UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Asymptotic properties of solutions of equations in Banach spaces. Schulzer, Michael 1959

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1959_A8 S25 A8.pdf [ 1.97MB ]
Metadata
JSON: 831-1.0080638.json
JSON-LD: 831-1.0080638-ld.json
RDF/XML (Pretty): 831-1.0080638-rdf.xml
RDF/JSON: 831-1.0080638-rdf.json
Turtle: 831-1.0080638-turtle.txt
N-Triples: 831-1.0080638-rdf-ntriples.txt
Original Record: 831-1.0080638-source.json
Full Text
831-1.0080638-fulltext.txt
Citation
831-1.0080638.ris

Full Text

ASYMPTOTIC PROPERTIES OF SOLUTIONS OF  EQUATIONS IN BANACH SPACES  by MICHAEL SCHULZER B.A., U n i v e r s i t y  A  of British  Columbia,  1958  THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS ' in  t h e Department of MATHEMATICS  We a c c e p t required  THE  this  t h e s i s as conforming  to the  standard.  UNIVERSITY OF BRITISH COLUMBIA September,  1959  I  ABSTRACT  Certain the  equation  properties  Pu  =  v  It w i l l  scribed  element  of  defined  on  the  mapping, which val  of  and  a  establishing  the  u(?0  Under  asymptotic  solutions  and  that  they  determined  those  of  P  and  types  of  spaces,  functions  and  interval.  by  and  will  consisting  a  a real  contraction v a r i a b l e A.  positive  be  inter-  proved, of  the  and  v  possess  i t will  be  P  they  are  asymptotic  a recursive  obtained  spaces,  Euclidean  of  pre-  shown  that  asymptotically expansions  process  from  v.  results  Banach  that  possess  be  The  that  exist,  may  and  uniqueness  a s A, — ^ 0,  which  is a  = v(A.) .  the hypothesis  expansions  space  on  a theorem and  be  v  the half-open  o f P(A,)u(?0  asymptotic  unique,  i n the  existence  of  i s a transformation  v depend  over  Then  Q  P  u  will  that  transformation  and  values  space  assumed  subset  P  0 < A, < X m  be  space,  linear  that  assumes  solution  the  a closed  sum  solution  i n a Banach  investigated.  of  of the  spaces  such of  continuous  will as  be  applied  to p a r t i c u l a r  finite-dimensional  Lebesgue-square-summable functions  over  a  closed  In p r e s e n t i n g the  this thesis  i n partial fulfilment of  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  of B r i t i s h Columbia, I agree that it  freely  agree that for  available  the Library  f o r r e f e r e n c e and s t u d y .  I further  permission f o r extensive copying of t h i s  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my  D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . that  s h a l l make  copying or p u b l i c a t i o n  I t i s understood  of this thesis  for financial  g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  Department o f  M ^Ue-wvflXt c $  The U n i v e r s i t y o f B r i t i s h Vancouver 8 , Canada.  Columbia,  permission.  TABLE OF CONTENTS  Page No. 1.  INTRODUCTION  1  2.  BANACH SPACES  4  3.  ASYMPTOTICALLY CONVERGENT  4.  ASYMPTOTIC SUMS AND EXPANSIONS  5.  TRANSFORMATIONS  6.  SEQUENCES AND SERIES OF TRANSFORMATIONS  15  7.  EQUATIONS IN BANACH SPACES AND THEIR SOLUTIONS  17  8.  ASYMPTOTIC SOLUTIONS OF EQUATIONS AND THEIR  IN BANACH  SEQUENCES  SPACES  5 8 13  REPRESENTATIONS  22  APPLICATIONS TO REAL EQUATIONS  28  10.  APPLICATIONS TO SYSTEMS OF NUMERICAL EQUATIONS  31  11.  APPLICATIONS TO INTEGRAL EQUATIONS  33  12.  REFERENCES  39  9.  ACKNOWLEDGEMENTS  My without  whose  friendly thesis on and  those I  Research  goes  to Dr. Charles  invaluable  assistance  would  theory  during  gratitude  never  were  guidance  in this have  based  o n some  Council  a l l o f my  also  like  of Canada graduate  Swanson,  continued  o f 1959  materialized.  on a p p l i c a t i o n s should  summer  and  A.  The  of h i s previous  this sections work,  followed  his  t o thank  the National  f o rtheir studies.  suggestions.  financial  support  -  1.  INTRODUCTION The  problem  under  existence  o f an e l e m e n t  d o m a i n X)  i n a Banach  Pu  = v, u n d e r  element on  c o n s i d e r a t i o n concerns u  i n some  space,  and P  fixed,  satisfying  the assumptions  i n t h e space  that  v  the  closed  the  equation  i s a prescribed  i s a transformation  defined  .25 . This  ively  type  of problem  i n the past [17],  McFarland references is  1 -  be  of p a r t i c u l a r  blems  integral  found  when  i t may  [5]/  usual  method  extens-  [13]-  Further  The e q u a t i o n  be r e m a r k e d equations  many  give  rise  of e s t a b l i s h i n g  sequence  of elements  by  means  process,  and showing  [14]  iterative  i n ~J} s a t i s f y i n g obtains  Newton s T  McFarland  such  method [17]  an  to  Banach  the existence  d e f i n i n g an a p p r o p r i a t e  u  pro-  [8] •  f o r a general  converges  integral  that  of  sequence  Pu = v  i n t e r p r e t e d a s an  a solution  o f some  [14],  as K a n t o r o v i c h  of  this  u  discussed  and Kaazik  to differential  equations  The  authors  i n [14].  interest  In t u r n ,  related  such  B a r t l e [2]  may  equation.  by  has been  i n t h e Banach  the equation iterative  for solving generalizes  space  norm  Pu = v .  process  real  u^ i n that  t o an  element  Kantorovich  by g e n e r a l i z i n g  equations,  the sequence  consists  while  defined  b y an  - 2 ~  i n f i n i t e continued  fraction.  We  s h a l l employ a  r e c u r s i v e p r o c e s s based on the f a m i l i a r Neumann sequence of s u c c e s s i v e  Liouville-  approximations [ 1 2 ] ,  [15], [16]. We  s h a l l suppose t h a t P i s the sum  transformation  [18]  and t h a t P and  v  and  a c o n t r a c t i o n mapping [15]  A  over a h a l f - o p e n  positive  an e x i s t e n c e  and uniqueness theorem  for  the s o l u t i o n u (A,) of P(A,)u(A.) = v(A.) w i l l be  The  d i s c u s s i o n i s modeled a f t e r t h a t of van der  for  D  : 0 < A, < A, . — o  In s e c t i o n 7,  [ 1 9 ] , who  on  vary over s u i t a b l e s e t s as a r e a l  v a r i a b l e A. assumes v a l u e s interval  of a l i n e a r  has  established a s i m i l a r existence  r e a l and complex e q u a t i o n s (see s e c t i o n  proved. Corput  theorem 9)«  In s e c t i o n 8, the concept of an " a s y m p t o t i c s o l u t i o n " w i l l be i n t r o d u c e d .  