has the genus s e m i c o n t l n u i t y p r o p e r t y in E 1 at the point c. Then ~p a l s o has the g e n u s s e m i e o n t i n u i t y p r o p e r t y at c in E 2. P r o o f . 1. We f i r s t p r o v e that f o r a n y c we have Hr{r > c}E 1 = Hr{r > C}E 2 (with the obvious notation). F i r s t o f all, s i n c e I { ~ C } E , ~ { q ) ~ a } E , we have Hr{cp > C}E 1 --< Hr{r > C}E2, i . e . , the s e c o n d p r o p e r t y of the g e n u s is p r e s e r v e d h e r e . Now let K ~ { ( p N ~ c } E ~ be a c o m p a c t set of the s a m e genus as ~r > C}E2. We take a finite H - i n v a r i a n t c o v e r i n g S(U?, e 7) of the c o m p a c t set K and v e c t o r s rlu such that uv ~- r~.~ ~ E, (we do 7 not w o r r y about the i n c r e a s e of ~p). Next, r e p e a t i n g the a r g u m e n t s used in c o n s t r u c t i n g ~ ( u ) , we f o r m a p a r t i t i o n of unity d(u, 7) and obtain a m a p p i n g of the c o m p a c t s e t TK by the f o r m u l a Tu = ~ d ( u , "~)(U v + rvv ). C l e a r l y , TK~E~;

since t h e r e e x i s t s an e > 0 such that S (K, e)E~ ~ {~N~C}E.., if e7 and the v e c t o r s rlu.y a r e

s m a l l enough, the i m a g e TK r e m a i n s in S(K, e)E2, i . e . , i n f ~ ( u ) ~ c .

Thus, T K ~ { ~ C } E , , i . e . ,

Hr{go > C}Ei ->

TK

H{ep > C}E2, f r o m which the e q u a l i t y Hr{cp > e}E i = Hr{~p > C}E2 follows. 2. L e t GE2 be an open set c o n t a i n i n g {r >- C}E 2. Then f o r an open G ~ { ~ > / C } E , t h e r e e x i s t s a set G1 s u c h that GLcG, Gi ~ {~ >/C}E,, I G~ ~ GE~ (this is obvious). T h u s H r G i -< H r G E 2, and since GE2 is a r b i t r a r y , we have H r * {r -> C}E1 s Hr*{~p -> e]E 2. But b y the c o n d i t i o n s of the l e m m a and p a r t 1,

Hr {e~ > c -- e}E, = Hr {~ > c -- e}E, = Hr* {~ >/C}E, f o r s m a l l e > 0, i . e . , Hr{go > c - e]E2 -< Hr*{~p >- C}E 2. The r e v e r s e inequality is c l e a r , so that f o r s m a l l e > 0 we have the e q u a l i t y Hr{~p > c - e]E2 = H r * { r >- C}E 2, as r e q u i r e d . If E is a Montel s p a c e and the functional g is such that v a ~ ( 0 , co) the s e t ~g _< a} is bounded, then the s e m i c o n t i n u i t y of the genus is v e r i f i e d e a s i l y . Indeed, if CN(U) z e > 0, then by condition (h) of the i n t r o d u c tion and the p r o p e r t i e s of the function c3, the point u lies in s o m e ~g -< a(c) < .o}, t . e . , in s o m e c o m p a c t set. On the o t h e r hand, it follows f r o m r >- c that f(u) >- c, i . e . , u lies at a n o n z e r o d i s t a n c e f r o m 0. Thus f o r all c > 0 {r z c } is a c o m p a c t s e t which does not c o n t a i n 0. All the p r o p e r t i e s of the g e n u s a r e valid h e r e . By the fifth p r o p e r t y t h e r e e x i s t s an s > 0 such t h a t Hr*{ cflN- -> c} = HrS({ ~PN -> c}, e). But { ~N c - Y 0} is c o m p a c t f o r s m a l l N0 > 0 and goN is continuous, and t h e r e f o r e t h e r e e x i s t s a Y0 > 0 s u c h that { W ~ c--~0} ~ S ( { ~ / > c } , e), i.e., Hr*(gPN ->c} --Hr{~P N > c - y ~ >Hr*(~P N ->e}, as r e q u i r e d . The c o m p a c t n e s s of (g0N -> c} a l s o i m p l i e s condition (j).

Conditions (i'), (j') f o r ~N a r e v e r i f i e d a n a l o -

gously. We t u r n to the t h e o r e m s on the c r i t i c a l points. + == sup inf TN (u), c~.~ = g e n u s not l e s s than p, cp,~ +

Let 9~'~ be the c l a s s of all c o m+p a c t s e t s in E ' \ e with inf sup ~ . ( u ) . The n u m b e r s Cp, N' Cp, N u s e d below a r e

c o n s i d e r e d f o r those p f o r which 9g~=/:~, i . e . , Cp, N a r e defined.

