Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 932831, 22 pages
doi:10.1155/2008/932831
Research Article
Asymptotic Representation of the Solutions of
Linear Volterra Difference Equations
István Győri and László Horváth
Department of Mathematics and Computing, University of Pannonia, 8200 Veszprém,
Egyetem u. 10, Hungary
Correspondence should be addressed to István Győri, gyori@almos.vein.hu
Received 26 February 2008; Accepted 4 April 2008
Recommended by Elena Braverman
This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations.
Some sufficient conditions are presented under which the solutions to a general linear equation
converge to limits, which are given by a limit formula. This result is then used to obtain the exact
asymptotic representation of the solutions of a class of convolution scalar difference equations,
which have real characteristic roots. We give examples showing the accuracy of our results.
Copyright q 2008 I. Győri and L. Horváth. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The literature on the asymptotic theory of the solutions of Volterra difference equations is
extensive, and application of this theory is rapidly increasing to various fields. For the basic
theory of difference equations, we choose to refer to the books by Agarwal 1, Elaydi 2, and
Kelley and Peterson 3. Recent contribution to the asymptotic theory of difference equations
is given in the papers by Kolmanovskii et al. 4, Medina 5, Medina and Gil 6, and Song
and Baker 7; see 8–19 for related results.
The results obtained in this paper are motivated by the results of two papers by
Applelby et al. 20, and Philos and Purnaras 21.
This paper studies the asymptotic constancy of the solution of the system of
nonconvolution Volterra difference equation
zn 1 n
i0
Hn, izi hn,
n ∈ Z ,
1.1
2
Advances in Difference Equations
with the initial condition
1.2
z0 z0 ,
where z0 ∈ Rd , Hn, i0≤i≤n and hnn≥0 are sequences with elements in Rd×d and Rd ,
respectively.
Under appropriate assumptions, it is proved that the solution converges to a finite limit
which obeys a limit formula. Our paper develops further the recent work 20. The distinction
between the works is explained as follows. For large enough n, in fact n ≥ 2m 2, the sum in
1.1 can be split into three terms
m
Hn, izi i0
n−m−1
Hn, izi m
Hn, n − jzn − j,
1.3
j0
im1
since
n
Hn, izi in−m
m
Hn, n − jzn − j.
1.4
j0
In 20, Theorem 3.1 the middle sum in 1.3 contributed nothing to the limit limn→∞ zn,
since it was assumed that
n−m
Hn, i O.
1.5
lim lim sup
m→∞
n→∞
im
In our case, we split the sum in 1.1 only into two terms, and the condition 1.5 is not
assumed. In fact, we show an example in Section 4, where 1.5 does not hold and hence in 20,
Theorem 3.1 is not applicable. At the same time our main theorem gives a limit formula. It is
also interesting to note that our proof is simpler than it was applied in 20.
Once our main result for, the general equation, 1.1 has been proven, we may use it for
the scalar convolution Volterra difference equation with infinite delay,
Δxn Axn n
Kn − jxj gn,
n ∈ Z ,
1.6
j−∞
with the initial condition,
xn ϕn,
n ∈ Z− ,
1.7
where A ∈ R, and Knn≥0 , gnn≥0 and ϕnn≤0 are real sequences.
Here, Δ denotes the forward difference operator to be defined as usual, that is, Δxn :
xn 1 − xn, n ∈ Z .
If we look for a solution xn λn0 , n ∈ Z λ0 ∈ R \ {0} of the homogeneous equation
associated with 1.6, we see that λ0 is a root of the characteristic equation
λ0 1 A ∞
i0
Kiλ−i
0 .
1.8
I. Győri and L. Horváth
3
We immediately observe that λ0 ∈ R is a simple root if
∞ −1 −i
λ0 iKiλ0 < 1.
1.9
i1
In the paper 21 see also 22, it is shown that if λ0 > 0 satisfies 1.8 and 1.9, and
the initial sequence ϕnn≤0 is suitable, then for the solution x of 1.6 and 1.7 the sequence
zn : λ−n
0 xn, n ∈ Z is bounded. Furthermore, some extra conditions guarantee that the
limit z∞ : limn→∞ zn is finite and satisfies a limit formula.
In our paper, we improve considerably the result in 21. First, we give explicit necessary
and sufficient conditions for the existence of a λ0 ∈ R \{0} for which 1.8 and 1.9 are satisfied.