I t w i l l be shown t h a t , under  s u i t a b l e h y p o t h e s e s , a s y m p t o t i c s o l u t i o n s e x i s t , and the set of a l l such s o l u t i o n s i s c o n t a i n e d  i n an  that  equivalence  c l a s s of a s y m p t o t i c a l l y e q u a l mappings from the i n t e r v a l i n t o the domain ID w i l l be o b t a i n e d  .  Moreover, a r e p r e s e n t a t i o n  f o r any  theorem  typical solution in this class:  under the assumption t h a t the p r e s c r i b e d q u a n t i t i e s P v  have a s y m p t o t i c e x p a n s i o n s , the s o l u t i o n  u  also  and  J\  -  possesses by  a  such  recursive By  spaces  our  can  exist.  asymptotic  in  sections It  a  i f the  process  found  This  may  be  real  noted  situation,  special spaces 9),  the  spaces val  and  been  u(A.)  v.  solutions ordinary the  use  in  Banach  solutions of  the  theory  is defined  and  discussed  of  9-H  continuous  a  f  Banach will  or  A,  i s in a  variety  arrived  element  and  of  limit  investigated  T2~space  (Hausdorff  the  functions  of  applications  to  will  be  presented.  The  space  of  real  (section  Euclidean  square-summable  has  an  be  complex.  spaces be  by  neighbourhoods E r d e l y i £9]  i n which  n-dimensional  Lebesgue.  and  i s replaced  i n which  is real  considered  of  P  derivable  s i m i l a r r e s u l t s can  A,  defined.  sections  types  of  by  which  that  variable  such  In  when' no  of  is  3-6.  have  and  asymptotic  convergence,  latter  those  i s accomplished  points  space)  the  from  even  t o p o l o g i c a l space  a  -  expansion;  method,  be  of  at  an  3  space over  functions  a  numbers  (section closed, in the  10),  and  bounded  sense  of  the  inter-  - 4 -  2.  BANACH  SPACES  Definition elements (i)  2.1.  u, &  A  v , w,  is a  (a) S  (b)  linear  vector  i s an  additive  admits  of  (iii)  or complex),  (3)  (ap)u =  (4)  a(u +  e  l l l l > °#  (*>)  I |u  (c)  j|au||  u  a n d  + v| | <  with  u  =  |a|  in  u  u  in  IB  norm  Thus,  —^0  i f u =  0.  ||v | | .  any Cauchy  an  as n  f  u  - u n  c  n  l  norm  i s a sequence  I I — ^ 0 as n, m  of t o an of  —=  m' '  element —  sequence  i n t h e Banach  i f  , and l l u  exists  i f and o n l y  ||u||. space;  .  in3  1 1  element  the Banach  ll ll = 0  converges  of  n  every  a number  1 1  u -  + av.  | |u| | +  i s a complete  there  v £ !B , t h e n  that:  (a)  elements  i f u,  a , (3i,  + 0u  v) = a u  i s associated  element  scalars  a(3u)  a norraed s p a c e ;  elements  conditions:  !B  ( a + 3 ) u = au  such  abstract  group.  so t h a t ,  (2)  u,  then  abelian  a  there  the following  m u l t i p l i c a t i o n by  u  of  space;  (1)  iBis  i s a s e t !B  space  satisfying  (real  (ii)  Banach  ~ °  u  i n JB  SO  that  It  i s well-known  functions) sense In  of  3C.  and  9-11  the  spaces  (of square-summable  Lebesgue)  sections  that  are  Banach  applications  spaces  (of cont inuous  functions  i n the  [18], [ l l ] ,  [l],  to these  spaces w i l l  [20],  be  considered.  3-  ASYMPTOTICALLY L e t /\.  A.  be  is a positive  q  be  a  real  CONVERGENT the  positive  real  single-valued  SEQUENCES interval  number.  function  Let  q  Cp^(n  from  into  =  0,  the  1,  where 2,  ...)  positive  numbers.  Definition  3.1.  asymptotic  sequence  <ip  (i)  The  sequence  a s A, —>  [ ^ } is said <  >  n  eA  f o r a l l A,  (A,) = 1  (ii)  For Cp (A,) = A,  N  Let of  denote  as A , — >  , = o(Cp ) n+1 n  example,  N  sequence  <p  i s an {A, },  the  positive the  sequence  0  =  0,  1,  an  [<j? l  2,  o for a l l  n  a s y m p t o t i c sequence n  t o be  0 i f  ^ o  JA.  0 < A, < A, ,  be  n.  defined  by  a s A, — ^  0.  a  non-increasing  real  numbers,  and  f o r each  half-open  interval  0 <  A, <  A, .  n  let  - 6 -  Let Banach Let from  (u (A.)}  be  n  -A  the  Definition  3B  a  of  -A.  The  of  (1) for  each  (u)  is said  integer  sequence In  to  [(  u n  converge  i  n  n  <  is said  to  a  n  l  sequence so  converge  mapping [ Cp  },  and  that  >^ A £  n  n  n.  be  an  asymptotic  limit  of  the  ))• sequence  asymptotically  j|u 0O||  Q  A,—^U^CA,),  single-valued  a <jp (\),  p a r t i c u l a r , the  and  a  asymptotic  n  -A. .  f o r e a c h A, £  {(u^)j  exists  u (A.)|| <  in a  n.  sequence  , an  elements  mapping:  p o s i t i v e numbers -  (2)  IB  of  defined  n  i f there  into  o  sequence  u (?0  , f o r each  3.2.  asymptotically (u)  sequence  single-valued  into  o  a  «B , w i t h  space  (u )  be  n  ^ ^ n ^ ) '  to e  {(u^)]  to  zero i f  A»  f o r each  n  f o r p o s i t i v e numbers  is said  a  n  . n  The der  Corput We  sequence \  £  A . q  above  terminology follows  that  used  by  van  [19]. may  need The  note not  here  that  converge  fact  that  an  asymptotically  i n the j|u(A.) -  usual  sense  u (\)j| n  <  a  convergent for  any  <J?  00  n  -  does  not imply  that  {(u )]  t h e sequence  numbers,  defined  to  zero,  as c a n be e a s i l y  in  the ordinary  de  Bruijn  in  by p a r t s ,  not converge  [9] •  a given  a n d shows  real  Again,  integral  by  that the  asymptotically  but  diverges  ( u ) a n d ( v ) o n Jf\  Two m a p p i n g s equal  i f , f o reach  are said  n, t h e r e  v and a p o s i t i v e i n t e r v a l 'n  exists  A  a  so t h a t n  I |u(A.) - v(A.) | | < Y < P ( ? 0 , A. e \ * n  n  {(u )} n  be an a s y m p t o t i c a l l y  ( u ) b e an a s y m p t o t i c another  such  equal,  since,  ||u(Aj  - v(A.)|| < —  if  evaluates  of  asymptotically  but does  0[19],  f o r a n y A. >  For  sense.  3.3.  