DOI 10.1007/s10958-015-2466-6 Journal of Mathematical Sciences, Vol. 208, No. 5, August, 2015

ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS V. M. Evtukhov1 and M. A. Talimonchak1

UDC 517.925.44

We establish asymptotic representations for a class of solutions of nonlinear systems of ordinary differential equations asymptotically close to cyclic systems.

1. Statement of the Problem and Auxiliary Notation Consider a system of differential equations yi0 D fi .t; y1 ; : : : ; yn /; where fi W Œa; !Œ�

Yn

iD1

(1.1)

i D 1; n;

�Y 0 �! R; i D 1; n are continuous functions, �1 < a < ! � C1; 2 �Y 0 ; i

i

i 2 f1; : : : ; ng; are unilateral neighborhoods of Yi0 ; and Yi0 is equal either to 0 or to ˙1:

Definition 1.1. A solution .yi /niD1 of system (1.1) defined on the interval Œt0 ; !Œ� Œa; !Œ is called a P! .ƒ1 ; : : : ; ƒn�1 / -solution, where �1 � ƒi � C1; i D 1; n � 1; if it satisfies the following conditions: for

yi .t / 2 �Y 0 i

lim

t "!

t 2 Œt0 ; !Œ;

0 yi .t /yiC1 .t /

yi0 .t /yiC1 .t /

lim yi .t / D Yi0 ;

t "!

D ƒi ;

(1.2)

i D 1; n;

(1.3)

i D 1; n � 1:

Earlier, the asymptotics of P! .ƒ1 ; : : : ; ƒn�1 / -solutions for a cyclic system of differential equations of the form3 yi0 D ˛i pi .t /'iC1 .yiC1 /;

i D 1; n;

where ˛i 2 f�1; 1g ; i D 1; n; pi W Œa; !Œ!�0; C1Œ; i D 1; n; are continuous functions, 'i W �Y 0 !�0I C1Œ;

i D 1; n; are continuously differentiable functions satisfying the conditions lim

yi !Y 0 i yi 2� 0 Y i

yi 'i0 .yi / D �i ; 'i .yi /

i D 1; n;

n Y

iD1

i

�i ¤ 1;

1

Mechnikov Odessa National University Ukraine, 65000, Odessa, 2 Dvoryanskaya Str. Assume that a > 1 for ! D C1 and a > ! � 1 for ! < C1: 3 Here and in what follows, for all functions and parameters with index n C 1; we assume that they are in one-to-one correspondence with the corresponding quantities with index 1. 2

Translated from Neliniini Kolyvannya, Vol. 17, No. 2, pp. 213–227, April–June, 2014. Original article submitted January 3, 2014 . 1072-3374/15/2085–0535

c 2015 �

Springer Science+Business Media New York

535

V. M. E VTUKHOV

536

AND

M. A. TALIMONCHAK

were investigated in [1–5]. The aim of the present paper is to establish necessary and sufficient conditions for the existence of P! .ƒ1 ; : : : ; ƒn�1 / -solutions to systems of differential equations of a more general form (1.1) and asymptotic relations for these solutions as t " ! in the case where ƒi ; i D 1; n � 1; are nonzero real constants. In this case, we can find a nonzero constant ƒn D lim

t "!

yn .t /y10 .t / yn0 .t /y1 .t /

that gives the relationship between the first and n th components of the P! .ƒ1 ; : : : ; ƒn�1 / -solution. For this constant, we get � � 0 y2 .t /y10 .t / yn .t /yn�1 yn .t /y10 .t / .t / 1 D D lim 0 ::: 0 : ƒn D lim 0 y2 .t /y1 .t / ƒ1 : : : ƒn�1 t "! yn .t /y1 .t / t "! yn .t /yn�1 .t /

(1.4)

Setting

�i D

8 1 ˆ ˆ ˆ ˆ <

if Yi0 D C1

or Yi0 D 0

ˆ �1 if ˆ ˆ ˆ : or

and

Yi0 D �1 Yi0 D 0 and

�Y 0 is a right neighborhood of 0; i

�Y 0 is a left neighborhood of 0; i

we note that the numbers �i ; i D 1; n; specify the signs of components of the P! .ƒ1 ; : : : ; ƒn�1 / -solution in a certain left neighborhood of !: We study the problem of existence of P! .ƒ1 ; : : : ; ƒn�1 / -solutions of system (1.1) for fixed values of ƒi 2 R=f0g; i D 1; n � 1; and the asymptotics of these solutions as t " ! in the case where this system is close, in a certain sense, to a cyclic system with regularly varying nonlinearities. Definition 1.2. We say that the system of differential equations (1.1) satisfies the condition N.ƒ1 ; : : : ; ƒn�1 /; where ƒi 2 R=f0g; i D 1; n � 1; if, for each k 2 f1; : : : ; ng; there exist a number ˛k 2 f�1; 1g; �! �0; C1Œ of order a continuous function pk W Œa; !Œ �! �0; C1Œ; and a continuous function 'kC1 W �Y 0 kC1

0 such that, for any functions yi W Œa; !Œ �! �Y 0 ; i D 1; n; satisfying �kC1 regularly varying as ykC1 ! YkC1 i conditions (1.2) and (1.3), the representation

fk .t; y1 .t /; : : : ; yn .t // D ˛k pk .t /'kC1 .ykC1 .t //Œ1 C o.1/�

as

t "!

