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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 College A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s thesis as conforming to the pegulred standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © BITA JENAB , 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 ABSTRACT The asymptotic behaviour of n o n o s c i l l a t o r y solutions of second order nonlinear ordinary d i f f e r e n t i a l equations i s studied. Necessary and s u f f i c i e n t conditions are given for the existence of 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 at i n f i n i t y . Existence of n o n o s c i l l a -tory solutions i s established using the Schauder-Tychonoff f i x e d point theorem. Techniques such as f a c t o r i z a t i o n of l i n e a r disconjugate operators are employed to reveal the s i m i l a r nature of asymptotic solutions of non-l 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. Some examples i l l u s t r a t i n g the asymptotic theory of ordinary d i f f e r e n t i a l equations are given. - i i -TABLE OF CONTENTS Page Abstract i i Acknowledgements v Introduction 1 Prel i m i n a r i e s 4 1. Introduction 4 2. The Schauder-Tychonoff Fixed Point Theorem 4 3. F a c t o r i z a t i o n of Disconjugate Operators 9 Chapter I - Asymptotic Properties of Semilinear Ordinary D i f f e r e n t i a l Equations 13 1. Introduction 13 2. Bounded Asymptotic Solutions 13 3. Unbounded Asymptotically Linear Solutions 21 4. Asymptotic Properties of the Emden-Fowler Equation 25 5. A More General Case 27 Chapter II - Asymptotic Properties of Quasilinear Ordinary D i f f e r e n t i a l Equations 30 1. Introduction 30 2. S u f f i c i e n t C r i t e r i a for Existence of Bounded Solutions ... 30 3. Necessary Conditions f o r Existence of Bounded Solutions .. 36 4. Summary 39 Chapter III - Asymptotic Solutions of Semilinear Ordinary D i f f e r e n t i a l Equations with Factorized Linear Part ... 41 1. Introduction 41 2. Existence of P o s i t i v e Solutions of Semilinear Equations .. 42 3. A Sp e c i a l Case 50 4. Examples 51 - i i i -TABLE OF CONTENTS (Continued) Page Appendix 57 Conclusions 60 Bibliography 62 - i v -ACKNOWLEDGEMENTS I am greatly indebted to my advisor, Dr. C A . Swanson, for suggesting the topi c , f o r h i s guidance and sound c r i t i c i s m , and most of a l l f o r h i s endless patience with me throughout the preparation of t h i s work. I also thank Dr. J.G. Heywood f o r reading the manuscript. I express my gratitude to the University of B r i t i s h Columbia and the Natural Sciences and Engineering Research Council of 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 i s to study the asymptotic behaviour of sol u -t i o n s of nonlinear second order d i f f e r e n t i a l equations. Necessary and s u f f i c i e n t conditions w i l l be established for the existence of p o s i t i v e solutions and e x p l i c i t asymptotic behaviour w i l l be given f o r both bounded and unbounded p o s i t i v e solutions. In the past decade, there has been an increasing i n t e r e s t i n studying the existence and q u a l i t a t i v e behaviour of solutions of nonlinear d i f f e r e n -t i a l equations. The subject i s of great importance not only i n theory but also i n a p p l i c a t i o n . For example, applications of the singular semilinear equation y" + a ( t ) y " X = 0 , X > 0 (1) ar i s e from boundary layer theory of viscous f l u i d s and has been studied by C a l l e g a r i and Nachman [6,7]. Mathematicians such as Hammett [16], Kusano and Onose [19]., and Singh [33,34] have studied o s c i l l a t i o n theory for fun c t i o n a l d i f f e r e n t i a l equations of the type ( r ( t ) y ' ( t ) ) ' + a ( t ) y ( t - T ( t ) ) = f ( t ) . (2) 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 of equation with J ^ ^ = <*> can be found i n var i a b l e mass problems. Another popular d i f f e r e n t i a l equation i s the Emden-Fowler equation y" + a(t) |y| Y sgn y = 0 (3) 2. The study of the Emden-Fowler equation originated around the turn of the century i n theories of gas dynamics i n astrophysics. At that time, the fundamental problem was to i n v e s t i g a t e the equilibrium configuration of the mass of s p h e r i c a l clouds of gas i n s t e l l a r structures. Lord K e l v i n [38] i n 1862 proposed that the gaseous cloud i s under convective equilibrium. A few years l a t e r , Lane [24] modelled t h i s phenomenon by introducing the equation U 2 £> • y" - o . (4, In 1907, Emden published his famous t r e a t i s e Gaskugeln [11] on the study of s t e l l a r configurations governed by equation (4) . Ever since, t h i s equation has been coined as the Lane-Emden equation. The mathematical foundation for the study of such an equation and also of the more general equation (3) was made by Fowler i n a series of four papers [12]-[15] during 1914-1931. The i n t e r e s t e d reader may r e f e r to an e x c e l l e n t summary i n Bellman's book [5, Chapt. VI I ] . The f i r s t serious i n v e s t i g a t i o n concerning the asymptotic properties of the generalized Emden-Fowler equation was made by Atkinson [1,2,3]. A f t e r that, the importance of t h i s equation and i t s various a p p l i c a t i o n s were widely recognized. The Emden-Fowler equation appears i n many d i f f e r -ent f i e l d s such as gas dynamics and f l u i d mechanics; see e.g. Sansone [31, p. 431] and the survey a r t i c l e by Conti, G r a f f i and Sansone [9]. More recentl y , a p p l i c a t i o n s of the Emden-Fowler equation have a r i s e n i n the study of r e l a t i v i s t i c mechanics, nuclear physics and chemically reacting systems. Readers interes t e d i n the p h y s i c a l aspects of such studies may wish to consult Wong [41] for an extensive review and bibliography. 3. The study of asymptotic properties of solutions of d i f f e r e n t i a l equa-tions has also won considerable attention from a t h e o r e t i c a l standpoint. Recently, mathematicians have applied the asymptotic theory of ordinary 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 the realm of nonlinear p a r t i a l d i f f e r e n t i a l equations are now seen i n a new l i g h t o r i g i n a t i n g 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. Mathematicians such as Kusano and Swanson [20,21,22], Noussair [27] are to be considered the pioneers of t h i s new approach. This work incorporates r e s u l t s found by those mentioned above but focuses only on the ordinary d i f f e r e n t i a l equations aspect. The backbone of t h i s study i s the Schauder-Tychonoff f i x e d point theorem [32,40], used over and over i n proving existence of p o s i t i v e solutions of the d i f f e r e n -t i a l equation under consideration. In the next chapter we w i l l s t ate t h i s theorem a f t e r giving the necessary preliminary d e f i n i t i o n s . PRELIMINARIES 4. 1. Introduction The purpose of t h i s chapter i s twofold: To provide the preliminary material that leads to the Schauder-Tychonoff fixed point theorem, and to discuss the techniques we employ f o r proving existence of so l u t i o n s of the d i f f e r e n t i a l equation under consideration. 2. The Schauder-Tychonoff Fixed Point Theorem Before s t a t i n g the Schauder-Tychonoff f i x e d point theorem, we present the following d e f i n i t i o n s [ 2 9 , 3 0 ] : 0.1 D e f i n i t i o n : A metric space <X,p> i s a nonempty set X of elements together with a real-valued function p defined on X x X such that f o r a l l x, y, and z i n X: i ) P(x,y) > 0; i i ) p(x,y) = 0 i f f x=y; i i i ) p(x,y) = p(y,x); iv) p(x,y) < p(x,z) + p(z,y). The function p i s c a l l e d a metric. 0.2 D e f i n i t i o n : A topological space <X,T> i s a nonempty set X of points together with a family T of subsets ( c a l l e d open) possessing the following properties: i ) X e T, <|> e x; i i ) Ox e T and 0 2 e T imply 01f\02 e T; i i i ) 0 £ x implies \J 0 e x . a The family T i s c a l l e d a topology f o r the set X. 0.3 D e f i n i t i o n . A set X of elements i s c a l l e d a vector space over the r e a l s i f there e x i s t s a function + on X x X and a function • on R x X to X which s a t i s f y the following conditions: i ) x + y = y + x; i i ) (x + y) + z = x +(y + z ) ; i i i ) There i s a vector 6 i n X such that x + 6 = x for a l l x i n X; iv) A(x + y) = Ax + Ay; A e R, x,y e X; v) (A + u)x = Ax + ux; A,u e R, x e X; v i ) A(ux) = (Au)x; A,u e R, x e X; v i i ) 0.x = 9, 1.x = x. 0.4 D e f i n i t i o n . A normed vector space i s a vector space X together with function II II : X •* R +, c a l l e d a norm, from X i n t o nonnegative r e a l s , with the properties: i ) llxll > 0, llxll = 0 <=> x = 0; i i ) llx+yll < llxll + llyll; i i i ) IIaxII = |a| llxll; f o r a l l x e X and a e R. A normed vector space becomes a metric space i f we define a metric p by p (x,y) = llx-yll. 6. 