Asymptotic forms of positive solutions of second-order quasilinear ordinary di¨erential equations with sub-homogeneity

31 (2001), 35±49

Asymptotic forms of positive solutions of second-order quasilinear ordinary di¨erential equations with sub-homogeneity

Ken-ichi Kamo and Hiroyuki Usami

(Received March 8, 2000)

Abstract. We give asymptotic forms of positive solutions of second-order quasilinear ordinary di¨erential equations. We obtain further the uniqueness of positive decaying solutions.

1. Introduction

We consider second-order quasilinear ordinary di¨erential equations of the form

…ju⁰j^{a 1}u⁰†⁰ˆp…t†juj^{l 1}u;

…1:1†

on which we impose the following assumptions throughout this paper:

(H1) a and l are positive constants satisfying the sub-homogeneity condition 0<l<a;

(H2) p:‰t0;y† ! …0;y† is a continuous function such that p…t†@t^s as t!y.

Henceforth the notation ``f…t†@g…t† as t!y'' means that lim_t!y f…t†=g…t†

ˆ1.

By a solution of (1.1) we mean a function u such that u and ju⁰j^{a 1}u⁰ are of class C¹, and u satis®es (1.1) near ‡y. Throughout this paper we shall con®ne ourselves to the study of those solutions that do not vanish identically near ‡y.

When aˆ1, equation (1.1) reduces to the well-known Emden-Fowler equation of sublinear type:

u⁰⁰ˆp…t†juj^{l 1}u; 0<l<1:

…1:2†

Qualitative theory for (1.2) has been studied in great detail by many authors;

see, for example [2, 5, 7, 8]. Therefore it is an interesting problem to show

2000 Mathematics Subject Classi®cation. 34D05, 34E05.

Key Words and Phrases. quasilinear equation, positive solution, asymptotic behavior.

how one can extend asymptotic theory obtained for (1.2) to the quasilinear equation (1.1). It was shown in [6] that some of known results can be extended surely to equation (1.1). In the earlier paper [4] the authors have considered (1.1) under the super-homogeneity condition 0<a<l, and have shown that asymptotic forms for positive solutions of (1.1) with aˆ1 are still valid for those of (1.1) with some obvious modi®cations. However, it seems to the authors that only a few of known results for (1.2) have been extended to (1.1) (under the sub-homogeneity assumption). We therefore in the present paper intend to study further asymptotic theory of positive solutions of (1.1).

In particular, we aim to obtain asymptotic forms of positive solutions of (1.1).

To make a survey for possible asymptotic behavior of positive solutions of equation (1.1) we consider the equation

…ju⁰j^{a 1}u⁰†⁰ˆq…t†juj^{l 1}u;

…1:3†

with 0<l<aandqAC…‰t₀;y†;‰0;y††, which has slight generality than (1.1).

Let us give a rough classi®cation of positive solutions of (1.3), as a ®rst step, according to their asymptotic behavior. It is known that for every positive solution u, exactly one of the following four asymptotic behavior is possible:

…D† …decaying solution†

t!ylim u⁰…t† ˆ lim

t!yu…t† ˆ0;

…AC† …asymptotically constant solution†

t!ylim u⁰…t† ˆ0 and lim

t!y u…t† ˆconstA…0;y†;

…AL† …asymptotically linear solution†

t!ylim u⁰…t† ˆ lim

t!y

u…t†

t ˆconstA…0;y†;

…ASL† …asymptotically superlinear solution†

t!ylim u⁰…t† ˆ lim

t!y

u…t†

t ˆ ‡y:

Necessary and/or su½cient conditions for the existence and nonexistence of solutions of types (D), (AC), (AL), and (ASL), respectively, have been obtained in [6]: Equation (1.3) has solutions of

…i† type …D† if

…_{y …y}

t q…s†ds

_1=a

dt<y;

…ii† type …AC† if and only if

…_{y …y}

t q…s†ds

_1=a

dt<y;

…iii† type …AL† if and only if …_y

t^lq…t†dt<y;

…iv† type …ASL† if and only if …_y

t^lq…t†dtˆy;

and has no solutions of type (D) if lim inf_t!yt^1‡aq…t†>0.

