Publicacions Ma em`a iques, Vol 39 (1995), 301–312.
ASYMPTOTIC EQUIVALENCE
OF VOLTERRA DIFFERENCE SYSTEMS
J. Mo chalo
Abs ac
The pu pose o his pape is o gi e some esul s on he asymp o ic
ela ionship be ween he solu ions o a linea diffe ence equa ion
and i s pe u bed nonlinea equa ion.
1. In oduc ion
The p oblem o he asymp o ic equi alence o sys ems o o dina y
diffe en ial equa ions has been s udied by many au ho s, as e.g. B aue
[3], B aue and Wong [4], Boundo ides and Geo giou [2], Lowell Lo elady
[11], Mo cha#lo [13], ˘
S ec [17], Szufla [18], and o he s.
The p oblem o he asymp o ic equi alence o in eg odiffe en ial equa-
ions has been s udied by Mo cha#lo [14], Razapo [16], Talpala u [19].
The p oblem o he asymp o ic beha io o solu ions o o dina y di -
e ence equa ions has been s udied by Benzaid [1], Conffman [5], D oz-
dowicz, Popenda [6], Elaydi, Gyo i [8], Li [10] and Pin o [15].
In his pape , we shall conside some esul s on he asymp o ic ela-
ionship be ween he solu ions o a linea Vol e a diffe ence equa ion
and i s pe u bed nonlinea equa ion. The au ho knows only he wo ks
o Talpala u [19], Ved and Go#lo ina [21], Ved and Kap agae [20], deal-
ing wi h he abo e p oblem o special case.
2. No a ions and Defini ions
He e N(n0)={n0,n
0+1,...}, whe e n0is a na u al numbe o ze o;
Rk- he k- dimensional eal euclidean space wi h he no m
|x|=
k

i=1
|xi|,x=(x1,... ,x
k);
302 J. Mo chalo
Mk- he space o all k×kma ices D=(dij) wi h he no m |D|=
max
j
k

i=1
|dij|,I- iden i y ma ix.
We deno e by Φ(N,Rk) he space o all unc ions om N(n0)in oRk.
Le Φ1=Φ
1(N,Rk) be he Banach space in Φ o all bounded unc ions
u:N(n0)→Rkwi h no m x=|x(n)|Φ1= sup{|x(n)|:n∈N(n0)}.
In his pape we conside he ollowing sys ems o diffe ence equa ions
(2.1) x(n+1)=[A+B(n)]x(n)+
n

=0
[K(n− )+Q(n, )]x( ),
(2.2) x(n+1)=Ax(n)+
n

=0
K(n− )x( )+ (n)+F(n, x(n))
hough e as pe u ba ions o
y(n+1)=Ay(n)+
n

=0
K(n− )y( ),(2.1’)
y(n+1)=Ay(n)+
n

=0
K(n− )y( )+ (n)(2.2’)
whe e x,y, a e k-dimensional ec o s,
A- is a cons an ma ix k×k,
B, K :N(n0)→Mk
Q:N(n0)×N(n0)→Mk,
F:N(n0)×U→Rkis o any n∈N(n0)
con inuous as a unc ion o x∈U
(U- a egion in Rk).
We define he esol en ma ix R(n, m) o he equa ion
(2.3) x(n+1)=[A+B(n)]x(n)+
n

=0
[K(n− )+Q(n, )]x( )+ (n)
as he unique solu ion o he ma ix diffe ence equa ion [7]
(2.4) R(n+1,m)=[A+B(n)]R(n, m)
+
n

=m
[K(n− )+Q(n, )]R( , m),n≥m,
Asymp o ic equi alence o Vol e a di e ence sys ems 303
wi h R(m, m)=I.
Using he esol en ma ix R(n, m) we can es ablish he ollowing e-
la ion [7] (Va ia ion o Cons an s Fo mula)
(2.5) x(n, 0,x
0)=R(n, 0)x0+
n−1