0.5 D e f i n i t i o n . A normed vector space which i s complete with respect to 11.11 i s a Banach space. 0.6 D e f i n i t i o n . A vector space X with a topology x on i t i s c a l l e d a t o p o l o g i c a l vector space i f a d d i t i o n i s a continuous fu n c t i o n from X x X into X and m u l t i p l i c a t i o n by scalars i s a continuous function from R x X i n t o X. 0.7 D e f i n i t i o n . A subset K of a vector space X i s said to be convex i f whenever i t contains x and y, i t also contains the e n t i r e 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 D e f i n i t i o n . A to p o l o g i c a l vector space X i s l o c a l l y convex i f there i s a l o c a l base B whose members are convex sets. 0.9 D e f i n i t i o n . A complete metrizable l o c a l l y convex space i s c a l l e d a Frechet space. 0.10 D e f i n i t i o n . A to p o l o g i c a l space X i s said to be compact i f every open covering U of X has a f i n i t e subcovering. 0.11 D e f i n i t i o n . A subset K of a topological space X i s c a l l e d compact i f i t i s compact as a subset of X. 0.12 D e f i n i t i o n . A space X i s said to be sequentially compact i f every i n f i n i t e sequence from X i n t o X has a convergent subsequence. 7. 0.13 D e f i n i t i o n . A space X i s said to be countably compact i f every countable open covering has a f i n i t e subcovering. 0.14 Lemma [29, p. 163]. For a metric space the notions of compactness, countable compactness and sequential compactness are equivalent. We are now i n the p o s i t i o n to quote the Schauder-Tychonoff f i x e d point theorem [17, p. 405]. Theorem 0.1 (Schauder-Tychonoff). Let B be a l o c a l l y convex, t o p o l o g i c a l vector space. Let Y be a compact, convex subset of B and T a continuous map of Y into i t s e l f . Then T has a fixed point y e Y, i . e . Ty = y. Theorem 0.1 was f i r s t proved by Schauder [32] under the assumption that B i s a Banach space and t h i s case of the theorem i s usually c a l l e d "Schauder's f i x e d point theorem". For a proof of Theorem 0.1, see Tychonoff [40], Although Theorem 0.1 i s the standard Schauder-Tychonoff f i x e d point theorem, we f i n d i t more applicable to use the following c o r o l l a r y instead. C o r o l l a r y 0.2 [17, p. 405]. Let B be a l i n e a r , l o c a l l y convex, t o p o l o g i c a l , complete Hausdorff space (e.g., l e t B be a Banach or Frechet space). Let Y be a closed, convex subset of B and T a continuous map of Y i n t o i t s e l f such that the image TY of Y has a compact closure. Then T has a fixed point y e Y. To see how Coro l l a r y 0.2 i s obtained from Theorem 0.1, the intere s t e d reader may wish to consult Hartman [17, p. 405]. From now on, we w i l l r e f e r to Cor o l l a r y 0.2 as the Schauder-Tychonoff f i x e d point theorem. 8. In order to v e r i f y that the hypotheses of the Schauder-Tychonoff f i x e d point theorem are met we use two other theorems; namely Lebesgue's dominated convergence theorem to show T i s a continuous mapping and the As 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. Let g be integrable over E and l e t ( f n ) be a sequence of measurable functions such that | f n | < g on E for every n = 1,2,.. and f(x) = lim f (x) for almost a l l x i n E. Then n n-»-°° / f = lim / f . E E 0.15 D e f i n i t i o n . A family F of r e a l valued functions i n 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 ) i s c a l l e d uniformly bounded i n I i f and only i f there exists M > 0 (independent of f) such that | f ( x ) | < M for a l l x e I and f e F. 0.16 D e f i n i t i o n . A family F of functions from a topological space X into R i s c a l l e d equicontinuous at the point x e X i f given e > 0 there i s an open set 0 containing x such that |f(x) - f ( y ) | < e for a l l y i n 0 and a l l f e F. The family i s said to be equicontinuous on X i f i t i s equicontinuous at each point x i n X. Theorem 0.4 - A s c o l i - A r z e l a Theorem Let F be an equicontinuous uniformly bounded family of r e a l valued functions 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 ) . Then F contains a 9. uniformly convergent sequence of functions f ^ , converging to a function f e C(I) where C(I) denotes the space of a l l continuous bounded functions on I. Thus any sequence i n F contains a uniformly bounded convergent subsequence on I and consequently F has compact closure i n C(I). 3. F a c t o r i z a t i o n of Disconjugate Operators In the course of t h i s t h e s i s , we adopt two techniques for proving existence of bounded and unbounded solutions of second order semilinear d i f f e r e n t i a l equations of the general type: ( r ( t ) y ' ) ' + h(t,y) = 0 , t > 0 . (0.1) The f i r s t technique gives necessary and s u f f i c i e n t conditions for a l l so l u t i o n s of the d i f f e r e n t i a l equation above to have a s p e c i f i c asymptotic behaviour. The second technique, which i s somewhat an i n d i r e c t approach, f i r s t considers 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 = 0 . The operator L i s factored into two r e a l f i r s t order operators and i t i s l a t e r shown that the nonlinear d i f f e r e n t i a l equation under study has solutions y ^ ( t ) , y 2 ( t ) with the same asymptotic behaviour as the solutions of the l i n e a r equation. Therefore, the necessary and s u f f i c i e n t conditions for existence of such solutions are given i n terms of solutions Z j ( t ) , z 2 ( t ) of Lz = 0. Here, we explain the f a c t o r i z a t i o n technique of second order l i n e a r operators. Consider the l i n e a r equation Ly = y" + • 1 ( x ) y ' + <(>2(x)y = 0 10. (0.2) where each <j>^  Is real-valued and continuous i n I = (0,°°). Our goal here i s to show how L can be expressed as a product of f i r s t order l i n e a r operators. An operator L of 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 at 0 0 i f and only i f each n o n t r i v i a l solution of (0.2) has at most one zero i n some i n t e r v a l [ a , ~ ) . The Wronskian of n functions u.^  e C n ''"(I), i = l , . . . , n i s defined as usual by W = d e t ( u i ( j " 1 ) ) , i , j = l , . . . , n , and w i l l also be denoted by [u^-.-.u ]. I t i s e a s i l y seen that the operator L can be uniquely recovered from a fundamental set {z 1,z 2} of solutions of (0.2) by the formula [ z x , z 2 , y ] * • -fiTT^r • (0-3) We say that L corresponds to the fundamental set {z 1,z 2}» Cl e a r l y to a system of functions z^(x), z 2 ( x ) defined on I there corresponds some operator L with c o e f f i c i e n t s that are continuous i n I i f and only i f e C(I) (i=l,2) and [ z j , z 2 ] ( x ) * 0 i n I. We can now explain the f a c t o r i z a t i o n of the nono s c i l l a t o r y (disconjugate) operator L [25,28,39]. 1 1 . Theorem 0 . 5 . The operator L i s no n o s c i l l a t o r y at 0 0 i f and only i f i t has a f a c t o r i z a t i o n L - P2 _ 1 h P I " 1 k P O " 1 co-*) i n some i n t e r v a l [a, 0 0), where P 0 = zi» Pi = (.z2/z1)\ p 2 = ( P Q P ^ - 1 , ( 0 . 5 ) Lz = 0 , i = 1 , 2 , zj > 0 and [ z 1 , z 2 ] * 0 i n [a,°») Proof. If L i s no n o s c i l l a t o r y at °°, solutions z^ of ( 0 . 2 ) e x i s t as described above and each p^ , i n ( 0 . 5 ) i s well defined i n some i n t e r v a l 2 — i [a, 0 0), with p i e C [a, 0 0), i = 0 , 1 , 2 . For any y e C 2[a,°°), c a l c u l a t i o n shows that d y [ Z l ' y l 7 - - ; ( 0 . 6 ) dx zx 2 j [zi>y] z1[z1,z2,y] T - i r = • (°- 7> d X l Z l » Z 2 l [ Z l , z 2 ] 2 These are, i n f a c t , s p e c i a l cases of a well-known property of Wronskians [ 2 8 ] . Combination of ( 0 . 3 ) , ( 0 . 6 ) , and ( 0 . 7 ) y i e l d s [zltz2] d [z,,y] z2 ' d Z l 2 d y ^ z^ dx [ z i } z 2 ] Z l ^ z ^ dx ^•[z 1,z 2] dx ^ z ^ ^ proving ( 0 . 4 ) . The converse i s obvious. th Generalizations to n order l i n e a r operators L appear i n [ 2 5 , 3 9 ] 12. 