(1.5)

is true. By virtue of the definition of regularly varying functions (see [6, pp. 9, 10], Chap. I, Sec. 1.1), every function 'i ; i 2 f1; : : : ; ng; admits a representation 'i .z/ D jzj�i Li .z/;

(1.6)

where Li W �Y 0 �! �0; C1Œ is a function continuously varying as z ! Yi0 ; i.e., such that i

lim

z!Y 0 i z2� 0 Y i

Li .�z/ D1 Li .z/

for any

� > 0:

(1.7)

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

537

It is known (see [6, pp. 10–15], Chap. I, Sec. 1.2) that the limit relation (1.7) is true uniformly in � on any segment Œc; d � 2 �0; C1Œ (Property M1 ) and there exists a continuously differentiable function Li0 W �Y 0 �! �0; C1Œ slowly varying as z ! Yi0 (property M2 ) and satisfying the conditions lim

z!Y 0 i z2� 0 Y i

Li .z/ D 1; Li0 .z/

lim

z!Y 0 i z2� 0 Y i

i

zL0i0 .z/ D 0: Li0 .z/

(1.8)

Moreover, the function 'i 0 .z/ D jzj�i Li0 .z/

(1.9)

is continuously differentiable in the interval �Y 0 and such that i

lim

z!Y 0 i z2� 0 Y i

'i .z/ D 1; 'i0 .z/

lim

z!Y 0 i z2� 0 Y i

0 z'i0 .z/ D �i : 'i0 .z/

(1.10)

We now assume that, for some ƒi ; i 2 f1; : : : ; n � 1g; system (1.1) satisfies the condition N.ƒ1 ; : : : ; ƒn�1 / and, in addition, the orders �k ; k D 1; n; of the functions 'k are such that n Y

kD1

(1.11)

�k ¤ 1:

This enables us to introduce some auxiliary definitions. Q By virtue of (1.4), we get niD1 ƒi D 1: According to condition (1.11), the expression 1 � ƒi �i C1 is nonzero for at least one value i 2 f1; : : : ; ng . Let I D fi 2 f1; : : : ; ng W 1 � ƒi �iC1 ¤ 0g;

IN D f1; : : : ; ngnI

and let l be the minimum element of the set I : In view of the choice of l; we introduce auxiliary functions Ii ; i D 1; n; and nonzero constants ˇi ; i D 1; n; by setting 8 t R ˆ ˆ pi .� / d � ˆ ˆ ˆ < Ai Ii .t/ D ˆ ˆ Rt ˆ ˆ ˆ : Il .� /pi .� / d � Ai

for i 2 I ; for i 2 IN ;

8 1 � ƒi �iC1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˇl < ˇi D ƒl : : : ƒi�1 ˆ ˆ ˆ ˆ ˆ ˆ ˇl ˆ ˆ : ƒl : : : ƒn ƒ1 : : : ƒi�1

for i 2 I ; for i 2 fl C 1; : : : ; ngnI ; for i 2 f1; : : : ; l � 1gnI ;

where each integration limit Ai 2 f!; ag is chosen to guarantee that the corresponding integral Ii tends either to zero or to 1 as t " !:

V. M. E VTUKHOV

538

AND

M. A. TALIMONCHAK

In addition, we introduce the numbers A�i

D

º

1 for Ai D a; �1 for Ai D !;

(1.12)

i D 1; n:

Then sign Ii .t / D

8 � < Ai :

for i 2 I ;

t 2�a; !Œ:

A�i A�l for i 2 IN ;

By using (1.2), we conclude that, in the case where the condition N.ƒ1 ; : : : ; ƒn�1 / is satisfied, for the existence of P! .ƒ1 ; : : : ; ƒn�1 / -solutions of system (1.1), it is necessary that the inequalities for Yi D ˙1;

˛i �i > 0

˛i �i < 0

for Yi D 0

(1.13)

be true for i D 1; n . 2. Main Results Theorem 2.1. Let ƒi 2 R n f0g; i D 1; n � 1; let the system of differential equations (1.1) satisfy the condition N.ƒ1 ; : : : ; ƒn�1 /; and let the orders of regularly varying functions 'k ; k D 1; n; in representations (1.5) guarantee the validity of conditions (1.11). Moreover, let l D min I : Then, for the existence of P! .ƒ1 ; : : : ; ƒn�1 / -solutions of system (1.1), it is necessary and if the equation algebraic in � n Y

iD1

0 @

i�1 Y

j D1

1

ƒj C �A �

n Y

iD1

0

@� i

i�1 Y

j D1

1

ƒj A D 0

(2.1)

does not have roots with zero real part, then it is also sufficient that lim

Ii .t /Ii0C1 .t /

0 t "! Ii .t /IiC1 .t /

D ƒi

ˇiC1 ˇi

(2.2)

for each i 2 f1; : : : ; ng and the sign conditions A�i ˇi > 0

for

Yi D ˙1;

A�i ˇi < 0

for

Yi D 0;

� � sign ˛i A�i ˇi D �i

(2.3) (2.4)

be satisfied. Moreover, each solution of this kind admits, as t " !; the asymptotic representations yi .t / D ˛i ˇi Ii .t /Œ1 C o.1/� 'iC1 .yiC1 .t //

for

i 2 I;

(2.5)

