Analytic Solutions of second order Nonlinear Difference Equations all of whose Eigenvalues are 1

Analytic Solutions of Second Order Nonlinear Difference Equations all of whose Eigenvalues are 1

MAMISUZUKI(*)

ABSTRACT- For nonlinear difference equations, it is difficult to obtain analytic so- lutions. Especially, when all the absolute values of the equation are equal to 1, it is quite difficult.

We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations,

x(t‡1)ˆX(x(t);y(t));

y(t‡1)ˆY(x(t);y(t));

(

whereX(x;y)ˆl1x‡ P

i‡j^2cijxⁱy^j,Y(x;y)ˆl2y‡ P

i‡j^2dijxⁱy^jand we assume some conditions. For these equations, whenjl1j 6ˆ1or jl2j 6ˆ1, we have ob- tained analytic solutions in earlier studies. In the present paper, we will prove the existence of an analytic solution and obtain analytic solutions of the differ- ence equations for the casel1ˆl2ˆ1:

1. Introduction.

We start by considering the following second order nonlinear differ- ence equation,

u(t‡1)ˆU(u(t);v(t));

v(t‡1)ˆV(u(t);v(t));

( (1:1)

whereU(u;v) an dV(u;v) are entirefunctionsforuandv. Wesuppose thatthe

(*) Indirizzo dell'A.: Department of Mathematics and Science, College of Liberal Arts, J. F. OberlinUniversity. 3758 Tokiwa-cho, Machida-City, Tokyo, 194-0294, Japan.

E-mail: m-suzuki@obirin.ac.jp

2000Mathematics Subject Classification. 39A10, 39A11, 39B32.

equation(1.1) admits anequilibrium point (u;v): u v

ˆ U(u;v) V(u;v)

. We canassume, without loss of generality, that (u;v)ˆ(0;0):Furthermore we suppose thatUandVare writteninthe following form

u(t‡1) v(t‡1)

ˆM u(t) v(t)

‡ U₁(u(t);v(t)) V₁(u(t);v(t))

;

whereU1(u;v) an dV1(u;v) are higher order terms ofuandv, an dMis a constant matrix. Letl₁,l₂be the characteristic values of the matrixM. For some regular matrixPdetermined byM, put u

v ˆP xy , thenwe can transform the system (1.1) into the following simultaneous system of first order difference equations (1.2):

x(t‡1)ˆX(x(t);y(t));

y(t‡1)ˆY(x(t);y(t));

( (1:2)

whereX(x;y) an dY(x;y) are supposed to be holomorphic and expanded in a neighborhood of (0;0) inthe form

X(x;y)ˆl1x‡ X

i‡j^2

cijxⁱy^jˆl1x‡X1(x;y);

Y(x;y)ˆl₂y‡ X

i‡j^2

d_ijxⁱy^jˆl₂y‡Y₁(x;y);

8>

><

>>

(1:3) : or

X(x;y)ˆlx‡y‡ X

i‡j^2

c⁰_ijxⁱy^jˆlx‡X₁⁰(x;y);

Y(x;y)ˆly‡ X

i‡j^2

d⁰_ijxⁱy^jˆly‡Y₁⁰(x;y);

8>

><

>>

(1:4) :

where, inthe second case,lˆl₁ˆl₂.

In this paper we consider analytic solutions of difference system (1.2).

In [9] and [8], already, we have obtained general analytic solutions of (1.2) inthe case jl1j 6ˆ1 or jl2j 6ˆ1. However inthe case jl1j ˆ jl2j ˆ1, it is difficult to prove the existence of analytic solution or obtain an analytic solution of the equation. For a long time we have not been able to treat equation (1.2) under the latter condition.

For analytic solutions of nonlinear first order difference equations, Kimura [2] has studied the cases inwhich one eigenvalue is equal to 1, furthermore Yanagihara [10] has studied the cases in which the absolute value of the eigenvalue is equal to 1. Here we shall study analytic solutions

of nonlinear second order difference equations in which the absolute values of the eigenvalues of the matrixMare both equal to 1.

As anexample of the equation(1.3), we consider the following ``popu- lationmodel''

u(t‡2)ˆau(t‡1)‡bu(t‡1) au(t)

au(t) ; fort> 1;

whereaˆ1‡r>0,b>0 are constants,ris the net (births minus death) endogenous population growth rate. This model was proposed by Prof. D.

