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C o m p u t i n g 2 8 , 4 3 - - 5 1 1 9 8 2)

omoutino

9 by Springer-Verlag1982

S p u r i o u s S o L u t i o n s f o r D i s c r e t e S u p e r l i n e a r

B o u n d a r y V a l u e P r o b l e m s

W . - J . B e y n a n d J . L o r e n z , K o n s t a n z

Received May 22, 1981

Abstract - - Zusammenfassung

Spurious Solutions for D iscrete Sup erlinear Bound ary Value Problem s. W e cons ider f in i t e d imens iona l

non l ine ar e igenva lue p rob lems o f t he type

A u = 2 F u

where A is a matr ix and

( F u ) i = f ( u i ) , i = 1 . . . . m .

These m ay be though t o f a s d i sc r e ti za tions o f a co r r espond ing bounda ry va lue p rob lem. W e show tha t

posit ive, sp ur ious solut ion bran ches o f the discrete eq uat ions (which have been observed in some cases in

[I , 7]) typically arise if f increases sufficiently stron g an d if A -1 has at least two positive c olu m ns of a

cer ta in type. W e t reat in m ore de tai l the cases f ( u ) = eu and f ( u ) = u ~ where also discrete bifurcation

diagram s are given.

K ey w o rd s . spur ious solut ions, d iscretizat ions, non l inea r e igenvalue prob lems, super l inear funct ions.

A M S S u b j e c t C l a s s i fi c a t io n :

34 B 15, 65 H 10, 65 L 10.

Zusgtzliche LiJsungen yon Diskretisierungen superlinearer Rand wertaufgaben . Es

werden end l i ch

d imens iona le , n i ch t l inea re E igenwer tp rob leme der Form A u = 2 F u mit e ine r M at r ix A und e inem Fe ld

( F u ) , = f ( u ) ,

i= 1 . . . . . rn bet rachtet . Diese k6 nn en als Diskret i s ierung eines e ntspreche nden

Ra ndw er tproblem s angesehen werden. W ir zeigen, dab diese diskreten Gleich ungen dan n zusfi tz t iehe,

posi t ive L6sungszweige (welche in [1, 7] heobac htet w urden) aufweisen, w en nfh inre ich end stark w~ichst

un d A -1 m indestens zwei posit ive Spal ten yon einem bes t imm ten Typ besi tz t. A usf i ihr licher werden die

F N i e

f ( u ) = e

~ u n d

f ( u ) = u ~

beh and elt , f t ir die auc h diskrete Verzweigungsdiagramme ang egeb en

werden.

1 .

I n t r o d u c t i o n

R e c e n t l y , v a r i o u s a u t h o r s h a v e o b s e r v e d t h a t d i s c r e t e a n a l o g u e s o f n o n l i n e a r

b o u n d a r y v a l u e p r o b l e m s m a y h a v e s p u r io u s s o l u t i o n s t h a t d o n o t c o n v e r g e t o a n y

o f t h e c o n t i n u o u s s o l u t i o n s ( cf. G a i n e s [ 4 ] , A l l g o w e r [ 1 ] , B o h l [2 -] , P e i t g e n , S a u p e ,

S c h m i t t [ 7 ] , D o e d e l , B e y n [ 3 ] ) .

L e t u s c o n s id e r , fo r e x a m p l e , t h e n o n l i n e a r e ig e n v a l u e p r o b l e m

- u = A f ( u ) i n [ 0 ,1 ] , 2 > 0 , u ( 0 ) = u ( 1 ) = 0

a n d a d i sc r e te a n a l o g u e o f t h e t y p e

A u = 2 F u , u ~ ,

2 > 0 ( 1 )

w h e r e A ~ L I N ] i s a n m x m - m a t r i x a n d F i s t h e d i a g o n a l f ie ld

F u )) i = f u i ) , i - - 1 . . . , m .

0 0 1 0 - 4 8 5 X / 8 2 / 0 0 2 8 / 0 0 4 3 / $ 0 1 . 8 0

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44 W .-J, Bey n and J. Loren z:

T h e n t h e r e a r e e s s e n ti a ll y t w o t y p e s o f s p u r i o u s s o l u t i o n b r a n c h e s f o r t h e d is c r e t e

e q u a t i o n ( t ) :

T y p e I :

S o l u t io n b r a n c h e s w h i c h o c c u r f o r l a r g e 2 a n d w h i c h t e n d t o s o l u t io n s o f F u = 0 a s

) , - , oo .

