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. It has been our i n t e n t i o n to survey a modern aspect of t h i s progress. 62. BIBLIOGRAPHY 1. F.V. Atkinson, On l i n e a r perturbations of nonlinear d i f f e r e n t i a l equations, Canad. J. Math., 6(1954), pp. 561-571. 2. , The asymptotic solutions of second order d i f f e r e n t i a l equations, Ann. Mat. Pura. Appl., 37 (1954), pp. 347-378. 3. , On second order nonlinear o s c i l l a t i o n s , P a c i f i c J . Math., 5 (1955), pp. 643-647. 4. S. Belohorec, O s c i l l a t o r y solutions of c e r t a i n nonlinear d i f f e r e n t i a l equations of second order, Mat. - Fyz. Vasopis Sloven. Akad. Vied., 11 (1961), pp. 250-255. 5. R. Bellman, S t a b i l i t y Theory of D i f f e r e n t i a l Equations, McGraw-Hill, New York, 1970. 6. A.J. C a l l e g a r i and A. Nachman, Some singular, nonlinear d i f f e r e n t i a l equations a r i s i n g i n boundary layer theory, J . Math. Anal. Appl., 64 (1978), pp. 96-105. 7. , A nonlinear singular boundary value problem i n the theory of pseudoplastic f l u i d s , SIAM J. Appl. Math., 38 (1980), pp. 275-281. 8. C.V. Coffman and J.S.W. Wong, O s c i l l a t i o n and n o n o s c i l l a t i o n theorems for second order ordinary d i f f e r e n t i a l equations, Funkcial. E k v a c , 15 (1972), pp. 119-130. 9. R. Conti, D. G r a f f i and G. Sansone, The I t a l i a n c ontribution to the theory of nonlinear ordinary d i f f e r e n t i a l equations and to nonlinear mechanics during the years 1951-1961, Q u a l i t a t i v e Methods i n the Theory of Nonlinear Vibrations, Proc. Internat. Sympos. Nonlinear Vi b r a t i o n s , Vol. I I , 1961, pp. 172-189. 10. R.E. Edwards, Functional Analysis, Holt, Rinehart and Winston, Inc., 1965. 11. R. Emden, Gaskugeln, Anwendungen der mechanischen Warmen - theorie auf Kosmologie und meterologische Probleme, B.G. Teubner, L e i p z i g , 1907, Chapt. XII. 12. R.H. Fowler, The form near i n f i n i t y of r e a l , continuous solutions of a c e r t a i n d i f f e r e n t i a l equation of the second order, Quart. J. Math., 45 (1914), pp. 289-350. 13. , Some re s u l t s on the form near i n f i n i t y of r e a l , continuous solutions of a c e r t a i n type of second order d i f f e r e n -t i a l equations, Proc. London Math. S o c , 13 (1914), pp. 341-371. 63. 14. , The s o l u t i o n of Emden's and s i m i l a r d i f f e r -e n t i a l equations, Monthly Notices Roy. Astro. S o c , 91 (1930), pp. 63-91. 15. , Further studies of Emden's and s i m i l a r d i f f e r e n t i a l equations, Quart. J . Math., 2 (1931), pp. 259-288. 16. M.E. Hammett, N o n o s c i l l a t i o n properties of a nonlinear d i f f e r e n t i a l equation, Proc. Amer. Math. S o c , 30(1971), pp. 92-96. 17. P. Hartman, Ordinary D i f f e r e n t i a l Equations, John Wiley, New York, 1964. 18. K. Kr e i t h and C A . Swanson, Asymptotic solutions of semilinear e l l i p t i c equations, J . Math. Anal. Appl., 98 (1984), pp. 148-157. 19. T. Kusano and H. Onose, O s c i l l a t i o n s of fun c t i o n a l d i f f e r e n t i a l equations with retarded arguments, J . D i f f e r e n t i a l Equations, 15 (1974), pp. 269-277. 20. T. Kusano and C A . Swanson, Asymptotic properties of semilinear e l l i p t i c equations, Funkcial. E k v a c , 26 (1983), pp. 115-129. 21. , Asymptotic theory of singular semilinear e l l i p t i c equations, Canad. Math. B u l l . , 27 (1984), pp. 223-232. 22. , Growth properties of stationary Klein-Gordon equations, SIAM J. Math. Anal., 16 (1985), pp. 440-446. 23. T. Kusano, C A . Swanson and H. Usami, Pairs of p o s i t i v e solutions of qu a s i l i n e a r e l l i p t i c equations i n e x t e r i o r domains, P a c i f i c J . Math., to appear. 24. I.J. Homer Lane, On the t h e o r e t i c a l temperature of the sun under the hypothesis of a gaseous mass maintaining i t s volume by i t s i n t e r n a l heat and depending on the laws of gases known to t e r r e s t i a l experiment, Amer. J . S c i . and Arts, 4 (1869-1870), pp. 57-74. 