Dendrios [1], and may be transformed into a system as in (1.2). In [6], we have investigated properties of a solution assuming its existence when aˆ1, i.e.rˆ0. However if both eigenvalues of the equation are equal to 1, i.e. under the conditionrˆ0, the existence of a solution of the model is not established. Inthe next paper, we will show a solutionof the population model inthe caserˆ0.

Moreover we have studied some economic models in the form (1.2), but we had to exclude the casejl1j ˆ jl2j ˆ1. For example, in[7], we consider the following ``duopoly system''

x(t‡1)ˆay(t)(1 y(t)) y(t‡1)ˆbx(t)(1 x(t));

(

whereaandbare constants. For this system, we havel₁ˆ l₂. In[7], we consider the system under the condition such that 05l²₁ˆl²₂51.

Therefore we will investigate the equation (1.2) in the case jl1j ˆ

ˆ jl2j ˆ1.

In this present paper, making use of theorems in [2] and [5], we will prove the existence of a solution and obtain an analytic solution of (1.2) under the conditionsl1ˆl2ˆ1, inwhichX,Yare defined by (1.3), i.e., we suppose that

X(x;y)ˆx‡ X

i‡j^2;i^1

c_ijxⁱy^jˆx‡X1(x;y);

Y(x;y)ˆy‡ X

i‡j^2;j^1

dijxⁱy^jˆy‡Y1(x;y):

8>

><

>>

(1:5) :

Here we suppose thatX1(x;y)60 orY1(x;y)60.

Next we consider a functional equation

C(X(x;C(x)))ˆY(x;C(x));

(1:6)

whereX(x;y) an dY(x;y) are holomorphic functions injxj5d₁,jyj5d₁. We assume thatX(x;y) an dY(x;y) are expanded there as in (1.5).

Now we will consider the meaning of equation (1.6).

Consider the simultaneous system of difference equations (1.2). Suppose (1.2) admits a solution(x(t);y(t)). Ifdx

dt 6ˆ0 for somet0, thenwe canwrite tˆc(x) with a functioncina neighborhood ofx0ˆx(t0), and we can write

yˆy(t)ˆy(c(x))ˆC(x);

(1:7)

there. Thenthe functionCsatisfies the equation(1.6).

Conversely we assume that a functionCis a solutionof the functional equation(1.6). If the following first order difference equation

x(t‡1)ˆX(x(t);C(x(t)));

(1:8)

has a solutionx(t), we puty(t)ˆC(x(t)). Thenthe (x(t);y(t)) is a solutionof (1.2). Hence if there is a solutionCof (1.6), thenwe canreduce the system (1.2) to a single equation (1.8).

This relation is important in order to derive analytic solutions of non- linear second order difference equations which are written as in (1.2). We have proved the existence of solutionsC of (1.6) in[3] (or [4]) and [8] for other conditions onXandY.

Hereafter we considertto be a complex variable, and concentrate on the difference system (1.2). We define a domainD1(k0;R0) by

D₁(k₀;R₀)ˆ ft : jtj>R₀;jarg[t]j5k₀g;

(1:9)

wherek0 is any constant such that 05k0%p

4, an dR0is a sufficiently large numberwhichmaydependonXandY.FurtherwedefineadomainD(k;d) by

D(k;d)ˆ fx : jarg[x]j5k;05jxj5dg;

(1:10)

where d is a small constant, and k is a constant such that kˆ2k0, i.e., 05k%p

2.

Our aim inthis paper is to prove the following Theorem 1.

THEOREM1. Suppose X(x;y)and Y(x;y)are expanded in the forms (1.5)such that X1(x;y)60or Y1(x;y)60,and put Aˆc20.

(1)Suppose that kc₂₀ ˆd₁₁for some k2N, k^2,then we have a formal solution x(t)of(1.2)the following form

1

At 1‡ X

j‡k^1

^q_jkt ^j logt

t

_k! ₁

; (1:11)

where^q_jkare constants defined by X and Y.

(2) Suppose kc20ˆd1150 for some k2N, k^2,and R1 ˆ

ˆmax (R0;2=(jAjd)),then there is a solution x(t)of(1.2)such that (i)x(t)is holomorphic and x(t)2D(k;d)for t2D1(k0;R1), (ii)x(t)is expressible in the following form

x(t)ˆ 1 At 1‡b

t;logt

t

! ¹

; (1:12)

where b(t;logt=t)is asymptotically expanded for t2D1(k0;R1)such that b

t;logt

t

X

j‡k^1

^q_jkt ^j logt

t _k

; as t! 1through D₁(k₀;R₁).