478

F o r a c y c l i c g r o u p H, e . g . , it can often be

9

:k

p r o v e d t h a t HrE \ 8 = co (this f o l l o w s f r o m the r e s u l t s of [8]), i . e . , Cp, N a r e d e f i n e d f o r a l l p. f r o m c o n d i t i o n s (j), (j') t h a t c~,~.~ Im %v, T H E O R E M 1.

L e t c p+ , N = c p+~ g , N = c > 0 , w h e r e q - > 0 .

T h e n H r * A N , c - q>

It f o l l o w s

+!.

P r o o f . F o r a n y 6 > 0 we h a v e H r ( ~ N > c - 6} -> p + q b y the h y p o t h e s i s of the t h e o r e m . T h u s , Hr*{~ N -> c} >- p + q. But Hr{~ N > c} < p b y h y p o t h e s i s , i . e . , b y L e m m a 1 Hr*AN, c -> q + 1. An a n a l o g o u s r e s u l t h o l d s f o r Cp, N" T H E O R E M 2.

F o r a n y e > 0 t h e r e e x i s t at l e a s t J[~. . . . . +~ s e t s Hu of s o l u t i o n s of Eq. (1) in the d o m a i n

Hr.d[~ P r o o f . We c o n s i d e r h e r e o n l y the c a s e Hr.J[~
T h e a n a l o g o u s c a s e f o r Hr.J[~=oo (the s e t A N u n -

L e t Hr.J[~-~n, Hr*{%-~0} = k . B y L e m m a 1, k - H r { r N > 0} -< Hr*AN, 0. A t the s a m e t i m e t h e r e e x i s t s c > 0 s u c h t h a t Hr{~p N > 0} = Hr*{~ N ~ c} = k 1. T h e g e n u s s e m i c o n t i n u i t y p r o p e r t y is v a l i d in the d o m a i n {~N -> c} f o r a l l l e v e l s not l e s s t h a n c. T h e r e a r e f i n i t e l y m a n y n u m b e r s c + ~ in the d o m a i n {r -> c}, and l e t t h e m be c 1 > . . . c m = c. It is c l e a r t h a t Hr*{cpN >- Cm} = Hr*{~0 N - c} and H r * A N , c m -> kl - Hr{~N > Cm}. H r { r N > c m} = H r * { r N ~ Cm_l}, i . e . , H r * A N , Cm_~ -> Hr{~ N > Cm} - H r { r N > Cm_l}, e t c . obta in

> c m, w h e r e Furthermore, A s a r e s u l t , we

Hr*A~. o + ~ Hr*AN,e i ~ k -- Hr* {~%,~ 0}. i:t

T h u s , t h e r e a r e at l e a s t k s e t s Hu in A N and A + N,0" It f o l l o w s f r o m the d e f i n i t i o n of CN t h a t {r

~ Jg[~,~],

w h e r e 0 < a ~ -< a ~ < ~ and a ~ -~ a ,

2 a N -~ a a s N - " ~ , s o that c h o o s i n g N s u f f i c i e n t l y l a r g e , we a c h i e v e t h a t the s e t A + N,0 is c o n t a i n e d in the d o m a i n ~'~ . . . . . +~ with a n y e > O. Since n = Hr,J{~ ~ Hr*jl[~ -: Hr* [(j//~. ~ {~N ~:~ 0}) (j (J{~ N {r
We c o n s i d e r the d o m a i n {~N < 0}.

Hr*(J{~ fl {r

s e t {~N < 0} is open and c o n t a i n s J/= N{ff~.<0}, s o t h a t Hr{~ N < 0} ~ n ( s t a r t i n g with the b e g i n n i n g of Sec. 2) f o r the d o m a i n {~N < 0}, r e p l a c ~ g of ~N, we p r o v e t h a t t h e r e a r e at l e a s t n - k s e t s of p o i n t s in the s e t A ~ , c o u r s e , L e m m a 1 m u s t have the f o r m H r * A N , c >- Hr*{~~ -< c} - H r { ~ N T h e p r o o f d o e s not c h a n g e . ) M o r e o v e r , {~.~-~.0}~ J / ri n N~, a N~,~, where a~ ~a, ] s e t s Hu.

a ~ l "-*a a s N - ~ .

k. C a r r y i n g out a l l t h e c o n s t r u c t i o n s the i n c r e a s e of ~oN with the d e c r e a s e 0 -: {u : ~ { u ) = | ~N(U) < 0}. (Of < c} in the c a s e of t h e f u n c t i o n a l ~N. T h u s , A N N A \ c o n t a i n s at leas~ n

Since the p o i n t s in A,~U A.~ a r e s o l u t i o n s of Eq. (1) with k = ~Ne~(g(u) - a ) , the t h e o r e m i s p r o v e d .

