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
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Asymptotic properties of solutions of equations in Banach spaces. Schulzer, Michael 1959
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Title | Asymptotic properties of solutions of equations in Banach spaces. |
Creator |
Schulzer, Michael |
Publisher | University of British Columbia |
Date Issued | 1959 |
Description | Certain properties of the solution u of the equation Pu = v in a Banach space will be investigated. It will be assumed that v is a prescribed element of the space, P is a transformation defined on a closed subset in the space and consisting of the sum of a linear transformation and a contraction mapping, and that P and v depend on a real variable λ. which assumes values over the half-open positive interval 0 < λ ≤ λₒ. Then a theorem will be proved, establishing the existence and uniqueness of the solution u(λ) of P(λ)u(λ) = v(λ) . Under the hypothesis that P and v possess asymptotic expansions as λ→0, it will be shown that asymptotic solutions exist, that they are asymptotically unique, and that they possess asymptotic expansions which may be determined by a recursive process from those of P and v. The results obtained will be applied to particular types of Banach spaces, such as finite-dimensional Euclidean spaces, spaces of Lebesgue-square-summable functions and of continuous functions over a closed interval. |
Subject |
Equations Generalized spaces |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080638 |
URI | http://hdl.handle.net/2429/39917 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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