T y p e I I :

S o l u t i o n b r a n c h e s w h i c h o c c u r f o r s m a l l 2 a n d w h i c h t e n d t o i n f i n i t y ( i n t h e

m a x i m u m n o rm ) a s ) ~ 0 .

T y p e - I - s o l u t i o n s ar e w e ll u n d e r s t o o d a n d c a n b e a n a l y z e d b y m e a n s o f t h e re d u c e d

e q u a t i o n F u = 0 ,

w h i c h i s o b t a i n e d f r o m (1 ) i f w e d i v i d e b y 2 a n d l e t 2 ~ o o (c f. B o h l [ 2 ] , P e i t g e n ,

S a u p e , S c h m i t t [ 7 ] ) .

T y p e - I I - s o l u t i o n s a r e t y p i c a l f o r s u p e r l i n e a r f u n c t io n s f , b u t - a p a r t f r o m

n u m e r i c a l r e s u l t s - t h e i r e x i s t e n c e h a s o n l y b e e n p r o v e d i n s p e c i a l c a s e s ( s ee

A l l g o w e r [ 1 ] f o r t h e c a s e f ( u ) = u4, m = 3 a n d n o t e t h a t t h e u n b o u n d e d c o n t i n u a i n

P e i tg e n , S a u p e , S c h m i t t [ 7 , s e c t i o n 4 ] m a y a l l c o n s is t o f n u m e r i c a l l y r e l e v a n t

so lu t i ons ) .

T h e p u r p o s e o f t h is p a p e r is t o c l a ri fy t h e s i t u a t i o n f o r s p u r i o u s s o l u t io n s o f t y p e I I .

I n s e c ti o n 2 w e s h o w u n d e r a c e rt a in g r o w t h c o n d i t i o n o n f t h a t a n y p o s i ti v e c o l u m n

o f A - 1 w h i c h a t t a in s i ts m a x i m u m o n t h e d i a g o n a l l e a d s t o a p o s i ti v e b r a n c h o f t y p e -

I I - s o lu t i o n s . T h e s h a p e o f t h e s o l u t io n s o n t h is b r a n c h is g iv e n b y t h e r e s p e c t i v e

c o l u m n o f A - 1.

I n p a r t i c u l a r , t h e d i s c r e te G e l l a n d p r o b l e m ( G e l f a n d [ 5 ] ) w h e r e

2 - 1

- 1 2 - 1

A = ( m - .I - l ) 2 : ; ( 2 )

- I

, f ( u ) = e

2 - 1

- 1 2

h a s a t l e a s t m d i s t i n c t b r a n c h e s o f t h i s ty p e .

I f rn is o d d t h e n o n l y o n e o f t h e s e c o n t a i n s t h e s y m m e t r i c s o lu t i o n s w h i c h c o r r e s p o n d

t o s o l u ti o n s o f t h e b o u n d a r y v a l u e p r o b l e m . A ll t h e o t h e r b r a n c h e s a r e s p u r io u s .

M o r e o v e r - a s t h e n u m e r i c a l a n d t h e o r e t i c a l r e s u lt s o f s e c t io n 3 s h o w - t h e r e a r e

s t i l l f u r t h e r p o s i t i v e s p u r i o u s s o l u t i o n b r a n c h e s f o r t h i s e x a m p l e .

O u r g r o w t h c o n d i t io n o n f e x c lu d e s t h e im p o r t a n t c a s e

f ( u ) = u ~ , ~ _ > ,

w h i c h w i ll b e t r e a t e d i n s o m e d e t a i l i n s e c t i o n 4 .

I t i s s h o w n t h a t f o r ~ s u ff i c ie n t l y l a r g e , s a y ~ > ~1 > i, t h e s a m e s i t u a t i o n a s i n

s e c t i o n 2 p r e v a i l s .