These results yield necessary and su½cient conditions for (1.1) to have solutions of type (D), (AC), (AL) and (ASL), respectively: Equation (1.1) has a positive solution of

…I† type …D† if and only if s‡a‡1<0;

…II† type …AC† if and only if s‡a‡1<0;

…III† type …AL† if and only if s‡l‡1<0;

…IV† type …ASL† if and only if s‡l‡1V0:

It is not known how positive solutions of type (D) and type (ASL) behave near

‡y. In order to give asymptotic forms of all possible positive solutions it is essential to determine asymptotic forms near‡y of positive solutions of those types. We can settle this problem completely in this paper.

It is of another interest to see whether or not positive solutions of each of above-mentioned types are unique. We will show that equation (1.1) has at most one positive solution of type (D).

This paper is organized as follows: In § 2 we prepare auxiliary lemmas which will be employed later. In § 3 we give two theorems concerning asymptotic properties of positive solutions of type (ASL) and of type (D).

These theorems play crucial role in establishing our results. Asymptotic forms for all positive solutions of equation (1.1) are given in § 4, which is the main object of this paper. In § 5 we give uniqueness theorems of solutions of type (D) as an application of our asymptotic theory. Lastly in § 6 we will mention the duality between the results of super-homogeneous case obtained in [4] and those of sub-homogeneous case obtained in this paper.

2. Preliminary lemmas

In this section we collect preliminary lemmas which will be employed in the sequel.

Consider two di¨erential equations of the same kind …jy⁰j^{a 1}y⁰†⁰ˆf…t†jyj^{b 1}y;

…2:1†

…jY⁰j^{a 1}Y⁰†⁰ˆF…t†jYj^{b 1}Y;

…2:2†

where b>0 is a constant.

Lemma 2.1. Suppose that f and F are nonnegative and continuous and f…t†UF…t† for aUtUb, and y and Y are positive solutions on ‰a;bŠ of (2.1) and (2.2),respectively. If y…a†UY…a† and y⁰…a†<Y⁰…a† then y…t†<Y…t† for a<tUb.

A proof of Lemma 2.1 is found in [4].

Lemma 2.2. Let f AC¹‰T;y† be such that f⁰ is bounded,and„_y

jf…t†j^gdt

<y for some g>1. Then, lim_t!y f…t† ˆ0.

A proof of Lemma 2.2 is found in [3]. The following is a variant of l'Hospital's rule:

Lemma 2.3. Let f…t† and g…t† be continuously di¨erentiable functions de®ned near y and g⁰…t†00. Then,we have

lim inf

t!y

f⁰…t†

g⁰…t†Ulim inf

t!y

f…t†

g…t†Ulim sup

t!y

f…t†

g…t†Ulim sup

t!y

f⁰…t†

g⁰…t†

if either lim_t!yg…t† ˆy or lim_t!y g…t† ˆlim_t!y f…t† ˆ0 holds.

Lastly we consider the two dimensional ®rst-order di¨erential system dw

dt ˆ …A‡B…t††w;

…2:3†

where A is a constant matrix with simple characteristic roots and BAC‰T;y†

satis®es lim_t!yB…t† ˆ0.

Lemma 2.4. Let k be an arbitrary characteristic root of A. Then we can

®nd a solution w of (2.3) satisfying the inequalities c1exp …Rek†t d1

…_t

TkB…s†kds

Ukw…t†k

Uc₂ exp …Rek†t‡d_{2 …t}

TkB…s†kds

; for some positive constants c1;c2;d1 and d2.

A proof of Lemma 2.4 is found in [1, Chapter 2].

3. Asymptotic equivalence theorems

Let us consider the following two equations …jx⁰j^{a 1}x⁰†⁰ˆa…t†jxj^{l 1}x;

…3:1†

…jy⁰j^{a 1}y⁰†⁰ˆb…t†jyj^{l 1}y;

…3:2†

where 0<l<a, a and b are positive continuous functions near ‡y. The next theorem asserts that, if a and b have the same asymptotic behavior, then so do positive solutions of type (ASL) of equation (3.1) and of equation (3.2).

Theorem 3.1. Suppose that

t!ylim a…t†

b…t†ˆ1

holds. Let x and y be positive solutions of type (ASL) of (3.1) and (3.2), respectively. Then x…t†@y…t† as t!y.