number  Let  for  verified,  A..  space  converges  n  converges  be a s y m p t o t i c a l l y  (3)  be  sense  f o r any  i n t h e Banach  = nlA. ,  n  integration  the ordinary  positive  Let  by u (A.)  sequence  Definition to  {(u^)j  £ 7 , PP« 1 2 - 1 4 ]  resulting  converges  n  example,  successive  7 -  a  limit  limit.  f o revery <  Then  of this  convergent sequence.  sequence. L e t(v)  (u) and (v) a r e a s y m p t o t i c a l l y  n,  ||u(A.) - u ( \ ) | | n  • i|v(?0 -  u 00|| n  (p (AO + 0 to (A.) = y  n ^ n  A, £ "^"n"  n 'n  Conversely,  ( u ) i s an a s y m p t o t i c  '  <p (A.) 'n^n  i t can be e a s i l y  limit  verified  f o rthe a s y m p t o t i c a l l y  that, convergent  - 8 -  sequence (u),  ((u )},-  then  (v) i s another  Hence  with  respect  class  say that  )}  4.  ASYMPTOTIC  SUMS AND  ( u ) be  1,  ...  2,  Definition  4.1.  expansion  ^  equal to  of the  limits  sequence.  o f an  i s characterized equal  by a  mappings  f jp }' t  n  limit  of the sequence  unique.  EXPANSIONS  a mapping  a single-valued  n=0,  limit  of asymptotically  the asymptotic  i s asymptotically  Let  sequence  to the sequence  ((u  be  convergent  equivalence  We  asymptotic  the s e t of a l l asymptotic  asymptotically unique  and i f (v) i s a s y m p t o t i c a l l y  n  mapping  o n -A.  defined A  from  q  q  . ilB  into  Let  (  u n  )  , f o r each  . (u) i s s a i d  oo  ^ n^' U  i  f  i  s  t o have a  n  a s  y P" m  the t o  ' l t  asymptotic c  limit  of  n =0 the  sequence  N-1 [(  "V  u ) } , N = 1, n  2,  3#  •••  •  n=0 (u)  —  > Thus,  (u  n  J .  (u)  —  21  ( ) u  n  i  f  In t h i s  case  we  write,  -  9 -  N-l ||u(A.)  (1)  21  -  u  ^ ) ! l  n  ^  a  N^N  ' '  ( A  )  ^  £  N '  A  n=0 where We ^""(u  >  N  also  (u).  the  limit  follows  From  to find  criteria  F o r example,  of real a^  convergent  are,  series  uniqueness  u n  of  sequence, i t  sum i s a s y m p t o t i c a l l y  unique.  f o rthe asymptotic  ) # i . e . f o rthe existence  In t h e case  of o r d i n a r y  at times,  quite  i t i s well-known  o r complex  converges  the formal  of asymptotic  (  sum.  ... .  and has t h e asymptotic  a criterion  of a series  corresponding  that  asymptotically,  the property  an a s y m p t o t i c  space  case  the asymptotic  wish  convergence  verify.  a n d N = 1, 2, 3,  o f an a s y m p t o t i c a l l y  that We  0  say i n t h i s  ) converges  sum  of  a  numbers,  convergence,  difficult  that,  to  i n t h e Banach  the series  absolutely i f  n=l A  n  V I an 'I n - » c o lim  v  The  ,  a = k <  following  used  theorem  to determine  necessary  1, o r i f shows  asymptotic  and s u f f i c i e n t  to  possess  an a s y m p t o t i c  of  the sequence  f(u )} n  lim  n-^^  that  1  I*  +  a.  1  = k < 1.  n  a simpler  convergence.  condition  criterion In f a c t ,  f o rthe s e r i e s  sum i s t h e a s y m p t o t i c  to zero.  Similar  may b e a /  (u )  convergence  results  have  been  - 10 -  obtained van  by  Borel  der Corput  Theorem  The  sum  series  asymptotically  Proof.  L e t ( u ) be  positive  (2)  ^  i f and o n l y  verges  u(A.) e !B  .  number  to  ^ n ^ °^ U  [ 9 ] and  sum N,  and a p o s i t i v e  ]T n=0  ||u(\)  ^  PPi qs n  h a s an  {(u^)}  con-  n  there  interval  M  n  u (A,)||  of  ^ n^' U  exists -A  ^,  w:  "-*  n  a so  that  < a 0p (A.)  u (A.)||  N -  a  zero.  f o r any f i x e d  ll'uOO -  m  i f the sequence  the asymptotic  Then,  and  (3)  [4], Erdelyi  [ 19] •  4.2.  asymptotic  [ 3 ] , Carleman  < a  N  N + 1  cp  N + 1  ( A . ) < a ap (A.) N  N  n =0 for  \  e  A  . N  Then,  l|« 0O|| N  from  < I!  (2) a n d  *00 "  (3);,hy  21 " n n=0  < 2a Q> (A.) f o r X e A . N  Hence,  N  { ( u )}  converges  0 0  Minkowski's  "  UU) 11 +  "  inequality,  Zr n°°ll u  n=0  r  asymptotically  to  zero.  -  Conversely, {(u )}.  Then  n  11 -  suppose  there  zero  exist  i s an a s y m p t o t i c  positive  n u m b e r s (X^ a n d a n o n f  increasing  sequence  f^» } w h i c h  lA^},  so t h a t ,  f o r any X e A . ^ ,  vals (4)  ll»  n +1  is  satisfied.  (n  = 0, 1, 2,  0O||  <?JX)  -4  < a  + n  1  cp  n+  1  the step-function  1  1  m+n  (X) - 0  n  < X < X n - o f o r 0 < A, < X .  1  n  (A.) I —  u m+n  I n d e e d , when ' &  O"  f o r A.  f o r any p o s i t i v e  m+n  N  n  ...) a s f o l l o w s :  <X . (X) | j  (5)  of inter-  the inequality  n  v  Then,  a sequence  <i a 9 00  (A.)  Define  (0 I  defines  n  limit of  1  X  1  i n t e g e r s m, n ,  | < 2~  m  a Cp (x),  X e  n 'n  < X < X , t h e n , by m+n— — n' ' 3  a n d (5) i s s a t i s f i e d .  When, '  n  A.  definition, ' on t h e o t h e r  hand, 0 < X < X , t h e n , by c o n s t r u c t i o n , ' m+n ' '  ^ OO I N ^ "(*•)-I Im+n < ^^m +9 ' m+n n a  m+n  v  11  We  (6)  v  shall  1  show  that  . (A.) < 2 ~ a Cp (A.), ' — n ^ n ' m  v  v  (u), given  by t h e s e r i e s  uOO = 21 ^) c^) cr  is  j  an a s y m p t o t i c  u  j  sum o f  /  (u) . n  -  If follows  A,  tends  from  (4)  12  -  to a positive  limit  A,  as  n  —  i  t  that  i  j?"  CT.(A.)u,(A.)}  j=0  * is  a Cauchy  o f SB  completeness norm  given  by  asymptotic  denote cT  i+ i  situation  the  (A.)  I t may  be  the Banach  verified  hence,  =  •••  =  , whenever '  We  verify  A .  °-  £  X^  that  , and  i s that A  Q  ,  that  (u)  i s an  in which  l e t i = i(A,)  > X.  Hence  A. £ A  Then  the series  represents  an  (6) element  . o  that  Then  N  interest  integer with  u(A.) i n !B  and,  by  converges i n the  F o r any X  f o r a n y A, £ A  A. £  Then,  ( l ) and  real  .  0 0  largest  =^+2^)  of  terminates  Let  .  (6), satisfies  as n — >  =*" 0  sequence  .  sum.  The X  , this  e l e m e n t u(A.) i n »B  t o an  u(A.),  for 0 < X < X  sequence,  u(A,), 0" (A.) o  given =  in (6), satisfies ( l ) .  <7(A.) =  ...  =  W  =  1,  by ( 5 ) ,  lluOO -  u . (A.) I I  2 1  <or (A.)||u (A.)|| N  N  j=0 <  2a 9 (A.),  This  N  N  A, £  completes the  A . N  proof.  ^  N + 1  (A.)||u  N + 1  (A.)||  +  - 13 -  5.  TRANSFORMATIONS  Definition defined valued  5.1.  