We now c o n s i d e r the c a s e Hr.J[~.-:oo. H e r e t h i s c o n d i t i o n will be a s s u m e d and the t h e o r e m w i l l h a v e a c o n d i t i o n a l c h a r a c t e r . H o w e v e r , if we a s s u m e t h a t c o n d i t i o n (e) of the i n t r o d u c t i o n h o l d s , t h e n b y i n t r o d u c i n g the f u n c t i o n a l a (u) b y the f o r m u l a : f o r a l l u ~ 0 g ( a (u)u) = a , it is e a s y to p r o v e t h a t it is c o n t i n u o u s on E \ 0 and H - i n v a r i a n t (as in [3]). T h u s , the m a p p i n g u -~ a (u)u is c o n t i n u o u s f r o m E \ 0 into or

and c o m m u t e s with the a c t i o n of H, L e.,

H r . J L ~ o o if HrE \O~-c~. THEOt~EM 3.

•

If H r . J [ ~ c % then e i t h e r Hr*A N = ~ o r e l s e the s e t of n o n z e r o eq, N i s i n f i n i t e .

P r o o f . Indeed, l e t Hr*A N < ~ and Hr* {Jt'~ N { ~ N ~ 0}) -~cc f o r the s a k e of d e f i n i t e n e s s . T h e n H r { ~ N > 0} = :1: b y L e m m a 1. A s s u m e t h a t t h e r e a r e f i n i t e l y m a n y n o n z e r o l e v e l s Cq, N and l e t c 1 > . . . > c k be t h e s e l e v e l s . T h e n H r * { ~ N -> Ck} = ~ , f o r the g e n u s c a n c h a n g e o n l y on l e v e l s of the t y p e Cq, N. T h u s Hr{~ N > Ck} Hr*{~0 N _> Ckl - H r * A N , Ck = ~o e t c . A p p l y i n g t h i s a r g u m e n t k t i m e s , we g e t Hr{~0 N > c i ) = ~o, which is i m p o s s i b l e s i n c e cx = sup inf~n(,~) -~ sup q~N(u), i . e . , {r > c l } = ~ . T h e t h e o r e m is p r o v e d . A s t r o n g e r r e s u l t c a n be o b t a i n e d if E is a s s u m e d to be a B a n a c h s p a c e and c o n d i t i o n s (i), {j) a r e r e p l a c e d b y the P a l a i s - S m a l e c o n d i t i o n .

479

•

T H E O R E M 4. a s a l i m i t point.

U n d e r the above conditions, e i t h e r Hr*AN, 0 = .o o r e l s e the set of n o n z e r o Cq, N has z e r o

P r o o f . 1. H e r e we u s e the o b j e c t s f r o m the p r o o f of L e m m a 2 in Sec. 3. I n s t e a d of the r e s u l t (,) we c a n p r o v e the following: t h e r e e x i s t s a T > 0 such that f o r all u~{ep~>c--T} e i t h e r ~:~ (u) N {q~N~>c ~ ~} =/= ~ o r ~)~ (u)~ S(A+r 2e)=/= ~ ; we get this b y dividing:the n u m b e r T in (.) by two. In the definition of G1 we r e p l a c e the condition ~ ( u ) ~ { ( p ~ > c } b y ~ ( u ) ~ { ( p ~ > c - } - ? } . The s e t G c o n s t r u c t e d with this Gl r e t a i n s the p r o p e r t y ~)(~}~(7. We c a r r y out all the c o n s t r u c t i o n s of L e m m a 2 down to Go. It is c l e a r that the p r o p e r t i e s HrG0 = H r G and ~ - i (Go) U Go ~ Go a r e r e t a i n e d . In the s a m e w a y as in L e m m a 2, Go~{c--~}, i . e . , H r G = H r G o ~ Hr{cp N > c - T} -> Hr*{~ON >- c}. But H r G -< HrG1 + HrG~ -< I-Ir{~ N > c + T} + Hr*AN, c, whence

Hr{~p~.>c~'f} ~ H r * { q ~ c } - - H r * A . ~ .

o.

