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Asymptotic theory of second-order nonlinear ordinary differential equations Jenab, Bita 1985

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ASYMPTOTIC THEORY OF  SECOND-ORDER  NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS  by B i t a Jenab B.A., Mount Holyoke C o l l e g e  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department o f Mathematics  We accept  this  t h e s i s as conforming  to t h e p e g u l r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1985  © BITA JENAB , 1985  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the head o f  department o r by h i s o r her r e p r e s e n t a t i v e s .  my  It is  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department o f The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  written  ABSTRACT  The  asymptotic behaviour of n o n o s c i l l a t o r y  s o l u t i o n s o f second  nonlinear ordinary d i f f e r e n t i a l equations i s studied. sufficient  Necessary  c o n d i t i o n s a r e g i v e n f o r the e x i s t e n c e o f p o s i t i v e  with s p e c i f i e d asymptotic behaviour a t i n f i n i t y .  Techniques  and  solutions  Existence of n o n o s c i l l a -  t o r y s o l u t i o n s i s e s t a b l i s h e d u s i n g the Schauder-Tychonoff theorem.  order  fixed  point  such as f a c t o r i z a t i o n o f l i n e a r d i s c o n j u g a t e o p e r a t o r s  are employed to r e v e a l the s i m i l a r n a t u r e of a s y m p t o t i c s o l u t i o n s of nonl i n e a r d i f f e r e n t i a l equations to that of l i n e a r equations. illustrating  the a s y m p t o t i c  Some examples  theory of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s are  given.  - i i-  TABLE OF CONTENTS Page  Abstract  i i  Acknowledgements  v  Introduction  1  Preliminaries  4  1.  Introduction  4  2.  The Schauder-Tychonoff F i x e d P o i n t Theorem  4  3.  F a c t o r i z a t i o n of Disconjugate  9  Operators  Chapter I - Asymptotic P r o p e r t i e s of Semilinear  Ordinary  D i f f e r e n t i a l Equations  13  1.  Introduction  13  2.  Bounded Asymptotic S o l u t i o n s  13  3.  Unbounded A s y m p t o t i c a l l y  21  4.  Asymptotic P r o p e r t i e s of the Emden-Fowler E q u a t i o n  25  5.  A More G e n e r a l Case  27  Linear Solutions  Chapter I I - Asymptotic P r o p e r t i e s of Q u a s i l i n e a r  Ordinary  D i f f e r e n t i a l Equations 1.  Introduction  2.  Sufficient  3.  Necessary C o n d i t i o n s  4.  Summary  30 30  C r i t e r i a f o r Existence  of Bounded S o l u t i o n s  f o r E x i s t e n c e o f Bounded S o l u t i o n s  ...  30  ..  36 39  Chapter I I I - Asymptotic S o l u t i o n s of S e m i l i n e a r  Ordinary  D i f f e r e n t i a l Equations with Factorized Linear Part  ...  41  1.  Introduction  2.  Existence  3.  A S p e c i a l Case  50  4.  Examples  51  of P o s i t i v e  41 S o l u t i o n s of S e m i l i n e a r  - iii  -  Equations  ..  42  TABLE OF CONTENTS (Continued) Page  Appendix  57  Conclusions  60  Bibliography  62  - iv -  ACKNOWLEDGEMENTS  I am g r e a t l y i n d e b t e d to my a d v i s o r , Dr. C A . Swanson, f o r s u g g e s t i n g t h e t o p i c , f o r h i s guidance  and sound c r i t i c i s m , and most o f a l l f o r h i s  e n d l e s s p a t i e n c e with me throughout  the p r e p a r a t i o n of t h i s work.  I also  thank Dr. J.G. Heywood f o r r e a d i n g t h e manuscript. I express my g r a t i t u d e to the U n i v e r s i t y of B r i t i s h Columbia and t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada f o r t h e i r generous f i n a n c i a l  support.  -  v  -  1. INTRODUCTION  Our main i n t e r e s t here t i o n s o f n o n l i n e a r second sufficient  i s to study  order d i f f e r e n t i a l equations.  behaviour  of s o l u -  Necessary and  c o n d i t i o n s w i l l be e s t a b l i s h e d f o r the e x i s t e n c e o f p o s i t i v e  s o l u t i o n s and e x p l i c i t a s y m p t o t i c and  the a s y m p t o t i c  unbounded p o s i t i v e  behaviour w i l l be g i v e n f o r b o t h bounded  solutions.  In t h e p a s t decade, t h e r e has been an i n c r e a s i n g i n t e r e s t i n s t u d y i n g the e x i s t e n c e and q u a l i t a t i v e behaviour t i a l equations.  o f s o l u t i o n s of n o n l i n e a r d i f f e r e n -  The s u b j e c t i s o f g r e a t importance  also i n application.  n o t o n l y i n t h e o r y but  For example, a p p l i c a t i o n s o f the s i n g u l a r s e m i l i n e a r  equation  y" + a ( t ) y "  arise  from  = 0 ,  X > 0  (1)  boundary l a y e r theory of v i s c o u s f l u i d s and has been s t u d i e d by  C a l l e g a r i and Nachman [ 6 , 7 ] . and  X  Onose [19]., and Singh  Mathematicians  such as Hammett  [33,34] have s t u d i e d o s c i l l a t i o n  [ 1 6 ] , Kusano  theory f o r  f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s o f t h e type  (r(t)y'(t))' + a(t)y(t-T(t))  =  f(t) .  Many i n t e r e s t i n g a p p l i c a t i o n s of t h i s type o f e q u a t i o n w i t h J be  found  (2)  ^ ^  = <*> can  i n v a r i a b l e mass problems.  Another p o p u l a r d i f f e r e n t i a l e q u a t i o n i s the Emden-Fowler e q u a t i o n  y" + a ( t ) | y | sgn y = 0 Y  (3)  2. The  study of the Emden-Fowler e q u a t i o n o r i g i n a t e d around the t u r n o f  c e n t u r y i n t h e o r i e s of gas fundamental problem was  dynamics i n a s t r o p h y s i c s .  t o i n v e s t i g a t e the e q u i l i b r i u m  mass of s p h e r i c a l c l o u d s of gas 1862 few  At  in stellar  proposed t h a t the gaseous c l o u d years l a t e r , Lane [24]  the  c o n f i g u r a t i o n of  structures.  Lord K e l v i n  i s under c o n v e c t i v e  modelled t h i s phenomenon by  t h a t time,  the  the  [38]  equilibrium.  introducing  in  A  the  equation  U £> • y" - o . 2  In 1907,  Emden p u b l i s h e d  s t e l l a r configurations has  been coined  as  (4,  h i s famous t r e a t i s e Gaskugeln [11]  governed by e q u a t i o n ( 4 ) .  the Lane-Emden e q u a t i o n .  f o r the study o f such an e q u a t i o n and was  made by Fowler i n a s e r i e s of  The  i n t e r e s t e d r e a d e r may  The  on  the  study  Ever s i n c e , t h i s mathematical  of  equation  foundation  a l s o of the more g e n e r a l  equation  four papers [12]-[15] d u r i n g  (3)  1914-1931.  r e f e r to an e x c e l l e n t summary i n Bellman's book  [5, Chapt. V I I ] . The of the  first  s e r i o u s i n v e s t i g a t i o n c o n c e r n i n g the a s y m p t o t i c  generalized  After that,  Emden-Fowler e q u a t i o n was  ent  fields  p. 431]  such as gas  and  made by A t k i n s o n  the importance of t h i s e q u a t i o n and  were w i d e l y r e c o g n i z e d .  the  The  properties  i t s various  [1,2,3].  applications  Emden-Fowler e q u a t i o n appears i n many d i f f e r -  dynamics and  survey a r t i c l e  f l u i d mechanics; see  by C o n t i , G r a f f i and  e.g.  Sansone  Sansone [ 9 ] .  More  r e c e n t l y , a p p l i c a t i o n s of the Emden-Fowler e q u a t i o n have a r i s e n i n study of r e l a t i v i s t i c mechanics, n u c l e a r systems.  physics  and  chemically  the  reacting  Readers i n t e r e s t e d i n the p h y s i c a l a s p e c t s of such s t u d i e s  wish to c o n s u l t  Wong [41]  f o r an  extensive  review and  [31,  bibliography.  may  3. The study o f a s y m p t o t i c p r o p e r t i e s o f s o l u t i o n s o f d i f f e r e n t i a l equat i o n s has a l s o won c o n s i d e r a b l e a t t e n t i o n from a t h e o r e t i c a l s t a n d p o i n t . R e c e n t l y , mathematicians  have a p p l i e d t h e a s y m p t o t i c t h e o r y o f o r d i n a r y  d i f f e r e n t i a l equations to p a r t i a l d i f f e r e n t i a l equations.  Problems  posed  i n t h e r e a l m o f n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e now seen i n a new l i g h t  originating  from the r e s u l t s based on asymptotic behaviour of  solutions of ordinary d i f f e r e n t i a l equations. Kusano and Swanson [20,21,22], p i o n e e r s o f t h i s new  Mathematicians  such as  N o u s s a i r [27] a r e to be c o n s i d e r e d the  approach.  T h i s work i n c o r p o r a t e s r e s u l t s found by those mentioned focuses o n l y on the o r d i n a r y d i f f e r e n t i a l e q u a t i o n s a s p e c t . of t h i s study i s t h e Schauder-Tychonoff  above but The backbone  f i x e d p o i n t theorem [32,40],  used  over and over i n p r o v i n g e x i s t e n c e of p o s i t i v e s o l u t i o n s o f the d i f f e r e n t i a l e q u a t i o n under c o n s i d e r a t i o n .  In the next c h a p t e r we w i l l s t a t e  theorem a f t e r g i v i n g the n e c e s s a r y p r e l i m i n a r y  definitions.  this  4. PRELIMINARIES  1.  Introduction The purpose o f t h i s c h a p t e r  m a t e r i a l that leads  i s t w o f o l d : To p r o v i d e  to the Schauder-Tychonoff f i x e d  d i s c u s s t h e t e c h n i q u e s we employ f o r p r o v i n g differential  2.  equation  0.1  p o i n t theorem, and t o  existence of s o l u t i o n s o f the  consideration.  The Schauder-Tychonoff F i x e d P o i n t Theorem Before s t a t i n g  the  under  the preliminary  t h e Schauder-Tychonoff f i x e d p o i n t theorem, we  following definitions  Definition:  together  A metric  present  [29,30]:  space <X,p> i s a nonempty s e t X of elements  w i t h a r e a l - v a l u e d f u n c t i o n p d e f i n e d on X x X s u c h t h a t f o r a l l  x, y, and z i n X:  The  0.2  i)  P ( x , y ) > 0;  ii)  p(x,y) = 0  iii)  p(x,y) =  iv)  p(x,y) < p ( x , z ) + p ( z , y ) .  i f f  x=y;  p(y,x);  function p i s c a l l e d a metric.  Definition:  A t o p o l o g i c a l space <X,T> i s a nonempty s e t X of p o i n t s  t o g e t h e r w i t h a f a m i l y T o f subsets properties:  ( c a l l e d open) p o s s e s s i n g  the f o l l o w i n g  The  i)  X e T, <|> e x;  ii)  O  iii)  0  x  e T  and  0  £ x  implies  imply 01f\02  e T  2  \J a  0  e T;  ex.  f a m i l y T i s c a l l e d a t o p o l o g y f o r the s e t  0.3  Definition.  