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

Ii .t / yi .t / D ˛ i ˇi Œ1 C o.1/� 'iC1 .yiC1 .t // Il .t /

i 2 IN ;

for

539

.2:6l /

and explicit representations of the form 1

yi .t / D �i jIi .t /j ˇi

Co.1/

;

i D 1; n;

as

(2.7)

t " !:

Furthermore, a k -parameter family of these solutions exists in the case where the set of roots of the algebraic equation (2.1) contains k roots (with regard for multiplicities) for which the signs of their real parts are opposite to the sign of the number A�l ˇl : Remark 2.1. Equation (2.1) definitely has no roots with zero real part if

Qn

iD1 j�i j

< 1:

Indeed, if Eq. (2.1) has a root �0 D i �; where � 2 R; then, in view of (2.1) and the fact that all ƒj ; j D 1; n � 1; are nonzero real constants, we arrive at the inequality ˇ ˇ ˇ i�1 n ˇ i�1 Y Y ˇ ˇ ˇ ˇ Yˇ ˇ ˇ ˇ ˇ ˇ� ˇƒj ˇ ; ƒ C � ƒj D j�i j j 0 ˇ ˇ ˇ ˇ iD1 j D1 iD1 j D1 j D1 n Y

which implies that

i�1 Y

n Y

j�i j � 1:

n Y

j�i j < 1;

iD1

Hence, if

iD1

then Eq. (2.1) does not have roots with zero real part. Proof of the Theorem. Necessity. Let yi W Œt0 ; !Œ! �.Yi0 /; i D 1; n; be an arbitrary P! .ƒ1 ; : : : ; ƒn�1 / solution of system (1.1). Since system (1.1) satisfies the condition N.ƒ1 ; : : : ; ƒn�1 /; by virtue of Definition 1.2, for each k 2 f1; : : : ; ng; there exist a number ˛k 2 f�1; 1g; a continuous function pk W Œa; !Œ�!�0; C1Œ; and 0 such that a continuous function 'kC1 W �Y 0 �!�0; C1Œ of order �kC1 regularly varying as ykC1 ! YkC1 kC1 relation (1.5) is true for any k 2 f1; : : : ; ng . Hence, by virtue of (1.1), the following representation is true: yi0 .t / D ˛i pi .t /'iC1 .yiC1 .t //Œ1 C o.1/�;

i D 1; n;

as t " w:

This implies that yi0 .t / D ˛i pi .t /Œ1 C o.1/�; 'iC1 .yiC1 .t //

i D 1; n;

as t " !:

(2.8)

V. M. E VTUKHOV

540

AND

M. A. TALIMONCHAK

Integrating each of these relations for i 2 I from Bi to t; where Bi D ! if Ai D ! and Bi D t0 if Ai D a; we obtain Zt

yi0 .� / d � D ˛i Ii .t /Œ1 C o.1/�; 'iC1 .yiC1 .� //

i 2 I;

as t " !:

(2.9)

Bi

We compare the integral on the left-hand side with a function yi .t / ; 'iC10 .yiC1 .t //

zi .t / D

i D 1; n;

where 'iC10 W �YiC1 !�0; C1Œ are continuously differentiable functions satisfying conditions (1.10). If Bi D t0 ; then, by virtue of (2.9), lim

Zt

yi0 .� / d � D 1: 'iC1 .yiC1 .� //

t "! Bi

Thus, by the L’Hospital rule in the Stolz form and conditions (1.3) and (1.10), we find yi .t/ 'iC10 .yiC1 .t// lim Z t t "! yi0 .�/ d � 'iC1 .yiC1 .�// Bi

yi0 .t / 'iC10 .yiC1 .t //

D lim

t "!

�

0 0 yi .t /'iC10 .yiC1 .t //yiC1 .t / 2 .yiC1 .t // 'iC10

yi0 .t /

'iC10 .yiC1 .t //

D 1 � lim

t "!

0 yiC1 .t /'iC10 .yiC1 .t //

'iC10 .yiC1 .t //

D 1 � ƒi �iC1 D ˇi ¤ 0

lim

t "!

0 yi .t /yiC1 .t /

yi0 .t /yiC1 .t /

for i 2 I :

(2.10)

If Bi D !; then, by virtue of (2.9), lim

Zt

t "! Bi

yi0 .� / d � D 0: 'iC1 .yiC1 .� //

We now show that, in this case, the function zi has a finite (or equal to ˙1 ) limit as t " !: Since zi0 .t / D

� � 0 0 .yiC1 .t //yiC1 .t / yi .t /'iC10 yi0 .t / 1� ; 'iC10 .yiC1 .t // 'iC10 .yiC1 .t //yi0 .t /

by virtue of the first relation in (1.10) and (2.8), the derivative of the function zi preserves its sign in a certain left neighborhood of ! and, hence, has the limit as t " !; finite or equal to ˙1: We now show that lim t "! zi .t / D

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

541

0: Assume the contrary, i.e., let

lim zi .t / D

t "!

¼

either a nonzero constant

(2.11)

or ˙ 1:

By virtue of (2.8), we obtain the relation yi0 .t / ˛i pi .t / D Œ1 C o.1/� yi .t / zi .t /

as t " !:

Integrating this relation from t0 to t; we get jyi .t /j ln D ˛i jyi .t0 /j Here,

Z

Zt

t0

pi .t /dt Œ1 C o.1/� zi .t /

as t " !:

! t0

pi .t/ dt converges because Bi D Ai D ! and, by the assumption, lim

t "!