2. Proof of Theorem1.

In [2], Kimura considered the following first order difference equation w(t‡l)ˆF(w(t));

…D1†

whereFis represented in a neighborhood of1by a Laurent series F(z)ˆz

1‡X¹

jˆm

b_jz ^j

; b_mˆl6ˆ0:

(2:1)

He defined the following domains D(e;R)ˆ ft : jtj>R;jarg[t] uj5p

2 e;orIm(e^{i(u e)}t)>R;

orIm(e^i(u‡e)t)5 Rg;

(2:2)

D(e;^ R)ˆ ft : jtj>R;jarg[t] u pj5p

2 e orIm(e ^{i(u‡p e)}t)>R

orIm(e ^i(u‡p‡e)t)5 Rg;

(2:3)

where e is anarbitrarily small positive number,R is a sufficiently large number which may depend oneandF, an duis defined byuˆargl, (inthis present paper, we consider the caselˆ1 in(D1)). He proved the following Theorems A and B.

THEOREMA. -Equation(D1)admits a formal solution of the form t 1‡ X

j‡k^1

^qjkt ^j logt

t _k! (2:4)

containing an arbitrary constant,where^qjkare constants defined by F.

THEOREMB. -Given a formal solution of the form(2.4)of(D1),there exists a unique solution w(t)satisfying the following conditions:

(i) w(t)is holomorphic in D(e;R), (ii) w(t)is expressible in the form

w(t)ˆt 1‡b

t;logt t

!

; (2:5)

where the domain D(e;R)is defined by(2.2),and b(t;h)is holomorphic for t2D(e;R),jhj51=R,and in the expansion

b(t;h)X¹

kˆ1

b_k(t)h^k: Here b_k(t)is asymptotically developed into

b_k(t) X¹

j‡k^1

^ q_jkt ^j;

as t! 1through D(e;R),where^q_jkare constants which are defined by F.

Also there exists a unique solutionw which is holomorphic in^ D(e;^ R) and satisfies a condition analogous to (ii),where the domainD(e;^ R)is defined by(2.3).

In Theorems A and B, Kimura defined the functionFas in(2.1). Inour method, we do not have a Laurent series for the function F. Hence we derive the following Propositions.

Inthe following,Awill be a constant which is defined as in Theorem 1 such thatAˆc₂₀, wherec₂₀ is the coefficient in (1.5).

PROPOSITION2. -SupposeF(t)~ is formally expanded such that F(t)~ ˆt 1‡X¹

jˆ1

b_jt ^j

!

; b₁ˆl6ˆ0:

(2:6)

Then the equation

c(F(t))~ ˆc(t)‡l (2:7)

has a formal solution

c(t)ˆt 1‡X¹

jˆ1

qjt ^j‡qlogt t

!

; (2:8)

where q1can be arbitrarily prescribed while other coefficients qj(j^2)and q are uniquely determined by b_j,(jˆ1;2; ),independently of q1.

PROPOSITION3. -SupposeF(t)~ is holomorphic and expanded asymp- totically inft : 1=(At)2D(k;d);A50gas

F(t)~ t 1‡X¹

jˆ1

bjt ^j

!

; b1ˆl6ˆ0;

where D(k;d)is defined in(1.10).Then the equation(2.7)has a solution wˆc(t);which is holomorphic inft : 1=(At)2D(k=2;d=2);A50gand has an asymptotic expansion

c(t)t 1‡X¹

jˆ1

q_jt ^j‡qlogt t

!

;

there.

These Propositions are proved as in [2] pp. 212-222. SinceAˆc₂₀50 and k0 ˆk=2, we see that xˆ 1=(At)2D(k=2;d=2) equivalent to t2D1(k=2;2=(jAjd))ˆD1(k0;2=(jAjd)), whereD1(k0;R0) is defined in (1.9).

We define a functionfto be the inverse ofc, so thatwˆc ¹(t)ˆf(t).

Thenwe havefc(w)ˆw;cf(t)ˆt, furthermorefis a solutionof the following difference equation

w(t‡l)ˆF(w(t));~ …D†

whereF~is defined as inProposition2 (see pp. 236 in[2]). Hereafter, we put lˆ1. Sinceuˆ0, we then have the following Propositions 4 and 5, ana- logous to Theorems A and B.