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Spurious Solut ions for Discre te Supert inear Boundary Va lue Problems 45

N o t e h o w e v e r t h a t n o t a n y f ( u ) - u ~, a > 1 p r o d u c e s s p u r i o u s s o l u t i o n s . I t f o l l o w s

f r o m t h e r e s u lt s o f L o r e n z [ 6 ] t h a t i f A - ~ h a s o n l y p o s i t i v e e n t r ie s t h e r e c a n b e

c o m p u t e d a n u m b e r e o = % ( A - 1) > 1 f o r w h i c h t h e s y s t e m ( 1) w i t h f ( u ) = u~, [~ I < ~o

h a s e x a c t l y o n e p o s it i v e s o l u t io n b r a n c h . H e n c e , fo r a n y m a t r i x A o f t h is k i n d t h e r e is

a c r i t i c a l v a l u e e * , 1 < ~ o -< e * < e l < o % a t w h i c h s p u r i o u s s o l u t i o n s f o r t h e s y s t e m ( 1 )

w i t h f ( u ) = u ~* b e g i n t o e x i s t ( s e e F i g . 2 i n s e c t i o n 4 ).

2 S p u r i o u s S o l u t i o n s in t h e G e n e r a l C a s e

L e t H [J d e n o t e t h e m a x i m u m n o r m i n [~m .

I t w il l b e c o n v e n i e n t t o p a r a m e t r i z e s o l u t i o n b r a n c h e s o f e q u a t i o n (1 ) b y t h is n o r m ,

i . e . w e l o o k f o r s o l u t i o n s

( u , 2 ) = ( u ( r ) , 2 ( r ) ) E ~ + 1

o f th e s y s t e m

A u = 2 F u , IJ u I [ = r > - 0 . ( 3 )

L e t u s a s s u m e t h a t B = A - ~ e x is ts . T h e n o u r a n a l y s i s is b a s e d o n t h e f o l l o w i n g

r e f o r m u l a t i o n o f (3 )

v = f ( r ) - a F ( r B v ) , r >O , [IB y II = 1 . ( 4 )

I t i s e a s i l y v e r i f i e d t h a t t h e s o l u t i o n s ( u , 2 ) e 6 " + 1 o f ( 3 ) a n d ( v, # ) o f ( 4 ) a r e r e l a t e d b y

( u, 2 ) = ( r B v , r f ( r ) - I / ~ ) . ( 5)

A s w e w i ll s h o w , e q u a t i o n ( 4) l e a d s t o a r e a s o n a b l e r e d u c e d p r o b l e m i f w e l e t r - -, o o.

T h e a s s u m p t i o n s i n t h e f o l lo w i n g t h e o r e m a r e e a s il y se e n t o b e f u lf il le d b y e x a m p l e

(2 ) (c f . s ec t ion 3 ) .

T h e o r e m 1 :

( i) L et f : (0 , oo)--*(0, oo) be a C l- fun ct ion such that

r

s u p f ( z r ) ~ 0 , s u p f ' (z r ) ~ 0

f( r ) o<~<_t 7 (/ ' f o<~_<t

as r -+ co ho lds for every

t ~ (0, 1).

(ii) L e t B = A - 1 e x i s t a n d l et B j = ( B l j , . . . ,

B, , j ) r

be a co lumn o f B sa t i s f y ing

Bj j> Bi j> O ViA~j.

Then there ex is ts a co nt inous ly d([. ]brentiable branch o f po s i t ive solut ions (u

(r), 2 (r)),

r 2 ro of the sy s tem

(3 )

which satisf ies

r -1 u ( r ) ~ B ~ ~ B j, 2 ( r ) r - l f ( r ) ~ B ~ 1 a s

r ~ o 9 ( 6)

P r oo f :

L e t v ~ = B ~ 1 eJ, w h e r e d i s t h e j - t h u n i t v e c t o r i n N '~ . T h e n f r o m a s s u m p t i o n (ii) w e

h a v e a n e > 0 s u c h t h a t

1 > ( B y ) i > 0

V i~ : j, v e V = { v e ~ : II v - v ~

I I -<g}. (7)

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46 W . - J . B e yn a nd J . Lo re n z :

W e w i l l n o w a p p l y t h e i m p l i c i t f u n c t i o n t h e o r e m t o t h e e q u a t i o n

T ( v , s ) = O , ( v , s ) e V x

w h e r e T : V x R--* ~ " i s d e f i n e d b y

I v j f ( ( B v ) ~ l s [ - ~ ) f ( I s [ - ~ ) - l - v i , i+ -j , s O

T i v , s )- = - t - v i , i j , s = 0

[ ( B v ) j - 1 , i = j .