4. Examples We now give two examples to i l l u s t r a t e further how the described technique works. d 2 Example 0.6. The n o n o s c i l l a t o r y operator L = i s , of course, already i n dx 2 f a c t o r i z e d form. As a check of (0.4), take z^(x) = 1, z 2 ( x ) = x and obtain from (0.5) that p 0 = Pi = P 2 = 1. Example 0.7. The d i f f e r e n t i a l operator L defined by Ly = y" - k 2 y , for a p o s i t i v e constant k, i s n o n o s c i l l a t o r y since Lz = 0 has l i n e a r l y independent solutions / \ _kx kx z,(x) = e , z 2 ( x ) = e In t h i s case (0.5) becomes p 0(x) = e k X , p x(x) = 2 k e 2 k X , p 2(x) = ( 2 k ) _ 1 e ^ , and the f a c t o r i z e d form of L i s given by / T . . . kx r -2kx, kx , - i , (Ly)(x) = e [e (ye )'J' , unique up to a m u l t i p l i c a t i v e constant. A d d i t i o n a l examples, of le s s t r i v i a l type, are given i n Chapter 3, Section 4. Given t h i s background, we can now discuss the asymptotic behaviour of solutions of a class of second order d i f f e r e n t i a l equations. 13. CHAPTER I ASYMPTOTIC PROPERTIES OF SEMILINEAR ORDINARY DIFFERENTIAL EQUATIONS 1. Introduction Necessary and s u f f i c i e n t conditions w i l l be derived for the existence of an eventually p o s i t i v e s o l u t i o n of y" ± f(x,y) = 0 x > a (1.1±) under the following hypotheses: ( i ) f i s a continuous p o s i t i v e valued function i n [a,°°)x(0,«>) . ( i i ) f i s monotone i n y for each f i x e d x, i . e . , f i s e i t h e r nonincreasing or nondecreasing i n y for each fixed x e [a,°°). By a s o l u t i o n of equation (1.1±) we mean a function y(x) which i s continuous and s a t i s f i e s (1.1±) on some h a l f - l i n e [x0,°°). Such a s o l u t i o n i s c a l l e d o s c i l l a t o r y i f i t has a r b i t r a r i l y large zeros, otherwise i t i s c a l l e d 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 one sign for a l l s u f f i c i e n t l y large x. A p o s i t i v e s o l u t i o n y(x) of (1.1±) defined i n some h a l f - l i n e [x Q ,<*>) 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 . In this t h e s i s , we consider only n o n o s c i l l a t o r y solutions of the equation under study. 2. Bounded Asymptotic Solutions In t h i s s e c t i o n we obtain necessary and s u f f i c i e n t conditions for (1.1±) to have solutions which behave asymptotically l i k e nonzero constants. 14. Theorem 1.1. A necessary and s u f f i c i e n t condition f o r (1.1±) to have a po 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 Q ,«>) with a f i n i t e p o s i t i v e l i m i t as x •*• 0 0 i s that x / x f(x,c) dx < - (1.2) a holds for some constant c > 0. The proof w i l l be given f o r (1.1±) under the assumption that f(x,y) i s nonincreasing. Proof: We f i r s t prove the s u f f i c i e n c y of (1.2) i n the case of equation (1.1+). Choose x Q > a large enough so that / x f(x,c) dx < c . x0 Let B = C[x0,°°) be the space of a l l continuous bounded functions y i n [XQ ,») with norm IIyII = sup |y(x)|. Let Y be closed convex subset of B x>x Q defined by Y = {y e B: c < y(x) < 2c, x > x Q} . Let T: B + B be the mapping defined by (Ty)(x) = 2c - / (t-x) f ( t , y ( t ) ) dt; x > x n . (1.3) k0 x Any solution of the i n t e g r a l equation i s a solution of the d i f f e r e n t i a l equation (1.1+). If y i s a continuous s o l u t i o n of the i n t e g r a l equation (1.3), then 15. y'(x) = / f ( t , y ( t ) ) dt; x y"(x) = -f(x,y(x)) . Furthermore, y(x) decreases to the l i m i t 2c as x + °°. In order to apply the Schauder-Tychonoff f i x e d point theorem, we need to v e r i f y that: a) T maps Y i n t o Y; b) T i s a continuous 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 Q . Then f o r t > x > x Q oo (t-x) f ( t , y ( t ) ) < t f ( t , c ) . Therefore the i n t e g r a l / (t-x) f ( t , y ( t ) ) dt x converges for a l l x > x n by (2.1) and furthermore i t s a t i s f i e s / (t-x) f ( t , y ( t ) ) dt < / t f ( t , c ) dt < c. x x Q Hence oo 2c > (Ty)(x) = 2c - / (t-x) f ( t , y ( t ) ) dt > 2c - c = c, x proving that Tye Y. To prove (b), l e t {y n) be a sequence i n Y that converges i n the B norm to ye B. Since Y i s closed, y e Y. From the d e f i n i t i o n of T, |Ty n(x) - Ty(x)| = |/ (t-x) [ f ( t , y n ( t ) ) - f ( t , y ( t ) ) ] dtj . x But the integrand has l i m i t 0 as n •*• 0 0 for a l l t > x > x n since 16. sup |y (t) - y ( t ) | + 0 as n •*• » , t>x 0 n and f i s continuous. Also the integrand i s bounded i n absolute value by oo t f ( t , c ) (since c < y (t) < 2c), which has f i n i t e i n t e g r a l / t f ( t , c ) dt < c. x Hence lim IITy -Tyll = 0 by Lebesgue's dominated convergence theorem. T i s n-*-°° n therefore continuous. To prove ( c ) , we need to show that the family T Y i s uniformly bounded and equicontinuous. Obviously T Y i s uniformly bounded, as shown i n part ( a ) . To prove equicontinuity, l e t z = Ty E T Y. Then oo oo |z'(x)| = |J f ( t , y ( t ) ) dt| < / t f ( t , c ) dt < c , X X Q since y ( t ) > c and f ( t , y ( t ) ) i s nonincreasing i n y by assumption. For e > 0 l e t 6 = E / C , then f o r x, , x 2 i n [x 0,°°) with I x ^ x ^ < 6 the Mean-Value Theorem gives |z(x x) - z ( x 2 ) | = | z ' U ) | |x: - x 2 | < C ( E / C ) = £ . This proves equicontinuity since 6 i s independent of x 1 , x 2 and y E Y. By the A s c o l i - A r z e l a theorem, TY has compact closure. In view of (a), (b), (c) the Schauder-Tychonoff fixed point theorem shows that T has at l e a s t one f i x e d point y E Y, i . e . , Ty(x) = y(x) for a l l x > X Q . 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 equation Ty = y twice, as shown before, completes the proof that y(x) i s a p o s i t i v e s o l u t i o n of (1.1+) which decreases to the l i m i t 2c as x + °°. 17. Now we w i l l show that (1.2) i s a necessary condition f o r (1.1+) to have a po s i t i v e s o l u t i o n which behaves asymptotically l i k e a nonzero constant. Suppose that y(x) i s a p o s i t i v e s o l u t i o n of (1.1+) and also y(x) tends to a f i n i t e p o s i t i v e l i m i t . We w i l l prove that the lim y'(x) = 0. Suppose y'(x) < 0. Since y"(x) < 0, then y'(x) i s a negative decreasing function, so y'(x) < - | ot| for x > x Q . Now for x > x Q, by the Mean-Value Theorem, y(x) - y ( x Q ) — — — = y ' ( c ) < - | « | x 0 < 5 < x . x X Q Therefore y(x) < - |a|(x - x Q ) + y(x Q ) and we conclude that y(x) w i l l be eventually negative f o r x large enough. But t h i s 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), so y'(x) > 0 for x > x Q . Furthermore, since y'(x) i s a decreasing function, i t follows that lim y'(x) > 0 e x i s t s . I t i s impossible f o r lim y'(x) > 0 since then x->-°° y(x) - y ( x Q ) > lim y'(x) (x-x Q) x*°° forces lim y(x) = + <=. Thus lim y'(x) = 0. X+oo Integration of (1.1+) twice y i e l d s : y'(x) = -/ f ( t , y ( t ) ) dt , X oo y(x) - y( x Q ) = -/ / f ( s , y ( s ) ) ds dt x 0 s-t 18. Since y(x) i s bounded and decreasing the above has a f i n i t e l i m i t as y(x) •»• °°. Changing the order of in t e g r a t i o n , we have: X 0 0 oo g — < "J (/ f(s,y(s)) ds) dt = -/ (/ dt) f ( s , y ( s ) ) ds x o z x0 x0 0 0 = _/ (s-x Q) f ( s , y ( s ) ) ds . x0 0 0 Therefore, / s f ( s , y ( s ) ) ds < 0 0 since y(s) i s bounded, say x0 0 < y(s) < K for x > x Q and f ( s , y ( s ) ) i s nonincreasing i n y. I t follows that: 0 0 / s f(s,K) ds < °° x0 for some K > 0. In the case of (1.1-) the mapping (1.3) i s replaced by 0 0 (Ty)(x) = c + / (t-x) f ( t , y ( t ) ) dt x > x, x and v i r t u a l l y the same argument as before y i e l d s the existence of a po s i t i v e s o l u t i o n of (1.1-) with lim y(x) = c. The necessity part i s x->-°° proven s i m i l a r l y . It can be shown as before that lim y'(x) = 0, and hence i n t e g r a t i o n of (1.1-) and interchanging the order of i n t e g r a t i o n gives: 19. 