Second, we prove the existence of the limit z∞ and give a limit formula for z∞ under the
condition only λ0 /
0. These two statements are formulated in our second main theorem stated
in Section 3. The proof of the existence of z∞ is based on our first main result.
The article is organized as follows. In Section 2, we briefly explain some notation and
definitions which are used to state and to prove our results. In Section 3, we state our two main
results, whose proofs are relegated to Section 5.
Our theory is illustrated by examples in Section 4, including an interesting nonconvolution equation. This example shows the significance of the middle sum in 1.3, since only this
term contributes to the limit of the solution of 1.1 in this case.
2. Mathematical preliminaries
In this section, we briefly explain some notation and well-known mathematical facts which are
used in this paper.
Let Z be the set of integers, Z : {n ∈ Z | n ≥ 0} and Z− : {n ∈ Z | n ≤ 0}. Rd stands
for the set of all d-dimensional column vectors with real components and Rd×d is the space
of all d by d real matrices. The zero matrix in Rd×d is denoted by O, and the identity matrix
by I. Let E be the matrix in Rd×d whose elements are all 1. The absolute value of the vector
x x1 , . . . , xd T ∈ Rd and the matrix A Aij 1≤i,j≤d ∈ Rd×d is defined by |x| : |x1 |, . . . , |xd |T
and |A| : |Aij |1≤i,j≤d , respectively. The vector x and the matrix A is nonnegative if xi ≥ 0 and
Aij ≥ 0, 1 ≤ i, j ≤ d, respectively. In this case, we write x ≥ 0 and A ≥ O. Rd can be endowed
with any norms, but they are equivalent. A vector norm is denoted by · and the norm of
a matrix in Rd×d induced by this vector norm is also denoted by ·. The spectral radius of
the matrix A ∈ Rd×d is given by ρA : limn→∞ An 1/n , which is independent of the norm
employed to calculate it.
A partial ordering is defined on Rd Rd×d by letting x ≤ yA ≤ B if and only if y − x ≥
0B − A ≥ O. The partial ordering enables us to define the sup, inf, lim sup, lim inf, and so
forth for the sequences of vectors and matrices, which can also be determined componentwise
and elementwise, respectively. It is known that ρA ≤ ρ|A| for A ∈ Rd×d , and ρA ≤ ρB if
A, B ∈ Rd×d and O ≤ A ≤ B.
3. The main results
First, consider the nonconvolutional linear Volterra difference equation
zn 1 n
i0
Hn, izi hn,
n ∈ Z ,
3.1
4
Advances in Difference Equations
with initial condition
z0 z0 .
3.2
Here, we assume
H1 z0 ∈ Rd , H : Hn, i0≤i≤n and h : hnn≥0 are sequences with elements in Rd×d
and Rd , respectively;
H2 for any fixed i ≥ 0 the limit H∞ i : limn→∞ Hn, i is finite and ∞
i0 |H∞ i| < ∞;
H3 the matrix
V : lim
lim
n→∞
N→∞
n
3.3
Hn, j
jN
is finite;
H4 the matrix
W : lim
lim sup
N→∞
n→∞
n Hn, j
3.4
jN
is finite and ρW < 1;
H5 the limit h∞ : limn→∞ hn is finite.
By a solution of 3.1, we mean a sequence z : znn≥0 in Rd satisfying 3.1 for any
n ∈ Z . It is clear that 3.1 with initial condition 3.2 has a unique solution.
Now, we are in a position to state our first main result.
Theorem 3.1. Assume (H1 )–(H5 ) are satisfied. Then for any z0 ∈ Rd the unique solution z :
znn≥0 of 3.1 and 3.2 has a finite limit at ∞ and it satisfies
−1
lim zn I − V n→∞
∞
H∞ izi h∞ .
3.5
i0
Under conditions H3 and H4 |V | lim
N→∞
n
Hn, j ≤ W,
lim n→∞ jN
3.6
and hence ρW < 1 yields ρV < 1, thus I − V is invertible. On the other hand under our
conditions the unique solution z : znn≥0 of 3.1 and 3.2 is a bounded sequence, therefore
∞
i0 H∞ izi is finite, and 3.5 makes sense.
The second main result is dealing with the scalar Volterra difference equation
Δxn Axn n
j−∞
Kn − jxj gn,
n ∈ Z ,
3.7
I. Győri and L. Horváth
5
with the initial condition
xn ϕn,
n ∈ Z− ,
3.8
where A ∈ R, K : Z →R, g : Z →R, and ϕ : Z− →R are given.