X.  A set X of elements i s c a l l e d a v e c t o r  r e a l s i f t h e r e e x i s t s a f u n c t i o n + on X x X and to X which s a t i s f y  the  following  i)  x + y = y +  ii)  (x + y) + z = x +(y  0.4  R, x,y e X;  v)  (A + u)x = Ax +  ux;  A,u  e  vi)  A(ux)  A,u  e R,  with the  = (Au)x; 1.x  =  R, x e X;  x e  X;  x.  A normed v e c t o r  I : X •* R + ,  space i s a v e c t o r  space X together  with  c a l l e d a norm, from X i n t o nonnegative r e a l s ,  properties:  i)  llxll > 0,  ii)  llx+yll < llxll +  iii)  IIaxII =  f o r a l l x e X and we  6 i n X such that x + 6 = x f o r a l l x i n X; A e  function I  if  + z);  A(x + y) = Ax + Ay;  Definition.  • on R x X  conditions:  iv)  = 9,  a function  the  x;  i i i ) There i s a v e c t o r  v i i ) 0.x  space over  llxll = 0 <=>  |a|  0;  llyll;  llxll;  a e R.  define a metric  x =  A normed v e c t o r space becomes a m e t r i c  p by p (x,y)  =  llx-yll.  space  6. 0.5  Definition.  11.11 i s a Banach  0.6  Definition.  topological  A normed v e c t o r space which i s complete w i t h r e s p e c t t o space.  A v e c t o r space X w i t h a t o p o l o g y x on i t i s c a l l e d a  v e c t o r space i f a d d i t i o n  i n t o X and m u l t i p l i c a t i o n R x X into  0.7  by s c a l a r s  i s a continuous f u n c t i o n  from X x X  i s a continuous f u n c t i o n  from  X.  Definition.  A subset K of a v e c t o r space X i s s a i d  whenever i t c o n t a i n s x and y, i t a l s o  c o n t a i n s the e n t i r e  to be convex i f l i n e segment  L ( x , y ) = {z: z = Xx + ( l - X ) y : 0 < X < l} j o i n i n g x and y.  0.8  Definition.  A topological  v e c t o r space X i s l o c a l l y  i s a l o c a l base B whose members a r e convex  0.9  Definition.  Frechet  0.10  convex i f t h e r e  sets.  A complete m e t r i z a b l e l o c a l l y  convex space i s c a l l e d a  A topological  to be compact i f every open  space.  Definition.  space X i s s a i d  c o v e r i n g U o f X has a f i n i t e s u b c o v e r i n g .  0.11 it  Definition.  A subset K of a t o p o l o g i c a l  space X i s c a l l e d compact i f  i s compact as a s u b s e t of X.  0.12  Definition.  A space X i s s a i d  to be s e q u e n t i a l l y  i n f i n i t e sequence from X i n t o X has a convergent  compact i f every  subsequence.  7. 0.13  Definition.  A space X i s s a i d t o be c o u n t a b l y compact i f every  c o u n t a b l e open c o v e r i n g has a f i n i t e s u b c o v e r i n g .  0.14  Lemma [29, p. 163].  c o u n t a b l e compactness We theorem  a r e now  and s e q u e n t i a l compactness  compactness,  are equivalent.  i n the p o s i t i o n to quote the Schauder-Tychonoff f i x e d  point  [17, p. 405].  Theorem 0.1  ( S c h a u d e r - T y c h o n o f f ) . L e t B be a l o c a l l y  v e c t o r space. map  For a m e t r i c space the n o t i o n s of  convex,  topological  L e t Y be a compact, convex s u b s e t of B and T a c o n t i n u o u s  of Y i n t o i t s e l f . Theorem 0.1  was  t h a t B i s a Banach  Then T has a f i x e d  p o i n t y e Y, i . e .  f i r s t proved by Schauder  Ty = y.  [32] under the assumption  space and t h i s case of the theorem i s u s u a l l y  "Schauder's f i x e d p o i n t theorem".  F o r a p r o o f of Theorem 0.1,  called  see  Tychonoff [40], Although Theorem 0.1 theorem,  we  i s the s t a n d a r d Schauder-Tychonoff f i x e d  f i n d i t more a p p l i c a b l e to use the f o l l o w i n g c o r o l l a r y  C o r o l l a r y 0.2  [17, p. 405].  L e t B be a l i n e a r , l o c a l l y  point instead.  convex,  t o p o l o g i c a l , complete H a u s d o r f f space ( e . g . , l e t B be a Banach or F r e c h e t space).  L e t Y be a c l o s e d , convex subset of B and T a c o n t i n u o u s map  into i t s e l f a fixed  such t h a t the image TY o f Y has a compact c l o s u r e .  of Y  Then T has  p o i n t y e Y.  To see how reader may  C o r o l l a r y 0.2  i s o b t a i n e d from Theorem 0.1,  wish to c o n s u l t Hartman [17, p. 405].  r e f e r t o C o r o l l a r y 0.2  From now  the i n t e r e s t e d on, we  as the Schauder-Tychonoff f i x e d p o i n t  will  theorem.  8. In o r d e r point  t o v e r i f y t h a t t h e hypotheses o f t h e Schauder-Tychonoff f i x e d  theorem a r e met we use two other  theorems; namely Lebesgue's  dominated convergence theorem t o show T i s a c o n t i n u o u s mapping and t h e A s c o l i - A r z e l a theorem to show Ty has compact  closure.  Theorem 0.3 - Lebesgue's Dominated Convergence Theorem. L e t g be i n t e g r a b l e over E and l e t ( f ) be a sequence of measurable n  functions  such that  |f | < g n  on E f o r every n =  f ( x ) = l i m f (x) f o r almost a l l x i n E. n n-»-°°  /  f  =  E  0.15 D e f i n i t i o n .  Then  lim / f . E  A f a m i l y F of r e a l valued  ( f i n i t e or i n f i n i t e )  1,2,.. and  i s c a l l e d uniformly  functions  i n an i n t e r v a l I  bounded i n I i f and only  i f there  e x i s t s M > 0 (independent of f ) such t h a t  |f(x)| < M f o r a l l x e I and f e F.  0.16 D e f i n i t i o n .  A f a m i l y F of f u n c t i o n s  from a t o p o l o g i c a l space X i n t o  R i s c a l l e d equicontinuous a t the point x e X i f given an open s e t 0 c o n t a i n i n g x such that a l l f e F.  The f a m i l y  e > 0 there i s  | f ( x ) - f ( y ) | < e f o r a l l y i n 0 and  i s s a i d t o be e q u i c o n t i n u o u s on X i f i t i s  e q u i c o n t i n u o u s at each p o i n t x i n X.  Theorem 0.4 - A s c o l i - A r z e l a Theorem L e t F be an e q u i c o n t i n u o u s u n i f o r m l y functions  bounded f a m i l y o f r e a l  f on an i n t e r v a l I ( f i n i t e or i n f i n i t e ) .  valued  Then F c o n t a i n s  a  9. u n i f o r m l y convergent f  sequence o f f u n c t i o n s f ^ , c o n v e r g i n g t o a  e C ( I ) where C ( I ) denotes  on I .  Thus any  c o n s e q u e n t l y F has  F a c t o r i z a t i o n of D i s c o n j u g a t e In  of a l l c o n t i n u o u s  bounded f u n c t i o n s  sequence i n F c o n t a i n s a u n i f o r m l y bounded  subsequence on I and  3.  the space  convergent  compact c l o s u r e i n C ( I ) .  Operators  the c o u r s e o f t h i s t h e s i s , we  e x i s t e n c e of bounded and  function  adopt  two  techniques f o r proving  unbounded s o l u t i o n s of second  order s e m i l i n e a r  d i f f e r e n t i a l e q u a t i o n s o f the g e n e r a l type:  ( r ( t ) y ' ) ' + h(t,y)  The  first  technique g i v e s n e c e s s a r y and  =0,  t > 0 .  sufficient  (0.1)  conditions for a l l  s o l u t i o n s o f the d i f f e r e n t i a l e q u a t i o n above t o have a s p e c i f i c  asymptotic  behaviour.  approach,  first  The  second  t e c h n i q u e , which i s somewhat an i n d i r e c t  c o n s i d e r s the l i n e a r d i f f e r e n t i a l  equation  Lz = ( r ( t ) z ' ) ' + q ( t ) z =  The  operator L i s factored  into  two  real f i r s t  0 .  order o p e r a t o r s and  it is  l a t e r shown t h a t the n o n l i n e a r d i f f e r e n t i a l e q u a t i o n under study  has  solutions y ^ ( t ) ,  solutions  of  2  the l i n e a r e q u a t i o n .  for z (t) 2  y ( t ) w i t h the same asymptotic  behaviour  as the  T h e r e f o r e , the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s  e x i s t e n c e of such s o l u t i o n s are g i v e n i n terms of s o l u t i o n s of Lz = 0.  Here, we  e x p l a i n the f a c t o r i z a t i o n t e c h n i q u e of  order l i n e a r o p e r a t o r s . C o n s i d e r the l i n e a r  Zj(t),  equation  second  10. (0.2)  Ly = y" + • ( x ) y ' + <(> (x)y = 0 1  2  where each <j>^ I s r e a l - v a l u e d and continuous i n I = (0,°°).  Our g o a l here i s  t o show how L c a n be expressed as a product o f f i r s t o r d e r  linear  operators. An if  operator  L o f type (0.2) i s c a l l e d n o n o s c i l l a t o r y a t  each n o n t r i v i a l  s o l u t i o n o f (0.2) has a t most one zero  0 0  i f and o n l y  i n some i n t e r v a l  [a,~). The  Wronskian of n f u n c t i o n s u.^ e C  n  ''"(I), i = l , . . . , n i s d e f i n e d as  u s u a l by  W = det(u  and  will It  ( j i  "  1 )  ),  i,j=l,...,n ,  a l s o be denoted by [ u ^ - . - . u ] . i s easily  seen t h a t t h e o p e r a t o r  L can be u n i q u e l y  recovered  from a  fundamental s e t { z , z } of s o l u t i o n s o f (0.2) by the formula 1  2  [z ,z ,y] x  *  2  • -fiTT^r •  (0  We say t h a t L corresponds to the fundamental s e t { z , z } » 1  2  -  3)  C l e a r l y to a  system o f f u n c t i o n s z ^ ( x ) , z ( x ) d e f i n e d on I t h e r e corresponds some 2  operator C(I)  L with c o e f f i c i e n t s that are continuous i n I i f and o n l y i f  ( i = l , 2 ) and [ z j , z ] ( x ) * 0 i n I . 2  We c a n now e x p l a i n the f a c t o r i z a t i o n o f the n o n o s c i l l a t o r y (disconjugate)  operator  L  [25,28,39].  e  11.  Theorem 0 . 5 .  The o p e r a t o r L i s n o n o s c i l l a t o r y a t  0 0  i f and o n l y i f i t has a  factorization -  L  i n some i n t e r v a l  =  Proof.  0,  h  _ 1  k  PI"1  PO"  co-*)  1  [ a , ) , where 00  P  Lz  P2  =  z  0  i»  Pi  (.z /z )\  =  2  i = 1 , 2 , zj > 0  p  1  and  =  2  ( P Q P ^  -  ,  1  [ z , z ] * 0 i n [a,°») 1  2  I f L i s n o n o s c i l l a t o r y a t °°, s o l u t i o n s z^ o f ( 0 . 2 )  d e s c r i b e d above and each p^, i n ( 0 . 5 ) 2  [ a , ) , with p 00  i  —  e x i s t as  i s w e l l d e f i n e d i n some i n t e r v a l  i  C  e  ( 0 . 5 )  [a, ), i = 00  For any y  0,1,2.  C [a,°°),  calculation  2  e  shows t h a t d  y  7-  [  l '  Z  dx z  d X  These a r e , i n f a c t ,  [z z ] lt  ^  proving  ( 0 . 4 ) .  2  z^  1  th  7  Z l  2  cases of a well-known p r o p e r t y o f Wronskians  dx [ z  Generalizations to n  (°- >  [ ,z ]2  and  [z,,y]  The converse  2  •  Z  ( 0 . 3 ) , ( 0 . 6 ) ,  d  1  =  l l» 2l  special  Combination o f  [ 2 8 ] .  z [z ,z ,y]  r  Z  ( 0 . 6 )  2  [zi>y]  T - i  l  ;  x  j  y  -  i }  ( 0 . 7 )  z  z ] 2  Z  2  yields  ' d  Z  l  d  2  y  l ^ z ^ dx ^ • [ z , z ] dx ^ z ^ ^ 1  2  i s obvious. o r d e r l i n e a r o p e r a t o r s L appear i n [ 2 5 , 3 9 ]  12. 