0 0 y'(x) = / f ( t , y ( t ) ) dt X oo oo y(x) - y ( x Q ) = / (/ f ( s , y ( s ) ) ds) dt = / (s-x Q) f ( s , y ( s ) ) ds x Q s=t x Q As before, y(s) i s bounded and f(s,y) i s nonincreasing, so / sf(s,K) ds < / s f ( s , y ( s ) ) ds < 0 0 for s > x Q  x0 x0 Now, i f we replace the 'nonincreasing' assumption on f(x,y) by 'nondecreasing', the following changes must be made. We have to choose x Q > a large enough so that / xf(x,c) dx < x0 and replace Y by Y = {y e B: T < y(x) < c} A p a r a l l e l argument as before establishes f i x e d points i n Y of the mappings (TY)(x) = c - / ( t - x ) f ( t . y ( t ) ) dt x > x Q ; x 0 0 (TY)(x) = f + / (t-x) f ( t , y ( t ) ) dt x x > x Q , 20. y i e l d i n g p o s i t i v e s olutions of (1.1+), (1.1-), r e s p e c t i v e l y , s a t i s f y i n g c lim y(x) = c and lim y(x) = y. If f(x,y) i s nondecreasing and y(x) i s a p o s i t i v e s o l u t i o n with p o s i t i v e f i n i t e l i m i t as x-*-°°, then y(x) > K > 0 for x > x Q . Therefore f ( s , y ( s ) ) > f(s,K) f o r s > x Q and the necessity proof i s completed i n the same way as before. Example 1.2: Consider the d i f f e r e n t i a l equation ^ (x + xf(x,y) = 0 x > a > 1 (1.4) where f(x,y) i s continuous, p o s i t i v e and nonincreasing i n y. We w i l l prove that the above equation 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 [xQ,°°) i f and only i f oo / x log x f(x,c) dx < o° (1«5) a for some c > 0. Proof: It w i l l be shown that L i o u v i l l e ' s change of va r i a b l e s x = e , y(x) = Y(s) transforms (1.4) into the form (1.1+), and accordingly Theorem 1.1, can be applied. By the chain rule , dy _ dY j ds ^ 1 dY dx ds dx s ds ' e Then the d i f f e r e n t i a l equation (1.4) reduces to 21. + e 2s f ( e S , Y(s)) = 0 , s > log a . (1.6) The change of v a r i a b l e x = e transforms (1.5) in t o 0 0 / 2s g f(e ,c) ds < 0 0 (1.7) s e log a for some p o s i t i v e constant c. 2 s s Let F(s,Y) = e f ( e ,Y). Cle a r l y F(s,Y) i s nonincreasing i n Y since 2s s e i s independent of Y and f ( e ,Y) i s nonincreasing i n Y. In the case of equation (1.6), condition (1.2) has the form (1.7), and hence Theorem 1.1 shows that (1.7) i s necessary and s u f f i c i e n t f o r (1.6) to have a bounded po s i t i v e s o l u t i o n Y(s) i n some i n t e r v a l [ S g , 0 0 ) , i . e . , for (1.4) to have a 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 Asymptotically Linear Solutions The purpose of t h i s section i s to obtain necessary and s u f f i c i e n t conditions for (1.1±) to have solutions which behave asymptotically l i k e cx (c * 0) as x •+• °°. Theorem 1.2. A necessary and s u f f i c i e n t condition for (1.1±) to have a 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 [x0,°°) with lim y(x)/x p o s i t i v e and x>°° f i n i t e i s that 0 0 / f(x,cx) dx < «> a (1.8) 22. holds f o r some p o s i t i v e constant c. Proof: The proof w i l l be given f i r s t for (1.1-) under the assumption that f(x,y) i s nonincreasing. Choose XQ > a large enough so that 0 0 / f(x,cx) dx < c . X Q Let Y = {y e B: cx < y(x) < 2cx, x > x Q} where B = C [ X Q , ° ° ) i s the space of a l l continuous functions on [ X Q , » ) such that y(x)/x i s bounded with norm llyll = sup |y(x)/x|. C l e a r l y Y i s a closed, convex subset of B. Let T: B -»• X > X Q B be the mapping defined by x 0 0 (Ty)(x) = 2cx - / / f ( s , y ( s ) ) ds dt , x > x Q . (1.9) x Q t We need to v e r i f y that T maps Y into Y. Let y e Y so cx < y(x) < 2cx for x > X Q . Then x oo x 0 0 x 0 < / / f ( s , y ( s ) ) ds dt < / / f ( s , c s ) ds dt < c / dt < cx , X Q t X Q X Q XQ Hence cx < Ty(x) < 2cx so Ty e Y. Proceeding as i n Theorem 1.1, we can e a s i l y v e r i f y that T i s a continuous mapping and TY = {Ty: y e Y} has compact closure. The Schauder-Tychonof f f i x e d point theorem then shows that T has a f i x e d point y e Y, (Ty)(x) = y(x) for a l l x > x 0. 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 proof that y(x) i s a p o s i t i v e s o l u t i o n of (1.1-) with the property that lim y(x)/x = 2c. To show condition (1.8) i s necessary, l e t y(x) be a p o s i t i v e s o l u t i o n of (1.1-) i n [x0,°°) such that y(x)/x has a f i n i t e p o s i t i v e l i m i t at «>. Then, there e x i s t p o s i t i v e constants Ki and K, such that K :x < y(x) < K 2x for x > x x , (1.10) where x^ > x Q i s s u f f i c i e n t l y large. Since y" > 0, y'(x) i s p o s i t i v e and increasing and we have lim y'(x) < lim y(x)/x < K 2. x* 0 0 x+°° Integration of (1.1-) y i e l d s x / f ( s , y ( s ) ) ds = y'(x) - y ' ( X l ) < y»(x) ; x > x x , x l and since lim y'(x) < lim < 0 0 by assumption, the above implies that x->-oo x->-°° oo / f (s,y(s)) ds < 0 0 . x l Since f(x,y) i s nonincreasing, y(x) < K 2x implies that f(x,y(x)) > f ( x , K 2 x ) , so OO 00 / f ( s , K 2 s ) ds < / f ( s , y ( s ) ) ds < °° , x l x l 24. proving the necessity part of the theorem. In the case of (1.1+), replace (1.9) by x 0 0 (Ty)(x) = cx + / / f ( s , y ( s ) ) ds dt , x Q t and complete the proof as before. The necessity part i s v i r t u a l l y the same. Remark: Because of the nonincreasing property of f(x,y) as a function of y, con d i t i o n (1.2) implies (1.8) and consequently (1.2) guarantees the existence of two p o s i t i v e solutions y^(x) and y 2 ( x ) such that both y ^ x ) and y 2 ( x ) / x have f i n i t e p o s i t i v e l i m i t s at 0 0 and so i n p a r t i c u l a r lim y 1 ( x ) / y 2 ( x ) = 0. On the other hand, i f (1.8) i s s a t i s f i e d but not (1.2), then every proper p o s i t i v e s o l u t i o n y(x) i s unbounded with lim y(x)/x f i n i t e and p o s i t i v e . x+°° The r e s u l t s of theorem 1.2 can be proved s i m i l a r l y i n the case that f(x,y) i s nondecreasing i n y. To prove necessity, the f i r s t i n e q u a l i t y i n (1.10) instead of the second i n e q u a l i t y should be used. To prove the s u f f i c i e n c y part, choose x Q > a large enough so that 0 0 f(x,cx) dx < y . x 0 Let Y = {y e B: x < y(x) < cx}. The Schauder-Tychonoff fixed point theorem should be applied to the mappings 25. x 0 0 (Ty)(x) = cx - / / f ( s , y ( s ) ) ds dt, x > x Q ; x Q t x 0 0 (fy)(x) - |2L+ / / f(s,y(s)) ds dt, x > x Q ; x 0 t to y i e l d p o s i t i v e solutions of (1.1-), (1.1+), re s p e c t i v e l y , with the property that lim y(x)/x = c, lim y(x)/x = x->-°° x->-°° 4. Asymptotic Properties of the Emden-Fowler Equation The generalized Emden-Fowler 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±) for which f(x,y) = p(x)y Y where y i s a r e a l constant and p: [a,°°) + (0,«>) i s a continuous p o s i t i v e valued function. We note that f(x,y) i s monotone increasing i f y > 0 and monotone decreasing i f y < 0. Furthermore, equation (1.11±) may be c l a s s i f i e d as singular, sublinear or superlinear according to whether Y<0, 0 < y < 1 or T > 1, r e s p e c t i v e l y . Conditions (1.2) and (1.8) reduce to, r e s p e c t i v e l y , x / xp(x) dx < • ; (1.12) a x / x Yp(x) dx < » . (1.13) a 26. These r e s u l t s are a c t u a l l y w e l l known theorems of Atkinson [3] and Belohorec [4]. Theorem 1.3 [Atkinson]: Let y > 1. A l l solutions of (1.11±) are non-o s c i l l a t o r y i f and only i f oo / x p(x) dx < 0 0 . Theorem 1.4 [Belohorec]: Let 0 < y < 1. A l l solutions of (1.11±) are no n o s c i l l a t o r y i f and only i f oo / x^ p(x) dx < «> . Remark 2. Upon examining these theorems and our r e s u l t s i n Theorems 1.1 and 1.2 we note that i f y > 1, then (1.13) implies (1.12). Hence, (1.13) guarantees the existence of two proper p o s i t i v e solutions y,(x) and y 2 ( x ) such that both y^(x) and y 2 ( x ) / x 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 conditions (1.12) and (1.13) become dual to each other and we have the following properties: (a) the superlinear equation has an asymptotically constant solution; (b) the sublinear equation has an unbounded asymptotically l i n e a r s o l u t i o n . These r e s u l t s can be summarized as c o r o l l a r i e s to Theorems 1.1 and 1.2. Co r o l l a r y 1.5. Let y > 0. A necessary and s u f f i c i e n t condition for (1.11±) to have a bounded asymptotically constant s o l u t i o n i s that 27. / x p(x) dx < » a holds. C o r o l l a r y 1.6. Let y > 0. A necessary and s u f f i c i e n t condition for (1.11±) to have an unbounded asymptotically l i n e a r s o l u t i o n i s that oo / x Y p(x) dx < » a holds. 5. A More General Case In the previous section, we established the asymptotic propeties of the s p e c i a l case (1.11±). Here, we s h a l l consider a more general form of (1.1±). Consider the ordinary d i f f e r e n t i a l equation ( r ( x ) y ' ) ' ± f(x,y) = 0 ; x > a (1.14±) under the following hypotheses: i i i ) r: [a,°°) -»• (0 ,°o ) i s continuous and s a t i s f i e s lim R(x) = oo where x-»-oo X ds R ( x ) = ^ 7 ( 1 7 ' a n d a iv) f: [a, 0 0) x (0,») •*• (0,oo) i s continuous and nonincreasing i n y. Conditions (1.2) and (1.4) reduce to, r e s p e c t i v e l y , 00 / R ( x ) f ( X , c ) d x < oo ; (1.15) 28. / f(x,cR(x)) dx < » . (1.16) We have the following a l t e r n a t i v e to Theorems 1.1 and 1.2. Theorem 1.5. Equation (1.14±) has a proper 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 [x0,°°) such that 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 for some po s i t i v e constant c. Theorem 1.6. Equation (1.14±) has an eventually p o s t i i v e s o l u t i o n y(x) such that 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 at °° i f and only i f 00 / f(x,cR(x))dx < oo holds for some po s i t i v e constant c. Remark 3• Because of the nonincreasing property of f ( x , y ) , condition (1.15) implies condition (1.16) and consequently (1.15) guarantees the existence of two po s i t i v e solutions y ^ x ) and y 2 ( x ) of (1.14±) such that both l i m i t s lim y x ( x ) = K : , • lim y 2(x)/R(x) = K £ X+°o X-+-00 exist and are p o s i t i v e . 