By a solution of the Volterra difference equation 3.7 we mean a sequence x : Z→R
satisfies 3.7 for any n ∈ Z .
In what follows, by S we will denote the set of all initial sequence ϕ : Z− →R such that
for each n ∈ Z
−1
Kn − jϕj
3.9
j−∞
exists.
It can be easily seen that for any initial sequence ϕ ∈ S, 3.7 has exactly one solution
satisfying 3.8. This unique solution is denoted by xϕ : Z→R and it is called the solution of
the initial value problem 3.7, 3.8.
The asymptotic representation of the solutions of 3.7 is given under the next condition.
A There exists a λ0 ∈ R \ {0} for which
λ0 1 A ∞
Kiλ−i
0 ,
3.10
i0
∞ −1 −i
λ0 iKiλ0 < 1.
3.11
i1
From the theory of the infinite series, one can easily see that condition A yields
1/n
r : lim sup Kn
3.12
n→∞
is finite. Moreover, the mapping G : r, ∞→0, ∞, defined by
⎧
∞ ⎪
⎪
⎪ iKiμ−i1
⎪
⎪
⎪
⎨ i1
∞ Gμ : ⎪
iKir −i1
⎪
⎪
⎪
⎪
i1
⎪
⎩
∞
if μ > r,
if μ r > 0,
3.13
if μ r 0,
is real valued on r, ∞. It is also clear see Section 5 that if there is an n0 ≥ 1 such that
Kn0 /
0, and if Gr ≥ 1 then the equation
Gμ 1
3.14
has a unique solution, say μ1 .
Now we formulate the following more explicit condition:
B either Kn 0, n ≥ 1, and
1 A K0 / 0,
3.15
6
Advances in Difference Equations
or there is an n0 ≥ 1 with Kn0 /
0, and
i r defined in 3.12 is finite,
ii if Gr ≥ 1, then the constant A satisfies either
A > μ1 − 1 −
∞
Kiμ−i
1
3.16
i0
or
A < −μ1 − 1 −
∞
−1i Kiμ−i
1 ,
3.17
i0
iii if Gr < 1, then the constant A satisfies either
A≥r−1−
∞
Kir −i
3.18
i0
or
∞
A ≤ −r − 1 −
−1i Kir −i .
3.19
i0
0 for some n0 ≥ 1. It will be proved
Remark 3.2. Let K : Z →R be a sequence such that Kn0 /
in Lemma 5.7 that there is at most one λ0 ∈ C \ {0} satisfying 3.10 and 3.11. It is an easy
consequence of this statement that if λ0 ∈ C \ {0} satisfies 3.10 and 3.11, and λ1 ∈ C \ {0, λ0 }
is a solution of 3.10, then |λ1 | < |λ0 |, thus λ0 is the leading root of 3.10. Really, from the
condition |λ1 | ≥ |λ0 | we have
∞ ∞ −i1 −i1
iKiλ1 ≤
iKiλ0 < 1,
i1
3.20
i1
that is 3.11 holds for λ1 instead of λ0 , and this contradicts the uniqueness of λ0 .
Now, we are ready to state our second result which will be proved in Section 5. This
result shows that the implicit condition A and the explicit condition B are equivalent and
the solutions of 3.7 can be asymptotically characterized by λn0 as n→∞.
Theorem 3.3. Let A ∈ R, K : Z →R, g : Z →R, and ϕ ∈ S be given. Then
α Condition (A) holds if and only if condition (B) is satisfied.
β If condition (A) or equivalently condition (B) holds, moreover
∞
−1
−i
λ0
Ki − jϕj gi
i0
3.21
j−∞
is finite, then for the solution xϕ of 3.7, 3.8 the limit cϕ : limn→∞ λ−n
0 xϕn is finite and it
obeys
cϕ λ0 −
∞
i1
iKiλ−i
0
−1 ∞
i0
λ−i
0
−1
j−∞
Ki − jϕj gi
λ0 ϕ0 .
3.22
I. Győri and L. Horváth
7
4. Examples and the discussion of the results
In this section, we illustrate our results by examples and the interested reader could also find
some discussions.
Example 4.1. Our Theorem 3.1 is given for system of equations, however the next example
shows that this result is also new even in scalar case.