1 D const: zi .t /

Hence, the right-hand side of this relation has a finite limit as t " !; whereas its left-hand side has an infinite limit as t " ! by virtue of the second condition in (1.2), which is a contradiction. Thus, lim t "! zi .t / D 0: By using the L’Hospital rule, we also get (2.10). Therefore, in both cases, the relation Zt

yi0 .� / d � yi .t / D Œ1 C o.1/�; 'iC1 .yiC1 .� // ˇi 'iC10 .yiC1 .t //

i 2 I;

as t " !

Bi

is true. Hence, in view of (2.9) and the first relation in (1.10), we obtain the asymptotic representations (2.5). Further, it follows from (2.5) and (2.8) that, for i 2 I ; I 0 .t / yi0 .t / D i Œ1 C o.1/� yi .t / ˇi Ii .t /

as t " !:

(2.12)

We now multiply each relation (2.8) with i 2 IN by Il .t / and integrate it from Bi to t; where Bi are chosen in the same way as above. As a result, we obtain Zt

Bi

yi0 .� /Il .� / d � D ˛i Ii .t /Œ1 C o.1/�; 'iC1 .yiC1 .� //

i 2 IN ;

as t " !:

(2.13)

V. M. E VTUKHOV

542

AND

M. A. TALIMONCHAK

Repeating the reasoning presented above and using the L’Hospital rule in the Stolz form and conditions (1.3), (1.10), and (2.12), we find

yi .t/Il .t/ 'iC10 .yiC1 .t// lim Z t t "! yi0 .�/Il .t/ d � 'iC1 .yiC1 .�// Bi

D lim

yi0 .t/Il .t / 'iC10 .yiC1 .t //

C

yi .t /Il0 .t / 'iC10 .yiC1 .t //

�

0 0 yi .t /Il .t /'iC10 .yiC1 .t //yiC1 .t / 2 'iC10 .yiC1 .t //

yi0 .t /Il .t /

t "!

'iC1 .yiC1 .t //

D 1 C lim

t "!

yi .t /Il0 .t / yi0 .t /Il .t /

D 1 C ˇl lim

t "!

� lim

'iC10 .yiC1 .t //

t "!

yi .t /yl0 .t /

yi0 .t /yl .t / "

0 yiC1 .t /'iC10 .yiC1 .t //

lim

t "!

0 yi .t /yiC1 .t /

yi0 .t /yiC1 .t /

� �iC1 ƒi D 1 � ƒi �iC1

0 .t /ylC2 .t / yl0 .t /ylC1 .t / ylC1

0 .t /yi .t / yi�1 : : : C ˇl lim 0 0 yi�1 .t /yi0 .t / t "! yl .t /ylC1 .t / ylC1 .t /ylC2 .t /

D

ˇl D ˇi ¤ 0 ƒl ƒlC1 : : : ƒi�1

#

for i 2 fl C 1; : : : ; ngnI

and yi .t/Il .t/ 'iC10 .yiC1 .t// lim Z t t "! yi0 .�/Il .t/ d � 'iC1 .yiC1 .�// Bi

D lim

yi0 .t/Il .t/ 'iC10 .yiC1 .t//

C

yi .t /Il0 .t / 'iC10 .yiC1 .t //

t "!

yi .t /Il0 .t / yi0 .t /Il .t /

D 1 C ˇl lim

t "!

� lim

yi .t /yl0 .t / yi0 .t /yl .t / "

0 0 .yiC1 .t //yiC1 .t / yi .t /Il .t /'iC10 2 'iC10 .yiC1 .t //

yi0 .t /Il .t / 'iC1 .yiC1 .t //

t "!

D 1 C lim

�

t "!

0 yiC1 .t /'iC10 .yiC1 .t //

'iC10 .yiC1 .t //

lim

t "!

0 yi .t /yiC1 .t /

yi0 .t /yiC1 .t /

� �iC1 ƒi D 1 � ƒi �iC1

0 .t /ylC2 .t / yl0 .t /ylC1 .t / ylC1

0 .t /yi .t / yi�1 yn0 .t /y1 .t / y10 .t /y2 .t / C ˇl lim : : : : : : 0 0 0 0 yn .t /y1 .t / y1 .t /y2 .t / yi�1 .t /yi0 .t / t "! yl .t /ylC1 .t / ylC1 .t /ylC2 .t /

D

ˇl D ˇi ¤ 0 ƒl ƒlC1 : : : ƒn ƒ1 : : : ƒi�1

#

for i 2 f1; : : : ; l � 1gnI :

In view of (2.13) and the first relation in (1.10), we arrive at the asymptotic relations ( 2:6l ). Thus, by virtue of (2.8), the asymptotic relations (2.12) remain true for i 2 IN : Since relations (2.12) are true for i D 1; n and the analyzed solution satisfies the last limit representation in the definition of P! .ƒ1 ; : : : ; ƒn�1 / -solutions, conditions (2.2) are satisfied for each i 2 f1; : : : ; ng .