29. In the case of s p e c i a l i z a t i o n ( r ( x ) y ' ) ' = p(x)y -X X > 1 (1.17) of (1.14-), where r i s as before and p: [a,»)-»-(0,<») i s continuous, conditions (1.15), (1.16) reduce to, re s p e c t i v e l y , 00 / R(x) p(x) dx < OO (1.18) oo / [ R ( x ) ] " X p(x) dx < 00 (1.19) We have the following c o r o l l a r i e s to Theorems 1.5 and 1.6. Co r o l l a r y 1.7. Condition (1.18) i s necessary 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 p o s i t i v e l i m i t at <*>. Cor o l l a r y 1.8. Condition (1.19) i s necessary and s u f f i c i e n t f o r (1.17) to have an eventually p o s i t i v e s o l u t i o n y(x) such that y(x)/R(x) has a f i n i t e p o s i tive l i m i t at °°. The below c o r o l l a r y i s an immediate consequence of Remark 3. C o r o l l a r y 1.9. Condition (1.18) i s s u f f i c i e n t f o r (1.17) to have two eventually p o s i t i v e proper solutions y ^ x ) and y 2 ( x ) such that both y ^ x ) and y 2(x)/R(x) have f i n i t e l i m i t s at 0 0. In the next chapter we s h a l l study asymptotic solutions 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 of second order. 30. CHAPTER II ASYMPTOTIC PROPERTIES OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS 1. Introduction Necessary and s u f f i c i e n t conditions w i l l be derived f o r the existence of bounded p o s i t i v e solutions of quasilinear ordinary d i f f e r e n t i a l equations of the type y" + ygU.y.y') = 0 x > 0 (2.1) under the hypotheses l i s t e d below. 1) g( x>y»P) i s continuous and nonnegative f o r 0 < x < ° ° , 0 < y < ° ° and -°° < p < 0 0 • i i ) g(x,y,p) i s e i t h e r nondecreasing or nonincreasing i n y f o r each x,p. i i i ) g(x,y,p) i s nondecreasing i n p i f p > 0 for each x,y. The theorems [36] of t h i s chapter extend r e s u l t s of Coffman and Wong [8], Nehari [26] and others for the semilinear case y" + yg(x,y) = 0. 2. S u f f i c i e n t C r i t e r i a f o r Existence of Bounded Solutions In t h i s section, s u f f i c i e n t conditions are given f o r (2.1) to have a po s i t i v e s o l u t i o n y(x) which tends to a f i n i t e l i m i t as x-*-°°. The Schauder-Tychonoff f i x e d point theorem w i l l be used i n the proof. Theorem 2.1. Equation (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 0 ,°°) i f there e x i s t p o s i t i v e constants A and B such that 00 / xg(x,A,B) dx < 0 0 . (2.2) Proof: Case I. g(x,y,p) i s nonincreasing i n y for each x,p. Let C be a number s a t i s f y i n g A < C < A + B and choose x 1 > 0 large enough so that 00 C / tg(t,A,B) dt < C - A . x l Let C 1 = C 1 [x 1 ,°°) denote the l o c a l l y convex vector space of a l l continuously d i f f e r e n t i a b l e functions i n [x^, 0 0) with the topology of uniform convergence of functions and t h e i r f i r s t d erivatives on compact subintervals of [ x , , 0 0 ) , i . e . we have the convergence y •*• y (as n > °°) i n 1 n the topology of C 1 i f and only i f y n ( x ) y ( x ) and ^ r i ^ X ^ * ^ ' ^ a s n * °° uniformly on every compact subinterval of [ x ^ , 0 0 ) . Consider the set of continuously d i f f e r e n t i a b l e functions S = {y e C 1: A < y(x) < C and 0 < y'(x) < C-A for x > . C l e a r l y S i s a closed convex subset of C 1. Define the mapping T on S by 00 (Ty)(x) = C + / ( x - t ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t , x > ^ . (2.3) x We need to v e r i f y T maps S into S. Let y e S, that i s A < y(x) < C, 0 < y'(x) < C-A. Note that 00 oo 0 < / ( t - x ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t < C / tg(t,A,B)dt < C-A . X x^ 32. Hence, C > (Ty)(x) - C + / ( x - t ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t > C-(C-A) = A. (2.4) x Furthermore, 0 < (Ty)'(x) = / y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t < C / tg(t,A,B)dt < C-A (2.5) x x. for a l l x > x x . Therefore Ty e S. To show T i s a continuous mapping, l e t ( y n ) he a convergent sequence of functions i n S to y e S i n the topology of C 1. Then 00 |(Ty n)(x) - (Ty)(x)| < / ( t - x ) | y n ( t ) g ( t , y n ( t ) , y ^ t ) ) x " y ( t ) g ( t , y ( t ) , y ' ( t ) ) | dt . The integrand has uniform l i m i t zero on compact subintervals of [ X j , 0 0 ) and i s bounded above by Ctg(t,A,B) for t > X j , which has f i n i t e i n t e g r a l . By Lebesgue's dominated convergence theorem, lim (Ty )(x) = (Ty)(x) S i m i l a r l y , lim (Ty )'(x) = (Ty)'(x) on every compact subinterval. Therefore, T: S •* S i s continuous i n the C ^ X j , 0 0 ) topology. 33. To show that 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 to show that TS i s uniformly bounded and equicontinuous on [ X j , 0 0 ) . The uniform boundedness of {(Ty)(x)} and {(Ty)'(x)} i n [x l t«) i s obvious from (2.4) and (2.5). It remains to show that {(Ty)(x)} and {(Ty)'(x)} are equicontinuous i n [xj,°°) . For a l l y e S and x e [Xj,»), 0 0 |(Ty)'(x)| - 1/ y(t) g ( t , y ( t ) , y ' ( t ) ) dt| < C-A x by (2.5). By the Mean Value Theorem, (Ty)(x) i s equicontinuous i n [ X j , 0 0 ) . S i m i l a r l y , |(Ty)"(x)| = |-y g ( t , y ( t ) , y ' ( t ) ) | < C g(t,A,C-A) since y(t) > A, y'(t) < C-A and g(x,y,p) i s nondecreasing i n p and non-increasing i n y. This implies that (Ty)'(x) i s equicontinuous i n [ x ^ , 0 0 ) . By A s c o l i ' s theorem (extended to °°), there e x i s t s a convergent subsequence of {(Ty)(x)}, say {Ty (x)}, such that lim (Ty )(x) = z(x) i n [ X j , * ) , the n-»-°° convergence being uniform on any compact subinterval. For t h i s subsequence (y }, consider {(Ty )'(x)}, also uniformly bounded and equicontinuous. n n This has a convergent subsequence {(Ty )'(x)} such that lim (Ty )(x) = z'(x) uniformly on any compact subinterval of [Xj ,°°) . Therefore TS i s r e l a t i v e l y compact. By the Schauder-Tychonoff fixed point theorem, T has a fix e d point y e S, i . e . (Ty)(x) = y(x) f o r a l l x > X j . 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 equation Ty = y twice complete the proof that y(x) i s the required bounded p o s i t i v e s o l u t i o n of (2.1) which tends to the l i m i t C as 34. Case I I . g(x,y,p) i s nondecreasing i n y for each fixed x,p. Let C be a number s a t i s f y i n g max(A-B,0) < C < A and choose X j large enough so that 00 A / tg(t,A,B) dt < A - C . x l Let S = {y e C 1: C < y(x) < A and 0 < y'(x) < A-C for x > x,) . Define T: C 1 + C 1 by x (Ty)(x) - C + / ( t - X l ) y(t) g(t,y(t),y»(t)) dt x l OO + ( x - X l ) / y(t) g ( t , y ( t ) , y * ( t ) ) dt . x To show T maps S into S, l e t y e S. Then for x > x OO (Ty)(x) < C + / ( t - x 1 ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) dt x l OO < C + A / t g(t,A,B) dt x l < C + A - C = A . Therefore, C < (Ty)(x) < A. 35. S i m i l a r l y , 0 < (Ty)'(x) = / y(t) g ( t , y ( t ) , y ' ( t ) ) dt < A / g(t,A,B) dt < A-C x x from which Ty e S. To show T i s a continuous mapping, l e t be a convergent sequence of functions i n the topology of C 1 to y e S. Then, (Ty n)(x) - (Ty)(x)| < |/ ( t - X l ) [ y n ( t ) g ( t , y n ( t ) , y ; ( t ) ) - y(t) g [ t , y ( t ) , y ' ( t ) ) ] dtj x l 0 0 + | ( x - X l ) / [ y n ( t ) g ( t , y n ( t ) , y ; ( t ) ) - y(t) g ( t , y ( t ) , y ' ( t ) ) ] dt| . x Since the integrand has uniform l i m i t zero i n [x^ ,<*>) and i s bounded above by Atg(t,A,B) f o r t > Xj , i t follows from (2.2) and Lebesgue's dominated convergence theorem that and s i m i l a r l y , lim (Ty n)(x) = (Ty)(x) lim (Ty n)'(x) = (Ty)'(x) n-*-°° on every compact subinterval. T: S + S i s thus continuous i n the C 1 [ x 1 , 0 0 ) topology. In order to show that TS i s r e l a t i v e l y compact, we have to v e r i f y that the set of functions {Ty: y e s} is.uniformly bounded and equicontinuous on [x^o"). Obviously, {(Ty)(x)} and {(Ty)'(x)} are uniformly bounded. As 36. before, one can e a s i l y check that {(Ty)(x)} and {(Ty)'(x)} are equi-continuous. Thus, TS i s r e l a t i v e l y compact by A s c o l i ' s theorem. Applica-t i o n of the Schauder-Tychonoff 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 equation Ty = y twice completes the proof that y(x) i s the required bounded p o s i t i v e s o l u t i o n of (2.1). 