4.  Examples We  now  technique  g i v e two  examples to i l l u s t r a t e f u r t h e r how  the d e s c r i b e d  works.  Example 0.6.  f a c t o r i z e d form. from (0.5)  d  The n o n o s c i l l a t o r y o p e r a t o r L =  that p  Example 0.7.  2  dx  i s , of course, a l r e a d y i n 2  As a check of ( 0 . 4 ) , take z^(x) = 1, z ( x ) = x and 2  = 0  The  Pi  =  P  obtain  1.  = 2  d i f f e r e n t i a l o p e r a t o r L d e f i n e d by  Ly  f o r a p o s i t i v e constant k,  =  y" - k y 2  ,  i s n o n o s c i l l a t o r y s i n c e Lz = 0 has  linearly  independent s o l u t i o n s / \ kx kx z,(x) = e , z (x) = e _  2  In t h i s case (0.5)  p (x) 0  and  becomes  = e  k X  ,  p (x)  = 2ke  x  2 k X  , p (x) 2  = (2k) e _ 1  ^  ,  the f a c t o r i z e d form of L i s g i v e n by .. . (Ly)(x) / T  unique up  to a m u l t i p l i c a t i v e  =  e  kx  r  -2kx, kx , -i, [e (ye )'J' ,  constant.  A d d i t i o n a l examples, of l e s s t r i v i a l type, a r e g i v e n i n Chapter Section  3,  4.  Given  t h i s background, we  can now  d i s c u s s the asymptotic  s o l u t i o n s of a c l a s s of second order d i f f e r e n t i a l  equations.  behaviour  of  13. CHAPTER I ASYMPTOTIC PROPERTIES OF  1.  SEMILINEAR ORDINARY DIFFERENTIAL EQUATIONS  Introduction Necessary and  of an  eventually  s u f f i c i e n t c o n d i t i o n s w i l l be d e r i v e d  existence  p o s i t i v e s o l u t i o n of  y" ± f ( x , y ) = 0  under the  f o r the  following  x > a  (1.1±)  hypotheses:  (i)  f i s a c o n t i n u o u s p o s i t i v e v a l u e d f u n c t i o n i n [a,°°)x(0,«>) .  (ii)  f i s monotone i n y f o r each f i x e d x,  i.e., f i s either  nonincreasing  or n o n d e c r e a s i n g i n y f o r each f i x e d x e [a,°°).  By  a s o l u t i o n of e q u a t i o n (1.1±) we  continuous and  satisfies  (1.1±) on  i s c a l l e d o s c i l l a t o r y i f i t has called  some h a l f - l i n e  arbitrarily  [x ,°°).  large zeros,  half-line consider  l a r g e x.  [ x ,<*>) Q  only  A p o s i t i v e s o l u t i o n y(x)  otherwise i t i s one  sign for a l l  of (1.1±) d e f i n e d  i s c a l l e d a proper p o s i t i v e s o l u t i o n .  i n some  In t h i s t h e s i s ,  n o n o s c i l l a t o r y s o l u t i o n s of the e q u a t i o n under s t u d y .  Bounded A s y m p t o t i c  Solutions  In t h i s s e c t i o n we  o b t a i n n e c e s s a r y and  s u f f i c i e n t conditions  (1.1±) to have s o l u t i o n s which behave a s y m p t o t i c a l l y constants.  which i s  Such a s o l u t i o n  0  n o n o s c i l l a t o r y , i . e . , a n o n o s c i l l a t o r y s o l u t i o n has  sufficiently  2.  mean a f u n c t i o n y(x)  like  nonzero  for  we  14. Theorem 1.1.  A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r (1.1±) t o have a  p o s i t i v e s o l u t i o n i n some i n t e r v a l x •*•  00  [ x ,«>) with a f i n i t e p o s i t i v e l i m i t as Q  i s that x / a  (1.2)  x f ( x , c ) dx < -  holds f o r some constant c > 0. The p r o o f w i l l be g i v e n f o r (1.1±) under the assumption  that f(x,y) i s  nonincreasing.  Proof: (1.1+).  We f i r s t prove t h e s u f f i c i e n c y of (1.2) i n t h e case o f e q u a t i o n Choose x  > a l a r g e enough so t h a t  Q  / x  x f ( x , c ) dx < c .  0  L e t B = C[x ,°°) be the space of a l l continuous 0  [XQ ,») w i t h norm IIyII = sup | y ( x ) | . x>x  bounded f u n c t i o n s y i n  L e t Y be c l o s e d convex subset o f B  Q  d e f i n e d by Y  =  {y  e  B: c < y ( x ) < 2c, x > x } . Q  Let T: B + B be the mapping d e f i n e d by  ( T y ) ( x ) = 2c - / x  Any  ( t - x ) f ( t , y ( t ) ) d t ; x > x0 k  n  .  (1.3)  s o l u t i o n of the i n t e g r a l e q u a t i o n i s a s o l u t i o n of the d i f f e r e n t i a l  e q u a t i o n (1.1+). (1.3),  then  I f y i s a continuous  s o l u t i o n o f the i n t e g r a l  equation  15.  y'(x)  =  / x  f ( t , y ( t ) ) dt;  y"(x)  =  -f(x,y(x)) .  Furthermore, y(x) d e c r e a s e s to the l i m i t 2c as x + °°. In  o r d e r t o a p p l y the Schauder-Tychonoff f i x e d  to v e r i f y  p o i n t theorem, we  need  that:  a)  T maps Y i n t o  Y;  b)  T i s a c o n t i n u o u s mapping on Y ( i n the B norm);  c)  TY i s r e l a t i v e l y  compact where, TY = {Ty: y e Y}.  To prove (a) l e t y e Y so c < y(x) < 2c  for  x > x .  Then f o r t > x > x  Q  Q  oo  (t-x)  f(t,y(t))  converges  < t f(t,c).  for a l l x > x  / x  n  T h e r e f o r e the i n t e g r a l  / (t-x) f ( t , y ( t ) ) dt x  by (2.1) and f u r t h e r m o r e i t s a t i s f i e s  ( t - x ) f ( t , y ( t ) ) dt < / x  t f ( t , c ) d t < c. Q  Hence oo  2c > ( T y ) ( x )  proving  that Tye  =  2c - / x  ( t - x ) f ( t , y ( t ) ) dt > 2c - c = c,  Y.  To prove ( b ) , l e t { y ) be a sequence n  to ye B.  S i n c e Y i s c l o s e d , y e Y.  i n Y t h a t converges i n the B norm  From the d e f i n i t i o n  of T,  | T y ( x ) - T y ( x ) | = |/ ( t - x ) [ f ( t , y ( t ) ) - f ( t , y ( t ) ) ] d t j . x n  n  But the i n t e g r a n d has l i m i t 0 as n •*•  00  for a l l t > x > x  n  since  16. sup |y ( t ) - y ( t ) | + 0 t>x  as  n •*• » ,  n  0  and  f i s continuous.  Also  the  integrand  i s bounded i n a b s o l u t e  value  by  oo  t f ( t , c ) ( s i n c e c < y ( t ) < 2 c ) , which has  f i n i t e i n t e g r a l / t f ( t , c ) dt < x  Hence l i m IITy -Tyll = 0 by Lebesgue's dominated convergence theorem. n-*-°°  c.  T is  n  therefore To and (a).  continuous.  prove ( c ) , we  equicontinuous.  need to show t h a t the f a m i l y T Y i s u n i f o r m l y Obviously  T Y i s uniformly  To prove e q u i c o n t i n u i t y , l e t z = Ty  E T Y.  oo  |z'(x)| =  |J  f ( t , y ( t ) ) dt|  < / X  x  2  =  2  proves e q u i c o n t i n u i t y  0  assumption.  Ix^x^  For  < 6 the Mean-Value  shows t h a t T has  <  2  C(E/C)  2  =  £  and  .  y E Y.  By  compact c l o s u r e . Schauder-Tychonoff f i x e d p o i n t  f i x e d p o i n t y E Y,  D i f f e r e n t i a t i o n of the  which d e c r e a s e s to the  - x |  1  a t l e a s t one  completes the p r o o f  :  s i n c e 6 i s independent of x , x  In view of ( a ) , ( b ) , (c) the  before,  i n y by  i n [x ,°°) w i t h  | z ' U ) | |x  the A s c o l i - A r z e l a theorem, TY has  x > XQ.  t f ( t , c ) dt < c ,  gives  |z(x ) - z ( x ) |  This  Then  Q  f ( t , y ( t ) ) i s nonincreasing  e > 0 l e t 6 = E / C , then f o r x, , x Theorem  shown i n p a r t  oo  X  s i n c e y ( t ) > c and  bounded, as  bounded  i . e . , Ty(x)  = y(x)  i n t e g r a l e q u a t i o n Ty = y twice,  t h a t y(x)  theorem for a l l  as shown  i s a p o s i t i v e s o l u t i o n of (1.1+)  l i m i t 2c as x + °°.  17. we w i l l show t h a t (1.2) i s a n e c e s s a r y c o n d i t i o n f o r (1.1+) t o  Now  have a p o s i t i v e s o l u t i o n which behaves a s y m p t o t i c a l l y  l i k e a nonzero  constant. i s a p o s i t i v e s o l u t i o n o f (1.1+) and a l s o y(x)  Suppose t h a t y(x) to a f i n i t e  positive limit.  y'(x)  S i n c e y"(x)  < 0.  so y'(x)  We w i l l  < 0, then y'(x)  < - | ot| f o r x > x .  y(x) -  Therefore eventually  Q  =  y'(c)  < -|«|  x  0  < 5 < x .  XQ  and we conclude t h a t y(x)  Q  f o r x l a r g e enough.  p o s i t i v i t y o f y ( x ) , so y'(x)  impossible  function,  y(x )  Q  a decreasing  decreasing  Suppose  Q  y(x) < - |a|(x - x ) + y ( x ) negative  i s a negative  = 0.  Now f o r x > x , by the Mean-Value Theorem,  Q  — — — x  prove that the l i m y'(x)  tends  > 0  But t h i s i s a c o n t r a d i c t i o n t o the  for x > x . Q  Furthermore, s i n c e y'(x) i s  f u n c t i o n , i t f o l l o w s t h a t l i m y'(x)  f o r l i m y'(x) x->-°°  > 0 since  > l i m y'(x) x*°°  Q  f o r c e s l i m y(x) = + <=.  > 0 exists.  then  y(x) - y ( x )  Thus l i m y'(x)  (x-x ) Q  = 0.  X+oo  I n t e g r a t i o n o f (1.1+) twice y i e l d s :  y'(x)  = -/  - y(x )  = -/  X  y(x)  Q  x  0  w i l l be  f ( t , y ( t ) ) dt ,  oo  / s-t  f ( s , y ( s ) ) ds d t  It i s  18. S i n c e y ( x ) i s bounded and d e c r e a s i n g y(x) •»• °°.  t h e above has a f i n i t e l i m i t as  Changing the order of i n t e g r a t i o n , we have:  X  oo  0 0  — < "J (/ f(s,y(s)) x o  g  d s ) d t = -/  z  x  0  (/ x  d t ) f ( s , y ( s ) ) ds  0  00  = _/ x  ( s - x ) f ( s , y ( s ) ) ds . Q  0  00  Therefore,  / x  s f ( s , y ( s ) ) ds <  0 0  s i n c e y ( s ) i s bounded, say  0  0 < y(s) < K  for  x > x  and  Q  f ( s , y ( s ) ) i s nonincreasing  i n y. I t  follows that:  00  / x  s f ( s , K ) ds < °°  0  f o r some K > 0. In t h e case o f (1.1-) the mapping  (1.3) i s r e p l a c e d by  00  (Ty)(x) = c + / x  and  virtually  ( t - x ) f ( t , y ( t ) ) dt  the same argument as before y i e l d s  p o s i t i v e s o l u t i o n o f (1.1-) w i t h l i m y ( x ) = c. x->-°° proven s i m i l a r l y .  x > x,  the e x i s t e n c e of a The n e c e s s i t y p a r t i s  I t can be shown as b e f o r e t h a t l i m y'(x) = 0, and hence  i n t e g r a t i o n o f (1.1-) and i n t e r c h a n g i n g t h e o r d e r o f i n t e g r a t i o n g i v e s :  19. 00  y'(x)  =  /  =  / x  f ( t , y ( t ) ) dt  X  y(x) - y ( x ) Q  Q  oo  oo  (/ f ( s , y ( s ) ) ds) d t = / s=t x  ( s - x ) f ( s , y ( s ) ) ds Q  Q  As b e f o r e , y ( s ) i s bounded and f ( s , y ) i s n o n i n c r e a s i n g , so  / x  Now,  s f ( s , K ) ds < /  0  x  s f ( s , y ( s ) ) ds <  0 0  for s > x  i f we r e p l a c e the ' n o n i n c r e a s i n g ' assumption on f ( x , y ) by  'nondecreasing', the f o l l o w i n g changes must be made. x  Q  We have t o choose  > a l a r g e enough so t h a t  / x  and  Q  0  x f ( x , c ) dx <  0  r e p l a c e Y by  Y = {y e B:  T  < y ( x ) < c}  A p a r a l l e l argument as b e f o r e e s t a b l i s h e s f i x e d p o i n t s i n Y o f the mappings  (TY)(x) = c - / ( t - x ) f ( t . y ( t ) ) dt x  x > x  Q  ;  x > x  Q  ,  00  (TY)(x) = f + / ( t - x ) f ( t , y ( t ) ) dt x  20. y i e l d i n g p o s i t i v e s o l u t i o n s o f (1.1+), ( 1 . 