=0
R(n, +1) ( ),
whe e x(n, 0,x
0) is he unique solu ion o he equa ion (2.3) sa is ying
x(0,0,x
0)=x0.
Le Y(n) deno e he undamen al ma ix o he sys em (2.1’) [7]. No-
ice ha Y(0) = Iand y(n, 0,y
0)=Y(n)y0is he unique solu ion o
(2.1’) wi h y(0,0,y
0)=y0.
Mo eo e ,
(2.6) Y(n+1)=AY (n)+
n

=0
K(n− )Y( ).
Rema k [7]. We ema k he e ha he esol en ma ix R(n, m) o
equa ions o noncon olu ion ype is closely ela ed o he undamen-
al ma ix Y(n). By uniqueness o solu ions, i is easy o see ha
o equa ions o con olu ions ype such as (2.1’), R(n, 0) = Y(n) and
R(n, m)=Y(n−m).
In his pape we conside he no ion o asymp o ic equi alence gi en
by,
Defini ion. We say ha he equa ions (2.1) and (2.1’) o (2.2), (2.2’)
a e asymp o ically equi alen i , co esponding o each solu ion x=x(n)
o (2.1), ((2.2)), he e exis s a solu ion y=y(n) o (2.1’), ((2.2’)) wi h
he p ope y
(2.7) lim[x(n)−y(n)] = 0 as n→∞and con e sely.
3. Asymp o ic equi alence
We s a e he ollowing lemma.
Lemma 3.1. I
1. ϕ(n)is bounded on N(n0)and lim
n→∞ ϕ(n)=ϕ(∞)exis s,
2.
∞

k=n0
|g(k)|<∞,
304 J. Mo chalo
hen
lim
n→∞
n

k=n0
ϕ(n−k)g(k)=ϕ(∞)
∞

k=n0
g(k).
Theo em 3.2. Assume ha
1. all solu ions o he sys em (2.1’) end o fini e limi s as n→∞,
2.
∞

=0 |B( )|+

s=0
|Q( , s)|<∞,
3. de P=0, whe e P= lim Y(n)as n→∞,Pis a cons an
ma ix,
4. q=
∞

=0 |B( )||R( , 0)|+

s=0
|Q( , s)||R(s, 0)|<1.
Then,
a) co esponding o each solu ion x=x(n)∈Φ1o (2.1), he e exis s
a solu ion y=y(n)∈Φ1o (2.1’) such ha (2.7) is sa isfied
p o ided ha Condi ions 1, 2 hold,
b) in Rela ion (2.7) he solu ion y=y(n)o (2.1’) is unique i Con-
di ions 1, 2 and 3 a e sa isfied,
c) o each non-ze o solu ion x=x(n)∈Φ1o (2.1) he e co esponds
in Rela ion (2.7) a non-ze o solu ion y=y(n)∈Φ1o (2.1’), i
Condi ions 1, 2 and 4 hold and con e sely,
d) in Rela ion (2.7) he solu ion x=x(n)o (2.1) is unique i Con-
di ions 1, 2, 3 and 4 a e sa isfied.
P oo : By Fo mula (2.5) he solu ions x(n) o (2.1) and y(n) o (2.1’)
can be w i en as
(3.1) x(n)=Y(n)x0+
n−1

=0
Y(n− −1) B( )x( )+

s=0
Q( , s)x(s)
and
(3.2) y(n)=Y(n)y0,n∈N.
Fu he mo e, om he Rela ions (3.1) and (3.2) we ob ain
(3.3) x(n)−y(n)=Y(n)[x0−y0]
+
n−1

=0
Y(n− −1) B( )x( )+

s=0
Q( , s)x(s).
Asymp o ic equi alence o Vol e a di e ence sys ems 305
F om Assump ions 1, 2 and (3.1) we ob ain
|x(n)|≤|Y(n)||x0|
+
n−1