(7)

2. As in T h e o r e m 3, we a s s u m e Hr*{~p N >- 0} = ~o and Hr*AN, 0 < *~. T h e n I,ir{q N > 0}= 0% If c o is the + g r e a t e s t l o w e r bounded of the n o n z e r o Cq, N, then I,ir*{~pN -> Co} = 0o (cf. T h e o r e m 3). The set AN, Co has finite g e n u s b e c a u s e it is c o m p a c t ( P a l a i s - S m a l e condition). Thus, by Eq. (7) t h e r e e x i s t s a 3/ > 0 s u c h that I'Ir{cPN > Co + T} = 0% which c o n t r a d i c t s the definition of c 0. The t h e o r e m is p r o v e d . w

Eigenfunctions

of the

Pair

(f',

g')

o n A~

We r e m a r k f i r s t of all that if we s t r e n g t h e n the s m o o t h n e s s r e q u i r e m e n t s on the functionals f and g, when c o n s t r u c t i n g the m a p p i n g ~ ( u ) we c a n m a k e it i n c r e a s i n g in f and such that it does not leave d/~. This c a n be a c h i e v e d b y c h o o s i n g v e c t o r s rlu, along which f i n c r e a s e s , which lie in the tangent s p a c e to dt'~ at the point u. A f t e r the field r u is c o n s t r u c t e d , the shifts u ~ u + t r u a r e p r o j e c t e d onto d/~ by the m a p p i n g u ~ ~(u)u (of. Sec. 3). If we a s s u m e that conditions (i), (j) hold f o r the functional f on d/~, T h e o r e m s 1-3 c a r r y o v e r to this c a s e a u t o m a t i c a l l y . H o w e v e r , we will not i n c r e a s e the s m o o t h n e s s of the f u n c t i o a a l s f and g. A s s u m i n g that E is a B a n a c h s p a c e , that c o n d i t i o n s (i) and (j) a r e r e p l a c e d by the P a l a i s - Smale condition, and that conditions (d), (e) of the i n t r o d u c t i o n (hitherto unused) a r e added, we obtain the s a m e r e s u l t s in the l i m i t as N ~ r f r o m the t h e o r e m s of Sec. 3. We i n t r o d u c e the notation cp•, c~ ~ lim C p:t= , N ., {f >/>c}~ = {u:u ~ ~ ,

/(u)>/c}, and ~f = c}c~ a r e defined a n a l o -

g o u s l y , etc.; A is the s e t of solutions of Eq. (1) on ~ ' : , Ag = A N {f ----c}~, etc.; ~ p the c l a s s of c o m p a c t s e t s on ~'~ with genus at l e a s t p (just as in Sec. 3, we a s s u m e all the c l a s s e s 3ir a p p e a r i n g below a r e nonempty); c+ = sup inf ](u), c~- = inf sup /(h); A~.~p = lira U AN,~p. The l a s t l i m i t is n o n e m p t y since (J AN.~p is K ~ .~p

"a~

- - ~- - ~T2CZ u ~ l':

c o m p a c t b y the P a l a i s

-

N ~ oo N :~/ N ,

Smale condition.

N >.~N o

We o n l y c o n s i d e r the d o m a i n {~PN -> 0} and o m i t the + sign in c + p,

N-

I t is c l e a r that {~- I> c}~{] >I c}~, and hence c o N >- Cv f o r all N, Cp, oo -> Cp. Let Cp, .o - Cp = e > 0. ~in?a1 f o r l a r g e N the s e t {r ~ / 0 } c J / r 1 2 1 , w h e r e I ~ ' U ~ I and l a~l - ~ t a r e s m a l l , the v a r i a t i o n in the f u n c t ' o n

[

N,

N]

u n d e r the m a p p i n g u --* a ( u ) u (cf. Sec. 3) of the d o m a i n {~N -> 0} into ~ :

g

.[ (g' (su), u) ds

is u n i f o r m l y s m a l l .

is u n i f o r m l y s m a l l .

Thus,

By condition (e) of the introduction, this m e a n s that In(u) - 11 is uni-

i

f o r m l y s m a l l , and

17

I

(/' (su) u> ds is u n i f o r m l y s m a l l by condition (d). We c o n c l u d e f r o m this that the c h a n g e

in f is u n i f o r m l y s m a l l u n d e r the m a p p i n g u ~ a (u)u. oo s u c h that f o r all N - N(6)

We state this r e s u l t ;

f o r any 6 > 0 t h e r e e x i s t s an N(6) <

I/(~(u)~)-/(~)l~6.