B y a n e l e m e n t a r y d i s c u s s io n u s i n g (7 ) a n d a s s u m p t i o n (i) w e o b t a i n

0 T

T , n - - - ~ C ( V x N , N ' ) , k = l . . . . , m

C/ k

a s w e t l a s

H e n c e

i - - cSik i =~j

i (vO, 0) = (k = , . . . , m ) .

O V k ~. B j k , i = j

[ 6 T 0 0 ) )

de t ~ , ~ - v ( v , , = ( _~1) m-1 B # # 0 ( 8)

a n d w e f i n d a c o n t i n u o u s s o l u t i o n b r a n c h g ( s) s V , [ s ] < 6 s u c h t h a t 6 ( 0 ) = v ~ S i n c e

0 T

0 s

t h i s b r a n c h i s a l s o 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 o r s # 0.

N o w l e t u s d e f i n e

v ( r ) = f f ( r - ~ ) , # ( r ) = v j ( r ) i f r > 6 - 1 ,

th e n (v ( r) ,/1

r ) ) i s

a s o l u t i o n o f (4 ) - n o t e t h a t

( B v ( r ) ) j = t > ( B v ( r ) ) > 0 V i : j

a n d

v j ( r) = ( r) = ( r ) f ( r ) - i f ( r ( B v

(r))j).

M o r e o v e r

v ( r ) ~ v ~ ~ (r ) ~ B ~ 1 a s r - , o e

w h i c h p r o v e s ( 6) i f ( u (r ), 2 ( r) ) a r e d e f i n e d v i a t h e r e l a t i o n ( 5).

3 . T h e C a s e f u ) = e ~

A s a n a p p l i c a t io n o f t h e o r e m l e t u s c o n s i d e r th e d is c re t e G e l ' la n d p r o N e m (3 )

w h e r e A a n d f a r e d e f i n e d b y ( 2).

I n t h is c a s e B = A - 1 is g i v e n b y

[ i ( m + 1 - j ) , i _ < j

B i j = ( m +

t ) - 3 4 ( j ( m +

l - - i ) , i > _ j .

(9)

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Spur ious Solut ion s for Discrete Supe r l inear Bo und ary Vaiue Problem s 47

A s s u m p t i o n (i) o f t h e o r e m 1 is o b v i o u s l y sa t is f ie d a n d a s s u m p t i o n (ii) h o l d s f o r e a c h

c o l u m n o f B .

H e n c e t h e r e a r e a t l e a s t m d i s t i n c t p o s i t i v e s o l u t i o n b r a n c h e s

(u~(r),2J(r)), r>_ ro , j= 1 , . . . ,m

o f th e d i s c r e te G e l ' f a n d p r o b l e m ( 3 ), (2 ) a n d t h e s e s a t is f y th e a s y m p t o t i c r e l a t io n s

i - i i <_j (m + 1)3

R7

r )

{(m + l - i ) ( m + l - j ) -~ , i > j ' 2 J ( r ) ~ r e - j (m + 1 - j )

I n t h e c a se m = 9 w e h a v e c o m p u t e d t h e se b r a n c h e s n u m e r i c a l l y a n d o b t a i n e d t h e

p ic ture a s g iven in F ig . 1 .

o /

2 0 s p u r b u s b r a n c h e s

O

J J P I _ _ I ~

0 d O a O ~ O O O 8 0 0 0 x e r

Fig. i

T h e n u m b e r s j = 1 . . . . 5 in d i c a te t h e a b o v e b r a n c h es . T h e b r a n c h 5 c o n t a i n s t h e

s y m m e t r i c s o l u t i o n s ( i.e . u i = u ~ o - i , i = 1 , . . . , 9 ) w h i c h c o r r e s p o n d t o t h e s o l u t i o n s o f

t h e b o u n d a r y v a l u e p r o b le m . T h e s p u r i o u s b r a n c h e s w i t h i n d ic e s j = 6 . .. . . 9 a r e

r e l a t e d t o t h o s e w i t h j = I . . . . , 4 b y

u{(r)=ul~ i = 1 . . . .. 9, j = 6 , . . . , 9 .