PROPOSITION4. -SupposeF(t)~ is formally expanded as in(2.6).Then the equation(D)has a formal solution

wˆf(t)ˆt 1‡ X

j‡k^1

^q_jkt ^j logt t _k!

; (2:9)

where^q_jkare constants which are defined byF as in Theorem A.~

PROPOSITION5. - Suppose a functionf is the inverse of c such that wˆc ¹(t)ˆf(t). Given a formal solution of the form(2.9)of (D),there exists a unique solution w(t)ˆf(t)which is holomorphic and admits an asymptotic expansion for t2D₁(k₀;2=(jAjd))such that

wˆf(t)ˆt 1‡b

t;logt t

!

; (2:10)

where

b t;logt t

X

j‡k^1

^qjkt ^j logt t _k

:

This functionf(t)is a solution of difference equation of(D).

In[5], we proved the following Theorem C.

THEOREMC. -Suppose X(x;y)and Y(x;y)are defined in(1.5).Then we have following:

(1)If kc206ˆd11for any k2N, k^2,the formal solutionC(x)of(1.6)of the following form

C(x)ˆX¹

mˆ1

a_mx^m; (2:11)

is identical to0,i.e.,a₁ˆa₂ˆ ˆ0.

(2)If kc₂₀ˆd₁₁for some k2N, k^2,we have a formal solutionC(x)of (1.6)of the following form

C(x)ˆX¹

mˆk

amx^m; (2:12)

i.e.,a1ˆa2ˆ ˆak 1ˆ0.

(3)Suppose

kc₂₀ˆd₁₁50 for some k2N; k^2:

(2:13)

For anyk,05k% p

2and smalld,we define the following domain D(k;d), D(k;d)ˆ fx : jarg[x]j5k;05jxj5dg:

(1:10)

There is a constantd>0and a solutionC(x)of(1.6),which is holomorphic

and can be expanded asymptotically in D(k;d)such that C(x)X¹

jˆk

a_jx^j: (2:14)

PROOF OFTHEOREM 1. - We will first prove (1) of Theorem 1. From Theorem C (2), we have a formal solution C(x) of (1.6) which canbe for- mally expanded so that one has

C(x)ˆX¹

jˆk

a_jx^j: (2:15)

On the other hand puttingw(t)ˆ 1

Ax(t)in(1.8), sinceAˆc₂₀, we have w(t‡1)ˆ 1

Ax(t‡1)ˆ 1

AX x(t);C(x(t))

ˆ 1

AX 1

Aw(t);C

1 Aw(t)

!: (2:16)

From (1.5), we have X x(t);C(x(t))

ˆx(t)‡ X

i‡j^2;i^1

c_ijx(t)ⁱ(C(x(t))_j

ˆx(t)

1‡ X

i‡j^2;i^1

c_ijx(t)^{i 1}(C(x(t))_j :

Thus we have 1 X x(t);C(x(t))

ˆ 1

x(t)n

1 P

i‡j^2;i^1 cijx(t)^{i 1}(C(x(t))jo

ˆ 1 x(t)

"

1‡

X

i‡j^2;i^1

cijx(t)^{i 1}(C(x(t))_j

‡

X

i‡j^2;i^1

cijx(t)^{i 1}(C(x(t))_j2

‡

X

i‡j^2;i^1

cijx(t)^{i 1}(C(x(t))_j3

‡

# :

Sincew(t)ˆ 1

Ax(t), we have 1

X x(t);C(x(t))ˆ Aw(t)

"

1‡ X

i‡j^2;i^1

cij

1 Aw(t)

_{i 1}

C 1 Aw(t)

^j!

‡ X

i‡j^2;i^1

c_ij 1 Aw(t)

_{i 1}

C 1

Aw(t) ^j!₂

‡

X

i‡j^2;i^1

c_ij 1 Aw(t)

_{i 1}

C 1 Aw(t)

^j₃

‡

# : Since C(x) is a formal solutionof (1.6) such that C(x)ˆC 1

Aw ˆ

ˆ P¹

mˆka_j 1

Aw _m

, (k^2), we have 1

AX x;C(x)ˆw

1‡c₂₀1

Aw ¹‡X

k^2

~c_k(w) ^k

; (2:17)

where~c_kare defined byc_ijanda_k(i‡j^2,i^1,k^2). From (2.17) and the definition ofA, we canwrite (2.16) inthe following form

w(t‡1)ˆF(w(t))~ ˆw(t)

1‡w(t) ¹‡X

k^2

~c_k(w(t)) ^k

: (2:18)

On the other hand, puttinglˆ1 an d mˆ1 in(2.1), i.e.uˆarg[l]ˆ

ˆarg[1]ˆ0, and making use of Proposition 4, we have the following for- mal solutionw(t) (2.19) of the first order difference equation (2.18),

w(t)ˆt 1‡ X

j‡k^1

^q_jkt ^j logt

t _k!