Thus, Let  (D—^  map  P  X  A  i s said jD  of elements  u  f o r any  u £ T>  iB ) denote  into  .  i n tB  , i f  in  A  i s a  single-  into  , A u = w,  with  w £ B  .  the set of a l l transformations  ), depending  determined X,  on t h e v a r i a b l e  ^ e t (A) be a s i n g l e - v a l u e d  mapping  which  where A  from  Q  ( X > — ) .  into  We  define  Definition  the class  5.2.  Lip(P  . Cp )  in  , with  u ^ v, t h e r e  bounded  Lip(lD  The mapping  class  a  transformation  L e t A = A(A.) b e a u n i q u e l y  i n ( i) ~ ^ S  £ "^Q*  SPACES t o be a  domain  on a c l o s e d mapping  element  IM BANACH  positive  , Cp ) a s  (A) i s s a i d  i f ,f o r a l l pairs  function  exists  follows: t o be  i n the  of elements  a positive  Cp , d e f i n e d  u, v  number  f o r X £ -A.  a and , such  that  u, v £ It  15 follows  | |A Q O u  for X  -  that,  A(A,)v| I < a Cp(A.) | |  any two e l e m e n t s  £ A , Q  we  f o r (A) i n L i p ( J >  say that  u, v i n i ) .  u  -  , cp ) , we  v| | ,  X  I f a(p(A,) <  A(A,) i s a c o n t r a c t i o n  £  A  have ,  1 for  mapping:  a l l  -  Definition on  5.3.  A(A,)  , i f / f o r any | |A(?Ou -  with  A(7v.)u It  and  n'  A  U  has  ' ^^  that  any  a unique  e n  of  space,  and  A(A.)u  =  theorem  u  a  and  be  If the  point  the  A  o  usual  of  n  ^  and  the  forms  a  way  u  on  as  A  a  is a  Thus, T)  is a  element of  u this  is  where  whenever  well-known  complete  since  proof  oo,  ^oo,  space.  X)  mapping  n—^  n  It on  space,  •  as  .  a unique  in  0  then,  t  metric  a  u || — 0  mapping  T)  1.  contraction  on 2)  mapping  v| | ,  I I — ^  metric  space  i f A(A,) is a  closed  complete in  is  ^  metric so  fixed  that  point  [l5j« "linear  sections.  whole  is a  ~  u  a contraction  3 <  a  i n the  exists  found  notion  again  that  mapping  Q  on  , and  ~  U  f o r a l l A, £ y\.  may  S| |u -  seen i f |I  complete  bounded,  on  ,  in  inJD  mapping  in following  defined  v  II ^ n  fixed  there  The used  t o be  contraction  contraction  subset  in  Thus,  is a contraction  fact  a  ^  £  A(A,)v  -  is said  A(A.)v| j <  is clearly  continuous. U  u,  14  space set  linear  A <!$3  of  Banach  transformation"  .  linear  will  transformation  It i s additive,  a l l such  also  linear  is  homogeneous,  transformations  space  transformation,  [18], [15]:  '  we  be  define  | J A II o'  1 1  1  -  15 -  "'ifefr  M .II A  w e w t 0 In  general,  ( D — > B ), (A  we  define  = A(Bu),  S E Q U E N C E S AND Let  section  {^ J# n  mapping  (A  A .  n  said  (A),  (P  from  > A  n  is  Lip(X>  *  such  that  , Cp ) f o r e a c h  ^  ^  e  ^ i  e  a sequence  n  e  ^  into A  , A (A.) o ' n  A, Q  (A) £ n.  H  ( ~&—^ £  a  i  s  n  of elements i n  n, l e t (A ) d e n o t e  a  single-  ).  (  {(A^)},  asymptotically  forA,£  follows:  X> .  ^^n^  be  j\  B in  TRANSFORMATIONS  The s e q u e n c e  to converge  B),  OF  a n c  (A.) , A ,e '  6.1.  defined  —>  ^^n^  F o r each  valued  Definition  SERIES  A,  u £ T) .  u , Bu £  Let JA^CA,)}  3»  (2>-*!B). ):  ( A + B) a n d ( A . B ) a s  + B ) u = Au + B u ,  (A.B)u  6.  f o r any two t r a n s f o r m a t i o n s  Thus,  D  &  n = 0,  i f there  (A): A.—>A(A,),  Lip(T> , <? ) . Q  ) .  1,  exists  2, a mapping  with A(A.)  £  and (A - A ) R  £  - 16 -  is said  ( A )  This  definition In  of  formal  to  have  series  with of  6.2.  the  asymptotic  be  an  asymptotic  i s analogous  analogy  Definition  to  section  ( A ) be  asymptotic of  f  definition l+, we  ^  { ( A ^ ) } .  3*2 sums  follows:  d e f i n e d on  expansion  the  of  define asymptotic  t r a n s f o r m a t i o n s as  Let  limit  to  limit  ( A ) is  (A^),  i f ( A ) is  said an  sequence  )  N _ 1  -a n=0 22 V r *  n  =  i  ' ' ' ••• • 2  3  N-1  Thus,  ( A )  -  2_  '  ( A  n  )  i  f  (  ~  A  21 n=  (A.)  i s then  state  sum  a  theorem  in Lip  each  said  be  to  ,  ( J>  to  the  Qp) Q  an  asymptotic  effect  i f and  that  only  sum  n  A  E  )  0 of  ^  / ( A ) «— n  i f ( A  r  )  P^2>  Li  e  (A^) .  has  an  We  asymptotic  Lip(X> , G f ) n  for  n. oo  Theorem  6.3.  The  series  ^>  (^ ) n  has  an  asymptotic  sum  n=0 ( A )  each  Lip(X>  in  Q p  o  )  i f and  only  i f  ( A  N  )  e  Lip(X>  , C p  n  ) f o r  n. The  that  ,  of  proof  theorem  of 4«2.  this  theorem  i s omitted,  as  i t  follows  17  -  7.  EQUATIONS  IN  BANACH S P A C E S [14]  Kantorovich obtaining adapted the  the r e a l  = 0,  Px  into  another, The  X ,  X  Q  q  roots  of P  P(x) One  =  1  Newton's  method  equations,  x  o f an  of  and  has  equation  of  is a  twice-Frechet-differentiable  from  a linear  at x  o  .  selecting  normed  space  B  initial by  the  estimate  Frechet  Thus,  X )  Q  Q  Q  =  0.  to set -  o  general,  an  P(x) -  P ( x ) + P' ( X ) ( X -  ~  x  of  replacing  proceeds x  In  consists  , and  differential  P  SOLUTIONS  .  method  £ |B  the s o l u t i o n  [14],  t 0  THEIR  of numerical  where [10],  transformation  AND  has m o d i f i e d  i t to finding  type  -  u  [P» ( x ) ] ~ o  1  J  P(x ). o  x  a r e c u r s i v e method  i s thus  obtained,  with x  The in and  sequence  t h e B a n a c h norm i s unique  establishes of  - ~ x  n+1  - [P'(x ) ] n  n  u  i ] x  n  v  thus  J  i n a neighbourhood  the s o l u t i o n  x.  1  P(x ).  n'  generated  to the exact  the best  _  i s shown  solution of X  q  x,  i n !B  to  which •  p o s s i b l e c o n d i t i o n s f o r the  converge exists  Kantorovich existence  -  McFarland, fractions Banach  shall  space  + E(A.), and  be a f i x e d , u  positive  in  in a  A  ).  that  that  , where  ^  is a  e x i s t s , with  O  f o r every  (ii)  (E) e Lip(2) ,  (iii)  F o r any p o i n t  J | A~  z(A)  1  ,  A.  C^V.) I I <  L e t A ( A ) and q  e A  ,  a  +  in (  assumptions  i n J)  A,—>P(A.)  P(A,) = A ( A . )  m a p p i n g o n 1)  the following  A'^A.)  »  . a r e made:  fora >  0.  | | E(A,) Z (A.) | | = o ( l )  ^ 0.  