sup

Let Ni be such that ICp, N - Cp, ~o I -< e/4 for all N -> Ni and let K~-~ {cp~.>I Cp, N - ~/4} be a compact set with genus at least p. It is clear that inf / (u)~inf ~p(u) ~ cp.~ -- e/4 ~ Cp.~ -- e/2. If w e n o w choose 5 < e/2 and K N

9

a corresponding N(5) for N - N(6) the mapping u ~ a(u)u takes K N into a n e w c o m p a c t set Q lying in{f > Cp, ~o e}~, i.e., {f > ep}a, and having genus at least p, which is impossible. Thus, Cp, ~o = Cp. B y the Palais" S m a l e condition, the set A~o c

of points in AN, Cp c o n v e r g i n g to u ~ A...~p as N

480

- *

is not empty. W e find its points. Let {uN} be a sequence ~o.P B y the definition of AN, Cp, ( ~ N ( U N ) , v ) = 0.

The numbers 7tN = Nc~(g(uN) - a) = --~0. T h u s , b y condition (c), vv(f'(u)--kog'(u),v}~-O, i . e . , A~,c~Acp. We now c o n s i d e r the c a s e Cp = Cp+q, o r what is the s a m e , Cp, oo = Cp+q, Go. LEMMA 5. c - T}~--N----

F o r any c > 0 t h e r e e x i s t s a n u m b e r T > 0 not depending on N such that H r * A ~ c -> Hr{~N -> c + T} f o r all sufficiently l a r g e N.

P r o o f . Since by the P a l a i s - Smale condition the n u m b e r s /3 and d (cf. the p r o o f of L e m m a 2) can be c h o s e n independent of N (for l a r g e N), the n u m b e r T can be c h o s e n independent of N in r e s u l t (*) of L e m m a 2 also, f o r a given e. We r e f o r m u l a t e (*) in the s a m e way as in the proof of Eq. (7) by r e p l a c i n g S(AN, c, 2e) by the set S ( A + , c , 3e) containing S ( A ~ , c , 2e) f o r l a r g e N: t h e r e e x i s t s a T > 0 such that f o r all u ~ [ ~ . > c - - ~ } either ~e(u) r]{~>c+?}=#~ o r ~)~(u)NS(A*~,c, 3e)=/=~. It is c l e a r that T can be c h o s e n independent of N h e r e . T h e r e s t of the p r o o f c o n s i s t s of a r e p e t i t i o n of the p r o o f of Eq. (7) with the s e t S(A +, c' 3e) in place of S(A~, c, 2e). T H E O R E M 5. Under the a s s u m p t i o n s of the l a s t p a r a g r a p h , the t h e o r e m on c o a l e s c i n g c r i t i c a l points (in the t e r m i n o l o g y of [1]) holds f o r A~r ep: (cp = cp+q)=,- (Hr*A|162 q ~- 4~. P r o o f . F o r l a r g e N the set { r Cp + T} is m a p p e d by u --" a(u)u into {f > Cp + 6}a with s o m e 5 > 0. Thus, H r ~ l ~ > Cp + T} ~ Hr{f > Cp + 6}a < p b y the definition of Cp. On the o t h e r hand, Hr{(PN > c - T} "--" Hr{f>c~/}a > - p + q , i . e . , b y L e m m a 5 H r * A ~ Cp ->q + 1 w

Case

of Variable

a

We c o n s i d e r the q u e s t i o n of the b e h a v i o r of A ~ Cp when a v a r i e s . In this p a r a g r a p h the index a will be added to aft the notation: Cq, a , Cq, N, a , CN, ~, etc. T h i s notation is c l e a r , e, g., Cq, N, a 0 is the level Cq, N c a l c u l a t e d f o r the functional ~N, a 0 = f(u) - Nes(g(u) - no). The s a m e p r o p o s i t i o n s as in Seco 4 hold r e l a tive to f, g, E. As b e f o r e , we c o n s i d e r o n l y the d o m a i n s {~0N, a -> 0}. LEMMA 6.

Let~--a

0~0.

Thencp, N,a--Cp,

N, e0.

P r o o f . The p r o o f is by c o n t r a d i c t i o n . The set {Cp, N, a } is bounded f o r a c l o s e to a 0. L e t ~i "~ a0 be a s e q u e n c e such that Cp, N, a i - - Cp, N, a 0. It is e a s y to deduce that r cq(U) = ~N, a0(u) + ~(u, e0, a [ ) ( a i - a0), w h e r e the functional $(u, a0, a i ) is bounded in a b s o l u t e value.