Let us consider the scalar nonconvolution Volterra difference equation
zn 1 n
n − jα−1 j β−1
nαβ−1
j1
zj hn,
n ≥ 1,
4.1
with the initial condition
z1 z1 ,
4.2
where a1 α, β > 1 and real, and hnn≥1 is a real sequence such that its limit h∞ :
limn→∞ hn is finite.
Now, let the values z0 , h0 and the sequence H : Hn, i0≤i≤n be defined by
z0 : 0,
h0 : z1 ,
⎧
α−1 β−1
⎪
⎨ n − i i
, 1 ≤ i ≤ n,
Hn, i : nαβ−1
⎪
⎩0,
0 i ≤ n.
4.3
Then, it can be easily seen that problem 4.1, 4.2 is equivalent to problem 3.1, 3.2.
We find that H∞ i : limn→∞ Hn, i 0 for any fixed i ≥ 0.
It is known that
n
n
n − jα−1 j β−1
lim
Hn, j lim
Bα, β,
4.4
n→∞
n→∞
nαβ−1
j1
j1
where Bα, β is the well-known Beta function at α, β defined by
1
Bα, β : tα−1 1 − tβ−1 dt < 1.
4.5
0
Using the nonnegativity of Hn, i, 0 ≤ i ≤ n, and Lemma 5.2 we have that
n
n
Hn, j lim
Hn, j Bα, β < 1.
O ≤ V W lim lim
N→∞
n→∞
n→∞
jN
4.6
j0
Now, one can easily see that for the sequences h : hnn≥0 and H : Hn, i0≤i≤n all
of the conditions of Theorem 3.1 are satisfied. Thus, by Theorem 3.1 we get that the solution
z : znn≥1 of the initial value problem 4.1, 4.2 satisfies
lim zn n→∞
h∞
.
1 − Bα, β
On the other hand, we know see 23 that
n−N
n−N
n − jα−1 j β−1
Hn, j lim lim
Bα, β /
0,
lim lim
n→∞
n→∞
N→∞
N→∞
nαβ−1
jN
jN
and hence in 20, Theorem 3.1 is not applicable.
4.7
4.8
8
Advances in Difference Equations
Example 4.2. Let m ≥ 1 and 0 < τ1 < · · · < τm be given integers, and assume Ki /
0 if i ∈
{τ1 , . . . , τm } and Ki 0 if i ∈ Z \ {τ1 , . . . , τm }. Then,
n
Kn − ixi i−∞
∞
Kixn − i i0
m
Kτk xn − τk ,
n ∈ Z .
4.9
k1
Thus, 3.7 reduces to the delay difference equation
Δxn Axn m
Kτk xn − τk gn,
n ∈ Z ,
4.10
k1
and for any sequence ϕ : Z− →R, ϕ ∈ S holds.
Since Kn 0 for any large enough n, r : limn→∞ |Kn|1/n 0, moreover the function
G : 0, ∞→0, ∞ defined in 3.13 satisfies
⎧m
⎪
⎨ τk Kτk μ−τk 1 ,
Gμ k1
⎪
⎩
∞,
if μ > 0,
4.11
if μ 0.
Let μ1 > 0 be the unique value satisfying
Gμ1 m
−τ 1
τk Kτk μ1 k 1.
4.12
k1
Now, statementα of Theorem 3.3 is applicable and so the next statement is valid.
Proposition 4.3. For an A ∈ R there is a λ0 ∈ R \ {0} such that
λ0 A 1 m
k
K τk λ−τ
0 ,
k1
m
−1 −τ
λ0 τk K τk λ0 k < 1
4.13
k1
hold if and only if either
A > μ1 − 1 −
m
k
K τk μ−τ
1
4.14
k1
or
A < −μ1 − 1 −
m
k1
is satisfied.
k
−1τk Kτk μ−τ
1
4.15
I. Győri and L. Horváth
9
0. Then μ1 l|Kl|1/l1 , moreover 4.14
Now, let m 1, τ1 : l ∈ {1, . . .} and Kl /
and 4.15 reduce to
−l/l1
1/l1
− 1 − Kl lKl
,
A > lKl
4.16
1/l1
−l/l1
A < − lKl
− 1 − −1l Kl lKl
.
If especially l 1 and K1 ≥ 0, then μ1 K1, moreover 4.14 and 4.15 are equivalent to
the condition
A/
K1 − 1 − K1 − K1 − 1 K1 −1.
4.17
Example 4.4. Let q ∈ R \ {0} and Ki : qi , i ∈ Z . Then, 3.7 has the following form:
Δxn Axn n
qn−j xj gn,
n ∈ Z .