A ^ ( A ) V(A.) Q  A.  , consisting  and l e t (P) :  Assume  determined  (i)  A.  i n !B  (A.) i s a l i n e a r t r a n s f o r m a t i o n  be u n i q u e l y  as  sphere  number.  (P~->JB  where  element i n  J |u| | <  with  q  into  Q  closed  P = P(A.) f o r A, e - A ,  Suppose  (iv)  continued  Pu = v i n t h e  v i s a given  E(A.) i s a c o n t r a c t i o n  E (A.)  of  of a quadratic  the equation  , where  X*  Let  map-A  the solution  discuss  a l lelements  fixed,  up a m e t h o d  space.  Let of  i n [17]/ sets  to obtain  We Banach  18 -  i s an i n t e r i o r  point  of  , f o r every  e -A . o  The positive  following  interval  A  theorem  shows  that  , a n d an e l e m e n t  there  exists  u(A) i n D  , so  a that,  - 19 -  for  A, £ /\.,  v(A.) .  Moreover,  u(A.)  is  unique  T> .  in  Theorem  7.1.  exists !J} =  P(A.)u(A.) =  a positive  , so  that,  v(A.) ..  a s A.  Pu  «  , and A  (i)-(iv), an  , u(A.)  in  , and  there  element  u(A.)  satisfies |ju(A,)  -  in  P(A,)u(A.)  A~^A,) v(A,) |  |—^0  A  u o  u  P  = A  q  + E^" ^  the  written  i n the  form  + Eu  =  equation  v.  assumption = w  Let  1  v be  with  + w  u  o  ( i ) , (2)  is equivalent  to  Fu, = A ~ o  the u  (l)  j\  interval  i s unique  Since  By  (3)  assumptions  — 0 .  can (2) '  the  w h e n e v e r A. £  u(A.)  Proof. (1)  Under  1  v and  sequence =  F  i  u  =  -A~^"E. o  l  t>  ,  n  n  e  defined  recursively  as  follows  w  . = w n +1  + Fu  n  = 0,  1,  2,  ...  E l e m e n t s u , v , w, ... i n IB and transformations A , E , P, d e p e n d on A.. However, i t i s c o n v e n i e n t to s u p p r e s s t h i s dependence i n the n o t a t i o n , when c o n f u s i o n d o e s n o t a r i s e . Q u a n t i t i e s independent o f A. w i l l be c a l l e d " f i x e d . In t h i s r e g a r d , we f o l l o w van d e r C o r p u t ' s c o n v e n t i o n [19]• 1 1  -  Then the  this  Banach  positive Then,  (4)  -  will  be  to the d e s i r e d  ( i v ) i t follows  number.  L e t x, y  seen  to converge  solution,  that be  3 =  f o r s u i t a b l e A..  "V) -  a n y two  in  | |w| J  elements  is a i n X-  •  1  by ( i ) ,  1 | Fx But,  -  Fyj|  by  interval  (5)  sequence  norm  From  20  = |  ( i i ) and  so t h a t ,  | |Ex -  Eyj | <  U'^Ex  -  (iii),  we  can choose .A  f o r a n y A. £  y  a _ 1  < <x| | E x  Ey)| |  - Eyj| .  a positive  ,  1 I* - y | |  and (6)  | |E Z for  I | < ycT  any  Hence,  (7)  w  ||u  (4)  and  = 3(1  n  +  1  -  strong  u j | <  n  +  ,  1  n  ( 6 ) we  have,  Ilu.  - u II  = I IFu • I I < a  1  <  1.  for n  =  £A .  that  = 0,  1,  2,  ...  0,  IIEu  || < Y #  since  u  £  that  I|u - u I I < Y 11 + i m — ' j  X  induction  Y  + 3)"  (5),  From  1  1  m  +  1  1  for a l lm <  n.  m  It In  from  show by  Assume  (9)  , where y  £ X>  j I F x - F y | I < y | | X - y | |, We  (8)  z  1  fact,  follows f o r any  that j <  n,  U  q  , u^,  by  are a l li n  the hypothesis  of  .  induction  (  ||u.  -  u ||<||u o  <Y  -  1  u || o  Y  +  -  21  -  +  ||u  2  •••  +  Y  +  ujl  -  J  Y(  <  *  ...  +  ||  u.^U  -  U j  - Y)"~ •  1  Hence  I Njl I < II j u  Thus,  by 1  This  lI  G  (7)  u n +., 1 - un I I = 1  u  1 1  in  T)  .  to  an  element  is  a  y  and  I I F un  <  1, ID  Since  =  lim u  - F un-1 -II the  the  < YI I  =  n —> o o  X e A  0  u n  I S Y  _11  I } u  and  + Fu  converges  n  e  -A.  .  u  ,)  = w  n-1'  Moreover,  + F  lim u n  f o r any  u'  (l).  (10)  u'.  = w n-1  +  Fu  e X>  , with  Pu*  =  v,  u  is  l l u-  Using  u  we  have  u  u T  unique  ll«  Finally, A v|  |  _1  II  (iii)  F  oo  — ^  Furthermore,  u  =  norm  hence,  0  satisfies  , since,  u  sequence  i n the  ||u - u»|| = | |Fu - Fu* j j < y | | ~ ' l l < l l Hence  n +1  .  Thus D  n —>  n  [ u ^ j i s a Cauchy  , for a l lX  lim(w  ~  u  induction.  sequence  i n X)  n  w  (9),  i s complete,  u  1 +  G  continuous transformation u  in  u  completes  Since  for  Il l I < Yd - Y)"  +  O  1  the  in turn,  we  | |u -  This  completes  the  E  1  II  1  have | <  from  of  A  \  ul  I.  1 1  and  assumptions  (10),  a | |Eu| I  proof.  o  1 1  II  linearity  A^vl  (11)  = I I F u I I = MA"  = o(l)  as  A.—3*0.  ( i ) and  -  8.  ASYMPTOTIC We  SOLUTIONS  consider  asymptotic  sum  next  22 -  OF  EQUATIONS  the case  of a formal  AND  that  series  THEIR  REPRESENTATIONS  the mapping  of  ( E ) i s an  transformations,  oo n=l Let  ),  N  p r e s c r i b e d mapping  linear  fying  8.1.  ( v ) ^*-  A  as  into  ^n  1 1  ( X> — > B  ....  H  .  ) , and  L e t ( v ) be  Suppose  A  q  is a  before.  A mapping ^>  /  into  o  n = 1, 2, 3,  from  transformation,  Definition  (A.) map A n  : A,—>A  x  (A ) £ L i p ( S > . 9  let a  (A ) n '  ^ *  (u) from  S  s a  * *  A.  *° ^  c  e  a  into n  3D  satis-  asymptotic  solution  n =0 of  the relation  The  next  existence Let  (v)  ^ ^ _ (  theorem  establishes the conditions  o f an a s y m p t o t i c  u n  x  A  i s a linear  o  with  •I il  A  0  - 1  !I II  for a >  (ii)  (A ) £ L i p ( P  ,°? ),  (iii)  F o r any p o i n t  z  n  n  A  n  i s an  II  —  (v)— '  f o rthe  (A u ) . n  made: so t h a t  A  o  ^  exists, '  0.  = 1,  2,  3,  ...  in T) ,  I IA z I I < a ©  1 1  be  transformation,  <— a  n  )  s o l u t i o n of  the f o l l o w i n g assumptions  (i) '  (iv)  A  n  interior  '  n  (A.), A. £ A '  point  of D  n  , a n  > 0.  , f o r a l l A, £ A .  .  - 23 -  Theorem  8.2.  Under  t h e above  assumptions,  (v) ^  ^ n  U  n=0 has  an a s y m p t o t i c a l l y  j |u - A ^ v l | — > 0  Proof. by  From  E = y  unique  solution  (u).  as A.—>0.  ( i i )i t follows  ^ n ^ n ' '"'  s a  n  a s  by theorem  y P"' otic m  sum  :  6.3  of  (l)  v =  v  an  (E), given  n=l  (E) £ L i p ( X ^  By  that  ^  n=l with  Furthermore.  (iii),  asymptotic  ,Gp^).  Consider the equation  A u + Eu. o  i t follows  sum  of  from  (^  z n  theorem  )  ^  as  A , — ^ 0.  