1 - ) , r e s p e c t i v e l y ,  satisfying  c l i m y(x) = c and l i m y(x) = y . If f ( x , y ) i s nondecreasing positive f i n i t e  and y ( x ) i s a p o s i t i v e s o l u t i o n w i t h  l i m i t as x-*-°°, then y ( x ) > K > 0  f(s,y(s)) > f(s,K) f o r s > x  Q  for x > x . Q  Therefore  and t h e n e c e s s i t y p r o o f i s completed i n the  same way as b e f o r e .  Example 1.2:  Consider  ^  the d i f f e r e n t i a l  (x  equation  + xf(x,y) = 0  where f ( x , y ) i s continuous,  x > a > 1  p o s i t i v e and n o n i n c r e a s i n g i n y.  (1.4)  We w i l l  prove  t h a t t h e above e q u a t i o n has a bounded p o s i t i v e s o l u t i o n i n some i n t e r v a l [x ,°°) i f and o n l y i f Q  oo / a  x l o g x f ( x , c ) dx < o°  (1«5)  f o r some c > 0.  Proof:  I t w i l l be shown t h a t L i o u v i l l e ' s change o f v a r i a b l e s x = e , y ( x )  = Y ( s ) transforms  (1.4) i n t o the form (1.1+), and a c c o r d i n g l y Theorem 1.1,  can be a p p l i e d .  By  the c h a i n r u l e ,  dy dx  _  dY ds  j  ds ^ dx  1 e  dY s ds '  21. Then t h e d i f f e r e n t i a l e q u a t i o n  + e  2s  f(e , S  The change of v a r i a b l e x = e  (1.4) reduces t o  Y(s))  log  for  some p o s i t i v e constant Let F(s,Y) = e  e  2s  2s  0 ,  s e  2s  a  f(e  g  ,c) ds <  (1.7)  0 0  c.  s f ( e ,Y). C l e a r l y F(s,Y) i s nonincreasing  s i s independent o f Y and f ( e ,Y) i s n o n i n c r e a s i n g  equation  (1.6)  s > log a .  t r a n s f o r m s (1.5) i n t o  00  /  =  i n Y.  i n Y since  In t h e c a s e o f  ( 1 . 6 ) , c o n d i t i o n (1.2) has the form ( 1 . 7 ) , and hence Theorem  1.1  shows t h a t (1.7) i s n e c e s s a r y and s u f f i c i e n t f o r (1.6) t o have a bounded positive  s o l u t i o n Y ( s ) i n some i n t e r v a l  [ S g , ) , i . e . , f o r (1.4) to have a 0 0  bounded p o s i t i v e s o l u t i o n y ( x ) i n [exp S g , ) . 0 0  3.  Unbounded A s y m p t o t i c a l l y L i n e a r  Solutions  The purpose o f t h i s s e c t i o n i s t o o b t a i n n e c e s s a r y and s u f f i c i e n t conditions  f o r (1.1±) to have s o l u t i o n s which behave a s y m p t o t i c a l l y  like  cx (c * 0) as x •+• °°.  Theorem 1.2.  A n e c e s s a r y and s u f f i c i e n t  p o s i t i v e s o l u t i o n y ( x ) i n some i n t e r v a l  c o n d i t i o n f o r (1.1±) to have a [x ,°°) w i t h l i m y ( x ) / x p o s i t i v e and x>°° 0  f i n i t e i s that  00  / f ( x , c x ) dx < «> a  (1.8)  22. holds f o r some p o s i t i v e c o n s t a n t  Proof:  The  proof  f(x,y) i s  w i l l be g i v e n  c.  first  f o r (1.1-) under the assumption  that  nonincreasing.  Choose X Q > a l a r g e enough so  that  00  / X  Let Y =  {y e B:  cx < y(x)  < 2cx,  a l l c o n t i n u o u s f u n c t i o n s on  llyll = sup X>X  B be  |y(x)/x|.  f ( x , c x ) dx < c  x > x} Q  C[XQ,°°)  i s the  space o f  i s bounded w i t h norm  C l e a r l y Y i s a c l o s e d , convex subset of B.  Let T:  B -»•  Q  the mapping d e f i n e d  by  ( T y ) ( x ) = 2cx - / x  need to v e r i f y  cx < y(x)  where B =  such t h a t y ( x ) / x  [XQ,»)  x  We  .  Q  X  Q  x  / f ( s , y ( s ) ) ds dt < / t X  Hence cx < Ty(x)  < 2cx  so  Ty e  TY =  {Ty:  00  (1.9)  Q  so  / f ( s , c s ) ds  x dt < c /  XQ  Q  for a l l x > x . 0  dt < cx  ,  XQ  we  can  e a s i l y v e r i f y that T i s a  y e Y} has  compact c l o s u r e .  Tychonof f f i x e d p o i n t theorem then shows t h a t T has ( T y ) ( x ) = y(x)  x > x .  Y.  P r o c e e d i n g as i n Theorem 1.1, continuous mapping and  Let y e Y  ,  Then  oo  x  0 < /  / f ( s , y ( s ) ) ds dt t  that T maps Y i n t o Y.  for x > X Q .  < 2cx  Q  00  The  Schauder-  a f i x e d p o i n t y e Y,  D i f f e r e n t i a t i o n of t h i s  i n t e g r a l equation  twice completes the p r o o f the  property To  t h a t y(x)  that l i m y(x)/x =  show c o n d i t i o n (1.8)  i s a p o s i t i v e s o l u t i o n of (1.1-) w i t h  2c.  i s n e c e s s a r y , l e t y(x)  of (1.1-) i n [x ,°°) such t h a t y ( x ) / x  has  Then, t h e r e e x i s t p o s i t i v e c o n s t a n t s  Ki  0  K x  < y(x)  :  where x^  > x  Q  i n c r e a s i n g and  we  a finite positive limit and K,  < K x  large.  Since  at «>.  such t h a t  for  2  is sufficiently  be a p o s i t i v e s o l u t i o n  x > x  y" > 0,  x  y'(x)  ,  (1.10)  is positive  and  have  lim x*  y'(x)  < lim y(x)/x x+°°  00  <  K. 2  I n t e g r a t i o n of (1.1-) y i e l d s  x / f ( s , y ( s ) ) ds = y'(x) x  and  - y'(  X l  ) < y»(x)  ;  x > x  x  ,  l  s i n c e l i m y'(x)  < lim x->-°°  x->-oo  <  00  by assumption, the above i m p l i e s  oo  / x  Since  f ( s , y ( s ) ) ds <  2  .  l  f(x,y) i s nonincreasing,  f(x,K x),  00  y(x)  < K x 2  implies  that f ( x , y ( x ) ) >  so OO  / x  l  00  f(s,K s) 2  ds < / x  l  f ( s , y ( s ) ) ds < °° ,  that  24. proving  t h e n e c e s s i t y p a r t o f the theorem.  In the case o f (1.1+), r e p l a c e  (1.9) by  x ( T y ) ( x ) = cx + / / x t  0 0  f ( s , y ( s ) ) ds d t ,  Q  and  complete the proof  as b e f o r e .  The n e c e s s i t y p a r t i s v i r t u a l l y t h e  same.  Remark:  Because of the n o n i n c r e a s i n g  y, c o n d i t i o n (1.2) i m p l i e s existence  of f ( x , y ) as a f u n c t i o n o f  (1.8) and consequently (1.2) guarantees the  o f two p o s i t i v e s o l u t i o n s y ^ ( x ) and y ( x ) such t h a t both y ^ x ) 2  and  y (x)/x  have f i n i t e p o s i t i v e l i m i t s a t  lim  y (x)/y (x)  2  1  (1.2),  property  2  = 0.  On the other  0 0  and so i n p a r t i c u l a r  hand, i f (1.8) i s s a t i s f i e d  but not  then every p r o p e r p o s i t i v e s o l u t i o n y ( x ) i s unbounded w i t h  lim y(x)/x x+°° The  finite  and p o s i t i v e .  r e s u l t s o f theorem 1.2 can be proved s i m i l a r l y  f ( x , y ) i s nondecreasing i n y.  To prove n e c e s s i t y ,  (1.10) i n s t e a d o f t h e second i n e q u a l i t y should sufficiency  p a r t , choose x  Q  i n the case that  the f i r s t  be used.  inequality i n  To prove the  > a l a r g e enough so t h a t  00  f ( x , c x ) dx < y . x  Let Y = {y e B: theorem s h o u l d  0  x < y ( x ) < cx}.  The Schauder-Tychonoff f i x e d  be a p p l i e d t o the mappings  point  25. x ( T y ) ( x ) = cx - / x  Q  x ( f y ) ( x ) - |2L+ / x  0  00  / t  f ( s , y ( s ) ) ds d t ,  x > x  Q  ;  f ( s , y ( s ) ) ds d t ,  x > x  Q  ;  0 0  / t  to y i e l d  p o s i t i v e s o l u t i o n s of ( 1 . 1 - ) , (1.1+), r e s p e c t i v e l y , with t h e  property  t h a t l i m y ( x ) / x = c, l i m y ( x ) / x = x->-°° x->-°°  4.  Asymptotic  P r o p e r t i e s of the Emden-Fowler  The g e n e r a l i z e d Emden-Fowler  Equation  equation  y" ± p ( x ) y  Y  = 0,  x > a  (l.H±)  i s a s p e c i a l case of (1.1±) f o r which  f(x,y) = p(x)y  where y i s a r e a l constant valued f u n c t i o n .  Y  and p: [a,°°) + (0,«>) i s a continuous  positive  We note t h a t f ( x , y ) i s monotone i n c r e a s i n g i f y > 0 and  monotone d e c r e a s i n g i f y < 0.  Furthermore, e q u a t i o n  (1.11±) may  be  c l a s s i f i e d as s i n g u l a r , s u b l i n e a r o r s u p e r l i n e a r a c c o r d i n g t o whether Y<0,  0 < y < 1 or T > 1, r e s p e c t i v e l y . C o n d i t i o n s (1.2) and (1.8) reduce t o , r e s p e c t i v e l y ,  x / xp(x) dx < • ; a  (1.12)  x / a  x p ( x ) dx < » . Y  (1.13)  26. These r e s u l t s a r e a c t u a l l y w e l l known theorems o f A t k i n s o n  [3] and  Belohorec [ 4 ] .  Theorem 1.3  [Atkinson]:  L e t y > 1.  A l l s o l u t i o n s of (1.11±) a r e non-  o s c i l l a t o r y i f and only i f  oo  /  Theorem 1.4  x p(x) dx <  L e t 0 < y < 1.  [Belohorec]:  00  .  A l l s o l u t i o n s of (1.11±) a r e  n o n o s c i l l a t o r y i f and o n l y i f  oo  /  Remark 2. and  x^ p(x) dx < «> .  Upon examining these theorems and our r e s u l t s i n Theorems  1.2 we note t h a t i f y > 1, then (1.13) i m p l i e s  guarantees the e x i s t e n c e  Hence,  (1.13)  of two proper p o s i t i v e s o l u t i o n s y , ( x ) and y ( x ) 2  such t h a t both y ^ ( x ) and y ( x ) / x 2  have f i n i t e p o s i t i v e l i m i t s as x+°°.  Consequently, the i n t e g r a l c o n d i t i o n s other  (1.12).  1.1  and we have the f o l l o w i n g  (1.12) and (1.13) become dual  to each  properties:  (a)  the s u p e r l i n e a r e q u a t i o n has an a s y m p t o t i c a l l y  constant  (b)  the s u b l i n e a r e q u a t i o n has an unbounded a s y m p t o t i c a l l y  solution; linear  solution. These r e s u l t s can be summarized as c o r o l l a r i e s t o Theorems 1.1 and  C o r o l l a r y 1.5.  L e t y > 0.  A necessary and s u f f i c i e n t  (1.11±) t o have a bounded a s y m p t o t i c a l l y  constant  1.2.  condition for  s o l u t i o n i s that  27.  / x p(x) dx a  < »  holds.  L e t y > 0. A necessary  C o r o l l a r y 1.6.  and  sufficient  to have an unbounded a s y m p t o t i c a l l y l i n e a r  c o n d i t i o n f o r (1.11±)  solution i s that  oo  / a  x  Y  p(x)  dx < »  holds.  5.  A More G e n e r a l  Case  In the p r e v i o u s s e c t i o n , we the s p e c i a l case  (1.11±).  e s t a b l i s h e d the asymptotic  Here, we  shall  p r o p e t i e s of  c o n s i d e r a more g e n e r a l form of  (1.1±). Consider  the o r d i n a r y d i f f e r e n t i a l  (r(x)y')'  under the  iii)  equation  ± f(x,y) = 0 ;  x > a  (1.14±)  f o l l o w i n g hypotheses:  r : [a,°°) -»•  (0,°o)  i  s  continuous  and  satisfies  l i m R(x)  =  oo  where  x-»-oo X  R ( x )  =  ^  a  iv)  ds  7(17  '  a n d  f : [a, ) x (0,»)  •*•  C o n d i t i o n s (1.2)  and  00  (0,oo)  i  (1.4)  s  continuous  and  nonincreasing  i n y.  reduce t o , r e s p e c t i v e l y ,  00  /  R ( x ) f ( , c ) d x < oo ; X  (1.15)  28.  /  We have the f o l l o w i n g  Theorem 1.5. interval  f ( x , c R ( x ) ) dx < » .  (1.16)  a l t e r n a t i v e to Theorems 1.1 and 1.2.  E q u a t i o n (1.14±) has a p r o p e r p o s i t i v e s o l u t i o n y ( x ) i n some  [x ,°°) such that 0  y ( x ) has a f i n i t e p o s i t i v e l i m i t  as X-H» i f and  only i f  00  /  R(x)f(x,c)dx < »  holds f o r some p o s i t i v e c o n s t a n t c . Theorem 1.6.  