; (2:19)

where^q_jkare defined byF~in(2.18). From (2.17) and (1.6),F~is defined byX andY. Hence^qjkare defined byXandY.

Fromx(t)ˆ 1

Aw(t), we have a formal solutionof (1.2) such that x(t)ˆ 1

At 1‡ X

j‡k^1

^q_jkt ^j logt

t

_k! ₁ : (2:20)

From the relationof (1.2) and (1.8) with (1.6), we have proved (1) of Theorem 1.

Next we will show (2). From the assumptionthatkc20 ˆq1150 for some

k2N,k^2, we suppose thatR₀>Randk₀5p

4 e. Sinceuˆ0, we have D₁(k₀;R₀)D(e;R):

(2:21)

For ax2D(k;d), making use of Theorem C (3), we have a solutionC(x) of (1.6) which is holomorphic and can be expanded asymptotically inD(k;d) such that

C(x)X¹

jˆk

a_jx^j: (2;14)

SinceR₁ ˆmax (R₀;2=(jAjd)), making use of Proposition 5, we have a so- lutionw(t) of (2.18) which has anasymptotic expansion

w(t)ˆt 1‡b

t;logt t

!

;

in t2D₁(k₀;R₁). Therefore we have a solutionx(t) of (1.2) which has an asymptotic expansion

x(t)ˆ 1 At 1‡b

t;logt

t

! ¹

;

there. At first we take a small d>0. For sufficiently large R, since R₁^R₀>R, we will have

jx(t)j ˆ 1

At 1‡b

t;logt

t

1

5 1

jAjR(1‡1)5d;

(2:22)

fort2D1(k0;R1). SinceAˆc2050,

arg[x(t)]ˆarg

"

1 At 1‡b

t;logt

t

! 1#

ˆ arg[t] arg

1‡b

t;logt t

# : For sufficiently largeR₁, we thenhave

arg

"

1‡b

t;logt t

#5k0; fort2D1(k0;R1):

Hence we have

k₀ k₀arg[x(t)]k₀‡k₀:

From the assumptionofkˆ2k0, we have jarg[x(t)]j5k% p

2 for t2D₁(k₀;R₁):

(2:23)

From (2.22) and (2.23), we havex(t)2D(k;d) for a somek;

05k%p 2

. Hence we have aC(x(t)) which satisfies the equation(1.6).

Therefore from the existence of a solutionCof (1.6) and Proposition 5, we have a holomorphic solutionw(t) of first order difference equation (2.18) fort2D₁(k₀;R₁). Thus we obtaina solutionx(t) of (1.2) fortthere, which satisfies the following conditions:

(i) x(t) is holomorphic inD1(k0;R1), (ii) x(t) is expressible inthe form

x(t)ˆ 1 At 1‡b

t;logt

t

! 1

: (2:24)

Hereb(t;logt=t) is asymptotically expanded fort2D1(k0;R1) such that b

t;logt

t

X

j‡k^1

^q_jkt ^j logt

t _k

;

ast! 1throughD₁(k₀;R₁). p

Finally, we obtain a solution (u(t),v(t)) of (1.1) by the transformation u(t)v(t)

ˆP x(t) C(x(t))

:

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[4] M. SUZUKI,Holomorphic solutions of some system of n functional equations with n variables related to difference systems, Aequationes Mathematicae,57 (1999), pp. 21-36.

[5] M. SUZUKI,Holomorphic solutions of some functional equations II, South- east AsianBulletinof Mathematics,24(2000), pp. 85-94.

[6] M. SUZUKI,Difference Equation for a Population Model, Discrete Dynamics inNature and Society,5(2000), pp. 9-18.

[7] A. MATSUMOTO - M. SUZUKI, Analytic Solutions of Nonlinear Cournot- Duopoly Game, Discrete Dynamics in Nature and Society, 2005, issue 2 (2005), pp.119-133.

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[10] N. YANAGIHARA,Meromorphic solutions of some difference equations, Funk- cialaj Ekvacioj,23(1980), pp. 309-326.

Manoscritto pervenuto in redazione il 7 dicembre 2007