4.18
j−∞
It is clear that r : limn→∞ |qn |1/n |q|, and the function G defined in 3.13 is given by
⎧
|q|
⎪
⎨
2 , if μ > |q|,
Gμ 4.19
μ − |q|
⎪
⎩
∞,
if μ |q|,
moreover μ1 |q| |q| is the unique positive root of Gμ 1.
Thus statement α in Theorem 3.3 is applicable and as a corollary of it we obtain the
following.
Proposition 4.5. There is a λ0 ∈ R \ {0} such that 3.10 and 3.11 hold with the sequence Ki qi ,
i ∈ Z , if and only if either
|q| |q|
4.20
A > |q| |q| − 1 −
|q| |q| − q
or
A < − |q| |q| − 1 −
|q| |q|
.
|q| |q| q
4.21
Example 4.6. Let c ∈ 1, ∞ \ Z and let Ki : ci , i ∈ Z . Here ci is the extended binomial
coefficient, that is
cc − 1 · · · c − i − 1
c
:
4.22
, i ∈ Z .
i
i!
In this case, r limn→∞ nc 1/n 1 and by using the well-known properties of the
binomial series, we find
∞
1
1 c−1
c
Gμ i
μ−i1 2 c 1 , if μ ≥ 1.
4.23
i
μ
μ
i1
Thus, Gr G1 c2c−1 > 1, therefore by statement α of Theorem 3.3 we get the following.
10
Advances in Difference Equations
Proposition 4.7. There is a λ0 ∈ R \ {0} such that 3.10 and 3.11 hold with the sequence Ki ci , i ∈ Z , if and only if either
1 c
A > μ1 − 1 − 1 μ1
4.24
1 c
A < −μ1 − 1 − 1 −
,
μ1
4.25
or
where μ1 is the unique positive solution of the equation
1
1 c−1
c
1
1.
μ
μ2
4.26
Example 4.8. Let α > 3 and Ki : 1/2iα , i ≥ 1, and K0 : 0. Then, 3.7 reduces to the special
form
Δxn Axn n−1
1
α xj gn,
j−∞ 2n − j
n ∈ Z .
4.27
It is not difficult to see that r 1/2nα 1/n 1,
Gμ ∞
i1
1 −i1
μ
,
2iα−1
μ ≥ 1,
4.28
and G1 < 1. From statement α of Theorem 3.3 we have the following.
Proposition 4.9. There is a λ0 ∈ R \ {0} such that 3.10 and 3.11 hold with the sequence Ki 1/2iα , i ≥ 1, if and only if either
A>
∞
1
1
ςα
α
2i
2
i1
4.29
or
A < −2 −
∞
−1i
i1
2iα
−2 −
1
2α−1
−1
∞
1
1
−2
−
−
1
ςα,
2iα
2α−1
i1
where ς is the well-known Riemann function.
5. Proofs of the main theorems
5.1. Proof of Theorem 3.1
To prove Theorem 3.1 we need the next result from 20.
4.30
I. Győri and L. Horváth
11
Theorem A. Let us consider the initial value problem 3.1, 3.2. Suppose that there are M, N ∈
Z , M < N such that
n Hn, i
ρ sup
5.1
< 1,
n≥N iM
sup
M Hn, i < ∞.
5.2
n≥M i0
Assume also that supn≥0 |hn| < ∞. Then, there is a nonnegative matrix K ∈ Rd×d , independent of h
and z0 , such that the solution z of 3.1, 3.2 satisfies
zn ≤ K sup hm z0 ,
n ≥ 0.
5.3
m≥0
Now, we prove some lemmas.
Lemma 5.1. The hypotheses of Theorem 3.1 imply that the hypotheses of Theorem A are satisfied, and
hence the solution z : znn≥0 of 3.1, 3.2 is bounded.
Proof. Let > 0 be such that ρW E < 1. This can be satisfied because ρW < 1. Then, there
is an M0 ≥ 0 for which
n Hn, j < W E,
lim sup
n→∞
5.4
jM0
and hence for an M ≥ M0 , we have
n Hn, j < W E,
n ≥ M 1.
5.5
jM0
Thus,
n n Hn, j ≤
Hn, j < W E,
jM
n ≥ M 1,
5.6
jM0
therefore,
sup
n Hn, j ≤ W E.