E q u a t i o n (1.14±) has an e v e n t u a l l y  postiive solution y(x)  such t h a t y ( x ) / R ( x ) has a f i n i t e p o s i t i v e l i m i t a t °° i f and o n l y i f  00  /  f ( x , c R ( x ) ) d x < oo  holds f o r some p o s i t i v e c o n s t a n t c . Remark 3•  Because o f the n o n i n c r e a s i n g p r o p e r t y o f f ( x , y ) ,  (1.15) i m p l i e s  condition  e x i s t e n c e of two p o s i t i v e both  (1.16) and consequently (1.15) guarantees t h e solutions  y ^ x ) and y ( x ) o f (1.14±) such 2  limits lim  y (x) = K  X+°o  e x i s t and a r e p o s i t i v e .  x  condition  :  ,  • lim y (x)/R(x) = K 2  X-+-0  0  £  that  29. In t h e case o f s p e c i a l i z a t i o n  ( r ( x ) y ' ) ' = p(x)y  of  -X  X > 1  (1.17)  (1.14-), where r i s as b e f o r e and p: [a,»)-»-(0,<») i s c o n t i n u o u s ,  conditions  (1.15), (1.16) reduce t o , r e s p e c t i v e l y ,  00  /  R(x) p(x) dx <  (1.18)  OO  oo  /  We have the f o l l o w i n g  [R(x)]"  p(x) dx <  X  00  c o r o l l a r i e s to Theorems 1.5 and  (1.19)  1.6.  C o r o l l a r y 1.7.  C o n d i t i o n (1.18) i s n e c e s s a r y and s u f f i c i e n t f o r (1.17) to  have a p o s i t i v e  s o l u t i o n y ( x ) such that  y(x) has a f i n i t e  positive  limit  at <*>.  C o r o l l a r y 1.8.  C o n d i t i o n (1.19) i s n e c e s s a r y and s u f f i c i e n t f o r (1.17) to  have an e v e n t u a l l y positive  limit  p o s i t i v e s o l u t i o n y ( x ) such t h a t y ( x ) / R ( x ) has a  finite  at °°.  The below c o r o l l a r y i s an immediate consequence o f Remark 3.  Corollary  1.9.  eventually  C o n d i t i o n (1.18) i s s u f f i c i e n t f o r (1.17) to have two  p o s i t i v e proper s o l u t i o n s  and y ( x ) / R ( x ) have f i n i t e 2  In the next c h a p t e r we ordinary  differential  l i m i t s at  y^x) 00  and y ( x ) such t h a t b o t h 2  y^x)  .  s h a l l study asymptotic s o l u t i o n s o f q u a s i l i n e a r  e q u a t i o n s of second  order.  30. CHAPTER I I ASYMPTOTIC PROPERTIES OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS  1.  Introduction Necessary and s u f f i c i e n t c o n d i t i o n s w i l l be d e r i v e d f o r t h e e x i s t e n c e  of bounded p o s i t i v e s o l u t i o n s of q u a s i l i n e a r o r d i n a r y  differential  e q u a t i o n s o f t h e type  y" + y g U . y . y ' )  under  1)  the hypotheses  listed  =  0  x > 0  (2.1)  below.  g ( > y » P ) i s c o n t i n u o u s and nonnegative f o r 0 < x < ° ° , x  -°° < p <  00  0 < y < ° ° and  •  ii)  g(x,y,p)  i s e i t h e r n o n d e c r e a s i n g o r n o n i n c r e a s i n g i n y f o r each x,p.  iii)  g(x,y,p)  i s nondecreasing i n p i f p > 0 f o r each x,y.  The theorems [36] o f t h i s c h a p t e r extend r e s u l t s o f Coffman and Wong [ 8 ] , Nehari  2.  [26] and o t h e r s f o r the s e m i l i n e a r case y" + yg(x,y) = 0.  S u f f i c i e n t C r i t e r i a f o r E x i s t e n c e o f Bounded S o l u t i o n s In t h i s s e c t i o n , s u f f i c i e n t c o n d i t i o n s a r e g i v e n f o r (2.1) t o have a  positive  s o l u t i o n y ( x ) which  Schauder-Tychonoff  Theorem 2.1.  tends to a f i n i t e  f i x e d p o i n t theorem  limit  as x-*-°°.  The  w i l l be used i n t h e p r o o f .  E q u a t i o n (2.1) has a bounded p o s i t i v e s o l u t i o n y ( x ) i n some  i n t e r v a l ( x ,°°) i f t h e r e e x i s t p o s i t i v e c o n s t a n t s A and B such t h a t 0  00  /  Proof: Let  Case I.  xg(x,A,B) dx <  g(x,y,p) i s n o n i n c r e a s i n g  (2.2)  .  00  i n y f o r each  C be a number s a t i s f y i n g A < C < A + B and  choose x  1  x,p. > 0 l a r g e enough  so t h a t 00  C / x  Let  C  1  = C  1  dt < C - A  .  l  [ x ,°°) denote the l o c a l l y 1  continuously uniform  tg(t,A,B)  differentiable  convex v e c t o r space of a l l  f u n c t i o n s i n [x^, ) w i t h the topology  convergence of f u n c t i o n s and  s u b i n t e r v a l s of [ x , , ) , i . e . we 0 0  their f i r s t  d e r i v a t i v e s on  have the convergence y  of C  1  u n i f o r m l y on every  Consider  i f and  only i f y ( )  y ( ) and  x  x  n  compact s u b i n t e r v a l of  1  •*• y (as n > °°) i n  A < y(x) < C  and  C l e a r l y S i s a c l o s e d convex subset  ^ri^ ^  1  a  s  * °°  n  0 0  functions  0 < y'(x) < C-A  of C .  * ^ ' ^  X  [x^, ).  the s e t of c o n t i n u o u s l y d i f f e r e n t i a b l e  S = {y e C :  compact  n  1  the topology  of  00  Define  for  x >  .  the mapping T on S  by  00  (Ty)(x) = C + / x  We  (x-t)y(t)g(t,y(t),y'(t))dt,  need to v e r i f y T maps S i n t o S.  y'(x) <  C-A.  L e t y e S,  X  .  (2.3)  that i s A < y(x) < C, 0 <  Note t h a t  00  0 < /  x > ^  oo  (t-x)y(t)g(t,y(t),y'(t))dt  < C / t g ( t , A , B ) d t < C-A x^  .  32. Hence,  C > (Ty)(x) - C + / x  (x-t)y(t)g(t,y(t),y'(t))dt  > C-(C-A) = A.  (2.4)  < C / t g ( t , A , B ) d t < C-A x.  (2.5)  Furthermore,  0 < (Ty)'(x) = / x  for a l l x > x . x  To of  y(t)g(t,y(t),y'(t))dt  Therefore  Ty e  show T i s a continuous  S. mapping, l e t ( y ) he a convergent sequence n  f u n c t i o n s i n S to y e S i n the topology  of C . 1  Then  00  |(Ty )(x) - (Ty)(x)| < / ( t - x ) | y ( t ) x n  n  g(t,y (t), n  y^t))  " y ( t ) g ( t , y ( t ) , y ' ( t ) ) | dt .  The  i n t e g r a n d has  uniform  limit  zero on compact s u b i n t e r v a l s of  i s bounded above by Ctg(t,A,B) f o r t > X j , which has  [Xj, ) 0 0  finite integral.  Lebesgue's dominated convergence theorem,  lim  (Ty ) ( x ) =  (Ty)(x)  Similarly, lim  on every C^Xj, ) 0 0  (Ty  compact s u b i n t e r v a l . topology.  )'(x) =  (Ty)'(x)  T h e r e f o r e , T:  S •* S i s continuous  i n the  and By  33. To show t h a t TS = {Ty: y e S} i s r e l a t i v e l y compact, i t s u f f i c e s t o show t h a t TS i s u n i f o r m l y bounded and e q u i c o n t i n u o u s on [ X j , ) .  The  0 0  u n i f o r m boundedness o f {(Ty)(x)} and {(Ty)'(x)} i n [ x « ) i s obvious l t  (2.4) and ( 2 . 5 ) .  I t remains  e q u i c o n t i n u o u s i n [xj,°°) .  to show t h a t  from  {(Ty)(x)} and {(Ty)'(x)} a r e  F o r a l l y e S and x e [ X j , » ) ,  00  |(Ty)'(x)|  -  1/  y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t | < C-A x  by  (2.5).  By the Mean Value Theorem, ( T y ) ( x ) i s e q u i c o n t i n u o u s i n [ X j , ) . 0 0  Similarly,  | ( T y ) " ( x ) | = |-y g ( t , y ( t ) , y ' ( t ) ) |  < C g(t,A,C-A)  s i n c e y ( t ) > A, y ' ( t ) < C-A and g(x,y,p) i s nondecreasing increasing i n y. By A s c o l i ' s of  This implies that (Ty)'(x) i s equicontinuous i n [ x ^ , ) . 0 0  theorem (extended  to °°), there e x i s t s a convergent  subsequence  { ( T y ) ( x ) } , say {Ty ( x ) } , such t h a t l i m (Ty ) ( x ) = z ( x ) i n [ X j , * ) , the n-»-°°  convergence (y  i n p and non-  n  b e i n g u n i f o r m on any compact s u b i n t e r v a l .  F o r t h i s subsequence  }, c o n s i d e r {(Ty ) ' ( x ) } , a l s o u n i f o r m l y bounded and e q u i c o n t i n u o u s . n  T h i s has a convergent  subsequence {(Ty  )'(x)} such t h a t l i m (Ty  z'(x) u n i f o r m l y on any compact s u b i n t e r v a l o f [Xj ,°°) . relatively  compact.  By the Schauder-Tychonoff  fixed  T h e r e f o r e TS i s p o i n t theorem, T has a  f i x e d p o i n t y e S, i . e . ( T y ) ( x ) = y ( x ) f o r a l l x > X j . the i n t e g r a l e q u a t i o n Ty = y twice complete  )(x) =  D i f f e r e n t i a t i o n of  the proof that y ( x ) i s the  r e q u i r e d bounded p o s i t i v e s o l u t i o n o f (2.1) which tends t o the l i m i t C as  34.  Case I I . Let  g(x,y,p) i s n o n d e c r e a s i n g i n y f o r each f i x e d x,p.  C be a number s a t i s f y i n g max(A-B,0) < C < A and choose X j l a r g e enough  so t h a t  00  A / x  Let  tg(t,A,B) dt < A - C .  l  S = {y e C : C < y ( x ) < A  D e f i n e T: C  and 0 < y ' ( x ) < A-C  1  1  + C  1  f o r x > x,) .  by  x (Ty)(x) - C + / x  (t-  X l  )  y ( t ) g(t,y(t),y»(t)) d t  l  OO  + (x-  To show T maps S i n t o  X l  ) / x  y(t) g(t,y(t),y*(t))  S, l e t y e S.  dt .  Then f o r x > x  OO  (Ty)(x) < C + / x  (t-x )y(t)g(t,y(t),y'(t)) dt 1  l  OO  < C + A / x  t g(t,A,B) d t  l  < C + A - C = A  Therefore,  C < ( T y ) ( x ) < A.  .  35. Similarly,  0 < ( T y ) ' ( x ) = / y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t < A / g(t,A,B) d t < A-C x x  from which Ty e S. To show T i s a continuous  mapping, l e t  of f u n c t i o n s i n the topology of C  1  t o y e S.  be a convergent  sequence  Then,  (Ty )(x) - (Ty)(x)| n  < |/ x  (t-  X l  )  [y (t) g(t,y (t),y;(t)) - y(t) g[t,y(t),y'(t))] dtj n  n  l 00  +  |(x- ) X l  / x  [ y ( t ) g ( t , y ( t ) , y ; ( t ) ) - y(t) g(t,y(t),y'(t))] dt| . n  n  Since the i n t e g r a n d has uniform  limit  zero i n [x^ ,<*>) and i s bounded  above  by A t g ( t , A , B ) f o r t > X j , i t f o l l o w s from (2.2) and Lebesgue's dominated convergence theorem t h a t  lim and  ( T y ) ( x ) = (Ty)(x) n  similarly, lim (Ty )'(x) n-*-°° n  on every  compact s u b i n t e r v a l .  =  (Ty)'(x)  T: S + S i s thus continuous  i n the C [ x , ) 1  0 0  1  topology. I n o r d e r t o show t h a t TS i s r e l a t i v e l y the s e t of f u n c t i o n s {Ty: y e s} [x^o").  Obviously,  {(Ty)(x)}  compact, we have t o v e r i f y t h a t  i s . u n i f o r m l y bounded and e q u i c o n t i n u o u s  and {(Ty)'(x)} a r e u n i f o r m l y bounded.  As  on  36. before,  one c a n e a s i l y  continuous.  check t h a t  {(Ty)(x)}  Thus, TS i s r e l a t i v e l y  and {(Ty)'(x)}  compact by A s c o l i ' s  are equi-  theorem.  Applica-  t i o n o f t h e Schauder-Tychonoff theorem and d i f f e r e n t i a t i o n o f t h e i n t e g r a l equation  Ty = y twice completes the proof  t h a t y ( x ) i s the r e q u i r e d bounded  p o s i t i v e s o l u t i o n of (2.1).  3.  N e c e s s a r y C o n d i t i o n s f o r E x i s t e n c e o f Bounded S o l u t i o n s In t h i s s e c t i o n , we g i v e n e c e s s a r y  bounded p o s i t i v e  Theorem 2.2.  c o n d i t i o n s f o r (2.1) t o have a  s o l u t i o n i n some i n t e r v a l  A necessary  (XQ, ). 