Proof. For ®xed t₀g1, we can ®nd a su½ciently small number m>0 and a su½ciently large number M>0 such that

mx…t₀†Uy…t₀†UMx…t₀†; mx⁰…t₀†Uy⁰…t₀†UMx⁰…t₀†:

Furthermore there exists t₁>t₀ such that m^{a l}a…t†Ub…t†UM^{a l}a…t† for tVt1. Since the function x…t† ˆMx…t† is a positive solution of type (ASL) of the equation

…jx⁰j^{a 1}x⁰†⁰ˆ a…t†

M^{l a}jxj^{l 1}x;

we see that x…t†1Mx…t†Vy…t†, tVt₁ by Lemma 2.1. Similarly we see that mx…t†Uy…t†, tVt2. Put lˆlim inft!yx…t†=y…t†. We know from the above observation that 0<l<y. Employing Lemma 2.3, we obtain

lˆlim inf

t!y

x…t†

y…t† Vlim inf

t!y

x⁰…t†

y⁰…t†ˆlim inf

t!y

x⁰…t†^a y⁰…t†^a

_1=a

Vlim inf

t!y

‰x⁰…t†^aŠ⁰

‰y⁰…t†^aŠ⁰

_1=a

ˆlim inf

t!y

a…t†x…t†^l b…t†y…t†^l

!_1=a

ˆl^l=a;

hence lV1. Similarly we obtain lim sup_t!yx…t†=y…t†U1. This implies that limt!y x…t†=y…t† ˆ1; that is x…t†@y…t†. This completes the proof.

The next theorem asserts that if ain equation (3.1) and b in equation (3.2) have the same asymptotic behavior in some sence, then positive solutions of

type (D) of equation (3.1) and of equation (3.2) also have the same asymptotic behavior.

Theorem 3.2. Let s‡a‡1<0. Suppose that 0<lim inf

t!y

a…t†

t^s Ulim sup

t!y

a…t†

t^s <y and

t!ylim a…t†

b…t†ˆ1:

If x and y are positive solutions of (3.1) and of (3.2),respectively,of type (D), then x…t†@y…t†.

Proof. First we will show that there exists positive constants c₂, c₃, c₄ and c5 satisfying

c2tⁿUxUc3tⁿ; c4t^{n 1}U x⁰Uc5t^{n 1} for large t;

where nˆ …s‡a‡1†=…l a†<0. Integrating (3.1) fromt to ‡y, we obtain

‰ x⁰…t†Š^aUc1

…_y

t s^sx^lds …3:3†

U c₁

s 1t^s‡1x…t†^l; that is

x⁰x ^l=aUc6t^s‡1†=a for large t;

where c6>0 is a constant. Again integrating the above inequality from t to

‡y, we obtainxUc₇tⁿnear‡y, wherec₇>0 is a constant. Substituting this estimate into (3.3), we obtain x⁰…t†Uc₈t^{n 1}, where c₈ is a positive constant.

Next we will show that there exist positive constants c9 and c10 satisfying xVc₉tⁿ; x⁰Vc₁₀t^{n 1} for large t:

To this end let zˆx… x⁰†^a. Then z satis®es the identity z⁰ˆz … x⁰†

x ‡a…t†x^l … x⁰†^a

: Invoking Young's inequality, we have

z⁰Vc11z … x⁰† x

…la‡a†=…la‡2a‡1† a…t†x^l … x⁰†^a

…a‡1†=…la‡2a‡1†

ˆc11za…t†…a‡1†=…la‡2a‡1†x…l a†=…la‡2a‡1†… x⁰†^{la a2}†=…la‡2a‡1†

ˆc₁₁z…la‡a‡l‡1†=…la‡2a‡1†a…t†…a‡1†=…la‡2a‡1†;

that is,

z⁰z …la‡a‡l‡1†=…la‡2a‡1†Vc₁₁a…t†…a‡1†=…la‡2a‡1† for large t;

where c₁₁ is a positive constant. Integrating this inequality from t to y, we obtain