e}
Let c o < Cp, N, a 0, e. g,, Cp, N, a0 - co = 3e > 0. By the definition of Cp, N we have Hr{~N, a i > Cp, N, a i + and i f [ is such that tCp, N , a i - c 0 t < e, t h e n H r { r ai>c 0+2e}cp,N,a0-

e} < p, i . e . , Hr{u : CN, a0 (u) + r by the definition of Cp, N, a 0"

a0, a i ) ( a i - a0) > Cp, N, a 0 - e} < p, which is i m p o s s i b l e f o r s m a l l (ai - ~0)

NOw let e 0 - C p , N, a0 = 3e > 0. I n a n a l o g ] r w i t h t h e p r e c e d t n g , Hr{~N, a i > Cp, N , a 1 - e} - p.~ Hr{~Ni, a i > c o - 2e} - p, Hr{~N, a i > Cp, N, a0 + e} - p, and Hr{u: ~N, a0(u) + r a0, a i ) ( ~ i - a0) > Cp, N,c% + e} - p, which is i m p o s s i b l e f o r s m a l l (ai - a0). The l e m m a is proved. LEMMA 7,

Let0

T h e n C p , N , a - - C p , a as N - - ~ ,

u n i f o r m l y in

a~[al b].

P r o o f . Since f o r N~ < N2 we have eNd, a (u) -> q~N2' a (U), it follows that { ~ , , , .~ c} ~ {WN.-,~:> c}, w h e n c e Cp, NI (~ ~ C p , N2, a > ~ 2 9 ThenI~,~--a], I ~z. ~ - - c ~ a r e u n i f o r m l y - C p , oo,a =Cp, a 9 L e t { ~ x . ~ 0 } ~ , s m a l l in ~ f o r l a r g e N (this follows f r o m the p r o p e r t i e s of the function c3). Thus (cf. Sec. 4) u n d e r the m a p p i n g u "-" c~(u)u the functional f(u) v a r i e s by little, u n i f o r m l y in ~ and u f o r l a r g e N. If we indicate this v a r i a t i o n by AN, c~, u f, then (by the above u n i f o r m i t y ) t h e r e e x i s t s a function AN f - - 0 as N - - ~ such that JAN~ a , u fl ~ ~N"f, Since in~ ](u)~c, wehave d = inf inf )~(u), {%,v.a >~c} u~(~x.~>--cp,lv,c~-~ } ](a(u) u)>~c~,N,~--5--ANfo B u t s i n c e d = u~(u). {~N,c~ ~cp,~v,~_.5} and the s e t ~ ( u ) . { r ANforcp, N,~-Cp,~ as r e q u i r e d .

->cp,N,c~-5}for5 > 0 has genus at l e a s t p, d ~ c p , ( ~ . ~ANf+5. S i n c e 5 is a r b i t r a r y , we o b t a i n c p , N , a - C p ,

Thus, Cp,~ >-Cp, N , a - 5 a = JCp, N , ~ - e p , ~ } -
481

C O R O L L A R Y . L e t n ~ n 0 r 0. T h e n Cp, a - - Cp, a0. Indeed, l e t e > 0 be a r b i t r a r y and N s u c h that f o r a l l a in s o m e s e g m e n t [ a 0 - 6 , n 0 - ' 6 ] we have ICp, N , a - C p , a l -< e. C h o o s e 5 1 ~ ( 0 , 6 ) such t h a t I Cp, N , a Cp, N, a 0 I < e h o l d s f o r the N c h o s e n f o r a l l a ~ [ a o - - 6 1 , n 0 + 6 1 ] . Then Icp, n - c p , a 0 1 - Ip, a - c p , N, a l + I cp, N , a - Cp, N, a01 + I c p , N , a 0 - Cp, a01 - 3e. T H E O R E M 6. Proof.

Assume thata

Since e is a r b i t r a r y ,

--a 0andtkatfora

the c o r o l l a r y is p r o v e d .

~ a 0 , Cp, a = c p + q , a .

T h e n c p , a 0 = C p + q , a0.

The p r o o f f o l l o w s at o n c e f r o m the a b o v e c o r o l l a r v .

T h e c o n s i d e r a t i o n of the o p p o s i t e c a s e is s o m e w h a t m o r e c o m p l e x . L e t a ~ a0, Cp, n0 = e p + l , a 0 = . 9 9 = Cp+q, a0, but a s s u m e t h a t f o r a c a 0 , Cp, a ~ c p + i , n - . . 9 -> c p + q , oz0, w i t h s o m e of t h e i n e q u a l i t i e s s t r i c t . P u t q

q

B~ = U Aoo c 'k ~, Bz~-.~= kUoA~.%+n~. h=0

. ' P"t-

Palais-Smale

'