3. Necessary Conditions f o r Existence of Bounded Solutions In t h i s section, we give necessary conditions f o r (2.1) to have a bounded p o s i t i v e solution i n some i n t e r v a l (XQ, 0 0). Theorem 2.2. A necessary condition for (2.1) to 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 (XQ,°°)> X Q > 0, i s the existence of a p o s i t i v e number A such that oo / x g(x,A,0) dx < ~ . (2.6) Proof: Let y(x) be a pos i t i v e bounded sol u t i o n of (2.1). Since y"(x) < 0, y'(x) i s nonincreasing and nonnegative i n some i n t e r v a l [XQ, 0 0) (otherwise there i s a contradiction to the p o s i t i v i t y of y ( x ) ) . Therefore, there e x i s t s a p o s i t i v e number A such that A/2 < y(x) < A i n t h i s i n t e r v a l . Note that x y(x) - y(x Q) = / y ' ( t ) d t > y'(x)(x-x 0) . x0 Since y(x) i s nondecreasing and bounded, y'(x)(x-XQ) i s also bounded f o r x > x 0• Integration of (2.1) twice gives x y(x) = y ( x Q ) + (x-x Q) y'(x) + / (t-x 0)y(t)g(t,y(t),y«(t))dt , x0 and since y(x) and y'(x)(x-x n) are bounded i t follows that / ( t - x 0 ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t < -0 w0 x Now l e t X j = 2x Q SO that t - x Q > t/2 for t > X j . Then oo oo / t y ( t ) g ( t > y ( t ) , y , ( t ) ) d t < / ( t - x Q ) y ( t ) g ( t , y ( t ) , y ' ( t ) ) d t < «» . x l x0 By hypothesis ( i i i ) g(x,y,p) i s nondecreasing i n p. If i n a d d i t i o n g( x»y»p) i s nonincreasing i n y, we have g ( t , y ( t ) , y ' ( t ) ) > g(t,A,0) , from which we conclude that A / tg(t,A,0)dt < / ty(t)g(t,A,0)dt x l x l < / t y ( t ) g ( t , y ( t ) , y ' ( t ) d t < » . x l The f i r s t i n e q u a l i t y holds since y(x) > A/2. This proves the necessity of (2.6). If g(x,y,p) i s nondecreasing i n y the proof i s completed i n a s i m i l a r manner. 38. Example 3.3. Consider the d i f f e r e n t i a l equation y" + d>(x) y Y + K x ) ( y ' ) P = 0 (2.7) where Y,B are nonnegative constants and <J>,i|> are continuous p o s i t i v e valued functions. Equation (2.7) i s of the form (2.1) where Y 8 y g(*,y,v) = <t>(x)y' + K x ) v . Conditions (2.2) and (2.6) reduce to, res p e c t i v e l y , oo 3 / x [<Kx) A Y - 1 + K x ) | ] dx < 0= , ' 00 / x[(j>(x) A Y - 1 ] dx < oo . We have the following c o r o l l a r i e s : C o r o l l a r y 2.3. Equation (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 i n t e r v a l (xQ,°°) i f both 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) to 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 Q ,°°) i s that 00 / X(J>(x)dx < oo holds. 39. 4. Summary In t h i s section, we summarize previous r e s u l t s by g i v i n g one condition that i s both s u f f i c i e n t and necessary for a l l solutions of (2.1) to be po s i t i v e and bounded. In a d d i t i o n to hypotheses ( i ) , ( i i ) , ( i i i ) , suppose that lim sup[g(x,A,B)/g(x,A,0)] < « x-»-» for a l l p o s i t i v e constants A and B. Then, there exist p o s i t i v e constants M and a such that | & 3 $ < M for 11 » > . . and therefore condition (2.6) implies (2.2). Theorems 2.1 and 2.2 then imply the following c o r o l l a r y . C o r o l l a r y 2.5. Suppose lim sup[g(x,A,B)/g(x,A,0) ] < =° for a l l p o s i t i v e constants A and B. Then (2.6) i s a necessary and s u f f i c i e n t condition f o r Eq. (2.1) to have a bounded po s i t i v e solution i n some i n t e r v a l (x 0,°°), x Q > 0. Consider the following a l t e r n a t i v e to hypothesis ( i i i ) . ( i v ) g 0(x,y) < g(x,y,p) < <f>(p) g 0U,y) for a l l x > 0 , y > 0 , p > 0 where gQ(x,y) is nonnegative, continuous, and monotone i n y for each x > 0, and <Kp) ^ s positive, continuous, and nondecreasing for p > 0. 40. Corol l a r y 2.6. Under hypotheses ( i ) , ( i i ) and ( i v ) , Eq. (2.1) 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 (XQ, 0 0), x Q > 0 i f and only i f there e x i s t s a p o s i t i v e number A such that oo / xg Q(x,A)dx < * . / 41. CHAPTER I I I ASYMPTOTIC SOLUTIONS OF SEMILINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH FACTORIZED LINEAR PART 1. Introduction Our purpose i s to prove existence of p o s i t i v e solutions of the semi-l i n e a r ordinary d i f f e r e n t i a l equation Ly ± h(t,y) = 0 (3.1±) considered e a r l i e r i n Chapter I. Although the re s u l t s obtained here are of the same s p i r i t as previous r e s u l t s i n Chapter I, our method i s quite d i f f e r e n t . We represent the l i n e a r equation Lz = 0 i n i t s f a c t o r i z e d form and l a t e r w i l l give s u f f i c i e n t conditions f o r equation (3.1±) to have po s i t i v e solutions y^Ct), y 2 ( t ) with the same asymptotic behaviour as solutions Z j C t ) , z 2 ( t ) of Lz = 0. Consider the l i n e a r d i f f e r e n t i a l equation where p Q , p^, and p 2 are po s i t i v e continuous functions i n a p o s i t i v e i n t e r v a l [tg,°°). Two l i n e a r l y independent solutions of (3.2) are z x ( t ) = p 0 ( t ) , z 2 ( t ) = p Q ( t ) P x ( t ) for t > t n > where 42. t * l ( t > " / P i ( s ) ds . We assume that l i m P j ( t ) = °°, Z j ( t ) i s bounded above and z 2 ( t ) i s bounded away from zero i n [t0,°°) 2. Existence of P o s i t i v e Solutions of Semilinear Equations S u f f i c i e n t conditions w i l l be established f o r (3.1±) to have p o s i t i v e solutions 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 considered. Case I. Consider the following hypotheses on h ( t , y ) : ( H 1 ) . There e x i s t s a p o s i t i v e constant c such that h(t,u) i s continu-ous, nonnegative and nondecreasing i n u for 0 < u < c and for a l l t > t Q . (H 2 ) . There e x i s t s a p o s i t i v e constant c such that h(t,u) i s continu-ous, nonnegative and nonincreasing i n u for 0 < u < c and for a l l t > t Q . Theorem 3.1. Under hypotheses (Hj) or (H 2), a s u f f i c i e n t condition for y(t) equation (3.1±) to have a p o s i t i v e s o l u t i o n y(t) such that lim j £ — r — r - = t-M» Z l constant > 0 i s that / P x ( t ) p 2 ( t ) h ( t , K p 0 ( t ) ) dt < » (3.3) fc0 for a l l K e (o, —] where M = sup z ^ t ) . t > t 0 43. Proof: The proof w i l l be given f i r s t f o r (3.1+) under ( H j ) . For an a r b i t r a r y K e (0, ^] choose T = T(K) > t Q such that oo / P x ( t ) p 2 ( t ) h ( t , K p 0 ( t ) ] dt < | . (3.4) T The integrand i s c l e a r l y continuous and nonnegative since P 1 ( t ) , p 2 ( t ) , h ( t , K p Q ( t ) ) are a l l continuous and K p Q ( t ) < sup P 0 ( t ) < c. Let C = t>T C[T,°o) be the space of a l l continuous functions i n [T, 0 0) with the topology of uniform convergence on compact subintervals of [T,°°). Let Y be the closed convex subset of C defined as Y = {y e C: | p Q ( t ) < y ( t ) < K p Q ( t ) , t > T} . Let M be the mapping from C into C defined by oo s (My)(t) = Kp Q(t) - p Q ( t ) [/ (/ P l ( o ) da) p 2 ( s ) h(s,y(s)) ds], t > T . t t (3.5) We need to v e r i f y that ( i ) M maps Y in t o Y; ( i i ) M i s a continuous mapping on Y; ( i i i ) MY i s r e l a t i v e l y compact, and then apply the Schauder-Tychonoff f i x e d point theorem. To prove ( i ) l e t y e Y so |- p 0 ( t ) < y(t) < K p Q ( t ) for t > T. Then for t > T > t Q , 44. OO g OO / (/ Pl(o) da) p 2 ( s ) h(s,y(s)) ds < / P ^ s ) p 2 ( s ) h(s,Kp 0(s)) ds < t t t n by (3 .3) . Therefore the l e f t hand side converges for a l l t > T. Further-more, i t s a t i s f i e s oo s oo / (/ P l ( a ) do) p 2 ( s ) h(s,y(s)) ds < / P ^ s ) p 2 ( s ) h(s,Kp 0(s)) ds < | t t T by hypothesis. Hence, Kp 0 ( t ) > My(t) > Kp Q(t) - | p Q ( t ) = | p 0 ( t ) , so My e Y. To prove ( i i ) , l e t {y Q} be a convergent sequence of functions i n Y to y e i n the topology of C. From the d e f i n i t i o n of M |My n(t) - My(t)| OO g - IPoCO / (/ Pi<a> d a ) P 2 ( s ) [ h ( s . y n < s ) ) ~ h ( s , y ( s ) ) ] ds| t t The integrand has uniform l i m i t zero i n [T,<*>) and i s bounded i n absolute value by P j ( t ) p 2 ( t ) h ( t , K p Q ( t ) ) , which has f i n i t e i n t e g r a l . By Lebesgue' dominated convergence theorem |My R(t) - My(t)| -»• 0 uniformly as n-»-°°. Therefore lim My = My i n the topology of C and so M i s continuous. n>°° 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 to prove that the set of functions {My: y e Y} i s uniformly bounded and equicontinuous. {My} i s obviously uniformly bounded as shown i n ( i ) . To 45. show equicontinuity of {My}, one proceeds as before, i . e . one asserts that for a l l y e Y and t e [T,»)> oo g (My)'(t) = K P u ( t ) - p „ ( t ) / (/ P l ( o ) do) p 2 ( s ) h(s,y(s)) ds t t OO + p 0 ( t ) / P l ( t ) p 2 ( s ) h(s,y(s)) ds (3.6) t i s uniformly equibounded i n [T,°°). From the Mean Value theorem, My i s equicontinuous i n [T,°°) and by A s c o l i ' s theorem My i s r e l a t i v e l y compact. Applying the Schauder-Tychonoff fixed point theorem, we conclude that M has a f i x e d point y e Y; (My)(t) = y ( t ) f o r a l l t > T. 