=0
|Y(n− −1)||B( )||x( )|+

s=0
|Q( , s)||x(s)|.
Hence and diffe ence inequali y [9] we can easily ob ain ha all solu ions
o (2.1) a e bounded.
Thus
(3.4)
∞

n=0 
B( )x( )+

s=0
Q( , s)x(s)
<∞.
By Assump ion 1, Lemma 3.1 and Rela ions (3.3) (3.4) we ge
(3.5) lim
n→∞[x(n)−y(n)]
=Px0−y0+
∞

n=0 B(n)x(n)+
n

s=0
Q(n, s)x(s).
This shows ha o a bi a y solu ions x(n) and y(n) o (2.1), (2.1’)
espec i ely Rela ion (2.7) hold iff
(3.6) Px0−y0+
∞

n=0 B(n)x(n)+
n

s=0
Q(n, s)x(s)=0.
Equali y (3.6) defines a ela ion be ween all solu ions x(n), y(n)o
(2.1), (2.1’), espec i ely, o which (2.7) holds.
I P= 0, hen (3.6) means ha (2.7) holds o a bi a y solu ions
x(n), y(n) o (2.1), (2.1’) espec i ely.
On he o he hand i P= 0, hen o a bi a y solu ion x(n) o (2.1)
we ha e
(3.7) y0=x0+
∞

n=0 B(n)x(n)+
n

s=0
Q(n, s)x(s).
Hence o sui able solu ion y(n) o (2.1’) we conclude ha (2.7) holds.
Since Condi ion 3 holds, we claim ha he solu ion y(n) wi h he ini ial
condi ion y0defined by (3.7) is unique in (2.7).

306 J. Mo chalo
F om (2.5) o (n) = 0 and (3.7) we ha e
y0=x0I+
∞

n=0 B(n)R(n, 0) +
n

s=0
Q(n, s)R(s, 0)
o
(3.8) (I+P0)x0=y0
whe e
P0=
∞

n=0 B(n)R(n, 0) +
n

s=0
Q(n, s)R(s, 0).
Assume ha (I+P0)−1exis s and x(n)= 0 o n∈N(x0= 0). Then,
by (3.8), we ha e y0=0(y(n)= 0 o n∈N). Such a ma ix exis s i ,
o example, |P0|<1[12] (Banach Theo em’s). F om Assump ion 4 i
ollows ha |P0|<1.
Le he ini ial condi ion y0o solu ion y(n)=y(n, 0,y
0) be a bi a y.
F om (3.8) we ob ain
x0=(I+P0)−1y0.
Hence o e e y solu ion y(n)=0onN he e exis s a unique solu ion
x(n)=0onNsuch ha (2.7) holds and con e sely.
Rema k. I all solu ions o (2.1’) ends o ze o as n→∞and Con-
di ion 2 hold, hen all solu ions o (2.1) end o ze o as n→∞.
Now, we conside asymp o ic equi alence be ween Equa ions (2.2) and
(2.2’).
Lemma 3.3. Suppose ha he ollowing condi ions hold:
1. e e y solu ion o (2.2’) is bounded on N,
2. |F(n, x1)−F(n, x2)|≤g(n)||x1−x2| o n∈N,x1,x2∈U,
3.
∞

n=0
g(n)<∞and
∞

n=0
|F0(n)|<∞whe e F0(n)≡F(n, 0).
Then e e y solu ion o (2.2) is bounded on Nand
(3.9) |x(n)|≤LM(n),n∈N
Asymp o ic equi alence o Vol e a di e ence sys ems 307
whe e
Y0= sup
N
|Y(n)|,
L= sup
N
y0(n)+
n−1

=0
Y(n− −1)F0( )
<∞,
M(n) = exp Y0
n−1

=0
g( ),
y0(n)is a solu ion o (2.2’).
P oo : By he o mula (2.5) he solu ion o (2.2) can be w i en as
(3.10) x(n, 0,x
0)=y0(n)+
n−1

=0
Y(n− −1)F( , x( ))
whe e
y0(n)=Y(n)x0+
n−1

=0
Y(n− −1) ( ),n∈N.
Fu he mo e, i ollows om (2.5) and in iew 1 ha all solu ions
o (2.1’) a e bounded on N. Now, using he Rela ion (3.10) and he
Condi ion 2, we ge
(3.11) |x(n)|≤L+Y0
n−1