It is c l e a r t h a t the p o i n t w i s e l i m i t lira B~ ~ A~o ~D,~o (this f o l l o w s f r o m the ~o~

c o n d i t i o n ) a l t h o u g h the s e t

~§

o

B~ need not c o n t a i n Aoo, Cp, a 0-

L e t the f u n c t i o n N(a) be s u c h t h a t lim BN(~),~ = lim B~. Such a f u n c t i o n N(n) i N ( a ) - - oo a s n ~ n0] c a n 65~(% e

65~C~ o

a l w a y s be c h o s e n s i n c e Aoo is the l i m i t of A N. We fix e > 0 and t a k e a n u m b e r 5 > 0 s u c h t h a t f o r a l l cr n 0 + 6) in the d o m a i n {Cp, 0% n - 7 - (PN(a), n -< Cp, 0% a + 7} \ S(Bn, e) w e h a v e infJ[rp'N(a) ' n (u)l[ = ~ > 0. H e r e 7 > 0 is a n u m b e r s u c h t h a t AN(~L[%_v,%+vl~ S(B~, 3/4e). Carr~cing out a p r o o f a n a l o g o u s to t h a t of L e m m a 5, +

t a k i n g B n in p l a c e of AN, c a p p e a r i n g in t h a t t e m m a , we o b t a i n

HrS(B~, e) >1 Hr{~.~-(~, ~ c ~ . . . . --~'l}--Hr{%~-(~l. ~ > c ~ . . . . +71}

(8)

f o r s o m e 71 > 0. We r e m a r k t h a t H r S ( B a , e) c a n n o t be r e p l a c e d b y H r * B n h e r e , s i n c e t h e r e m a y f a i l to e x i s t a s i n g l e n u m b e r e f o r a l l a s u c h t h a t HrSCB a , e) = H r * B a . If we t a k e n c l o s e enough to a 0, we c a n a c h i e v e the f o l l o w i n g : ]c~+~. . . . --cp§

~ J < ~,/4, [c, . . . . --cp; .,-~, = [ <

"fx/2:Jcp. . . . --cp+q. . . . ] < ~ t / 4 , s i n e e b y h y p o t h e s i s Cp+q, a and ep, a have the s a m e l i m i t a s n ~ a 0. T h e n i n e q u a l i t y (8) m a y be r e w r i t t e n a s : HrS (B=, e) /> Hr{r ~ >cp+~ .,-(~. =--~, j2}--Itr{r162 ~>c~..~-(~. ~+~d2} ~ > q + l . We o b t a i n the f o l l o w i n g r e s u l t :

if Cp, a0 = Cp+q, a0, then f o r a n y e > 0 t h e r e e x i s t s a 5 > 0 so s m a l l t h a t

for all a~(ao--6, CZo+6) we have HrS(Ba, e ) - > q + l ,

i . e . , HS ih~oA~'~p+~'~'e~ ~ q + l "

T h i s p r o v e s t h a t the s e t s Acp+k ' a have a s u f f i c i e n t " c a p a c i t y " f o r a c l o s e to n 0. LITERATURE 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

482

CITED

L . A . L y u s t e r n i k , " T h e t o p o l o g y of f u n c t i o n a l s p a c e s and the v a r i a t i o n a l c a l c u l u s in the l a r g e , " T r . Mat. I n s t . , A k a d . Nauk SSSR, 19 (1947). R.S. Palais, "The Lusternik-Schnirelman t h e o r y of B a n a c h m a n i f o l d s , " T o p o l o g y , 5, No. 2, 115-132 (1966). F . E . B r o w d e r , " I n f i n i t e d i m e n s i o n a l m a n i f o l d s and n o n l i n e a r e l l i p t i c e i g e a v a l u e p r o b l e m s , " Ann. Math., 82, No. 3, 4 5 9 - 5 7 7 (1965). F . E . B r o w d e r , " N o n l i n e a r e i g e n v a l u e p r o b l e m s and g r o u p i n v a r i a n c e , " in: F u n c t i o n a l A n a l y s i s and R e l a t e d F i e l d s , S p r i n g e r - V e r l a g , New Y o r k (1970), pp. 1 - 5 8 . S. F u c i k , I. N e c a s , I. Soucek, and V. Soucek, S p e c t r a l A n a l y s i s of N o n l i n e a r O p e r a t o r s , L e c t . Notes M a t h . , Vol. 346, S p r i n g e r - V e r l a g , New Y o r k (1973). R. P a l a i s and S. S m a l e , "A g e n e r a l i z e d M o r s e t h e o r y , " Bull. A m . Math. Soc., 7.0, No. 1, 165-171

(1964). M.A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon (1964). A. S. Shvarts, "Some estimates for the series of a topological space in the sense of Krasnosel'skii," Usp. Mat. Nauk, 1_22, No. 4, 209-214 (1957). Hsia, Optimization [Russian translation], Mir, Moscow (1971). A . P . Robertson and W. Robertson, Topological Vector Spaces, Cambridge Univ. Press (1973). S . G . Suvorov, "Eigenvalues of some nonlinear operators," in: Math. Physics [in l~ussian], Vol. 11, Naukova Dumka, Kiev (1972), pp. 148-156.