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 equation My = y twice completes the proof that 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 the property that lim y ( t ) / p Q ( t ) = K, or t+oo equivalently lim y(t)/zj^(t) = K. t-V°o In the case of (3.1-) and h(t,y) s a t i s f y i n g (H^), replace (3.5) by oo s (My)(t) = | p 0 ( t ) + p 0 ( t ) [/ (/ P l ( o ) do) p 2 ( s ) h(s,y(s)) ds], t > T . t t The same argument as before y i e l d s the existence of a p o s i t i v e s o l u t i o n of (3.1-) with lim y ( t ) / z 1 ( t ) = j. Now consider (3.1+) and l e t h(t,y) s a t i s f y ( H 2 ) . For an a r b i t r a r y K e (0,^3 choose T > t Q such that oo / P : ( t ) p 2 ( t ) h ( t , K p 0 ( t ) ) dt < K . T Let Y be the closed convex subset of C defined as: Y = {y e C: Kp Q ( t ) < y(x) < 2Kp Q(t), t > T } 46. Let M be the mapping from C into C defined by oo g (My)(t) = 2Kp Q(t) - p Q ( t ) [/ (/ P l ( o ) da) p 2 ( s ) h(s,y(s)) ds], t > T . t t We need to v e r i f y ( i ) , ( i i ) , ( i i i ) as before. To prove ( i ) l e t y e Y so Kp Q ( t ) < y(t) < 2 K p Q ( t ) . Then f o r t > T; OO g oo / (/ P l ( a ) da) p 2 ( s ) h(s,y(s)) ds < / P ^ s ) p 2 ( s ) h(s,Kp 0(s)) ds < K . t t T Hence 2Kp Q(t) > My(t) > 2Kp Q(t) - Kp Q ( t ) = Kp Q(t) and My e Y. To prove ( i i ) l e t iy^ be a convergent sequence of functions i n Y to y e Y i n the topology of C. From the d e f i n i t i o n of M oo s I My (t)-My(t)| = | p Q ( t ) / (/ P l ( a ) da) p 2 ( s ) [h(s,y ( s ) ) -h ( s,y ( s ) ) ] ds| . t t The integrand has uniform l i m i t zero and i s bounded i n absolute value by Pj^(t) p 2 ( t ) h ( t , K p Q ( t ) ) , which has f i n i t e i n t e g r a l . Hence by Lebesgue's dominated convergence theorem lim My = My i n the topology of C and M i s n 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. A p p l i c a t i o n of the Schauder-Tychonoff fixed point 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 equation My = y twice completes the proof that 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 lim y ( t ) / p Q ( t ) = 2K, or equivalently, lim y ( t ) / z x ( t ) = 2K. t-»-°° t * 0 0 In the case of (3.1-) replace the previous mapping by 47. oo s (My)(t) - Kp 0(t) + p 0 ( t ) [J (/ P l ( o ) da) p 2 ( s ) h(s,y(s)) ds] t t and v i r t u a l l y the same argument as before y i e l d s the existence of a p o s i t i v e s o l u t i o n of (3.1-) with lim y ( t ) / z 1 ( t ) = K. £ + 0 0 Case I I . Consider the following hypotheses on h ( t , y ) : ( H 3 ) . There e x i s t s a p o s i t i v e constant c such that h(t,u) i s continuous, nonnegative and nondecreasing i n u for u > c and for a l l t > t Q . (H^). There e x i s t s a p o s i t i v e constant c such that h(t,u) i s continuous, nonnegative and nonincreasing i n u for u > c and for a l l t > t Q . Theorem 3.2. Under hypothesis (H 3) or (H^) a s u f f i c i e n t condition f o r equation (3.1±) to have a p o s i t i v e s o l u t i o n y(t) such that lim y ( t ) / z 2 ( t ) = t-*-°° constant > 0 i s that / p 2 ( t ) h ( t , K z 2 ( t ) ) dt < » (3.6) for a l l K > c/y where \i = i n f z 2 ( t ) . t > c o The proof w i l l be given f i r s t f o r (3.1+) with h(t,y) s a t i s f y i n g (H g). Proof: For an a r b i t r a r y K > c/\i choose T > t Q such that / p 2 ( t ) h ( t , K z 2 ( t ) ) dt < | T 48. The integrand i s c l e a r l y continuous and nonnegative since p 2 ( t ) and h ( t , K z 2 ( t ) ) are both continuous and K z 2 ( t ) > c/u i n f z 2 ( t ) > c for t > T. t>T Let Y be the closed convex subset of C defined as Y = {y e C: | z 2 ( t ) < y(t) < Kz 2(t)} . Let M be the mapping from C into C defined as: t 0 0 (My)(t) = | p 0 ( t ) PjCt) + p 0 ( t ) [/ P l ( s ) / p 2(o)h(a,y(a))d0 ds], t > T. T s (3.7) To show M maps Y into Y, l e t y e Y so — z 2 ( t ) < y(t) < K z 2 ( t ) . We have t oo t oo / P i ( s ) / p 2(o-)h(o,y(o))da ds < / P l ( s ) (/ p 2(a)h(o,Kz 2(a) ]da)ds T s T T < | / t p 1 ( s ) d s < | P x ( t ) . T Then | z 2 ( t ) < (My)(t) < | p 0 ( t ) P : ( t ) + j p Q ( t ) P : ( t ) = Kp Q(t) P L ( t ) = K z 2 ( t ) . and consequently My e Y. To prove M i s continuous, l e t {yfl} be a sequence i n Y that converges to y e Y i n the topology of C. Note that t oo |My n(t) - My(t)| = |p Q(t) / P l ( s ) / p 2(a)[h(o,y (a)) - h(o,y(o))]da ds| . T s 49. It follows r o u t i n e l y by previous arguments and the use of Lebesgue's dominated convergence theorem that lim My = My and thus M i s continuous. 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 that MY = {My: y e Y} i s r e l a t i v e l y compact. By the Schauder-Tychonoff fixed point theorem and d i f f e r e n t i a t i o n of My = y, we conclude that y(t) i s a p o s i t i v e s o l u t i o n of (1.2+) with the property that lim y ( t ) / p 0 ( t ) P 1 ( t ) = -j i . e . , lim y ( t ) / z 2 ( t ) = j. In the case of (3.1-) and h(t,y) s a t i s f y i n g (H 3) replace (3.7) by t <*> (My)(t) = K p 0 ( t ) PjCt) - p Q ( t ) [/ P l ( s ) / p 2 ( a ) h ( o , y ( 0 ) ) d a ds], t > T. T s As before, we can show that 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 lim y ( t ) / p 0 ( t ) P 1 ( t ) = K, or equivalently lim y ( t ) / z 2 ( t ) = K. Now suppose (H^) holds. For an a r b i t r a r y K > c/p choose T > t Q such that CD / p 2 ( t ) h ( t , K z 2 ( t ) ) dt < K . T Let Y be the closed convex subset of C defined as: Y = {y e C: K z 2 ( t ) < y(t) < 2 K z 2 ( t ) , t > T ] . A p a r a l l e l argument as before establishes fixed points of the mappings defined by 50. t » (My)(t) = K p Q ( t ) P x ( t ) + p Q ( t ) [/ P l ( s ) / p 2(a)h(o,y(a))da ds], t > T; T s t » (My)(t) = 2Kp Q(t) PjCt) - p Q ( t ) [/ P l ( s ) / p 2(o)h(a,y(a))da ds], t > T, T s y i e l d i n g p o s i t i v e solutions of (3.1+),(3.1-), re s p e c t i v e l y , with the property that lim y ( t ) / z 2 ( t ) = K, lim y ( t ) / z 2 ( t ) = 2K. We close t h i s section by gi v i n g the following summary. Theorem 3.1 guaranteed the existence of i n f i n i t e l y many bounded p o s i t i v e solutions y ( t ) of equation (3.1±) which behave asymptotically l i k e the s o l u t i o n z ^ t ) of Lz = 0, i . e . y(t) ~ z x ( t ) (as t * 0 0 ) . In Theorem 3.2, we gave s u f f i c i e n t conditions f o r the existence of i n f i n i t e l y many unbounded p o s i t i v e solutions that behave asymptotically l i k e z 2 ( t ) , i . e . y(t) ~ z 2 ( t ) as t-*-00. 3. A Special Case In t h i s section, we w i l l reveal the r e l a t i o n s h i p between the material of Chapter I and that of the present chapter. S p e c i f i c a l l y speaking, we w i l l show that equation (1.1±) i s a s p e c i a l case of (3.2±) and our r e s u l t s i n Chapter I I I reduce to previous r e s u l t s of Chapter I. Consider the s p e c i a l case Lz = z". Two l i n e a r l y independent solutions of Lz = 0 are z ^ t ) = 1, z 2 ( t ) = t. Furthermore, as shown i n the Introduction, L has the f a c t o r i z e d form where p 0 ( t ) = z x ( t ) = 1; P l ( t ) = [ z 2 ( t ) / z 1 ( t ) ] ' = 1; p 2 ( t ) = [ P o ( t ) P i C t ) ] " 1 = 1. We can e a s i l y show that the solutions y ^ t ) , y 2 ( t ) of y" ± h(t,y) = 0 have the same asymptotic behaviour as z ^ ( t ) , z 2 ( t ) as t-* 0 0. Conditions (3.3) and (3.6) of Theorems 3.1, 3.2 reduce to, re s p e c t i v e l y , th(t,K) dt < °° ; (3.8) h(t,Kt) dt < °° , (3.9) for some p o s i t i v e constant K. It may be noted that (3.8) and (3.9) are i d e n t i c a l to conditions (1.2) and (1.8) of Chapter I. As i n Chapter I, condition (3.8) i s s u f f i c i e n t for the existence of a pos i t i v e s o l u t i o n y ( t ) such that lim y(t) = constant > 0 and condition (3.9) i s s u f f i c i e n t f o r the existence of an unbounded s o l u t i o n y(t) such that lim y ( t ) / t = constant > 0. t. 4. Examples We close t h i s chapter by gi v i n g two examples of the general equation (3.1±). Example 3.3. Consider the ordinary d i f f e r e n t i a l equation I d , d z . , o „ L z = T d T ( t d t 0 ~ k 2 = 0 • (3.10) 52. where k i s a p o s i t i v e constant. Since (3.10) i s n o n o s c i l l a t o r y , there e x i s t l i n e a r l y independent, eventually p o s i t i v e solutions z ^ t ) and z 2 ( t ) such that z X ( t ) lim —y—r- = 0 . 2->(t) Moreover, the operator L i n (3.10) has the f a c t o r i z e d form (3.2) where Z l = Pfj' p l = ( ^ ^ J ' ' P2 = t p0 P l l " 1 ' The solutions z ^ t ) , z 2 ( t ) are given as before by z : ( t ) = p 0 ( t ) , z 2 ( t ) = p Q ( t ) P x ( t ) where t P x ( t ) = / P l ( s ) ds . An a p p l i c a t i o n of Theorems 3.1 and 3.2 w i l l e s t a b l i s h s u f f i c i e n t conditions f o r Ly ± h(t,y) = 0 to have p o s i t i v e solutions y^(t) ~ z ^ t ) and y 2 ( t ) ~ z 2 ( t ) as t*°°, where z^(t) and z 2 ( t ) are l i n e a r l y independent solutions of (3.10). The main task, here, i s to f i n d z 1 and z 2-Equation (3.10) i s a modified Bessel equation of order zero, that i s of the form z" + -i- z' - k 2 z = 0 (3.11) 53. Two asymptotically ordered s o l u t i o n s of (3.10), or equivalently (3.11), are of the form: Z l ( t ) = K Q ( k t ) ; z 2 ( t ) = I 0 ( k t ) , where I Q and K Q denote modified Bessel functions of order zero. In order to f i n d e x p l i c i t asymptotic behaviour of these solutions, we remove the f i r s t d e r i v a t i v e by l e t t i n g z ( t ) = t ~ 1 / 2 u ( t ) . Then (3.11) Is equivalent to u" + [ — - k 2 ] u - 0. 