0 0  c o n d i t i o n f o r (2.1) to have a bounded  s o l u t i o n i n some i n t e r v a l  (XQ,°°)> X  Q  positive  > 0, i s t h e e x i s t e n c e o f a p o s i t i v e  number A such t h a t oo  /  Proof: y'(x)  x g(x,A,0) dx < ~ .  (2.6)  L e t y ( x ) be a p o s i t i v e bounded s o l u t i o n o f ( 2 . 1 ) . i s n o n i n c r e a s i n g and nonnegative i n some i n t e r v a l  there i s a c o n t r a d i c t i o n to the p o s i t i v i t y  of y ( x ) ) .  e x i s t s a p o s i t i v e number A such t h a t A/2 < y(x)  S i n c e y"(x)  < 0,  [XQ, ) ( o t h e r w i s e 00  Therefore,  there  < A i n this interval.  Note  that x y(x) - y ( x ) = Q  / x  S i n c e y ( x ) i s nondecreasing x > x • 0  =  > y'(x)(x-x ) 0  .  0  and bounded, y ' ( x ) ( x - X Q ) i s a l s o bounded f o r  I n t e g r a t i o n o f (2.1) twice  y(x)  y'(t)dt  gives  x y ( x ) + ( x - x ) y'(x) + / (t-x )y(t)g(t,y(t),y«(t))dt , Q  Q  0  x  0  and  s i n c e y(x)  and  y'(x)(x-x )  /  (t-x )y(t)g(t,y(t),y'(t))dt 0  w 0  x0  l e t X j = 2x  Now  SO that  Q  are bounded i t f o l l o w s  n  t - x  Q  oo  / x  < -  for t > X j .  > t/2  Then  oo  ty(t)g(t y(t),y (t))dt ,  >  l  < / x  (t-x )y(t)g(t,y(t),y'(t))dt Q  g( »y»p) i s nonincreasing x  i n y, we  conclude  A/ x  >  g(t,A,0) ,  that  tg(t,A,0)dt < / l  x  < / x  first  If i n addition  have  g(t,y(t),y'(t))  from which we  < «» .  0  By h y p o t h e s i s ( i i i ) g(x,y,p) i s n o n d e c r e a s i n g i n p.  The  that  ty(t)g(t,A,0)dt l  ty(t)g(t,y(t),y'(t)dt  < »  .  l  i n e q u a l i t y holds s i n c e y(x)  > A/2.  T h i s proves the n e c e s s i t y  (2.6). I f g(x,y,p) i s n o n d e c r e a s i n g i n y the p r o o f manner.  i s completed i n a  similar  of  38. Example 3.3.  Consider the d i f f e r e n t i a l  y" + d>(x) y  where Y,B a r e nonnegative functions.  Equation  + Kx)(y')  Y  (2.7) i s of the form  =  (2.2) and (2.6) reduce  =  0  (2.7)  (2.1) where  Y 8 <t>(x)y' + K x ) v .  to, respectively,  oo  3  x[<Kx) A  /  P  c o n s t a n t s and <J>,i|> a r e c o n t i n u o u s p o s i t i v e v a l u e d  y g(*,y,v)  Conditions  equation  Y - 1  + K x ) | ] dx < 0= , '  00  /  x[(j>(x) A  Y - 1  ] dx < oo .  We have the f o l l o w i n g c o r o l l a r i e s :  C o r o l l a r y 2.3. interval  E q u a t i o n (2.7) has a bounded p o s i t i v e s o l u t i o n y ( x ) i n some  (x ,°°) i f both Q  oo  /  x<f>(x)dx <  0 0  ,  and 00  x\j;(x)dx < oo .  /  C o r o l l a r y 2.4.  A necessary  condition  f o r (2.7) t o have a bounded p o s i t i v e  s o l u t i o n i n some i n t e r v a l ( x ,°°) i s t h a t Q  00  / holds.  X(J>(x)dx < oo  39. 4.  Summary In  that  t h i s s e c t i o n , we summarize p r e v i o u s r e s u l t s by g i v i n g one c o n d i t i o n  i s both s u f f i c i e n t and n e c e s s a r y f o r a l l s o l u t i o n s  o f (2.1)  to be  p o s i t i v e and bounded. In a d d i t i o n  t o hypotheses ( i ) , ( i i ) ,  (iii),  suppose  that  lim sup[g(x,A,B)/g(x,A,0)] < « x-»-»  for and  a l l p o s i t i v e c o n s t a n t s A and B. a  such  that  | & 3 $  and  therefore  condition  imply t h e f o l l o w i n g  Corollary  2.5.  x  Q  (2.1)  (2.6)  for  for 1 1 » > . .  implies  (2.2).  Theorems 2.1 and 2.2 then  corollary.  Then (2.6)  i s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n  to have a bounded p o s i t i v e  for  s o l u t i o n i n some i n t e r v a l (x ,°°), 0  > 0.  Consider the following  (iv)  <M  Suppose l i m sup[g(x,A,B)/g(x,A,0) ] < =° f o r a l l p o s i t i v e  c o n s t a n t s A and B. Eq.  Then, there e x i s t p o s i t i v e c o n s t a n t s M  a l t e r n a t i v e to hypothesis  g ( x , y ) < g(x,y,p) < <f>(p) 0  (iii).  g U,y) 0  a l l x > 0 , y > 0 , p > 0 where g Q (x,y) i s nonnegative, continuous, and  monotone i n y for each x > 0, and <Kp) ^ s p o s i t i v e , continuous, and nondecreasing for p > 0.  40. C o r o l l a r y 2.6. bounded p o s i t i v e  Under hypotheses ( i ) , ( i i ) and ( i v ) , E q . (2.1) has a solution  i n some i n t e r v a l (XQ, ), x 00  t h e r e e x i s t s a p o s i t i v e number A such t h a t  oo  /  /  xg (x,A)dx < * . Q  Q  > 0 i f and o n l y i f  41. CHAPTER I I I ASYMPTOTIC SOLUTIONS OF SEMILINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH FACTORIZED LINEAR PART  1.  Introduction Our  purpose i s t o prove e x i s t e n c e o f p o s i t i v e s o l u t i o n s o f the semi-  linear ordinary d i f f e r e n t i a l  equation  Ly ± h ( t , y ) =  c o n s i d e r e d e a r l i e r i n Chapter  I.  Although  0  (3.1±)  the r e s u l t s o b t a i n e d here a r e o f  the same s p i r i t as p r e v i o u s r e s u l t s i n Chapter different.  I , our method i s q u i t e  We r e p r e s e n t the l i n e a r e q u a t i o n Lz = 0 i n i t s f a c t o r i z e d  and l a t e r w i l l g i v e s u f f i c i e n t c o n d i t i o n s f o r e q u a t i o n (3.1±) t o have p o s i t i v e s o l u t i o n s y ^ C t ) , y ( t ) with the same asymptotic 2  behaviour as  s o l u t i o n s Z j C t ) , z ( t ) o f Lz = 0. 2  C o n s i d e r the l i n e a r d i f f e r e n t i a l  where p , p^, and p Q  interval  [tg,°°).  2  a r e p o s i t i v e continuous  Two l i n e a r l y  independent  z (t) = p (t), x  for t > t  n  >  where  equation  0  functions i n a positive  s o l u t i o n s o f (3.2) a r e  z (t) = p (t)P (t) 2  Q  x  form  42. t *l(t>  " /  P i ( s ) ds .  We assume t h a t l i m P j ( t ) = °°, Z j ( t ) i s bounded above and z ( t ) i s bounded 2  away from z e r o i n [t ,°°) 0  2.  E x i s t e n c e of P o s i t i v e S o l u t i o n s of S e m i l i n e a r  Equations  S u f f i c i e n t c o n d i t i o n s w i l l be e s t a b l i s h e d f o r (3.1±) t o have p o s i t i v e s o l u t i o n s with s p e c i f i e d  asymptotic  behaviour.  D i f f e r e n t hypotheses on  h ( t , y ) w i l l be c o n s i d e r e d .  Case I.  (H ). 1  Consider  the f o l l o w i n g hypotheses on h ( t , y ) :  There e x i s t s a p o s i t i v e c o n s t a n t c such t h a t h ( t , u ) i s c o n t i n u ous,  nonnegative  and nondecreasing  i n u f o r 0 < u < c and f o r a l l  t > t . Q  (H ). 2  There e x i s t s a p o s i t i v e c o n s t a n t c such t h a t h ( t , u ) i s c o n t i n u ous,  nonnegative  and n o n i n c r e a s i n g i n u f o r 0 < u < c and f o r a l l  t > t . Q  Theorem 3.1. equation  Under hypotheses (Hj) or ( H ) , a s u f f i c i e n t 2  (3.1±) to have a p o s i t i v e s o l u t i o n y ( t ) such  condition for  y(t) that l i m — r — r - = j £  t-M»  Z  l  constant > 0 i s that  / fc  for  P ( t ) p ( t ) h ( t , K p ( t ) ) dt < » x  2  0  0  a l l K e (o, —] where M = sup z ^ t ) . >t t  0  (3.3)  43. Proof: For  The p r o o f w i l l be g i v e n f i r s t  f o r (3.1+) under ( H j ) .  an a r b i t r a r y K e (0, ^] choose T = T(K) > t  Q  such t h a t  oo  /  P ( t ) p ( t ) h ( t , K p ( t ) ] dt < | x  2  0  .  (3.4)  T  The  i n t e g r a n d i s c l e a r l y continuous and nonnegative s i n c e P ( t ) , 1  h ( t , K p ( t ) ) are a l l continuous and K p ( t ) < Q  sup P ( t ) < c. t>T  Q  0  p (t), 2  Let C =  C[T,°o) be the space o f a l l c o n t i n u o u s f u n c t i o n s i n [T, ) w i t h the t o p o l o g y 00  of  u n i f o r m convergence on compact s u b i n t e r v a l s of [T,°°).  L e t Y be the  c l o s e d convex s u b s e t o f C d e f i n e d as  Y  = {y e C: | p ( t ) < y ( t ) < K p ( t ) , Q  Q  Let M be the mapping from C i n t o C d e f i n e d  oo  t > T}  .  by  s  ( M y ) ( t ) = K p ( t ) - p ( t ) [/ (/ ( o ) da) p ( s ) h ( s , y ( s ) ) d s ] , t > T . t t Q  Q  P l  2  (3.5) We  need  to v e r i f y  that  (i)  M maps Y i n t o  (ii)  M i s a continuous mapping on Y;  ( i i i ) MY  Y;  i s r e l a t i v e l y compact,  and t h e n a p p l y the Schauder-Tychonoff f i x e d p o i n t To prove ( i ) l e t y e Y Then f o r t > T > t , Q  so  theorem.  |- p ( t ) < y ( t ) < K p ( t ) 0  Q  f o r t > T.  44. OO  by  OO  g  / (/ t t  (o)  Pl  da) p ( s ) h(s,y(s)) ds < / P ^ s ) p ( s ) h(s,Kp (s)) ds < t 2  2  0  n  (3.3).  T h e r e f o r e the l e f t hand s i d e converges f o r a l l t > T.  Further-  more, i t s a t i s f i e s  oo s / (/ ( a ) t t  oo  do) p ( s ) h ( s , y ( s ) ) ds < / T  P l  P ^ s ) p ( s ) h ( s , K p ( s ) ) ds < |  2  by h y p o t h e s i s .  2  0  Hence,  Kp (t) 0  > My(t)  > Kp (t) - | p ( t ) = | p (t) , Q  Q  0  so My e Y.  To prove ( i i ) , l e t { y } be a convergent  sequence o f f u n c t i o n s i n Y t o y e  Q  i n the topology o f C.  From the d e f i n i t i o n  ofM  |My (t) - My(t)| n  OO  -  The  g  IPoCO / (/ Pi< > t t a  i n t e g r a n d has uniform  d a  )  P  ( s ) 2  [ ( . y < ) ) ~ h ( s , y ( s ) ) ] ds| h  s  s  n  l i m i t zero i n [T,<*>) and i s bounded i n a b s o l u t e  v a l u e by P j ( t ) p ( t ) h ( t , K p ( t ) ) , which has f i n i t e i n t e g r a l . 2  Q  dominated convergence theorem Therefore  l i m My n>°°  By Lebesgue'  |My (t) - M y ( t ) | -»• 0 u n i f o r m l y as n-»-°°. R  = My i n the topology  of C and so M i s c o n t i n u o u s .  n  To show MY = {My: y e Y } i s r e l a t i v e l y  compact, i t i s s u f f i c i e n t t o  prove that the set of f u n c t i o n s {My: y e Y } i s u n i f o r m l y bounded and equicontinuous.  {My} i s o b v i o u s l y u n i f o r m l y bounded as shown i n ( i ) . To  45. show e q u i c o n t i n u i t y of f o r a l l y e Y and  {My},  one  proceeds as b e f o r e , i . e . one  t e [T,»)>  oo  (My)'(t)  =  asserts that  K  P  u  ( t )  -  p„(t)  g  / (/ t  P l  (o)  do)  p (s) h(s,y(s)) 2  ds  t  OO  + p (t) / t 0  i s uniformly  ( t ) p ( s ) h ( s , y ( s ) ) ds  Applying  i n [T,°°) and  From the Mean Value theorem, My  by A s c o l i ' s theorem My  the Schauder-Tychonoff f i x e d  M has a f i x e d p o i n t y e Y;  (My)(t)  i n t e g r a l e q u a t i o n My  (3.6)  2  equibounded i n [T,°°).  equicontinuous  of the  P l  i s relatively  p o i n t theorem, we  = y ( t ) f o r a l l t > T.  = y twice  completes the proof  p o s i t i v e s o l u t i o n of (3.1+) w i t h the p r o p e r t y  is  compact.  