T h e r e f o r e t h e b r a n c h e s j a n d 1 0 - j c o a l e s c e i n F ig . 1.

N o t e t h a t t h e s p u r i o u s b r a n c h e s j = l , .. ., 4 a r e n o t c o n n e c t e d w i t h e a c h o t h e r b u t

r a t h e r b e n d b a c k to i n f i n it y c r e a t i n g a n o t h e r t y p e o f s p u r i o u s s o l u t io n s ( d e n o t e d

bye

T h e s p u r i o u s s o l u t i o n s o n t h e f - b r a n c h e s s ti ll a t t a i n t h e i r m a x i m u m i n t h e j - t h

c o m p o n e n t . T h e i r s h a p e s u g g e s t s t h a t t h e y c a n b e r e p r e s e n t e d a s y m p t o t i c a l l y a s a

l i n e a r c o m b i n a t i o n o f t w o c o l u m n s o f B = A - 2 . T h i s o b s e r v a t i o n is m a d e p r e c i s e i n

t h e f o l l o w i n g t h e o r e m .

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48 W .-J. Bey n and J. Loren z:

T h e o r e m :

C o n s i d e r t h e s y s t e m (3) w h e r e f ( u ) = e ~. S u p p o s e t h a t t h e m a t r i x B = A - J e x i s t s a n d

h a s t w o c o l u m n s

B j - - ( B l j, . .. , B ~ ) T , B k = ( B l k , . .. , Bra g)T

w i t h t h e f o l l o w i n g p r o p e r t ie s

O < B i k < B k k V i ~ k , O < B i j < B j j Y i~ -.j, ( 1 0 a )

B~k < _ Bjg V i # k

or B ij<_Bkj V i~ j

(1 0 b )

e z- B ~ j < B ~ I - B j ~ l i 0 c

T h e n t h e r e e x i s t s a c o n t i n u o u s l y d i f fe r e n t i a b le b r a n c h o f p o s i t i v e s o l u t i o n s ( u

(r), 2 (r)),

r >_r o o f t h e e q u a t i o n s (3) which sa t i s f i e s

r -1 u ( r ) ~ e B i + f l B k , 2 ( r ) r - l e r ~ a s r ~ o o

w h e r e

~ = A - ' ( B k k - B j k ) , f l = A - 1 ( B j j -B k j ) , A = B j j B k ~ - B j k B , j . ( 1 )

P r o o f :

W e w i l l o n l y g i v e t h e m a i n s t e p s i n th e p ro o f a n d l e a v e t h e d e t a i l s ~o t h e r e a d e r .

T h e s y s t e m ( 4 ) m a y n o w b e w r i t t e n a s

v ~ = p e x p ( r ( ( B v ) ~ - 1)), i = 1 .. .. m, ii B v II = 1. (12)

L e t u s in t r o d u c e t h e o p e r a t o r P : ~ m + l x N - - ~ " b y

P i ( w , s ) = { w l, i4 =k e N d + l ,

w k , w , , + l s i = k ' w s 6 ~

a n d t h e o p e r a t o r T : R ~ + I • ~ E ~ + t b y

~- wj ex p(] s [-~ [ ( B V ( w , s ) ) ~ - 1]), ie {1 . . . , m}\{ j , k}

( B P ( w , s ) )j - - 1 , i = j

T / (w , s )= ( B P ( w , s ) ) k + w , , + t s _ l , i = k

W k - - W m 1 S - - W j e - w r n l~ i - - - m 1 .

T h e e x p o n e n t i a l t e r m i n t h i s d e f i n i t i o n is s e t t o z e ro i f s = 0 . I n s t e a d o f ( 12 ) w e

c o n s i d e r t h e e q u a t i o n

r ( w , s ) = 0 ( 1 3 )

i n a n e i g h b o u r h o o d o f

s = 0 , w = w ~ + f le k + y em + t G R + 1

w h e re e , f l > 0 a r e g i v e n b y (11 ) a n d y > 0 i s d e f i n e d b y

fl = o~e -~ . (14)

F r o m (1 1), (1 4) w e h a v e T ( w ~ a n d u s i n g ( 1 0 a , b ) w e f i n d a n e i g h b o u r h o o d

W = { (w , s ) e ~ ~ + x ~ : 1 w-w ~ I I + I s l < ~ }

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Spur ious So lu t ions fo r D is c re te Super l inea r Boun dary Va lue P rob lems 49

s uc h t ha t T e C 1 W , R m +l) a n d

ew,~+ a , (B P(w , s ) ) i~(O, 1) V i + j , k , ( w , s ) ~ W .