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

543

In addition, relations (2.12) yield the following explicit form of the P! .ƒ1 ; : : : ; ƒn�1 / -solution: 1

yi .t / D �i jIi .t /j ˇi

Co.1/

i D 1; n;

;

as t " !:

In view of the condition lim t "! yi .t / D Yi0 and the definitions of P! .ƒ1 ; : : : ; ƒn�1 / -solutions and the number A�i ; we get the sign conditions (2.3). The validity of the sign conditions (2.4) directly follows from (2.5) and .2:6l / if we take into account the signs of the functions yi and Ii ; i D 1; n; in the interval Œt0 ; !Œ . Sufficiency. Assume that, parallel with conditions (2.2)–(2.4), the algebraic equation (2.1) also does not have roots with real part equal to zero. We show that, in this case, the system of differential equations (1.1) possesses at least one P! .ƒ1 ; : : : ; ƒn�1 / -solution admitting the asymptotic representations (2.5), (2.6 l ), and (2.7) as t " ! and determine the number of these solutions. First, we consider a system of relations of the form yi D Qi .t /Œ1 C vi �; 'iC10 .yiC1 /

i D 1; n;

(2.14)

where

Qi .t / D

8 ˛i ˇi Ii .t / ˆ ˆ <

Ii .t / ˆ ˆ : ˛i ˇi Il .t /

for i 2 I ; for i 2 I

and 'iC10 is a continuously differentiable function regularly varying in the interval �Y 0 and satisfying condiiC1 tions (1.10). By virtue of the property M2 of regularly varying functions, this function exists. In exactly the same way as in [4], we show that, in the set D0 D Œt0 ; !Œ�V0 ; where t0 2 Œa; !Œ and V0 D fvN � .v1 ; : : : ; vn / W jvi j � N i D 1; n; of 1=2; i D 1; ng; it uniquely determines continuously differentiable implicit functions yi D Yi .t; v/; the form �i

N D �i jIl .t /j ˇl Yi .t; v/

Czi .t;v/ N

;

i D 1; n;

where �l D 1;

�i D

i�1 Y

ƒj ;

j D1

i 2 f1; ngnflg;

and the functions zi ; i D 1; n; are such that jzi .t; v/j N �

jLi j ; 2jˇl j

i D 1; n;

for .t; v/ N 2 D0

and N D0 lim zi .t; v/

t "!

uniformly in vN 2 V0 :

(2.15)

V. M. E VTUKHOV

544

AND

M. A. TALIMONCHAK

By virtue of the sign conditions (2.3) and (2.4), it follows from (2.15) that the vector function .Yi /niD1 has the following properties: Yi .t / 2 �Y 0 i

lim Yi .t; v/ N D Yi0

t "!

for t 2 Œt0 ; !Œ;

uniformly in vN 2 V0 ;

(2.16)

i D 1; n:

We now show that lim

t "!

.Yi .t; v// 1 N 0t Ii .t / D ; 0 Yi .t; v/I N i .t / ˇi

i D 1; n;

uniformly in vN 2 V0 :

(2.17)

By using (2.14), we obtain 0 N 0t 'iC10 .YiC1 .t; v// N Q0 .t / .YiC1 .t; v// N 0t .Yi .t; v// D i C ; Yi .t; v/ N Qi .t / 'iC10 .YiC1 .t; v// N

Multiplying (2.18) by

(2.18)

i D 1; n:

Ii .t / ; i D 1; n; we get Ii0 .t /

Qi0 .t /Ii .t / .Yi .t; v// N 0t Ii .t / D Yi .t; v/I N i0 .t / Qi .t /Ii0 .t / C

0 0 Ii .t /IiC1 .t / YiC1 .t; v/' N iC10 .YiC1 .t; v// N .YiC1 .t; v// N 0t IiC1 .t / ; Ii0 .t /IiC1 .t / 'iC10 .YiC1 .t; v// N YiC1 .t; v/I N iC1 .t /

Solving this system of algebraic equations with respect to

i D 1; n:

.Yk .t; v// N 0t Ik .t / ; k D 1; n; we find Yk .t; v/I N k0 .t /

� � n i�1 0 X N j0 C10 Yj C1 .t; vN / Qi0 .t /Ii .t / Y Ij .t /Ij C1 .t / Yj C1 .t; v/' .Yk .t; v// N 0t Ik .t / � � D Yk .t; v/I N k0 .t / Qi .t /Ii0 .t / Ij0 .t /Ij C1 .t / 'j C10 Yj C1 .t; v/ N j Dk

iDk

2

� � n Y Ij .t /Ij0 C1 .t / Yj C1 .t; v/' N j0 C10 Yj C1 .t; vN / � � � 41 C Ij0 .t /Ij C1 .t / 'j C10 Yj C1 .t; v/ N �

j Dk

0

�

i�1 0 N j0 C10 Yj C1 .t; vN Qi0 .t /Ii .t / Y Ij .t /Ij C1 .t / Yj C1 .t; v/' @ � � Qi .t /Ii0 .t / Ij0 .t /Ij C1 .t / Y ' .t; v/ N j C10 j C1 iD1 j D1 k�1 X

0

� @1

�

Ij .t /Ij0 C1 .t / Yj C1 .t; v/' N j0 C10 Yj C1 .t; vN � � � Ij0 .t /Ij C1 .t / ' .t; v/ N Y j C10 j C1 j D1 n Y

3 � 1�1 / A 7 5;

� 1 / A k D 1; n:

(2.19)

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

545

Here, in view of the form of the functions Qi ; i D 1; n; 8 0 Ii .t / ˆ ˆ ˆ ˆ < Ii .t /

Qi0 .t / D ˆ 0 Qi .t / ˆ I .t / Il0 .t / ˆ ˆ : i � Ii .t / Il .t /

and, hence,

for i 2 I ; for i 2 IN

8 < 1 for i 2 I ; Qi0 .t /Ii .t / D lim 0 : t "! Qi .t /Ii .t / 0 for i 2 IN :

(2.20)

In addition, by virtue of (2.16) and (1.10), lim

t "!