=0
g( )|x( )|,
which implies, by G onwall inequali y
|x(n)|≤Lexp Y0
n−1

=0
g( )=LM(n).
Because he unc ion M1(n) is bounded on N, we can conclude ha
he solu ion x(n) o (2.2) is also bounded on N.
Rema k. F om (3.9), we ha e
(3.12) |x(n)|≤[Y0|x0|+ 0+Y0F1]M(n)
whe e
0= sup
N
n−1

=0
Y(n− −1) ( )
<∞,
F1=
∞

n=0
|F0(n)|<∞.
308 J. Mo chalo
Theo em 3.4. Le
1. all solu ions o he sys em (2.1’) end o fini e limi s as n→∞,
2. Condi ions 2 and 3 o lemma 3.3 hold,
3. sup
n∈Nn−1

=0
|Z1(n, )||g( )|+
∞

=n
|Z2(n, )||g( )|<1whe e
Z1(n, )=Y(n− −1) −Y(n),Z
2(n, )=−Y(n).
Then o each solu ion x(n)o (2.2) he e co esponds a solu ion
y(n)o (2.2’) such ha (2.7) holds.
Mo eo e , suppose ha Condi ion 3 o Theo em 3.2 holds. Then
he solu ion y(n)o (2.2’) in Rela ion (2.7) is unique.
Le Condi ions 1-3 hold and
4. q1=Y0
∞

n=0
g(n)M1(n)<1.
Then o each solu ion o (2.2) wi h x0=0and
|x0|>(1 −q1)−1[F2+q1(F1+Y−1
0 0)]
whe e
F2=
∞

n=0
F0(n)
<∞
he e co esponds a solu ion y(n)o (2.2’) wi h y0=0such ha
(2.7) holds and con e sely.
I , in addi ion he Condi ion 3 o Theo em 3.2 holds, hen he
solu ion x(n)o (2.2) in Rela ion (2.7) is unique.
P oo : By 1 i ollows ha solu ions o (2.2’) a e bounded on N. Then,
he bounded p ope ies o solu ions o (2.2’) imply ha solu ions o (2.1’)
a e bounded oo. The fi s wo pa s o Theo em a e easily e ified (see
Theo em 3.2 and Lemma 3.3). Since P= 0, hen we can find a ini ial
condi ion y0o he solu ion y(n) o (2.2’) such ha
(3.13) y0=x0+
∞

n=0
F(n, x(n)),
whe e x(n) is a gi en solu ion o (2.2).
Asymp o ic equi alence o Vol e a di e ence sys ems 309
Le x0= 0, hen om (3.13), (3.12) we ha e
(3.14)
|y0|≥|x0|−
∞

n=0
F(n, x(n))
=|x0|−
∞

n=0
F(n, x(n)) −F(n, 0)] +
∞

n=0
F(n, 0)
≥|x0|−
∞

n=0
|F(n, x(n)) −F0(n)|−F2
≥|x0|−
∞

n=0
g(n)M1(n)[Y0|x0|+ 0+Y0F1]−F2
≥|x0|(1 −q1)−[(F1+Y−1
0 0)q1+F2]>0.
Hence y0=0.
Le he ini ial condi ion y0o he solu ion y(n) o (2.2’) be a bi a y
selec ion. Then by Condi ions 1, 2 and (3.13) he solu ion x(n) o (2.2)
be defined o all n∈Nand (2.7) be hold. By his means we gi e some
condi ions o exis ence and uniqueness o he solu ion x(n) o (2.2) in
Φ1which sa isfied (3.13).
Since he equa ion (2.2) wi h ini ial condi ion x0is equi alen o he
equa ion (3.10), hen subs i u ing o x0 om (3.13) in o (3.10)
(3.14) x(n)=y(n)+
n−1

=0
Z1(n, )F( , x( )) +
∞

=n
Z2(n, )F( , x( ))
whe e
y(n)=Y(n)y0+
n−1

=0
Y(n− −1) ( )
is a bi a y solu ion o (2.2’),
Z1(n, )=Y(n− −1) −Y(n)
Z2(n, )=−Y(n).
Le Tbe he ope a o defined o each x∈Φ1by he equa ion
Tx(n)=
n−1

=0
Z1(n, )F( , x( )) +
∞

=0
Z2(n, )F( , x( )).