12. 13.

THE

M . M . Postnikov, Introduction to Morse T h e o r y [in Russianl, Nauka, Moscow (1971). M . M . Vainberg, Variational Method and Method of Monotone O p e r a t o r s in the T h e o r y of Nonlinear Equations, Halsted P r e s s (1974).

THEORY

SPACES

V.

OF

SELF-ADJOINT

WITH

A

HERMITIAN

A.

Shtraus

OPERATORS

IN

BANACH

FORM

UDC 513.88

1. This article is a continuation of [1, 2] (cf. also [3]). The t e r m i n o l o g y and well-known r e s u l t s used by us without explanation m a y be found inthe s u r v e y s [4, 5]. By a (~9, Q ) - s p a c e we mean a complex Banach space or, which is defined a continuous Hermitian biliaear f o r m Q(x, y). In this paper we c o n s i d e r only regular [2, 4] (f9, Q)-spaces, and this is understood in the sequel without specific mention. An o p e r a t o r er acting in a (~9, Q ) - s p a c e is called Q-self-adjoint (Q-s. a.) if its domain ~(5r is dense in ~, Q ( ~ x , x) : Q ( x , ~ x ) and the relations 5 r Q(.;Cx, y ) : Q ( x , .gy) ( x ~ D ( M ) , y ~ ) ( . ~ ) ) imply that ~ r It is well known that the s p e c t r u m of a J - s . a. o p e r a t o r is s y m m e t r i c with r e s p e c t to the real axis R. This p r o p e r t y is retained by Q - s . a. o p e r a t o r s in the case of a (regular) reflexive (t9, Q)-space. We show that the r e q u i r e m e n t of reflexivity for ~ is essential. Example. such that

~

As initial spaces we take the following: I1 the set of doubly infinite s c a l a r sequences ~= {t~}_~,

[lh] j oo. m the set of sequences of the f o r m , ~ = ( m h } - ~ , sup {[m~I}
a n d t h e set c o r m

such that for ~ = { h}-~ ~c0, lim c~ = 0. The norm is defined in the standard way on each of these (complex) spaces. We put ~ = {c~+v] ~-~ (v = 0, i .... ), where c , = j l , k = 2 ~, n = 0 , t .... ; 0, k :_2, - - 9. n ., and consider e' equal to c. I. s. /co, {~,}0 } (c. I. s. denotes the closed linear span).

~o ~

As is e a s i l y seen,

c.1.~. {co, {~,.h }.

Let U be the right shift o p e r a t o r in I1, i.e., ~(U)=I~, U~ = {lk-d2~r

It is c l e a r that e' ~s an invariant

subspace for the adjoint o p e r a t o r U*, but U#c':#c ". We put* ~=1~4-c', Q(x, y ) = ~ (~(~)7.(.,~ , ~(u>_(~)~ I~)+ z~ ~'h H~'k " 7 t'k Ht'h J (X-.~-~ !~l~1, y~--i(y)+~)) ~=-~

0 ) "~r = 0 U = - - ~I " Thent i~p(~r and the point - i lies in the residual s p e c t r u m of the Q-s. a. o p e r a t o r ~r d e s i r e d example.

This c o n s t r u c t s the

The following results are important for the sequel (el. [2, Proposition 3]): 1) A Q-monotonic Q-weakIy convergent sequence of o p e r a t o r s is strongly convergent. 2) If the Q - s . a. o p e r a t o r s ~r and 9Y are such that for any x ~ ) : l Q ( N x where the constant C depends only on the (t9, Q)-space.

, x)[<.%Q(sCx, x), then It~II-%
* Cf. the general example given in [8[ of a r e g u l a r (~, Q)-space, the symbol 4 denoting direct sum. p(a~) and o(a~) are the set of r e g u l a r points and the s p e c t r u m of s~, respectively. T r a n s l a t e d f r o m Sibirskii Matematicheskii Zhurnal, Vol. 19, No. 3, pp. 685-692, May-June, 1978. Original article submitted November 22, 1976.

0037-4466/78/1903- 0483507.50 9 1979 Plenum Publishing C o r p o r a t i o n

483