4 t 2 A s o l u t i o n u(t) has the asymptotic form u(t) ~ e l 0^ t^ where to = /~qg = /kT = ±k by Thome's scheme [37]. Therefore, u(t) ~ exp (± kt) . By s u b s t i t u t i o n , 1 -kt Z i ( t ) ~ (constant) ( — e ) , / t 1 kt z 9 ( t ) ~ (constant) ( — e ) / t as t-*-00. Then p 2 = [PoP^ - 1 gives i— -kt P2(t) ~ (constant) / t e and conditions (3.3) and (3.6) reduce to 54. / / t e k h ( t , c K 0 ( k t ) ) dt < <» , fc0 / / t e h ( t , c l 0 ( k t ) ) dt < <» , c 0 r e s p e c t i v e l y , for some constant c > 0. Co r o l l a r y 3.3. A s u f f i c i e n t condition for (3.1±) ( i n the case that L i s given by (3.10)) to have a bounded p o s i t i v e s o l u t i o n y(t) such that lim y ( t ) / z : ( t ) = constant > 0 i s that / / t e h ( t , c K 0 ( k t ) ) dt < » for some po s i t i v e constant c. Co r o l l a r y 3.4. A s u f f i c i e n t condition for (3.1±) ( i n the case that L i s given 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 that lim y ( t ) / z 2 ( t ) = constant > 0 i s that / /t e k h ( t , c l 0 ( k t ) ) dt < » for some pos i t i v e constant c. Example 3.5. Consider the ordinary d i f f e r e n t i a l equation 55. L z = T o T ( t - p 2 fc2r 2 = 0 ( 3 - 1 2 ) where 2r i s a po s i t i v e integer and p i s a p o s i t i v e constant. Since (3.12) 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 2 ( t ) , p ^ t ) and p 0 ( t ) are given as before. A fundamental set { z ^ t ) , z 2 ( t ) } of eventually p o s i t i v e asymptotically ordered solutions of Lz = 0 has the asymptotic behaviour [35, p. 85] Z 2 M ~ t " ( r + 1 ) / 2 exP(p i^ i ) as t * 0 0 . Hence p 2 ( t ) ~ Ip" t ( 1 ~ r ) / 2 exp(-p ~ -^) as t-*- , P 2 ( t > Z 2 ( t ) 1 ( l - r ) / 2 , t*\ The conditions (3.3) and (3.6) reduce to, re s p e c t i v e l y , r+1 / t ( 1 ~ r ) / 2 exp(p hCt.Kz t^)) dt < - , 0 0 n - /o r+1 / t U T ) U exp(-p h ( t , K z 2 ( t ) ) dt < » , for some p o s i t i v e constant K. 56. C o r o l l a r y 3.5. A s u f f i c i e n t c o n d i t i o n f o r (3.1±) ( i n t h e c a s e t h a t L i s g i v e n by (3.12)) to have a d e c a y i n g p o s i t i v e s o l u t i o n y ( t ) w i t h l i m y ( t ) / z x ( t ) = c o n s t a n t > 0 i s t h a t 0 0 r+1 J t ( 1 _ r ) / 2 exp(p ^ T - h ( t , K z 1 ( t ) ) d t < -t„ h o l ds f o r some p o s i t i v e c o n s t a n t K. C o r o l l a r y 3.6. A s u f f i c i e n t c o n d i t i o n f o r (3.1±) ( i n the case t h a t L i s g i v e n by (3.12)) t o have an unbounded p o s i t i v e s o l u t i o n y ( t ) w i t h l i m t + o o y ( t ) / z 2 ( t ) = c o n s t a n t > 0 i s t h a t 0 0 n - W? r+1 / t U T ) U exp(-p l _ h ( t , K z 2 ( t ) ] d t < -r+1 2' h o l d s f o r some p o s i t i v e c o n s t a n t K. APPENDIX Recently, 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 to the semilinear ordinary d i f f e r e n t i a l equations (3.1±) to the quas i l i n e a r case Ly = h(t,y,y'). Although, Kusano et a l . , use the same technique ( f a c t o r i z a t i o n of disconjugate operators) to e s t a b l i s h existence c r i t e r i a , the outcome i s quite d i f f e r e n t . The re s u l t s obtained are much sharper than previous ones [36] and guarantee existence of global solutions instead of the usual l o c a l s o lutions. We s h a l l b r i e f l y discuss t h i s new work i n the l i g h t of the material of Chapter I I I . The operator L has the usual f a c t o r i z e d form with L z = p - ^ i F t p 7 ( T r a T ^ i - f t y ^ ' z e c 2 t t 0 , » ) ,Z2(tV i p 0 ( t ) = Z l ( t ) , P l ( t ) = [^-^y) , p 2 ( t ) = P o ( t ) p i ( t ) As before, z : ( t ) = p 0 ( t ) , z 2 ( t ) = p Q ( t ) P x ( t ) where t P : ( t ) = / Pi(s ) ds c0 are two l i n e a r l y independent, asymptotically ordered solutions of Lz = 0. Furthermore, a new function p(t) = p 0 ( t ) P 1 ( t ) / P 1 ( t ) + l p 0 ( t ) l 58. i s introduced. The following assumptions are made: 2 - i (Aj) Each p i ( t ) i s p o s i t i v e i n [t 0,»), p ± e C [t0,°°), i = 0,1,2 and lim P j ( t ) = + ». t->-oo (A 2) h: [t0,°°) x R+ x R -»• R, R + = [0,»), i s continuous and s a t i s f i e s |h(t,y,z)| < H(t,|y|,|z|) f o r a l l t e [tQ, 0 0), y e R+, where H(t,u,v) i s continuous i n [tp, 0 0) x R+ x R_j., nondecreasing i n u f o r each t,v and non-decreasing i n v for each t,u. (A 3) X - 1 H(f,Xu,Xv) i s a nondecreasing function of X e (0,») and lira X - 1 H(t,Xu,Xv) = 0 for each f i x e d (t,u,v) e [t0,°°) x R+ x R+. X>°° The following theorems give s u f f i c i e n t conditions f o r the existence of bounded and unbounded solutions of Ly = h(t,y,y') . (3.13) Theorem 3.7. A s u f f i c i e n t condition f o r equation (3.13) to have i n f i n i t e l y many p o s i t i v e (negative) solutions y(t) i n [tQ, 0 0) such that lim y ( t ) / z j ( t ) t+°° e x i s t s and i s p o s i t i v e (negative, r e s p e c t i v e l y ) i s that / P l ( t ) P x ( t ) H ( t , a p 0 ( t ) , b p ( t ) ) dt < 0 0 59. for some p o s i t i v e constants a and b. We remind the reader that the s u f f i c i e n t condition of Theorem 3.7 i s weaker than condition (2.2) of Chapter I I . Theorem 3.8. A s u f f i c i e n t condition for equation (3.13) to have i n f i n i t e l y many p o s i t i v e (negative) solutions y(t) i n [tQ, 0 0) such that lim y ( t ) / z 2 ( t ) e x i s t s and i s p o s i t i v e (negative, r e s p e c t i v e l y ) i s that oo / p 2 ( t ) H ( t , a p 0 ( t ) , b p ( t ) P x ( t ) ) dt < 0 0 for some p o s i t i v e constants a and b. As i n the semilinear case, the s u f f i c i e n t conditions established f o r the quasilinear d i f f e r e n t i a l equation (3.13) are i n terms of the solutions Z j ( t ) , z 2 ( t ) of Lz = 0. Equation (3.13) has p o s i t i v e (negative) solutions yi(t)» y~2(z) w i t n the same asymptotic behaviour as z 1 ( t ) , z 2 ( t ) . Once again, the asymptotic s i m i l a r i t y between sol u t i o n s of nonlinear equations and that of the l i n e a r part becomes evident. 60. CONCLUSIONS Nonlinear ordinary d i f f e r e n t i a l equations possess asymptotic solutions very s i m i l a r to solutions of t h e i r l i n e a r counterpart. We have examined t h i s phenomenon i n two ways. F i r s t , the nonlinear equation, may i t be semilinear or q u a s i l i n e a r , i s taken by i t s e l f . Necessary and s u f f i c i e n t conditions are established d i r e c t l y f o r existence of l o c a l s o l u t i o n s of the nonlinear d i f f e r e n t i a l equation under study. These conditions vary, depending on the nature of the s o l u t i o n (bounded or unbounded). The second method considers the l i n e a r part of the d i f f e r e n t i a l equa-t i o n . The kind of n o n l i n e a r i t y , i . e . semilinear or q u a s i l i n e a r , i s not of great importance at the outset. The l i n e a r operator i s f a c t o r i z e d i n t o a product of two l i n e a r operators, and the solutions of the l i n e a r equation are found. Usually, one s o l u t i o n Zj^(t) i s bounded above and below by p o s i t i v e constants or functions and the other s o l u t i o n z 2 ( t ) i s bounded away from zero. A f t e r v e r i f y i n g the existence of such s o l u t i o n s , the o r i g i n a l non-l i n e a r d i f f e r e n t i a l equation comes into the p i c t u r e . Conditions are found which not only guarantee existence of bounded and unbounded solutions of the nonlinear d i f f e r e n t i a l equation, but also o f f e r information about t h e i r asymptotic behaviour at i n f i n i t y . We have e x p l i c i t l y shown that the bounded s o l u t i o n behaves l i k e z^(t) and the unbounded sol u t i o n behaves l i k e z 2 ( t ) . In any event, the two methods are i n harmony with each other as shown i n numerous examples. The r e s u l t s of t h i s thesis can be sharpened by considering existence of global solutions. The appendix to Chapter III and the recent work of 61. Kusano, Swanson and Usami [23] discusses t h i s p o s s i b i l i t y . The subject of t h i s thesis has found a secure place among mathemati-cians 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 and i s d e f i n i t e l y progressing r a p i d l y since i t s early days near the middle of t h i s century. 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