z…a l†=…la‡2a‡1†Vc₁₂t…s…a‡1†‡la‡2a‡1†=…la‡2a‡1†; that is,

x^{a l}… x⁰†^{a…a l†}Vc13ts…a‡1†‡la‡2a‡1 for large t;

where c12 and c13 are positive constants. From the result x⁰Uc5t^{n 1} for large t as shown above, we see

x^{a l}t^{a…a l†…n 1†}Vc14ts…a‡1†‡la‡2a‡1; that is,

xVc₁₅tⁿ for large t;

where c₁₄ and c₁₅ are positive constants. Moreover some simple computation shows that x⁰Vc16t^{n 1}, where c16 is a positive constant. Similarly, we ®nd that

c₂tⁿUy…t†Uc₃tⁿ and c₄t^{n 1}U y⁰…t†Uc₅t^{n 1} hold near ‡y for some constants c_i>0, iˆ2;3;4;5.

Putlˆlim inft!yx…t†=y…t†. We know from the above consideration that 0<l<y. Invoking Lemma 2.3, we obtain

lˆlim inf

t!y

x…t†

y…t†Vlim inf

t!y

x⁰…t†

y⁰…t†ˆlim inf

t!y

… x⁰…t††^a … y⁰…t††^a

_1=a

Vlim inf

t!y

‰… x⁰…t††^aŠ⁰

‰… y⁰…t††^aŠ⁰

1=a

ˆlim inf

t!y

a…t†x…t†^l b…t†y…t†^l

!_1=a

ˆl^l=a:

This implies that lV1. Similarly we obtain lim sup_t!yx…t†=y…t†U1, and hence lim_t!yx…t†=y…t† ˆ1. This completes the proof.

4. Asymptotic forms of positive solutions

Possible asymptotic formulae for positive solutions of (1.1) are given here.

To get an insight into our problem, let us consider the typical case that p…t†1t^s:

…ju⁰j^{a 1}u⁰†⁰ˆt^sjuj^{l 1}u:

…4:1†

As was pointed out in [4], (4.1) may possess an exact positive solution of the form ct^l, c>0, lAR. In fact we can see that the function

u0…t† ˆ^ct^k with kˆs‡a‡1

a l and c^^{l a}ˆak…k 1†jkj^{a 1} …4:2†

is a positive solution of (4.1) ifk<0 ork>1. We further emphasize thatu₀ is of type (D) if k<0, and of type (ASL) ifk>1. This observation enables us to expect that positive solutions of type (D) and of type (ASL) behave like u0

in case of k<0 and k>0, respectively. We will ®nd that this conjecture is true. It is worth noting that the following equivalences hold:

k<0 ,s‡a‡1<0;

kˆ0 ,s‡a‡1ˆ0;

0<k<1,s‡l‡1<0<s‡a‡1;

kˆ1 ,s‡l‡1ˆ0;

k>1 ,s‡l‡1>0:

It will be found that asymptotic forms of positive solutions are a¨ected by the order relations of the three numbers 0, 1 and k.

Theorem 4.1. Let s‡l‡1>0 …k>1†. Then every positive solution u of equation (1.1) has the asymptotic form

u…t†@u₀…t†1ct^^k; where u0 is the function given by (4.2).

We can see this theorem by applying Theorem 3.1 to equations (1.1) and (4.1).

Theorem 4.2. Let s‡l‡1ˆ0 …kˆ1†. Then every positive solution u of equation (1.1) has the asymptotic form

u…t†@ a l a

_{1=…a l†}

t…logt†^{1=…a l†}:

To see this theorem, it su½ces to notice that the equation

…jv⁰j^{a 1}v⁰†⁰ˆ 1

t^l‡1 1‡ 1 …a l†logt

_{a 1}

1‡ 1‡l a …a l†logt

jvj^{l 1}v

has a positive increasing solution v explicitly given by v…t† ˆ a l

a

_{1=…a l†}

t…logt†^{1=…a l†}:

This simple observation and Theorem 3.1 give the validity of Theorem 5.2.

Theorem 4.3. Let s‡l‡1<0Us‡a‡1 …0Uk<1†. Then every positive solution u of equation (1.1) has the asymptotic form

u@c1t;

where c1 is a positive constant.

This is a simple consequence of the observation given in the Introduction.

