Asymptotic constancy of solutions of systems of difference equations

(1)

Internat. J. Math. & Math. Sci. VOL. 19 NO. 3 (1996) 501-506

501

ASYMPTOTIC

CONSTANCY

OF

SOLUTIONS

OF

SYSTEMS

OF DIFFERENCE

EQUATIONS

RIGOBERTOMEDINA Departamentode Ciencias UniversidaddeLosLagos Casilla933,Osorono,CHILE

MANUEL PINTO

Departamentode Matemiticas,Facultadde Ciencias Universidadde Chile,Casiila 653

Santiago, CHILE

(Received September 9, 1993 and in revised formOctober31, 1994)

ABSTRACT. Assuming only conditional summabilitywe study the convergence of the solutions of differencesystems

KEY WORDS AND PHRASES: DifferenceSystems,Weighted

Norms, Convergent

Solutions 1991AMSSUBJECTCLASSIFICATION CODES: 34K20

’1. INTRODUCTION

In [12],wehave studied theasymptoticequilibriumof ageneralnonlinear differenceequation

Az

f(n,z),

n N. ( )

After,wehave studiedtheexistenceof convergentsolutionsofnonlinearsystems whoselinearpart

has adichotomy See 13, 14, 16-18] Theseresults are obtainedunderabsolutelysummableconditions Motivated with theworkof Trench[7]ondifferentialequationsweshow thatsolutionsofasystem(1 ), approach constant vectors as n c, under assumptions which permit some orall of sum smallness conditions of

f

tobe stated interms ofconditional ratherthanabsolute convergence Through the paperthe conditionalconvergencewillbe simply calledconvergencewhiletheabsolute convergencewill

be explicitlymentioned

This kind of problem for ordinary differential equations has been widely investigated by many authors,forexamplesee 1,4,6]

It seems to usthat verylittle isknown about theconvergence of thesolutions of finite difference

equations(see 15]) Theonly results thatweknow concerningthisproblem forsecond orderdifference

equationsaregivenbyDrozdowiczandPopenda [2], CatilloandPinto[3], Szmanda[5],Handerson and

Peterson [8],Szafranskiand Szmanda

[9]

andMedina and Pinto 10,11 2. PRELIMINARIES

Consider the difference system (1.1), where N

{n0,n0

+

1,

...},

n0 is a given non-negative integer, x is an m-dimensional vector,

f:N

R’--,_R _{is a function and} _R _{denote the}

m-dimensionalrealEuclidean space,

A

isthe difference operator, e /xn x_l x, Throughoutthis

paperthe norm ofavector ormatrix isthe sum of theabsolute valuesof its elements

By

asolution of

Eq (1

1)we mean any functionx defined on

N,

which fulfills

Eq

(1 1) for all

sufficientlylargen Notethat the above definition of thesolution isdifferent fromthiswherexfulfills

Eq

(l.1)for allnEN

Throughoutthispaper,wewillsupposethefollowing

ASSUMPTION. Them rn matrtx

functton

VtsnonsingularonNand

(2)

502

and so nO

lim

V-(n)

exists(finite) (23)

LEMMA1.

If

q s anm-vector

funcnon

onNand V

(j)q

converges,then

E q

convergesand

3=n0 3=no

V(n)

Eql

-<

(l+K)p,

n_>no,

(24)

PROOF. With where

P’

E

V

(j)q.,

(2

6)

usingsummationby partsandthefundamental Theorem of sumcalculus,wehave

(2 7)

and

From (2.3),

(2.5)

and

(2

6),

IA(V-I(j))p,+I[

_<

pno[/l(v-l(j))l,

j

>

no;

lim

v-l(nl)pn,

0

hence,because of

(2.2),

we canletnl---)_ooin

(2.7)

weobtain

E

q.

V-1

(rl’)Pn

-Jr"

E/k(V-I(j))P:+I"

j=n j=n

Multiplyingby V

(n),

weobtain

V(r)

E

q. Pn-’k

V()

E

/k(V-I(j))P:+a"

j=n

Thus,by

(2.1), (2.5)

and

(2.6)

[V(n)’lq,

<

Ipl

/

[V(n)/k(v-l(J))I

Ip,+ll

3=n

_<

Il

+

Ipl"

[V(n)/k(v-l())[

3---n

_<(I+K);,

n_>no.