F i na l l y ,

d et ~ w W ~

a n d t h e i m p l i c it f u n c t io n t h e o r e m y i el d s a c o n t i n u o u s l y d i ff e r en t ia b l e b r a n c h o f

solu t ion s w s ), s ), I s l < 6 o f equ at i on 13) w hic h sa t i s fies w 0) = w~

N o w , a l l ou r a s s e r t ions f o l l ow f r om t he r e l a t ion 5 ) , be c a us e

v(19= P ( w ( r - l ) , r - ) , # ( r ) = w j ( r - l ) , r > M a x ~ , 6 -1 )

a r e s o l u t i ons o f t he s y s t e m 12) . [ ]

In th e spec ia l case where A an d B a re g iven by 2), 9 ) i t i s r ead i ly ve r i f i ed tha t ou r

a s s u m p t i o n s 1 0 a - c ) a r e sa ti sf ie d w i t h

m m

J < ~ - ,

k = j + l a nd j > ) - + l , k = j - 1 .

T he r e f o r e , t he o r e m 2 y ie l d s t he e x i st e nc e o f a t l e a s t m - 1 m odd ) , m - 2 m e ve n )

s p u r io u s s o l u t i o n b r a n c h e s d i s t in c t fr o m e a c h o t h e r a n d d i s ti n c t f r o m t h e b r a n c h e s

e s t ab l i sh e d b y t h e o r e m 1 . T h e c o r r e s p o n d i n g s o l u ti o n s a r e u n s y m m e t r i c .

4 . T h e C a s e f u ) = u s , ~ >-- 1

A s s u m pt i on i) o f t he o r e m 1 is obv i ous l y no t s a t is f ie d if

f ( u ) = u s, c~> 1.

H ow e ve r , in t h i s ca s e the s y s t e m 4 ) t a ke s t he s pe c i a l f o r m

v i = # ( B v ) ~ , i = 1 , .. ., m , I1 B y I1=1 15 )

w h i c h i s i nde pe n de n t o f r . T h e s y s t e m 15) c a n n ow be s o l ve d f o r v ,/2 ) w he r e e i s

c ons i de r e d a s a pa r a m e t e r .

T h e o r e m 3 :

Fo r ~ >_ 1 conside r the equ ation s

( A u) i = 2u~[ ( i =

1 .. .. m), ]1 u l[ = r 16)

and le t assumpt ion ( ii ) o f theorem 1 hold.

Then there e xis ts ~1 > 1 and for every ~ > ~1 a co nt inuou sly di f ferent iable pos i t ive

solut ion branch for 16):

u ( r ) = r u ~ , 2 r ) = r l- ~ 2 ~ , r > 0 .

He re u~ ~ ~" , 2~ ~ R are ind epende nt o f r an d sat is fy

u~--,B[j 1B j, 2 ~ - + B f j I a s ~ o o . 17)

P r oo f : A s i n t he p r oo f o f t he o r e m 1 l e t v~ B j~ ~ e a nd c hoos e e > 0 s uc h t h a t 7 )

holds .

4 Com puting 28 1

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5 0 W . - L B e y n a n d J . L o r e n z :

W e c o n s i d e r t h e e q u a t i o n

w h e r e T is n o w d e f in e d b y

T v , s ) = 0 , v .s )e x ~ ,

i 8 )

t v i - v j ( B v ) ~ I l s l , i j , s O

r i ( v , s ) = l vi, i ~ j , s = 0

(B v ) j - 1 , i= j .

W e h a v e T ~ C I ( V x ~ , R m) a n d a s i n 8 ) w e fi n d

d e t (v~ = - 1)~ - ~ B ; j + 0 .