0 .Y Yi .t; v/' N i0 N i .t; v// D �i ; 'i0 .Yi .t; v// N

i D 1; n;

uniformly in vN 2 V0 :

(2.21)

Thus, by using relations (2.2), (2.20), and (2.21) and the fact that ƒi �iC1 D 1 for i 2 IN ; we derive the limit relations (2.17) from (2.19). N niD1 satisfies conditions (1.2) and (1.3). ApAs follows from (2.16) and (2.17), the vector function .Yi .t; v// plying the transformation yi .t / D Yi .t; v1 .t /; : : : ; vn .t //;

(2.22)

i D 1; n;

to the system of differential equations (1.1) and using the fact that, for t 2 Œt0 ; !Œ and vN 2 V0 ; the vector function n N .Yi .t; v.t/// i D1 is a solution of the system of equations yi .t / D Qi .t /Œ1 C vi .t /�; 'iC10 .yiC1 .t //

(2.23)

i D 1; n;

we obtain N //; : : : ; Yn .t; v.t N /// fi .t; Y1 .t; v.t 'iC10 .YiC1 .t; v.t N /// �

0 N //'iC10 .YiC1 .t; v.t N ///fiC1 .t; Y1 .t; v.t N //; : : : ; Yn .t; v.t N /// Yi .t; v.t 2 'iC10 .YiC1 .t; v.t N ///

D Qi0 .t /Œ1 C vi .t /� C Qi .t /vi0 .t /;

(2.24)

i D 1; n:

Since the functions fi ; i D 1; n; satisfy the condition N.ƒ1 ; : : : ; ƒn�1 /; the vector function n N .Yi .t; v.t/// iD1 has properties (1.2) and (1.3) and the following representations are true as t " ! : fi .t; Y1 .t; v.t N //; : : : ; Yn .t; v.t N /// D ˛i pi .t /'iC10 .YiC1 .t; v.t N ///Œ1 C �i .t; v.t N //�;

i D 1; n;

V. M. E VTUKHOV

546

AND

M. A. TALIMONCHAK

where N // D 0 lim �i .t; v.t

t "!

uniformly in vN 2 V0 : By using (2.24), we obtain ˛i pi .t/Œ1 C �i .t; v.t N //� � ˛iC1 piC1 .t /

Qi .t /Œ1 C vi .t /� Œ�iC1 C ıiC1 .t; v.t N //� QiC1 .t /Œ1 C viC1 .t /� D Qi0 .t /Œ1 C vi .t /� C Qi .t /vi0 .t /;

i D 1; n;

where lim ıiC1 .t; v.t N // D 0

t "!

uniformly in vN 2 V0 : This implies that vi0 .t/ D

1 Œ˛i pi .t /Œ1 C �i .t; v.t N //� Qi .t/ � Qi .t /Œ1 C vi .t /� 0 � ˛iC1 piC1 .t / Œ�iC1 C ıiC1 .t; v.t N //� � Qi .t /Œ1 C vi .t /� ; QiC1 .t /Œ1 C viC1 .t /�

i D 1; n:

Separating the linear parts in the analyzed system, we get a system of differential equations of the form vi0 D

Il0 .t / ˇl Il .t /

Œqi .t / C bi i .t /vi C bi iC1 .t /viC1 C Vi1 .t; v/ N C Vi2 .t; v/� N ;

where N � �iC1 hiC1 .t / � gi .t /; qi .t / D hi .t /.1 C �i .t; v// hi .t / D

gi .t / D

ˇl Ii0 .t /Il .t / ; ˇi Ii .t /Il0 .t /

8 Ii0 .t /Il .t / ˆ ˆ ˇ ˆ l ˆ Ii .t /Il0 .t / ˆ < ˆ ˆ ˆ ˆ ˆ : ˇl

for i 2 I ;

Ii0 .t /Il .t / �1 Ii .t /Il0 .t /

bi i .t / D ��iC1 hiC1 .t / � gi .t /; Vi1 .t; v/ N D ��iC1 hiC1 .t /

i D 1; n;

!

N ; for i 2I

bi iC1 .t / D �iC1 hiC1 .t /; 1 C vi ıiC1 .t; v/; N 1 C viC1

i D 1; n;

(2.25)

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

547

Vi2 .t; v/ N D ��iC1 hiC1 .t /�i .v/; N �i .v.t N // D

1 C vi .t / � 1 � vi .t / C viC1 .t /; 1 C viC1 .t /

i D 1; n;

and, moreover, @�i .v.t N // D 0; jv1 jC:::Cjvn j!0 @vk .t / lim

i; k D 1; n:

In this system, in view of conditions (2.2), we have lim gi .t / D .1 � ƒi �iC1 / �i ;

lim hi .t / D �i ;

t "!

t "!

i D 1; n:

Thus, by virtue of conditions (2.22) and (2.23), we find lim qi .t / D 0;

t "!

lim Vi1 .t; v/ N D 0;

t "!

lim

jv1 jC:::Cjvn j!0

i D 1; n;

uniformly in .v1 ; : : : ; vn / 2 V0 ;

i D 1; n;