DEFINITION 1. Let

7-/(no)

be the Banach space

of

sequences h N

--

R such that Vh is

bounded,with norm

Ilhll

zup>_.olV(n)hl.

(2 8)

If

A

>

O, let

(3)

SOLUTIONS OFSYSTEMSOFDIFFERENCE EQUATIONS 503

Inaddition,weneedthe followingdefinition

DEFINITION 2. (See [7]) A vector tsaLtpschtzpoint

of

a vector

functton

f

thereare constantsr,c

>

0 such that

k(x)

s

de.fined

whenever

x

1

-<

r _{(2 9)}

and

(2 o)

From

(3.7)

and(3 8),

sup

II(n; 0)l

<

A/(1

+

K),

(3 S)

which ispossiblebecause of

(3 4)

andtheconvergenceof

(;

o)

II(n;h)l<A/(l+K)

if

n_>

and hE7-/x

(no).

If h E7-/x

(no),

defineyhby

yh=

-f(j,+h3),

n>na.

From

(3.2)

andLemma with q,

f(n,

+

h,),

3)hisdefined andsatisfies theinequality

IV(n)Yhl < (1

+

K)sup

II(e; h)l,

n

>

nl.

(3 9)

Now choosenl

no

sothat

tf

x-l

<

r; i=l,2.

3. MAIN RESULTS

The followingis our maintheorem

THEOREM 1. Foragivenvector

,

supposethere areconstant

A

>

0and

no

Nsuchthat the

function

f

ts

defined

ontheset

U

((n,x)II V(n)(x

)1

< A,n >

no},

(3 1)

and theseries

I(n; h)

E

V(3)f(j,

+

h3),

n

>_ no

(3 2)

converges

if

h

7"t

(no).

Suppose

alsothat

II(n;h’)-I(n;h2)l

<6llh

1-h211,

n>no,

(3 3)

wheneverh

1,

h

7-lx

(no),

where

o

<

_<

/(1

+

K)

(3 a)

Then

Eq. (1

1)hasasolution z whichis

defined

for

nsufficientlylargeand

sattsfies

lim

V(n)(z

)

0

(3.5)

Moreover,

tf

ytsany solutton

of

Eq.

(1 1)such that

dimooV(’)(u.

)

o,

(3 6)

then

z

y,

for

nsuffictentlylarge. PROOF. Ifh

7-/x(no)

thenfrom

(3.3)

II(n; h)l < [I(n; h)- I(n;

0)1

+

II(n;

0)1,

(4)

504

Fromthisand(3 9),

IV(n)Yh,,I

_<

,,

n

_>

Therefore

h

7-/A(n),

that is, 3) transforms

7"/x(nt)

into itsel Now, suppose h’ 7-lx(nl),

(i

1,2) ThenLema with qn

f(n,

+

h)

f(n,

+

h)

implies that

IV(n)(Yh

yh)[

<

(1

+

K)[I(n;h

)

(3 0) andso, from(2 8)and(3 3) (withn0 nl),

IlYh

Yh2ll

<

5(1

+

K)llh

h2ll.

Hence, from (3 4), 3) is a contraction mappings of

7-/A(n)

into itself, and therefore there is an

h

x

(nl)

such that h yh

,

thatis

From Lemma 1, with q,

f(n,(

+

h),

lirnV(n)h

0. Therefore the function x (

+

h

satisfies

Eq (1.1)

and(3 5) Ifysatisfies

Eq

(1.1)and

(3.6),

then h y (is in

x

(no)

forsome n2

>

n, and

[S0,

+

S0,

+

>_

By

anargumentlikethatwhichledto

(3

10),

lib

-

hll

_<

6(1

+

K)llh’

hll,

which implies that

h

forn

>

n2, because of(3

4)

Thisimplies thatx, y,, forn sufficiently

large.

WenowapplyTheorem tothe system

AXn

a(n)(Xn)

+

gn, n

>_ no.

(3

l)

THEOREM2.

Suppose A

isan mx matrix

function

and gisanm-vector.function, both

defined

onN,and isaLipschitzpoint

of

thef.-vecorjnction

.

Suppose

also that

V(j)[A(j)()

+

g,]

(3,12)

J=no

(3.13)

(3.14)

(3.15)

(3.16)

convergesand

IV(j)A(j)I

Iv-’(J)l

<

?=’no

Then the conclusion

of

TheoremIholds

for

Eq.

(3.11). PROOF. Let

which is finitebecause of

(2.3).