o  r  a  n  4-2  y  z  that '  e  (Ez) i s  a n c  *  "that  n =l  I I Ez j I = o ( l ) Thus,  assumptions  corresponding exists so  of our theorem  of theorem  T>  u (A.) e  show n e x t  ( u ) be 1  (v)~  another  2L( n A  that  )'  A  .  w  l  t  h  u  *  £  ^  '  X  A f  -A. ,  ( v ) — ' Z Z (An u ) . n=0 unique.  Then, 6  there  interval  Thus  solution.  imply the  Hence,  (u) i s a s y m p t o t i c a l l y  asymptotic u t  7«1«  and a p o s i t i v e  v = A u + E u w h e n e v e r A. £  We Let  assumptions  an e l e m e n t  that  (i)-(iv)  ^  - 24  We (2)  By  have  ||u  -  u«||  assumption  positive  -  -  MA;  A (U  1  -  Q  ( i i ) , there  intervals  exist  so  u')||  o||A (u  ut)||.  -  o  positive  that,  numbers a' n u and u in  for  N-1  and  T  N-1  || 21  (3)  <  (  V  -  A  n  U  ,  )  H  ^  Z I  n=l  'j V  "  A  n  U  H  ,  n=l  < 2a£ <p | Ju - u» | j x  < Then, u s i n g (4)  i a ' ^ l u  -  (2)  and  u'M  <  ^a" J|u  -  1  u»||.  (3),  ||A (u -  u»)||  o  -  21  ||  ( A  n  "  u  A  u n  t  )  H  n=l  <  (A^u n=0  <  A u»)||  -  n  n  N-1  N-1  2 1 v'l  i i v-  +  Mv  -  n=0  But  (v)  Hence, f r o m f o r A.  E  (4),  an  asymptotic  there  exist  sum  of  positive  n=0  / '  (A u) — n  and/  (A  -  '—  numbers 3^  so  u»). n  that,  A , N  Thus in  is  v  21  theorem  |  <  %.  -  (u)  is asymptotically  7.1/  u'|  3  I |u  N  N  =  1,  2, 3,  unique.  ...  Finally,  just  as  -  (5)  | |u - A ~ v | | This  We (u),  completes  wish  thus  <P (A.)  an  -  a s X ~ > 0 . proof.  asymptotic  solution  = A,  of  asymptotic  the f o l l o w i n g  representation for  ( v ) — '  f o r each  n  n  d e f i n e d i s an  over,  the  to f i n d  the asymptotic Let  0  25  n.  The  sequence  inequality  (A u ) . n  —  (section  clearly  {P l  sequence  (  n  3)»  More-  holds:  n  y<P.Qp . < Y Co f o r A, e A , v <L_ j ^ n - j — ' n T ' n n' n  (6)  1  > 0.  j=l Suppose are  fulfilled  that with  assumptions respect  Consider  the  equation  (7)  = w  + Fu.  u  (i)-(iv)  of theorem  to the asymptotic  8.2  sequence  {A. }. 11  Oo  Suppose  that  (v) ^  (v ), and t h a t  2_  (E) —  xx  n=0  (A )  n.  n=l  Then (8)  (w) ~ '  /  (w  n=0  n  ),  where  w  = A ^ v . n o n  n  C O  (F) y—'  / Z —  x  (F ), n' '  where  F  n  = -A  _ 1  A . o n  n=l We of  establish  (7) [ i . e . ,  i n the next  the asymptotic  theorem  solution  that  (u) of  the  solution  ( v ) ~ T~(A c—>  u)] n  - 26 -  has  an a s y m p t o t i c  method  of successive  Theorem  8.3.  asymptotic is  expansion,  an a s y m p t o t i c u  o  = w  the assumptions (u) of ( v ) " ^ ^  limit  (A U),  8 . 2 . th<  where  R  {(u )}  (v)'— / ( v ^ ) .  d e f i n e d by  n  o n  21  n  from t h e  of theorem  of the sequence  n u  directly  approximations.  Under  solution  obtained  W  j  71  +  j=0  Alternatively,  j  F  » - j  u  (  n  =  X  '  >  2  1>  •••)  j=l  O O  (u) has the asymptotic  expansion  /  (u  - u  n°0 with  u ^ = 0.  Proof. of  It suffices  positive  We it  numbers  ||u - u ^ J I  (9)  proceed  i s clear  (10)  t o show  from  |ju - U j . i l  s  <  that  \  n  n  by s t r o n g  I <  exists  a  sequence  that  o  6 A, ,  (5) that  there  z  A  ,  N  n = 1, 2, 3,  induction  on  (9) i s t r u e .  X  £  Aj*  f  n.  When  Suppose  o  r  e a c h  . . .  J  n = 1,  next  that  < n.  Then (11)  | |u - u J | <  <  21  || w • Fu -  I lw  W  k  -  21  n  2Iw || •. k  k=0  |i Fu -  Vn-k' I ,  p  ^ - P ^ u l j k k=l  •  ),  27  -  -  k=l From numbers  a  n  right-hand To  (8) i t f o l l o w s  there  so t h a t  each  side  ( l l ) i s bounded  complete  of  the proof,  term: i s a l s o  (12)  that  of order  of the f i r s t  we  must  A. "'". n+  exist two  above  show  that  positive  terms by  n +  n +  ^*  third  Now,  || Z I < V " V . ) l l < Z J I V n  i^*  a  the  on t h e  -  k  k=l  W - k l l  k=l  < a  I'V  "  Vn-kH-  k=l By it  c a n be  Then,  (13)  an  argument  . £ jD n—k assumption ( i i ) ,  seen  using  that  u  a  k=l  k  X  k=l  the hypothesis  i n theorem  1,  2,  ZIV- M» k  "  u  n-kH  '  (10).  = max(a. 6  Let £  1 from  (12)  and  (13)#  using  (6),  we  have  < k  ,  .)  k n-k+1  n Then,  7«1,  n.  k=l  n-k + 1  of induction  used  for k =  a 21 I I V " V n - k H ^ —  by  s i m i l a r to that  <  n  - 28 -  (14)  y~  ||  from ( l l ) ,  ||u  u  -  This  this  any  this under  (1)  TO  the  element  case  n+ 1  n  the  REAL  section,  to  the  A, £ A  6 _, A. , n+1 '  1  completes  specialized for  J] < n' —  APPLICATIONS In  J  1  Then, 1 1  9.  ( F k. u . - F.u ) J | < a en'n y + 1. ^ k n-k ' —  *L k=l  1 1  consideration  the  Banach  of  real  the  7^1  a  "Yj >  £ 0,  A, £ A  with  a  ||u||  0.  =  |u|  T)  is in  The  equation  , to  those  of  0  A. £ A  fixed. o  |E(X;A.) - E(y;A.)| for  be  are:  o  (ii)  with domain  corresponding assumptions  theorem (i)  The  will  form  Q  the  numbers,  | u | < Tj , w i t h  i s of  tB  space  space.  v(A.) = a u(A.) + E(u;A,), and  ^, . n+1  induction.  in this  interval  .  EQUATIONS  space  u  n + 1  1  any  x,  y  <Y^|x-y|  i n T)  , where y  (iii)  |E(z;A,)| = o ( l ) a s A . — > 0 ,  (iv)  Ivor ! < Tj .  >  and  f o r any z E D  .  .  1  Then,  by  solution  u(A.)  interval  J\. .  theorem of  7«1,  ( l ) i n X>  there  exists  , f o r A. i n a  a  unique  suitable  - 29 -  Assume asymptotic In  that  v(A.) a n d E(u;X,)  expansions  the case  that  as  ^P (^-)  possess  in definition A, , we  =  have, as i n  n  n  4-l«  4(1),  (2)  |v - 22  v  J  (  A  zyjfi  •  n=0 (3)  2^  |E -  n  a  -  )  u  n +  ^  1  Y]A  *•  £  A  N ,  n=l with theorem  8.3/  expansion  u n i f o r m l y f o r u £ X>  (3) h o l d i n g  -  the solution  2l_ / u  o  n  Under  the assumption  |a (\)| -  (4)  n  the  u  of  t h e same  r  .  Then,  ( l )possesses  type  an  by  asymptotic  as v.  that  0(A. ), n  relation  (5) v(A.) — a u + 22 (^) n=l  has (A  n  an a s y m p t o t i c a l l y ), defined  = CX (7^)u  A (A.)u n  n  theorem  8.2, An  van  by  1  Under  ,  which  equation  der Corput  ( i i ) :  n+ 1  (A ) n  E(u;A.) these  a n  Q  unique  solution.  : \—3-  A  satisfy  n  assumptions  of the type  [ 1 9 ] / who  valid  e  (i),  in this  of  o  the mappings , with (iii)  of  case.  d i s c u s s e d by  (iii),  function  he p r o v e d  -A  ( i i ) and  ( l ) has been  assumed  i s a continuous  assumptions,  Indeed,  (A.), f o r \  i s therefore  un+1  (iv) u  and  for u £jD  t h e e x i s t e n c e and  .  -  asymptotic We  may  uniqueness  note,  dition  representation  power  v  [ 6 , pp.  v •—' u this  in  T)  clear (A ) n  |v| <  case, into  that  1 1  , <f>) n  of  (6)  expansion  Another question so  ^  8.2  exists  asymptotic  «~"-^ v +  Theorem  .  a method  f o r convergent  of f i n d i n g  for a  -  3  a  prescribed  ...  A,  n  u  .  n+ 1  where v n 1  n.  =  N  Hence,  , with  3 j (A.) V^  f o r the  i n which  i s the problem  that  t  maps  i t  ( n + l)i, '  are satisfied, i n 2>  which  Moreover,  u  is  and  that  a l l of the  a n d an a s y m p t o t i c  |u -  formal substitution  example,  converge,  con-  has o b t a i n e d a  so t h a t ,  f o r each  of theorem  In g e n e r a l ,  u  1,  2  (n + l ) I  n  >0m  A,  i n X)  the problem  + 31 A . u  2  n  u  u  a  1,  I |A ul I < Y ^-# n — ' n '  £ Lip(P  solution  of  Inversion  |u| <  21 A,u  ( - l )  ( l ) .  i s t o o weak  A^CA.) i s t h e t r a n s f o r m a t i o n  1 1  assumptions  of  u, by e m p l o y i n g  consider  u, w i t h  satisfying  In  for  u  121-125].  an e x a m p l e ,  number  (6)  theorem f  series  (ii*)  van d e r Corput  to Lagrange s  As real  that  the uniqueness  Furthermore,  analogous  of the s o l u t i o n  however,  to insure  30 -  v| — > 0  will  lead  as t o an  solution.  a l l of the series i n of finding  u  in  31  -  (7)  -  v - u e*- , U  with of  1.  |v| <  7«1  theorem  conclude with  I t c a n be v e r i f i e d  that  are s a t i s f i e d a unique  an a s y m p t o t i c  iterative A  process  solution  expansion  be  10.  i n [7,  found  APPLICATIONS Let  u  may  assumptions  a n d we  exists  may  i n X)* *  be d e r i v e d  by an  8.3.  of the similar  problem  . as t — >  pp.  25-28].  TO S Y S T E M S OF  NUMERICAL  EQUATIONS  t h e Banach  space  under  discussion  Euclidean  space  V  F o r any  dimensional where  the  case,  o f (7)  which  discussion  , x t = xe may  in this  as i n theorem  different  that  i s t h e column  n  vector  have:  .  ( ^ ^ ) ,  u  i = 1,  be t h e nin V , n 2,  n,  we  ^  i=l Let  X)  be t h e s e t o f a l l e l e m e n t s  u  in V  , with n  I | |I ^  / where  u  linear  algebraic  (l)  v =  A  Q  U  >  0.  equations  + Eu,  C o n s i d e r t h e system  of  n  non-  -  where V  , n'  =  v = A  i = 1,  (v^),  i s the matrix  o  32  -  2,  n, 1 <  (a,,). i j  ^ 2.' ^ 2' * * "' ^ n ^ '  ^ = 1,  i s a given  element  i , i ,<  n, and Eu =  2,  n.  Assume  of  that  d e t C a . j ) £ 0.  (i)  Then,  (2)  v =  clearly,  the equation  u  A  o has  U  a solution  additional (ii)  Q  = A ^V.  We  Q  (iii)  F o r any z £  (iv)  u  7.1,  theorem u  interval  A.  For  (3)  these  solution  V  V  l  =  i l ^ l  a  2 = 21^1 a  satisfies thus i  and  u  2, = A  A. £ A  ||Ez||  assumptions  Moreover,  example, +  a  +  l2^2 a  22?2  *•  ^"v  A.—^0.  i t i s evident  A.  | |u -  i  °»  n  a  unique  i s in a  U |j q  ^0  that,  suitable as  A,—>0.  of equations  ^ l  ^ 2  +  assumptions  i nD  s  Y >  where  ( l )possesses  the pair +  ,  = o ( l ) as  i n J^> , w h e n e v e r .  x  .  the system  possesses  = 1,  X> ,  i s i n t e r i o r t o X^  Under by  following  x, y i n  | j E x - E y | |< y A,| | x - y | | ,  o  the  assumptions:  F o r a n y two e l e m e n t s  v  make  ( i i ) and ( i i i ) above,  a unique  solution  , provided  u =  and  (^ ^ ) ,  ( ^ j ) i snon-singular a  i s i n t e r i o r to  , where  v =( v . ) .  - 33 -  We *>y y  (3)' =  verify  that  F o r any  (£ ^ ) ,  i =  two  1,  ( i i i ) are indeed  elements  x,  2,  I5J  (4)  ( i i ) and  we  have,  inD  y  , with  satisfied x  =( $ ^ ) ,  clearly,  l?il<^.  Thus ||Ex  -  Eyj j - A ^ s i n ^  -  s i n ^ )  •  2  - ^ ^ f  ( ^  Now, |sin  5  I ? 2  X  "  *j < y |£ " < > J_ |  sin£  ~^2 ! —  x  I £  Y  3  2  -$2'*  W  h  e  r  , and  Y >  e  0.  Then  I I Ex  -  and  assumption  x =  (  I .)  Ey|  I  < Y A.  in  x  )  2  + $2^-2"  ( i i i ) is satisfied.  11.  APPLICATIONS  in  Banach  equations. of  y  |  J,  Finally,  f o r any  ,  Thus  this  x -  ( i i ) is fulfilled.  I I E x | I = A. | ( s i n \  In  II  TO  INTEGRAL  section,  spaces The  a l lcontinuous  will  be  Banach  = o(l)  EQUATIONS  the theory applied space  as A - * 0 .  53  f u n c t i o n s over  of asymptotic  to non-linear will  be  either  the closed  equations integral the  interval  space [0,  l ] ,  - 34 -  or  tSC (0, l ) o f s q u a r e - s u m m a b l e  t h e space  the  sense  will  of Lebesgue.  be r e g a r d e d  respectively, sphere  of ^ In  is  T>  Let  >  or of  t o be  and n o n - l i n e a r  over  the fixed  , t h e norm  integral  domain  Consider  the integral  w £ ~&  u(s) = v(s) +  and  K(s,t)  continuous  v £ "jfy .  , a  with  operators closed  Let  K  | jw| | < Tj ,  equation  1  with  and E  sup | u ( s ) | . 0<s< 1  ' (1)  o  | |u| | o f a n y u £  be t h e s e t o f a l l e l e m e n t s  0.  A  ^.  the space  defined  The t r a n s f o r m a t i o n s  as l i n e a r  defined  functions i n  1  j K ( s , t ) ' u ( t ) dt +  J  0  0 on t h e c l o s e d  square  be t h e t r a n s f o r m a t i o n  E ^ ( s , t ; u ( t ) ) dt  0 <  s, t <  defined  1,  by  1 Ku  J" K ( s , t )  =  u(t)  dt.  0 Then, — > ~@>) .  clearly, Again,  K  l e t E  i s a linear  transformation  be a t r a n s f o r m a t i o n  i n (^  with  1  Eu =  J E ^ C s ^ ; u ( t ) ) d t , X £ - A , u £ 2> . q  0 Then  (2)  ( l ) c a n be w r i t t e n  as  u = v + Ku + E u . The  following  assumptions  will  be made:  in —  )  - 35 -  (i)  ( I - K)"* e x i s t s , 1  (ii)  There  |E(s,t;  0 <  for  (iii)  (iv)  for l|u  A -  s, t <  1,  <  1  and  a,  ( I - K)  theorem  ( l ) has a unique  Under  | v  i n some (I - K ) "  1  7.1  solution  t <  that  i n "j)  solution  as  ±  0.  