Vi2 .t; v/ N D 0; jv1 j C : : : C jvn j

uniformly in t 2 Œt0 ; !Œ:

i D 1; n;

In addition, the matrix B0 obtained as the limit of the matrix B.t / formed by the coefficients of vk ; k D 1; n; in the square brackets on the right-hand sides of equations of the system takes the form 0

B B B B B B B0 D lim B.t / D B B t "! B B B @

���

0

�3 �3 � � �

0

��1

�2 �2

0

��2

:: :

:: :

:: :

::

0

0

0

���

��n�1

�1 �1

0

0

���

0

0

:

:: :

0

1

C C C C C :: C : : C C C C �n �n C A 0

��n

The corresponding characteristic equation det ŒB0 � �En � D 0; where En is the identity matrix of order n; has the form of Eq. (2.1). Hence, the matrix B0 does not have eigenvalues with real part equal to zero. Thus, we have proved that all conditions of Theorem 2.2 in [7] are satisfied for the system of differential equations (2.25). According to this theorem, the system of differential equations (2.25) has at least one solution fvi gniD1 W Œt1 ; !Œ! Rn ; t1 � t; approaching zero as t " !: Moreover, there exists a k -parameter family of these solutions if the set of roots of the characteristic equation (2.1) contains k roots (with regard for multiplicities) whose real parts have the sign opposite to the sign of the number A�l ˇl : In view of the change of variables (2.22), the system of equations (2.23) for the functions Yi .t; v1 .t /; : : : ; vn .t //; i D 1; n; and the condition N.ƒ1 ; : : : ; ƒn�1 / imposed on system (1.1), every solution of system (2.25) is associated with a solution

V. M. E VTUKHOV

548

AND

M. A. TALIMONCHAK

.y1 ; : : : ; yn / of the system of differential equations (1.1) admitting the asymptotic representations yi .t / D Qi .t /Œ1 C o.1/�; 'iC1 .yiC1 .t //

i D 1; n;

as t " !:

It remains to show that each of the indicated solutions of the system of differential equations (1.1) is a P! .ƒ1 ; : : : ; ƒn�1 / -solution. Since each of these solutions is associated with a solution .v1 .t /; : : : ; vn .t // of system (2.25) approaching N i D 1; n; conditions (1.2) zero as t " !; by virtue of the already established properties of the functions Yi .t; v/; are definitely satisfied. Moreover, for the indicated solutions of system (1.1), with regard for (2.23) and (2.2), we obtain lim

t "!

0 yiC1 .t /yi .t /

yiC1 .t /yi0 .t /

D

I 0 .t /Ii .t / ˇi D ƒi : lim i C1 ˇiC1 t "! IiC1 .t /Ii0 .t /

Hence, condition (1.3) from the definition of P! .Y10 ; : : : ; Yn0 ; �1 ; : : : ; �n / -solution is also satisfied. The theorem is proved. We now establish conditions under which the asymptotic representations (2.5) and (2. 6l ) can be improved. Definition 2.1 (see [8]). We say that a regularly varying function of order � of the form '.z/ D jzj� L.z/ satisfies condition S if, for any continuously differentiable function l W �Y 0 �!�0; C1Œ such that lim

z!Y 0 z2� 0 Y

z l 0 .z/ D 0; l.z/

the asymptotic relation '.zl.z// D L.z/Œ1 C o.1/�

as

z ! Y0

.z 2 �Y 0 /;

(2.26)

is true. The functions 'i ; i 2 f1; : : : ; ng; for which the function Li has a finite limit as z ! Yi0 definitely satisfy condition S: Functions of the form 'i .z/ D jzj� j ln zj�1 ;

'i .z/ D jzj� j ln zj�1 j ln j ln zjj�2 ;

where �1 ; �2 ¤ 0; and many other functions also satisfy condition S: Remark 2.2 (see [9]). If the function 'i ; i 2 f1; : : : ; ng; satisfies condition S and the function yi W Œt0 ; !Œ�! �Y 0 is continuously differentiable and such that lim yi .t / D Yi0 ;

t "!

yi0 .t / � 0 .t / Œr C o.1/� D yi .t / �.t /

as t " !;

A SYMPTOTIC B EHAVIOR OF THE S OLUTIONS OF N ONLINEAR S YSTEMS OF O RDINARY D IFFERENTIAL E QUATIONS

549

where r is a nonzero real constant and � is a real function continuously differentiable in a certain left neighborhood of ! for which � 0 .t / ¤ 0; then � � Li .yi .t // D Li j�.t /jr Œ1 C o.1/�

because, in the considered case,

where

yi .t / D z.t /l.z.t //;

as t " !

z.t / D �0 j�.t /jr ;

and lim

z!Y0 z2� 0 Y

z l 0 .z/ l.z/

D lim

t "!

z.t / l 0 .z.t // l.z.t //

� �0 � � i .t / z.t / yz.t �.t /yi0 .t / / � D lim � � 1 D 0: D lim t "! yi .t / z 0 .t / t "! r� 0 .t /yi .t / z.t /