conclude t h a t  Differentiation that y ( t ) i s a  t h a t l i m y ( t ) / p ( t ) = K, Q  or  t+oo  equivalently  lim y(t)/zj^(t) =  K.  t-V°o  In the c a s e of (3.1-) and h ( t , y ) s a t i s f y i n g  oo  (My)(t)  The  =  | p ( t ) + p (t) 0  0  same argument as b e f o r e  (3.1-) w i t h Now K e (0,^3  lim y ( t ) / z ( t ) = 1  c o n s i d e r (3.1+) and choose T > t  Q  (H^), r e p l a c e (3.5)  s  [/ (/ ( o ) t t P l  do)  p (s) h(s,y(s)) ds], 2  j. l e t h(t,y) s a t i s f y  (H ). 2  For an  such t h a t  P ( t ) p ( t ) h ( t , K p ( t ) ) dt < K :  t > T  .  y i e l d s the e x i s t e n c e of a p o s i t i v e s o l u t i o n of  oo  /  by  2  0  T L e t Y be the c l o s e d convex subset  of C d e f i n e d  as:  .  arbitrary  46. Y  =  {y e C: K p ( t )  < y(x) < 2 K p ( t ) , t > T }  Q  Q  L e t M be the mapping from C i n t o C d e f i n e d by  oo  (My)(t)  =  g  2 K p ( t ) - p ( t ) [/ (/ t t Q  Q  We need to v e r i f y  (i),  (ii),  ( o ) da) p ( s ) h ( s , y ( s ) ) d s ] , t > T .  P l  2  ( i i i ) as b e f o r e .  To prove ( i ) l e t y e Y so K p ( t ) Q  OO  P l  ( a ) da) p ( s ) h ( s , y ( s ) ) ds < / P ^ s ) p ( s ) h ( s , K p ( s ) ) T 2  2  Hence 2 K p ( t ) > My(t)  > 2Kp (t) - Kp (t) = Kp (t)  Q  Q  To prove ( i i ) l e t iy^ y e Y i n the topology  Q  of C.  i n t e g r a n d has uniform  and My e Y.  sequence o f f u n c t i o n s i n Y t o  s  P l  limit  (a)  da) p ( s ) 2  [h(s,y ( s ) ) - h ( s , y ( s ) ) ]  ds| .  zero and i s bounded i n a b s o l u t e v a l u e by  Pj^(t) p ( t ) h ( t , K p ( t ) ) , which has f i n i t e i n t e g r a l . 2  ds < K .  0  From the d e f i n i t i o n of M  oo Q  Q  be a convergent  I My ( t ) - M y ( t ) | = | p ( t ) / (/ t t  The  Then f o r t > T;  Q  oo  g  / (/ t t  < y(t) < 2Kp (t).  Q  dominated convergence theorem l i m My n-*-°°  Hence by Lebesgue's  = My i n the topology o f C and M i s n  continuous. Property ( i i i )  i s v e r i f i e d by A s c o l i ' s theorem.  Schauder-Tychonoff f i x e d  A p p l i c a t i o n o f the  p o i n t theorem and d i f f e r e n t i a t i o n of the i n t e g r a l  e q u a t i o n My = y twice completes the p r o o f t h a t y ( t ) i s a p o s i t i v e s o l u t i o n of (3.1+) with l i m y ( t ) / p ( t ) = 2K, or e q u i v a l e n t l y , l i m y ( t ) / z ( t ) = 2K. t-»-°° t* In t h e case o f (3.1-) r e p l a c e the p r e v i o u s mapping by Q  x  0 0  47. s  oo  (My)(t)  and  virtually  -  K p ( t ) + p ( t ) [J (/ t t 0  0  P l  (o)  da) p ( s ) h ( s , y ( s ) ) ds] 2  the same argument as b e f o r e y i e l d s  the e x i s t e n c e of a  p o s i t i v e s o l u t i o n of (3.1-) w i t h l i m y ( t ) / z ( t ) = K. 1  £ + 0 0  Case I I .  Consider  (H ).  There e x i s t s a p o s i t i v e c o n s t a n t c such t h a t h ( t , u ) i s  3  the f o l l o w i n g hypotheses  c o n t i n u o u s , nonnegative all (H^).  and  on  h(t,y):  nondecreasing  i n u f o r u > c and f o r  t > t . Q  There e x i s t s a p o s i t i v e c o n s t a n t c such t h a t h ( t , u ) i s c o n t i n u o u s , nonnegative all  Theorem 3.2.  and  n o n i n c r e a s i n g i n u f o r u > c and f o r  t > t . Q  Under h y p o t h e s i s  (H ) 3  or (H^) a s u f f i c i e n t  condition for  e q u a t i o n (3.1±) t o have a p o s i t i v e s o l u t i o n y ( t ) such t h a t l i m y ( t ) / z ( t ) = t-*-°° 2  constant > 0 i s that  /  p ( t ) h ( t , K z ( t ) ) dt < » 2  f o r a l l K > c/y where \i = i n f t > c  The  z (t). 2  o  p r o o f w i l l be g i v e n f i r s t  Proof:  f o r (3.1+) w i t h h ( t , y ) s a t i s f y i n g  For an a r b i t r a r y K > c/\i choose T > t  / T  Q  such t h a t  p ( t ) h ( t , K z ( t ) ) dt < 2  (3.6)  2  2  |  (H ). g  48. The  i n t e g r a n d i s c l e a r l y c o n t i n u o u s and nonnegative  s i n c e p ( t ) and 2  h ( t , K z ( t ) ) are both continuous and K z ( t ) > c/u i n f z ( t ) > c f o r t > T. t>T 2  2  2  L e t Y be the c l o s e d convex subset o f C d e f i n e d as  Y  =  {y e C: | z ( t ) < y ( t ) < K z ( t ) } 2  2  .  L e t M be the mapping from C i n t o C d e f i n e d a s : t ( M y ) ( t ) = | p ( t ) P j C t ) + p ( t ) [/ T 0  0  00  ( s ) / p ( o ) h ( a , y ( a ) ) d 0 d s ] , t > T. s  P l  2  (3.7)  To show M maps Y i n t o Y,  t / T  l e t y e Y so — z ( t ) < y ( t ) < K z ( t ) . 2  t  oo  Pi(s) / s  2  p ( o - ) h ( o , y ( o ) ) d a ds < / T 2  We  have  oo  P l  ( s ) (/ p ( a ) h ( o , K z ( a ) ]da)ds T 2  2  < | / p (s)ds < | P (t) . T t  1  x  Then  |  z ( t ) < (My)(t) < | p ( t ) P ( t ) + j p ( t ) P ( t ) = K p ( t ) P ( t ) = K z ( t ) . 2  and  0  c o n s e q u e n t l y My  :  Q  fl  to y e Y i n the topology of C.  t n  Q  L  2  e Y.  To prove M i s c o n t i n u o u s , l e t {y }  |My (t) - My(t)| =  :  |p (t) / T Q  P l  be a sequence i n Y t h a t  converges  Note t h a t  oo  ( s ) / p ( a ) [ h ( o , y ( a ) ) - h ( o , y ( o ) ) ] d a ds| s 2  .  49. I t f o l l o w s r o u t i n e l y by p r e v i o u s  arguments and  dominated convergence theorem that l i m My  the use  = My  and  of Lebesgue's  thus M i s c o n t i n u o u s .  n-M»  An  a p p l i c a t i o n of A s c o l i ' s theorem v e r i f i e s t h a t MY  relatively  compact.  By  d i f f e r e n t i a t i o n of My (1.2+) w i t h the  2  = y, we  0  0  h(t,y)  Q  can  satisfying  Now  [/ T  Pl  (H )  r e p l a c e (3.7)  3  <*> / p (a)h(o,y(0))da s  (s)  or e q u i v a l e n t l y l i m y ( t ) / z ( t ) =  1  suppose (H^)  2  holds.  For an a r b i t r a r y K > c/p  by  d s ] , t > T.  2  show that y ( t ) i s a p o s i t i v e s o l u t i o n of  l i m y ( t ) / p ( t ) P ( t ) = K, 0  1  j.  t  we  and  that l i m y ( t ) / p ( t ) P ( t ) = - j i . e . ,  (My)(t) = K p ( t ) PjCt) - p ( t )  before,  theorem  conclude that y ( t ) i s a p o s i t i v e s o l u t i o n of  In the case of (3.1-) and  As  y e Y} i s  the Schauder-Tychonoff f i x e d p o i n t  property  lim y ( t ) / z ( t ) =  = {My:  (3.1-) w i t h K.  choose T  > t  Q  such  that  CD  / T  L e t Y be  the  p ( t ) h ( t , K z ( t ) ) dt < K 2  2  c l o s e d convex subset of C d e f i n e d  Y  =  { y e C:  Kz (t)  A p a r a l l e l argument as before defined  by  2  .  as:  < y(t) < 2Kz (t), 2  t > T]  .  e s t a b l i s h e s f i x e d p o i n t s of the mappings  50. t (My)(t) = K p ( t ) P ( t ) + p ( t ) Q  x  [/ T  Q  (My)(t) = 2 K p ( t ) PjCt) - p ( t ) Q  Q  yielding positive solutions  P l  (s)  t [/ T  P l  » / p ( a ) h ( o , y ( a ) ) d a d s ] , t > T; s 2  » ( s ) / p ( o ) h ( a , y ( a ) ) d a d s ] , t > T, s 2  o f (3.1+),(3.1-), r e s p e c t i v e l y , with the  p r o p e r t y t h a t l i m y ( t ) / z ( t ) = K, l i m y ( t ) / z ( t ) = 2K. 2  2  We c l o s e t h i s s e c t i o n by g i v i n g t h e f o l l o w i n g guaranteed the e x i s t e n c e  = 0, i . e . y ( t ) ~ z ( t ) x  conditions solutions  3.  (as t * ) . 0 0  f o r the existence that  Theorem 3.1  of i n f i n i t e l y many bounded p o s i t i v e s o l u t i o n s y ( t )  of e q u a t i o n (3.1±) which behave a s y m p t o t i c a l l y Lz  summary.  l i k e t h e s o l u t i o n z ^ t ) of  I n Theorem 3.2, we gave  sufficient  o f i n f i n i t e l y many unbounded p o s i t i v e  behave a s y m p t o t i c a l l y  like z ( t ) , 2  i . e . y ( t ) ~ z ( t ) as t-*- . 00  2  A S p e c i a l Case In t h i s s e c t i o n , we w i l l r e v e a l  the r e l a t i o n s h i p between t h e m a t e r i a l  of Chapter I and that of the present c h a p t e r .  S p e c i f i c a l l y speaking, we  w i l l show t h a t e q u a t i o n (1.1±) i s a s p e c i a l c a s e o f (3.2±) and our r e s u l t s i n Chapter I I I reduce to p r e v i o u s r e s u l t s o f Chapter I . C o n s i d e r t h e s p e c i a l case Lz = z". of Lz = 0 a r e z ^ t ) = 1, z ( t ) = t . 2  Introduction,  Two l i n e a r l y  independent  Furthermore, as shown i n the  L has the f a c t o r i z e d form  solutions  where p (t)  =  z ( t ) = 1;  (t)  =  [ z ( t ) / z ( t ) ] ' = 1;  p (t)  =  [  0  P l  2  We can e a s i l y  x  2  P o  1  (t) PiCt)]"  = 1.  1  show t h a t the s o l u t i o n s y ^ t ) , y ( t ) o f y" ± h ( t , y ) = 0 2  have t h e same a s y m p t o t i c behaviour (3.3) and (3.6) of Theorems  as z ^ ( t ) ,  3.1, 3.2 reduce  z ( t ) as t-* . 00  2  Conditions  to, respectively,  t h ( t , K ) d t < °° ;  (3.8)  h ( t , K t ) d t < °° ,  (3.9)  t.  for  some p o s i t i v e c o n s t a n t K.  I t may be noted  t h a t (3.8) and (3.9) a r e  identical  t o c o n d i t i o n s (1.2) and (1.8) o f Chapter  condition  (3.8) i s s u f f i c i e n t  I.  As i n Chapter I ,  f o r the e x i s t e n c e o f a p o s i t i v e  solution y ( t )  such t h a t l i m y ( t ) = c o n s t a n t > 0 and c o n d i t i o n (3.9) i s s u f f i c i e n t e x i s t e n c e o f an unbounded  4.  s o l u t i o n y ( t ) such t h a t l i m y ( t ) / t  f o r the  = c o n s t a n t > 0.  Examples We c l o s e t h i s  c h a p t e r by g i v i n g two examples o f the g e n e r a l e q u a t i o n  (3.1±).  Example 3.3.  C o n s i d e r the o r d i n a r y d i f f e r e n t i a l  I d , L  z  =  T d T  , o  dz. (  t  dt  0  equation  ~  k  2  „ =  0  •  (3.10)  52. where k i s a p o s i t i v e  constant.  S i n c e (3.10) i s n o n o s c i l l a t o r y , t h e r e e x i s t l i n e a r l y  independent,  e v e n t u a l l y p o s i t i v e s o l u t i o n s z ^ t ) and z ( t ) such t h a t 2  z (t) l i m —y—r- = 0 . ->(t) X  2  Moreover, the o p e r a t o r L i n (3.10) has the f a c t o r i z e d  Z  l  Pfj'  =  p  l  ( ^ ^ J ' '  =  P2  =  t 0 p  form (3.2) where  P l l " ' 1  The s o l u t i o n s z ^ t ) , z ( t ) are g i v e n as b e f o r e by 2  z (t) = p (t), :  z (t)= p (t)P (t)  0  2  Q  x  where t P (t) = / x  An a p p l i c a t i o n of Theorems  P l  ( s ) ds .  3.1 and 3.2 w i l l  establish  sufficient  c o n d i t i o n s f o r Ly ± h ( t , y ) = 0 t o have p o s i t i v e s o l u t i o n s y ^ ( t ) ~ z ^ t ) and y ( t ) ~ z ( t ) as t*°°, where z ^ ( t ) and z ( t ) are l i n e a r l y  independent  s o l u t i o n s o f (3.10).  and z -  2  2  Equation  2  The main t a s k , here,  i s to f i n d z  1  2  (3.10) i s a m o d i f i e d B e s s e l e q u a t i o n of order zero, t h a t i s  of the form  z" +  -i-  z' - k z = 0 2  (3.11)  53. Two of  a s y m p t o t i c a l l y o r d e r e d s o l u t i o n s o f (3.10), o r e q u i v a l e n t l y (3.11), a r e the  form: (t)  = K (kt) ;  z (t)  = I (kt) ,  Z l  Q  2  where I Q and K  Q  0  denote m o d i f i e d B e s s e l f u n c t i o n s of order z e r o .  