under  .  these  .  the  u = u (s;A.) .  A. .  1  i n t e r i o r t o jD  (i)-(iv).  vl I  be  v  1  interval A  It will  s,  = o ( l ) as A,—>  positive >0  that  | < TJ , a n d A, £  establishes  assumptions  so  A|u - vj  |E(s,t;u)|  , with  following  (3)  v)| < y  number y ,  |u] <YJ,  ( l ) has a unique  theorem  K)" ]!  E(s,t;u) i s continuous f o r 0 <  11.1.  Proof. of  a positive  u) - E ( s , t ;  v £ ^  assumptions,  equation  exists  | u | <7J,  and  Theorem  | | ( l-  0.  a >  The  with  integral i n X>  .  Moreover.  A—>0.  v e r i f i e d that  are s a t i s f i e d .  a l l of the  Write  assumptions  ( l ) i n t h e form  v = u - Ku - E u . By  ( i ) and  (4)  ( i v ) , the l i n e a r  v = u -  has  a unique  and  ( i v ) of theorem  equation  Ku  solution  U  q  =  ( I - K)  1  v  7.1 a r e s a t i s f i e d .  i n jD  .  Again,  Thus ( i ) l e t x,y  e32>  - 36 -  By  (ii),  I I Ex  Thus  -  Ey||  <  val This  ;  A.| | x ,X),  u  theorem exists  also,  (a)  the  (ii)  (iii) may  u  of  may  thus  i n X)  ||u -  y(t))|  y || . and  Similarly  completes  Remarks:  E(s,t;  < Y  existence  -A  x(t))-  A. s u p | x ( t ) - y ( t ) | CKt<l  L i p ( S>  (E) e  solution  |E(s,t;  < Y  satisfied. the  sup s,t  be  be  7.1  is  verified.  applied, A,  , for |-| =  theorem  a  Since unique  in a positive  o(l)  inter-  X—>0.  as  proof.  Assumption  ( i ) may  be  replaced  by  the  following: (i ) 1  K(s,t)  i s continuous i n both  |K(s,t)| This  In  ||K||  Thus [15], Thus  fact, <  (i) is  an  t <  1,  f o r any  and  by  that  z  1. f o r assumption ( i ) in  ,  a well-known (I -  K)  with  has  | | Kz J j  theorem  a bounded  <  || z || .  [18], inverse.  satisfied, ( i i ) i s made  solution  [12],  s,  is sufficient  i t follows  Assumption the  1 for 0 <  condition  above.  (b)  <  arguments,  u  i n IP  assumption  of  to .  insure  the  By  theorem  the  continuity  on  uniqueness  E,  of  of  Osgood  without  a  - 37 -  the  Lipschitz  condition,  existence  of a one-parameter-family  in  .  In t h i s  regard,  [19],  and  section  9 of t h i s  A similar  theorem  t o 11.1  Hilbert  space  would  (0,l).  - {J  u||  s u f f i c e f o r the of  see a l s o  solutions  van d e r  Corput  work.  may  be  F o r any  established  u £ *L- , we  in  have.  |u(t)| dt|i 2  0 It (l) <jC  has  c a n be  a unique  , defined  are  shown  by  similar  solution  as  , w h e r e 2>  in  i n theorem  reasoning  11.1,  equation  i s a subset  i f the following  of  assumptions  made:  (i)  (I -  (ii)  |E (s,t;u)  K)"  exists,  1  x  for  0 <  G^(s,t;  s,  -  with  |j(I -  E^(s,t;v)| t <  1,  u,v) >  0  -  <  00 <  (iii)  ||  <  a,  u,  v <°o  -  a > v|  , where  ?  d t d s < y X.,  0  for  1  - 1  and  1  0  K)  G^(s,t;u,v)|u  J|G^(s,t; u ( t ) , v ( t ) ) T  (iv)  that  A, £  >  0 a n d any u,v  1  in  2  J  j  0  0  v £ £  |E^(s,t;  , and  u(t))|  ( I - K)  dtds  V  = o(l)  is interior  as A . — »  to  X>  0.  0.  - 38 -  It section Thus,  8  the  ( E ) ^  may  be  remarked  are  also  applicable  equation ( n^' A  u  n  o  -  here  v = e  *  r  n  u -  e  that to  Ku  -  the  theorems  of  integral  equations.  Eu  #  in  assumptions  of  with  theorem  8.3,  has  n= l a  unique  solution  asymptotic  u  in  , which  expansion obtainable  by  p o s s e s s e s an a  recursive  process.  -  12.  £l]  REFERENCES  S. B a n a c h . Monografje  |2]  R.6.  Theorie  Newt o n ' s  Amer. M a t h .  E. Borel.  S o c . 6.,  These,  £4]  T.G.T. Paris  [5]  E.A. C o d d i n g t o n  Carleman.  (1932).  i n Banach  5(1955),  spaces.  827-831.  points de^la  Annales  theorie des  de l ' E c o l e  (1895).  Normale  Lesfonctions quasi-analvtique.  (1926) C h a p t e r  differential  lineaires.  v o l . I , Warsaw  method  S u rquelques  fonct ions.  £6]  desoperations  Matematyczne,  Bartle.  Proc. [3]  39 -  5.  a n d N. L e v i n s o n .  equations.  Theory  McGraw-Hill,  of ordinary  New Y o r k  E.T. Copson. An i n t r o d u c t i o n t o t h e t h e o r y o f f u n c t i o n s o f a complex v a r i a b l e . Oxford University  Press  (1946).  £7]  N.G. d e B r u i j n . A s y m p t o t i c methods I n t e r s c i e n c e , New Y o r k (195*0 •  i n analysis.  £8]  G.F.D. D u f f . Partial U n i v e r s i t y of Toronto  equations.  £9]  (1955)  A. E r d e l y i .  York  (1956).  differential  (1956).  Asymptotic  expansions.  D o v e r , New  £10]  M. F r e c h e t .  £ll]  Ei. H a u s d o r f f . ZurTheorie R a u m e. J o u r n a l f u r Math.,  £12]  E.L. Ince. D o v e r (1956)  £13]  Y u . Y a . K a a z i k a n d E . E . Tamme. On a m e t h o d o f approximate solution of functional equations. Dokl. Akad. Nauk S S S R ( N . S . ) 101 (1955), 981-984; M.R. J J . , 647. (Russian).  Annales  de l ' E c o l e  Normale,  der linearen  167 (1932),  Ordinary d i f f e r e n t i a l C h a p t e r 3.  293  (1925).  metrischen  294-311.  equations.  -  40 -  [14]  L.V. K a n t o r o v i c h . F u n c t i o n a l a n a l y s i s and a p p l i e d mathematics. T r . by C D . Benster. National Bureau of S t a n d a r d s . U . C . L . A . (1952).  £15]  A.N. K o l m o g o r o f f a n d S.V. F o m i n . Elements of t h e theory of functions and f u n c t i o n a l a n a l y s i s I. Graylock, R o c h e s t e r , New Y o r k (1957).  [16]  W.V. L o v i t t .  New  York  Linear  (1950).  integral  equations.  Dover,  £l7]  J.E. McFarland. An i t e r a t i v e s o l u t i o n quadratic equation i n Banach s p a c e s . M a t h . S o c . 2., 5 ( 1 9 5 8 ) , 824-830.  of the Proc. Amer.  [18]  F . R i e s z a n d B. S z . - N a g y . F r e d e r i c k U n g a r , New Y o r k  analysis.  [19]  J.G. van d e r Corput. Asymptotic expansions I. Technical report I. C o n t r a c t A F - 18(600) 958. U n i v e r s i t y of C a l i f o r n i a , Berkeley (1954)•  [20]  A;C. Zaanen.  York  (1953).  Linear  Functional  (1955)•  analysis.  Interscience,  New  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080638/manifest

Comment

Related Items