Theorem 4.4. Let s‡a‡1<0 …k<0†. Then every positive solution u of equation (1.1) has one of the following asymptotic forms

u@u₀…t†1ct^^k; u…t†@c₁; u…t†@c₂t;

where c1 and c2 are positive constants,and u0 is given by (4.2).

To see this theorem we apply Theorem 3.2 to equations (1.1) and (4.1).

5. Uniqueness of positive decaying solutions

We now turn our attention to the problem of uniqueness of positive solutions of type (D).

Theorem 5.1. Under the assumption of Theorem 4.4,equation (1.1) does not have more than one positive solutions of type (D).

By this theorem we ®nd, for example, that the function u₀…t† is the only positive solution of type (D) of equation (4.1) if k<0.

Proof ofTheorem 5.1. Let x…t†andy…t†be positive solutions of (1.1) of type (D). By Theorem 4.4, we know that

x…t†;y…t†@^ct^k as t!y:

…5:1†

Furthermore, it is easy to see that

x⁰…t†;y⁰…t†@ckt^ ^{k 1} as t!y;

…5:2†

where c^ and k are given by (4.1).

Put z…t† ˆx…t† y…t†. It su½ces to show that z10. By using the mean value theorem, we know that there exist continuous functions x~₁…t† and x~₂…t†

such that

‰ax~₁…t†^{a 1}z⁰Š⁰ˆlp…t†x~₂…t†^{l 1}z;

…5:3†

where x~₁…t†, x~₂…t† lie between x…t† and y…t†, x⁰…t† and y⁰…t†, respectively. By (5.1) and (5.2) we see that x~₁…t†@^ckt^k, x~₂…t†@ct^^k. For the simplicity, we rewrite (5.3) in the form

‰x₁…t†z⁰Š⁰ˆlk…k 1†x₂…t†z;

…5:4†

where x₁…t†@t^{k 1†…a 1†} and x₂…t†@t^{s‡k…l 1†}.

The proof is divided into three cases according to the exponent …k 1†…a 1†. Here we will consider only the case …k 1†…a 1†<1, i.e.,

„_y

x₁…r† ¹drˆy because parallel arguments hold in the other cases. The change of variable sˆ„_t

x₁…r† ¹dr transforms (5.4) into z_ssˆlk…k 1†x₁…t†x₂…t†z;

…5:5†

where x₁…t†x₂…t†@1=‰1 …k 1†…a 1†Šs² as s!y. Hence we can rewrite (5.5) as

zssˆ lk…k 1†

‰1 …k 1†…a 1†Š²

1‡d…s†~ s² z;

…5:6†

where d…s†~ is a continuous function satisfying lim_s!y d…s† ˆ~ 0. Moreover the change of variable sˆe^r transforms (5.6) into



z z_ˆ lk…k 1†

‰1 …k 1†…a 1†Š²…1‡d…r††z;

…5:7†

where _ˆd=dr and d…r† is a continuous function satisfying limr!y d…r† ˆ0.

Now we reduce (5.7) to a ®rst-order system by introducing the new variables w₁ˆz, w₂ˆz:_

dw

dr ˆ …A‡B…r††w;

…5:8†

where

wˆ w₁ w₂

!

; Aˆ

0 1

lk…k 1†

‰1 …k 1†…a 1†Š² 1 0

B@

1 CA;

B…r† ˆ

0 0

lk…k 1†

‰1 …k 1†…a 1†Š² d…r† 0 0

B@

1 CA:

The eigenvalues of A are 1 2 1G



1‡ 4lk…k 1†

‰1 …k 1†…a 1†Š²

s !

:

For simplicity, we put

l₁ˆ1 2 1‡



1‡ 4lk…k 1†

‰1 …k 1†…a 1†Š²

s !

>0;

l₂ˆ1 2 1



1‡ 4lk…k 1†

‰1 …k 1†…a 1†Š²

s !