to f i n d e x p l i c i t a s y m p t o t i c behaviour  of t h e s e s o l u t i o n s , we  first  1  d e r i v a t i v e by l e t t i n g  z(t) = t ~  /  u(t).  2  In o r d e r  remove the  Then (3.11) Is e q u i v a l e n t  to u" +  [ — 4t  - k ] 2  A s o l u t i o n u ( t ) has the a s y m p t o t i c /kT  = ±k by Thome's scheme [37].  u(t)  By  u -  0.  2  form u ( t ) ~ e ^ ^ where to = /~qg l0  Therefore,  ~ exp  (± k t ) .  substitution, 1 -kt Z i ( t ) ~ (constant) ( — e ) , /t  z (t) 9  1 kt ~ (constant) ( — e ) /t  as t-*- . 00  Then p  2  = [PoP^  -1  gives  i—  P2(t) ~ (constant) / t e  and  c o n d i t i o n s (3.3) and  (3.6)  reduce  to  -kt  t  =  54.  / fc  /te  respectively,  Corollary  0  d t < <» ,  0  / c  h(t,cK (kt))  k  /te  h ( t , c l ( k t ) ) d t < <» , 0  0  f o r some constant c > 0.  3.3.  A sufficient  condition  f o r (3.1±) ( i n the case that L i s  g i v e n by (3.10)) t o have a bounded p o s i t i v e s o l u t i o n y ( t )  such  that  l i m y ( t ) / z ( t ) = c o n s t a n t > 0 i s that :  /  /te  h(t,cK (kt)) 0  dt < »  f o r some p o s i t i v e constant c .  Corollary  3.4.  A sufficient  condition  f o r (3.1±) ( i n the case that L i s  g i v e n by (3.10)) to have an unbounded p o s i t i v e s o l u t i o n y ( t ) such l i m y ( t ) / z ( t ) = c o n s t a n t > 0 i s that 2  /  for  /t e  k  h ( t , c l ( k t ) ) dt < » 0  some p o s i t i v e constant c .  Example 3.5.  C o n s i d e r the o r d i n a r y  differential  equation  that  55.  L z  =  T o T  (  -  t  p  2  fc2r  2  =  0  ( 3  where 2r i s a p o s i t i v e i n t e g e r and p i s a p o s i t i v e c o n s t a n t .  Since  i s n o n o s c i l l a t o r y i t has the f a c t o r i z e d form ( 3 . 2 ) , where p ( t ) , 2  p ( t ) a r e g i v e n as b e f o r e .  behaviour  ordered  [35, p. 85]  ( r + 1 ) / 2  2  conditions  ~ Ip" t ~  ( t )  P2  The  x (p i ^ i )  e  P  Hence  p  (1  Z  ( t >  2  (  t  )  1  r)/2  exp(-p ~^-) as t-*- ,  (l-r)/2  t*\  ,  (3.3) and (3.6) reduce t o , r e s p e c t i v e l y ,  /  ( 1 t  00  /  t  ~  r ) / 2  nU  T  /o )  U  f o r some p o s i t i v e constant  p ^ t ) and  s o l u t i o n s o f Lz = 0 has t h e a s y m p t o t i c  2  0 0  (3.12)  2  Z M ~ t" as t * .  1 2 )  A fundamental s e t { z ^ t ) , z ( t ) } o f e v e n t u a l l y  0  positive asymptotically  -  exp(p  exp(-p  K.  r+1  r+1  hCt.Kz^t)) dt < - ,  h ( t , K z ( t ) ) dt < » , 2  56.  Corollary given lim  3.5.  A sufficient  condition  by ( 3 . 1 2 ) ) t o have a d e c a y i n g  y ( t ) / z ( t ) = constant  positive  solution  y(t)  L i s  with  > 0 i s that  x  r+1  0 0  J  f o r (3.1±) ( i n t h e case t h a t  t  (  1  _  r  )  /  2  exp(p ^ - h ( t , K z ( t ) ) T  1  dt < -  t„  holds  f o r some p o s i t i v e  Corollary  3.6.  A  constant  sufficient  K.  condition  f o r (3.1±) ( i n the case  g i v e n by ( 3 . 1 2 ) ) t o h a v e a n u n b o u n d e d p o s i t i v e  solution  y(t)  that  L i s  with l i m t+oo  y(t)/z (t) 2  = constant  n -  0 0  /  holds  > 0 i s that  t  U  f o r some p o s i t i v e  T  W? )  U  r+1 exp(-p l _ h ( t , K z r+1  constant  K.  2  ( t ) ]  2'  dt < -  APPENDIX  R e c e n t l y , Kusano, Swanson and Usami  [23], have extended the r e s u l t s  p e r t a i n i n g t o the semilinear ordinary d i f f e r e n t i a l equations q u a s i l i n e a r case Ly = h ( t , y , y ' ) .  Although,  Kusano e t a l . , use the same  technique  ( f a c t o r i z a t i o n of disconjugate operators)  criteria,  the outcome i s q u i t e d i f f e r e n t .  sharper  (3.1±) t o t h e  to e s t a b l i s h existence  The r e s u l t s o b t a i n e d  a r e much  than p r e v i o u s ones [36] and guarantee e x i s t e n c e o f g l o b a l s o l u t i o n s  i n s t e a d of the u s u a l l o c a l  solutions.  We s h a l l b r i e f l y d i s c u s s t h i s new  work i n t h e l i g h t o f the m a t e r i a l o f Chapter I I I . The o p e r a t o r L has the u s u a l f a c t o r i z e d  L z  =  form  p-^iFtp7(TraT^i-fty^' z e c  2 t  t ,») 0  with  ,2 V Z  p (t) = 0  Z l  (t),  P  l  (t) =  i  (t  [^-^y)  ,  p (t) = 2  P  o  (  t  )  p  i  (  t  )  As b e f o r e ,  z (t) = p (t), :  0  z (t)= p (t) P (t) 2  Q  x  where t P (t) = / :  c  are two l i n e a r l y  P i ( s ) ds  0  independent, a s y m p t o t i c a l l y ordered  Furthermore, a new f u n c t i o n  s o l u t i o n s of Lz = 0.  58. p(t) = p ( t ) P ( t ) / P ( t ) + 0  is introduced.  1  1  lp (t)l 0  The f o l l o w i n g assumptions a r e made:  2-i ( A j ) Each p ( t ) i s p o s i t i v e i n [ t , » ) , i  p  0  ±  e C  [t ,°°),  i = 0,1,2 and  0  l i m P j ( t ) = + ». t->-oo  (A ) 2  h: [t ,°°) x R+ x R -»• R, R + = [ 0 , » ) , i s continuous and 0  satisfies |h(t,y,z)| < H(t,|y|,|z|) f o r a l l t e [ t Q , ) , y e R+, 00  where H(t,u,v) i s continuous i n  [ t p , ) x R+ x R_j., n o n d e c r e a s i n g i n u f o r each t , v and non00  decreasing (A ) 3  X  - 1  H(f,Xu,Xv) i s a n o n d e c r e a s i n g f u n c t i o n of X e (0,») and  lira X X>°° The  i n v f o r each t , u .  - 1  f o r each f i x e d ( t , u , v ) e [t ,°°) x R+  H(t,Xu,Xv) = 0  0  x R+.  f o l l o w i n g theorems g i v e s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f  bounded and unbounded s o l u t i o n s of  Ly = h ( t , y , y ' )  Theorem 3.7.  .  (3.13)  A s u f f i c i e n t c o n d i t i o n f o r equation  many p o s i t i v e ( n e g a t i v e )  (3.13) t o have  s o l u t i o n s y ( t ) i n [tQ, ) such t h a t l i m y ( t ) / z j ( t ) t+°° 00  e x i s t s and i s p o s i t i v e ( n e g a t i v e , r e s p e c t i v e l y ) i s t h a t  /  P l  infinitely  ( t ) P ( t ) H ( t , a p ( t ) , b p ( t ) ) dt < x  0  0 0  59. f o r some p o s i t i v e c o n s t a n t s We  a and  b.  remind t h e r e a d e r t h a t the s u f f i c i e n t c o n d i t i o n o f Theorem 3.7  weaker than c o n d i t i o n  Theorem 3.8.  (2.2)  A sufficient  many p o s i t i v e ( n e g a t i v e ) e x i s t s and  is  of Chapter I I .  condition for equation  (3.13) to have  s o l u t i o n s y ( t ) i n [tQ, ) such t h a t l i m 00  i s p o s i t i v e (negative,  infinitely y(t)/z (t) 2  respectively) i s that  oo  /  p (t) H(t,ap (t),bp(t) 2  0  f o r some p o s i t i v e c o n s t a n t s As the  i n the  z ( t ) of Lz 2  yi(t)» y~2( ) z  again,  wi  t n  = 0. the  00  x  b.  case, the s u f f i c i e n t c o n d i t i o n s e s t a b l i s h e d f o r  quasilinear differential  Zj(t),  and  semilinear  a and  P ( t ) ) dt <  e q u a t i o n (3.13) are  E q u a t i o n (3.13) has  i n terms of the  solutions  p o s i t i v e (negative)  solutions  same asymptotic b e h a v i o u r as z ( t ) , 1  z (t). 2  the a s y m p t o t i c s i m i l a r i t y between s o l u t i o n s of n o n l i n e a r  t h a t of the  l i n e a r p a r t becomes  evident.  Once equations  60. CONCLUSIONS  Nonlinear  ordinary d i f f e r e n t i a l  equations  possess  v e r y s i m i l a r t o s o l u t i o n s of t h e i r l i n e a r c o u n t e r p a r t . t h i s phenomenon i n two First, taken  by  We  solutions  have examined  ways.  the n o n l i n e a r e q u a t i o n , may  itself.  asymptotic  Necessary and  i t be s e m i l i n e a r o r q u a s i l i n e a r , i s  sufficient  c o n d i t i o n s are e s t a b l i s h e d  directly  f o r e x i s t e n c e of l o c a l s o l u t i o n s o f the n o n l i n e a r  equation  under study.  differential  These c o n d i t i o n s v a r y , depending on the nature  of  the s o l u t i o n (bounded o r unbounded). The tion.  second method c o n s i d e r s the l i n e a r p a r t o f the d i f f e r e n t i a l  The  k i n d of n o n l i n e a r i t y , i . e . s e m i l i n e a r or q u a s i l i n e a r , i s not  g r e a t importance a t the o u t s e t . product  of two  a r e found.  equa-  linear  l i n e a r operator i s f a c t o r i z e d into a  o p e r a t o r s , and  U s u a l l y , one  p o s i t i v e constants  The  the  s o l u t i o n s of the l i n e a r  s o l u t i o n Zj^(t) i s bounded above and  or f u n c t i o n s and  of  the other  equation  below by  s o l u t i o n z ( t ) i s bounded 2  away from z e r o . A f t e r v e r i f y i n g the e x i s t e n c e o f such s o l u t i o n s , the o r i g i n a l nonlinear differential  equation  comes i n t o the p i c t u r e .  which not o n l y guarantee e x i s t e n c e of bounded and the n o n l i n e a r d i f f e r e n t i a l asymptotic  behaviour  equation,  at i n f i n i t y .  We  but  C o n d i t i o n s are  unbounded s o l u t i o n s of  a l s o o f f e r i n f o r m a t i o n about  have e x p l i c i t l y  shown t h a t  z ^ ( t ) and  z (t).  methods a r e i n harmony w i t h e a c h o t h e r  In any  event,  the two  their  the  bounded s o l u t i o n behaves l i k e 2  found  the unbounded s o l u t i o n behaves  like  as  shown i n numerous examples. The  r e s u l t s of t h i s t h e s i s can be sharpened by c o n s i d e r i n g e x i s t e n c e  of g l o b a l s o l u t i o n s .  The  appendix to Chapter I I I and  the r e c e n t work of  61. Kusano, Swanson and Usami [23] The  discusses t h i s p o s s i b i l i t y .  s u b j e c t of t h i s t h e s i s has  found a secure  p l a c e among mathemati-  c i a n s i n t e r e s t e d i n nonlinear d i f f e r e n t i a l equations progressing  and  is definitely  r a p i d l y s i n c e i t s e a r l y days near the middle of t h i s  I t has been our i n t e n t i o n t o survey  a modern a s p e c t of t h i s  century.  progress.  62. BIBLIOGRAPHY  1.  F.V. A t k i n s o n , On l i n e a r p e r t u r b a t i o n s of n o n l i n e a r e q u a t i o n s , Canad. J . Math., 6(1954), pp. 561-571.  2.  , The asymptotic s o l u t i o n s of second o r d e r d i f f e r e n t i a l e q u a t i o n s , Ann. Mat. Pura. A p p l . , 37 (1954), pp. 347378.  3. Pacific J.  , On second order n o n l i n e a r Math., 5 (1955), pp. 643-647.  differential  oscillations,  4.  S. 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