25. A.J. Levin, N o n o s c i l l a t i o n of solutions of the equation x^ n^ + P l ( t ) x ( n - l ) + # > > + p n ( t ) x = 0, Russian Math. Surveys, 24 (1969), pp. 43-99. 26. Z. Nehari, On a class of nonlinear second order d i f f e r e n t i a l equations, Trans. Amer. Math. S o c , 95 (1960), pp. 101-123. 27. E.S. Noussair and C A . Swanson, P o s i t i v e solutions of q u a s i l i n e a r e l l i p t i c equations i n e x t e r i o r domains, J. Math. Anal. Appl., 75 (1980), pp. 121-133. 28. G. Polya, On the mean-value theorem corresponding to a given l i n e a r homogeneous d i f f e r e n t i a l equation, Trans. Amer. Math. S o c , 24 (1924), pp. 312-324. 64. 29. H.L. Royden, Real Analysis, 2nd ed., MacMillan Publishing Co., Inc., New York, 1968. 30. W. Rudin, Functional Analysis, McGraw-Hill Series i n Higher Mathematics, 1973. 31. G. Sansone, Equazioni D i f f e r e n z i a l i r e l Campo Reale, Vol. 2, 3rd ed., Z a n i c h e l l i , Bologna, 1963. 32. J . Schauder, Der Fixpunktsatz i n Funktionalraumen, Studia Math., 2 (1930), pp. 171-180. 33. B. Singh, Asymptotic nature of non o s c i l l a t o r y solutions of n order retarded d i f f e r e n t i a l equations, SIAM J . Math. Anal., 6 (1975), pp. 784-795. 34. , Necessary and s u f f i c i e n t conditions f o r maintaining o s c i l l a t i o n s and n o n o s c i l l a t i o n s i n general f u n c t i o n a l equations and t h e i r asymptotic properties, SIAM J . Math. Anal., 10 (1979), pp. 18-31. 35. L. S i r o v i c h , Techniques of Asymptotic Analysis, Springer-Verlag, New York - Heidelberg - B e r l i n , 1971. 36. C A . Swanson, Bounded p o s i t i v e solutions of q u a s i l i n e a r Schrodinger equations, Appl. Analysis, 14 (1983), pp. 179-190. 37. L.W. Thome, Zur Theorie der linearen, Differentialgleichungen, J . Reine. Agnew Math., 95 (1883), pp. 44-98. 38. W. Thompson (Lord K e l v i n ) , On the convective equilibrium of temperature i n the atmosphere, Manchester P h i l o s . Soc. P r o c , 2 (1860-62), pp. 170-176; r e p r i n t , Math, and Phys. Papers by Lord Kel v i n , 3 (1890), pp. 255-260. 39. W.F. Trench, Canonical forms and p r i n c i p a l systems for general disconjugate equations, Trans. Amer. Math. S o c , 189 (1974), pp. 319-327. 40. A. Tychonoff, E i n Fixpunktsatz, Math. Ann., I l l (1935), pp. 767-776. 41. J.S.W. Wong, On the generalized Emden-Fowler equation, SIAM Rev., 17 (1975), pp. 339-360.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Asymptotic theory of second-order nonlinear ordinary...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Asymptotic theory of second-order nonlinear ordinary differential equations Jenab, Bita 1985
pdf
Page Metadata
Item Metadata
Title | Asymptotic theory of second-order nonlinear ordinary differential equations |
Creator |
Jenab, Bita |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | The asymptotic behaviour of nonoscillatory solutions of second order nonlinear ordinary differential equations is studied. Necessary and sufficient conditions are given for the existence of positive solutions with specified asymptotic behaviour at infinity. Existence of nonoscillatory solutions is established using the Schauder-Tychonoff fixed point theorem. Techniques such as factorization of linear disconjugate operators are employed to reveal the similar nature of asymptotic solutions of nonlinear differential equations to that of linear equations. Some examples illustrating the asymptotic theory of ordinary differential equations are given. |
Subject |
Differential equations -- Asymptotic theory Differential equations, Nonlinear -- Asymptotic theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080164 |
URI | http://hdl.handle.net/2429/24690 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1985_A6_7 J45.pdf [ 2.55MB ]
- Metadata
- JSON: 831-1.0080164.json
- JSON-LD: 831-1.0080164-ld.json
- RDF/XML (Pretty): 831-1.0080164-rdf.xml
- RDF/JSON: 831-1.0080164-rdf.json
- Turtle: 831-1.0080164-turtle.txt
- N-Triples: 831-1.0080164-rdf-ntriples.txt
- Original Record: 831-1.0080164-source.json
- Full Text
- 831-1.0080164-fulltext.txt
- Citation
- 831-1.0080164.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080164/manifest