<0:

By Lemma 2.4, there exist solutions …w₁;w₂† and …w₁;w₂† of (5.8) satisfying c₁exp‰…l₁‡o…1††rŠUjw₁j ‡ jw₂jUc₂ exp‰…l₁‡o…1††rŠ;

c₃exp‰…l₂‡o…1††rŠUjw₁j ‡ jw₂jUc₄exp‰…l₂‡o…1††rŠ;

…5:9†

where c1, c2, c3 and c4 are positive constants. It is easily seen from (5.9) that f…w₁;w₂†;…w₁;w₂†g consists of a base of solutions of (5.8). Hence z is rep- resented as

zˆc₅w₁‡c₆w₁; …5:10†

_

zˆc₅w₂‡c₆w₂; …5:11†

where c₅ and c₆ are constants. We notice from (5.1) and (5.2) that z…r† ˆO exp kr

1 …k 1†…a 1†

; _

z…r† ˆO exp kr

1 …k 1†…a 1†

:

Below we will show that c5ˆc6ˆ0. If c500, then we ®nd from (5.10) and (5.11) that

jw1j ‡ jw2jUjzj ‡ jzj_ jc5j ‡ c₆

c5

…jw1j ‡ jw2j†:

…5:12†

By (5.9) we know that the left-hand side of (5.12) tends to y as r!y;

however, the right-hand side remains bounded. Hence we must have c₅ ˆ0;

that is

zˆc6w1; z_ˆc6w2: It follows therefore that

…jzj ‡ jzj†e_ ^{kr=…1 …k 1†…a 1††}ˆ jc₆j…jw₁j ‡ jw₂j†e ^{kr=…1 …k 1†…a 1††}: …5:13†

By the estimates for z and z, the left-hand side of (5.13) is bounded, while the_ right-hand side of (5.13) is estimated as

jc6j…jw1j ‡ jw2j†exp kr

1 …k 1†…a 1†

Vc3jc6jexp k

1 …k 1†…a 1†‡l2‡o…1†

r

:

Since 0<l<a, if c₆00, then the right-hand side of (5.13) diverges as r!y which is a contradiction. Hencec6ˆ0, i.e.,z10. This completes the proof.

If aˆ1, the condition for the uniqueness of positive decaying solutions can be relaxed. We consider the equation

u⁰⁰ˆg…t†juj^{l 1}u; 0<l<1;

…5:14†

where g is a positive function of class C¹ near ‡y.

Theorem 5.2. Let 0<l<1 and s< 2. Suppose that g…t† satis®es 0<lim inf

t!y

g…t†

t^s Ulim sup

t!y

g…t†

t^s <y;

…5:15†

and

0<lim inf

t!y

g⁰…t†

t^{s 1} Ulim sup

t!y

g⁰…t†

t^{s 1} <y:

…5:16†

Then (5.14) has at most one positive solution of type (D).

Proof. We ®nd from the proof of Theorem 3.2 that, for each positive solution u of (5.14) of type (D), there exist positive constants c₁, c₂, c₃ and c₄ satisfying

c1t^kUuUc2t^k; c3t^{k 1}U u⁰Uc4t^{k 1} for large t;

…5:17†

kˆs‡2 1 l:

Let x…t† and y…t† be positive decaying solutions of (5.14). Introduce the new functions w…t† ˆx…t†=y…t† and tˆ„_t

t0dr=y…r†². A computation gives w_ttˆg…t†y…t†^l‡3…w^l w†; tV0:

Moreover the change of variable sˆ

…_t

0 g…t…x††¹⁼²y…t…x††^l‡3†=2dx transforms this equation into

w f …s†w_ ˆw^l w; sV0;

…5:18†

where _ˆd=ds, and

f…s† ˆ ¹₂g…t† ³⁼²y…t† ^l‡1†=2‰g⁰…t†y…t† ‡ …l‡3†g…t†y⁰…t†Š:

From condition (5.15) and (5.16) we ®nd that 0<lim inf

s!y f…s†Ulim sup

s!y f…s†<y:

…5:19†

While from (5.17) and w_ ˆw⁰…t†dt

dt dt

ds ˆ …x⁰y xy⁰†g ¹⁼²y ^l‡3†=2; we know that

0<lim inf

s!y w…s†Ulim sup

s!y w…s†<y and w…s† ˆ_ O…1†:

…5:20†

Multiplying (5.18) by w_ and integrating the resulting equation on ‰s0;sŠ, we obtain

w_² 2

s s0

…_s

s0

f…r†w_²drˆ w^l‡1 l‡1

w² 2

s

s0

: …5:21†

By (5.19) and (5.20) this implies that w_ AL²‰s₀;y†. Since obviouslywˆO…1†

ass!y, Lemma 2.2 shows that lims!y w…s† ˆ_ 0. From Theorem 3.2 we see that lim_s!y w…s† ˆ1.