Theorem 2.2. Let ƒi 2 R n f0g; i D 1; n � 1; l D min I ; and let all functions 'i ; i D 1; n; satisfy condition S: Then every P! .ƒ1 ; : : : ; ƒn�1 / -solution (if it exists) of the system of differential equations (1.1) admits, as t " !; the asymptotic representations � �ˇ�i k n ˇ Y 1 ˇ ˇ ˇkC1 ˇ Œ1 C o.1/�; ˇ yi .t / D �i ˇ ˇQk .t /LkC1 �kC1 jIkC1 .t /j kD1

where

Qk .t/ D

8 ˛ ˇ I .t / ˆ ˆ < k k k

I .t / ˆ ˆ : ˛k ˇk k Il .t /

for k 2 I ; for k 2 I ;

�ik D

8 Q Q n ˆ �j jkD1 �j ˆ j Di C1 ˆ Q ˆ ˆ ˆ 1 � jnD1 �j < ˆ Qk ˆ ˆ ˆ j Di C1 �j ˆ ˆ Q : 1 � jnD1 �j

i D 1; n;

(2.27)

for k D 1; i � 1; for k D i; n:

Proof. As shown in the proof of Theorem 2.1, for the existence of P! .ƒ1 ; : : : ; ƒn�1 / -solutions of the system of differential equations (1.1), it is necessary that conditions (2.2)–(2.4) be satisfied and that each of these solution admits, as t " !; the asymptotic representations (2.5) and (2.6 l ). Moreover, these solutions admit the asymptotic representation (2.12). By virtue of this relation and Remark 2.2, we get �

Li .yi .t // D Li �i jIi .t /j

1 ˇi

�

Œ1 C o.1/�;

i D 1; n;

as t " !:

The asymptotic representations (2.5) and (2.6 l ) can be rewritten in the form � � 1 yi .t / ˇiC1 D Qi .t /LiC1 �iC1 jIiC1 .t /j Œ1 C o.1/�; jyiC1 .t /j�iC1

i D 1; n;

as t " !:

Resolving this system of algebraic equations with respect to y1 ; : : : ; yn ; we arrive at the asymptotic representations (2.27). The theorem is proved.

V. M. E VTUKHOV

550

AND

M. A. TALIMONCHAK

3. Conclusions In the present paper, for the system of differential equations (1.1), we introduce a class of so-called P! .ƒ1 ; : : : ; ƒn�1 / -solutions and study the problem of existence and asymptotics of these solutions in the case where ƒi ; i D 1; n � 1; are nonzero real constants. It is assumed that (condition N.ƒ1 ; : : : ; ƒn�1 / ), on solutions from this class, system (1.1) is asymptotically close to a cyclic system with regularly varying nonlinearities. Under this condition, we establish necessary and sufficient conditions for the existence of P! .ƒ1 ; : : : ; ƒn�1 / solutions of system (1.1) and the implicit asymptotic [as t " ! .! � C1/� relations for the components of these solutions. The explicit asymptotic relations for the components of these solutions are established under the assumption that all nonlinearities satisfy the condition S: Since, as !; we can choose any finite number from the range of the variable t in which the right-hand sides of system are defined, the results of the present paper enable us to describe the asymptotics not only of the regular solutions but also of various singular solutions of system (1.1) that cannot be extended to the right (see [10, pp. 238, 262]). REFERENCES 1. O. S. Vladova, “Asymptotic representations of the solutions of cyclic systems of differential equations with regularly varying nonlinearities,” Visn. Odes. Nats. Univ., Ser. Mat. Mekh., 15, Issue 19, 33–56 (2010). 2. E. S. Vladova, “Asymptotic behavior of the solutions of nonlinear cyclic systems of ordinary differential equations,” Nelin. Kolyvannya, 14, No. 3, 299–317 (2011); English translation: Nonlin. Oscillations, 14, No. 3, 313–333 (2011). 3. V. M. Evtukhov and E. S. Vladova, “Asymptotic representations of the solutions of essentially nonlinear two-dimensional systems of ordinary differential equations,” Ukr. Mat. Zh., 61, No. 12, 1597–1611 (2009); English translation: Ukr. Math. J., 61, No. 12, 1877–1892 (2009). 4. V. M. Evtukhov and O. S. Vladova, “Asymptotic representations of the solutions of essentially nonlinear cyclic systems of ordinary differential equations,” Differents. Uravn., 48, No. 5, 622–639 (2012). 5. V. M. Evtukhov and O. S. Vladova, “On the asymptotics of solutions of nonlinear cyclic systems of ordinary differential equations in special cases,” Mem. Different. Equat. Math. Phys., 54, 1–25 (2011). 6. E. Seneta, ”Regularly Varying Functions,” in: Lecture Notes in Math., Vol. 508, Springer, Berlin (1976). 7. V. M. Evtukhov and A. M. Samoilenko, “Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point,” Ukr. Mat. Zh., 62, No. 1, 52–80 (2010); English translation: Ukr. Math. J., 62, No. 1, 56–86 (2010). 8. V. M. Evtukhov and M. A. Belozerova, “Asymptotic representations of solutions of essentially nonlinear nonautonomous second-order differential equations,” Ukr. Mat. Zh., 60, No. 3, 310–331 (2008); English translation: Ukr. Math. J., 60, No. 3, 357–383 (2008). 9. V. M. Evtukhov and A. M. Samoilenko, “Asymptotic representations for the solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities,” Differents. Uravn., 47, No. 5, 628–650 (2011). 10. I. T. Kuguradze and T. A. Chanturiya, Asymptotic Properties of the Solutions of Nonautonomous Ordinary Differential Equations [in Russian], Nauka, Moscow (1990).