Let 6 beanynumber that satisfies

(3.4),

letcbeasin

(2.9),

andchoose

n2sothat

c IV(j)A(j)I

IV

-

(J)l

<-

6,

whichispossiblebecauseof

(3.13).

Henceforth,letn

_>

n2. Finally, let

A =r/a,

withras in

(2.9).

Wewillshow that)satisfies the requirementofTheorem 1, for

f(n,x)

A(n)b(x)

-F

(3.17)

(5)

SOLUTIONS OF SYSTEMS OFDIFFERENCE EQUATIONS 505

becauseof(3 14)and(3 16) Since is definedfor all zsatisfying (2 9),whileA and g are defined on

N,

itfollowsthat

f(n,x)

is defined on

U,

and

f(n,

+

h,)

isdefined forn

>

n

> no

ifh E

7"l(no)

Moreover,

ifh

,

h E

7-t

(n0),

then

by(2 8)and(2 1) This,(3 15)and(3 18)implythat

h h

[E

V(j)A[b( +

ha)

(

+

h)][

<

51[

I[,

n

>_

n2. (3 19)

With

f

as in (3 17), the functional in

(3

2)

becomes

I(n;

h)

E

V(j)[A(j)(

+

h3)

+

g3]

3---r

Fromthe convergence of(3 12),

I(n; 0)

exists This andthe convergenceof theseries in (3 19)with h hand h 0imply that

I(n; h)

existsforall h

7-/x(n0),

fn

>

n2

>

nl Knowing this,we can conclude from (3 19) that (3 3) holds whenever h

1,

h2E

7"/x(n0).

This completes the proof of

Theorem2

Stronger

resultsareavailableforalinearsystem

/kx,

A(n)x,.,

+

g,, n

_>

no.

(3 20)

THEOREM3.

Suppose

that

for

anm mmatrix

function

A

andan m-vector

functton

g

defined

OftN,

E

[V(j)A(j)V-I(J)[

<

oo,

(3.21)

and

3-E

V(j)[A(j)

+

g3]

(3 22)

converges

for

a given constantvector

.

Then

Eq. (3.20)

has aumqesolunon x which

sates.ties

(3.5). PROOF. Taking

(x)

x, theproofis similar tothatof Theorem2foragivenconstant vector

.

Thenexttheoremfollows fromthisandelementaryproperties oflinear differencesystems.

TIIEOREM4.

Suppose A

and g are

defined

on

N, (3.21)

holds and

E

V(j)A(j) and

E

V

(J)g3

converge. Then

Eq.

(3.20)

hasa unique solution which

satisfies

(3.5)for

anygivenconstant

vector

;

and everysolution

of

Eq. (3.22)

satisfies

(3.5)for

some

.

PROOF.

Any

constant vector is aLipschitz point of

(x)=

x.

Moreover,

if

E

V(j)A(j)

3:no

converges, then E V(j)A(j) converges, too. From this, the series

E V(j)[A(j)+93]

converges for any constant vector

.

Therefore, for everyconstant vector

,

Theorem3 ensuresthat there isauniquesolutionof

Eq. (3 22)

satisfying

(3.5).

The secondstatementof Theorem4follows the uniqueness givenby

(3.6)

inTheorem andproperties oflineardifference systems.

EXAMPLE

1. Thedifferencesystem

XlX2

--

₅

/kx2

n4(xl

x2)2

n n--2 x2 4-n

(3

23)

has the form

(3.11),

forn

>

1.

(6)

506

+

asn ooforall (4, provided

4

(

EXAMPLE

2. Wenow exhibit a system

z

A(n)x

whose solutionsall tendto constantvectors, eventhough

]A()J

Tothisend,weobsee that

if

( r,

2-e

o

co(eo)

_{converges for}

allr

>

0

Nowconsiderthe system

/x2 cn-5/2 dn-3/2 _x2

(324)

where a,b,c,d are constantsand b 0, sothat

E

IA(n)}

oo, and let

V(n)

diag(eun _ne

w)

_with

n:l

0

<

#

<

}

Here

V(n)A(n)

convergesand (3.21)holds, hence Theorem4implies that if

1

and

2

arearbitrary, then(3 24)hasa solutionsuch that

X

(7Z)

1

--+

as7Z OO

ACKNOWLEDGMENT. We thank the referees for their useful observations Thiswork has been

supported byFondecyt92-0148and Fondecyt1930839 REFERENCES

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