Multiplying (5.18) by w_ again and integrating the resulting equation on

‰s;y†, we obtain _ w²

2 …_y

s f…r†w_²drˆ 1 l‡1

1 2

w^l‡1 l‡1‡w²

2 : This formula shows that if w_20, then

1 l‡1

1

2<w^l‡1 l‡1

w²

2 for all s near y:

Since w>0, this inequality can not hold. Hence w_10, i.e., w11. Hence x1y. This completes the proof.

6. Duality between super-homogeneous case and sub-homogeneous case It is well known [7] that if aˆ1, i.e., for Emden-Fowler equations, there is a duality between the superlinear case (l>1) and the sublinear case (0<l<1). It is natural to expect that for equation (1.1) there exists similar duality between the super-homogeneous case (0<a<l) and the sub- homogeneous case (0<l<a). Hence we compare the results obtained in this paper and those obtained in [4]. To bring out the duality in full relief, we summarize our results in the following table:

Super-homogeneous case:

a<l Sub-homogeneous case:

l<a Relation of

parameters

Possible asymptotic

forms Relation of parameters

Possible asymptotic

forms k>1 s‡l‡1<0 u₀…t†;c₁t;c₂ s‡l‡1>0 u₀…t†

kˆ1 s‡l‡1ˆ0 c1 s‡l‡1ˆ0 Logarithmic growth 0<k<1 s‡a‡1<0

<s‡l‡1

c₁ s‡l‡1<0

<s‡a‡1

c₁t

kˆ0 s‡a‡1ˆ0 Logarithmic

decay s‡a‡1ˆ0 c1t k<0 s‡a‡1>0 u0…t† s‡a‡1<0 u0…t†;c1t;c2

In this table c₁ and c₂ are positive constants. When a<l and s‡a‡1ˆ0, every positive solution u of (1.1) has a logarithmic decay in the sense that

u…t†@a^{1=…l a†} a l a

_{a=…l a†}

…logt† ^{a=…l a†}:

References

[ 1 ] R. Bellman, Stability theory of di¨erential equations. McGraw-Hill, New York, 1953 (Reprint: Dover, 1969).

[ 2 ] T. A. Chanturiya, On the asymptotic representation of the solutions of the equation u⁰⁰ˆa…t†jujⁿsgnu, Di¨erentsial'nye Uravneniya8 (1972) 1195±1206. (in Russian) [ 3 ] K. Kamo, Classi®cation of proper solutions of some Emden-Fowler equations, Hiroshima

Math. J. 29 (1999) 459±477.

[ 4 ] K. Kamo and H. Usami, Asymptotic forms of positive solutions of second-order quasilinear ordinary di¨erential equations, Adv. Math. Sci. Appl. 10 (2000) 673±688.

[ 5 ] I. T. Kiguradze and T. A. Chanturiya, Asymptotic properties of solutions of non- autonomous ordinary di¨erential equations, Kluwer Academic Publisher, Dordrecht, 1992.

[ 6 ] M. Mizukami, M. Naito and H. Usami, Asymptotic behavior of solutions of a class of second order quasilinear ordinary di¨erential equations, preprint.

[ 7 ] M. Naito, Oscillation theory for ordinary di¨erential equations of Emden-Fowler type, SuÃgaku 37 (1985) 144±160. (in Japanese)

[ 8 ] J. S. W. Wong, On the generalized Emden-Fowler equations, SIAM Rev.17(1975) 339±

360.

Ken-ichi Kamo Department of Mathematics Graduate School of Science

Hiroshima University Higashi-Hiroshima 739-8526,Japan

kamo@math.sci.hiroshima-u.ac.jp Hiroyuki Usami

Department of Mathematics Faculty of Integrated Arts & Sciences

Hiroshima University Higashi-Hiroshima 739-8521,Japan

usami@mis.hiroshima-u.ac.jp