H e n c e , e q u a t i o n 1 8) c a n b e s o l v e d b y a C L f u n c t i o n

v s ) e m , I s l < a , v O ) = P .

W e the n o b t a in v ~ - 1 ), ~ = v2 0 r 1 )), u > a - 1 a s so lu t i on s o f 15) and ou~ as se r t i on s

h o l d w i t h

T h e v e c t o r u ~,) ,~ ) o f t h e o r e m 3 is a s o l u t i o n o f

Au)~

= ~Zu L i = 1 , , . . , m ,

il

1 .

S o m e n u m e r i c a l b r a n c h e s f o r t h i s e q u a t i o n i n t h e c a s e

m = 9 , m a t r i x A a s in 2 )

a r e g i v e n in F ig . 2 . N o t e t h a t a n y s o l u t io n i n t h i s d i a g r a m g i v es a c o m p l e t e b r a n c h

f o r p r o b l e m 1 6).

j . 1 - 7

/#a ~- f .- / / t

L

F

a~ seur~aus\tt\~\op~ncke

t

* 0

regc lar

r,#nck #

O lO o~ ZO 30 z,O

F i g . 2

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S p u r i o u s S o l u t io n s f o r D i s c re t e S u p e r l i n e ar B o u n d a r y V a l u e P r o b l e m s 5 1

The numbersj = 1,..., 5 denote the branches with the asymptotic behaviour (17) and

the branch 5 contains the symmetric solutions. Finally, the additional solutions on

the f branches are quite similar to those for the discrete Gel land problem as

established by theorem 2.

e ferences

[1] Al l gowe r , E . : O n a d i s c re t i z a t ion o fy + .~ yk=0 . In : P roc . C onf . R oy . I r i sh Ac a d . (M i l le r , J . J. H . ,

e d . ), pp . 1 -1 5 . Ac a d e mi c P re s s 1975 .

[2] B oh l , E . : On t he b i fu rc a t i on d i a g ra m o f d i s c re te a na l ogue s fo r o rd i na ry b i fu rc a t i on p rob l e m s . M a t h .

Me t h . App l . S c i . 1 , 566 -571 (1979) .

[3] Doe de l , E . J . , B e yn , W . - J . : S t a b i l i ty a nd mu l t i p l i c it y o f so l u t i ons t o d i s c re t i z a ti ons o f non l i ne a r

o rd i na ry d i f f e re n ti a l e qua t i ons . S IAM J . S ci . S t a t . C omp . 2 , 107 - -120 ( t 981 ) .

[4] Ga i ne s , R . : D i f f e re nc e e qua t i ons a s soc i a t e d w i t h bou nda ry v a l ue p rob l e m s fo r s e c ond o rde r

n o n l i n e a r o r d i n a r y d i f f e re n t ia l e q u a t io n s . S I A M J . N u m . A n a l . 11 4 1 1 - 4 3 4 ( 1 9 7 4 ) .

[5] Ge l ' f a nd , I . M . : S om e p rob l e ms i n t he t he o ry o f qua s i l i ne a r e qua t i ons . A me r . M a t h . S oc . T ra ns l . 29

2 9 5 - 3 8 1 ( 1 9 6 3 ) .

[6] Lo re nz , J . : Ap p l i c a t i ons o f Hi l b e r t ' s p ro j e c ti ve me t r i c i n t he s t udy o f non l i ne a r e ige nva l ue p rob l e ms .

P a r t I I ( t o a ppe a r ) .

[7] P e i t ge n , H . -O. , S a upe , D . , S c hm i t t , K . : No n l i ne a r e ll i p ti c bou nda ry va l ue p rob l e m s ve r sus t he i r

f i n i te d i f f e renc e a pp rox i ma t i ons : num e r i c a l l y i r r e l e va n t so lu t i ons . J . r e i ne a nge w. M a t he m a t i k 322

7 4 - 1 1 7 ( 1 9 8 1 ) .

Dr . W . - J . B e yn

Dr . J . Lo re nz

F a ku t t f i t f f i r Ma t he ma t i k

U n i v e r s i t/ i t K o n s t a n z

P os t fa c h 55 60

D - 7 7 5 0 K o n s t a n z 1

F e d e r a l R e p u b l i c o f G e r m a n y

4*