5.7
n≥M1 jM
But the matrices are nonnegative in the above inequality, thus
ρ
n Hn, j
sup
n≥M1 jM
≤ ρW E < 1,
5.8
12
Advances in Difference Equations
and this shows 5.1. Since condition H2 holds, we get
lim
n→∞
M M Hn, j H∞ j,
j0
5.9
j0
therefore,
sup
M Hn, j < ∞,
5.10
n≥M j0
thus 5.2 is satisfied.
In the next lemma we give an equivalent form of H3 .
Lemma 5.2. Let H : Hn, i0≤i≤n be a sequence of real d by d matrices which satisfies H2 . Then,
there exists a real d by d matrix V such that
n
Hn, j − V O
lim sup
n→∞ jN
lim
N→∞
5.11
if and only if
lim
N→∞
n
lim
n→∞
5.12
Hn, j
jN
is finite. In both cases
V lim
lim
n→∞
N→∞
n
lim
Hn, j
n→∞
jN
n
Hn, j −
j0
∞
H∞ j.
5.13
j0
If H satisfies H4 too, and 5.11 holds, then ρV < 1.
Proof. First we show that
lim
N→∞
lim
n→∞
n
5.14
Hn, j
jN
is finite if and only if
lim
n
n→∞
is finite, and in both cases
lim
N→∞
lim
n→∞
n
jN
Hn, j
5.15
Hn, j
j0
lim
n→∞
n
j0
Hn, j −
∞
j0
H∞ j.
5.16
I. Győri and L. Horváth
13
These come from H2 , since
n
Hn, j j0
N
n
Hn, j j0
n ≥ N 1, N ≥ 0.
Hn, j,
5.17
jN1
Suppose V is a real d by d matrix. Then, by H2 for every N ≥ 0
n
lim sup
Hn, j − V n→∞ jN1
max
lim sup
n→∞
n
N
Hn, j −
j0
H∞ j − V, −lim inf
n
n→∞
j0
Hn, j N
j0
j0
n
∞
5.18
H∞ j V
,
and hence
n
Hn, j − V lim sup lim sup
n→∞ jN1
N→∞
max
lim sup
n→∞
n
∞
Hn, j −
j0
H∞ j − V, −lim inf
n→∞
j0
Hn, j j0
5.19
H∞ j V
.
j0
Now, suppose that 5.11 holds. Then by 5.19, either
V lim sup
n→∞
−lim inf
n→∞
n
n
∞
Hn, j −
j0
H∞ j,
j0
Hn, j j0
∞
5.20
H∞ j V ≤ O,
j0
or
V lim inf
n→∞
lim sup
n→∞
n
n
Hn, j −
∞
j0
H∞ j,
j0
Hn, j −
j0
∞
5.21
H∞ j − V ≤ O.
j0
Both of the previous cases implies that
lim sup
n→∞
n
Hn, j ≤ lim inf
n→∞
j0
n
Hn, j
5.22
j0
which shows that
lim
n→∞
n
j0
Hn, j
5.23
14
Advances in Difference Equations
is finite and
n
V lim
n→∞
Hn, j −
∞
j0
H∞ j.
5.24
j0
As we have seen, this is equivalent with 5.12. If 5.12 is true or equivalently
n
lim
n→∞
5.25
Hn, j
j0
is finite, then by 5.19
V : lim
n→∞
n
Hn, j −
∞
j0
H∞ j
5.26
j0
satisfies 5.11. ρV < 1 follows from H4 . The proof is now complete.
Lemma 5.3. The hypotheses of Theorem 3.1 imply that
−1
cz : I − V ∞
H∞ jzj h∞
5.27
j0
is the only vector satisfying the equation
cz ∞
H∞ jzj V cz h∞.
5.28
j0
Proof. Since ρV ≤ ρW < 1 the matrix I − V is invertible, which shows the uniqueness part of
the lemma. On the other hand, by Lemma 5.1 we have that z is a bounded sequence, and hence
∞
j0 H∞ zzj is finite. Thus, cz is well defined and satisfies 5.28. The proof is complete.
Lemma 5.4. The vector defined by 5.27 satisfies the relation
cz N
Hn, jzj j0
n
Hn, jcz hn gn, N
5.29
jN1
for any n > N ≥ 0, where the sequence gn, N ∈ Rd , n > N, satisfies
lim
N→∞
lim gn, N
n→∞
0.
5.30
Proof. Let n > N ≥ 0 be arbitrarily fixed and
gn, N : N
j0
∞
n
H∞ jzj V −
Hn, j cz h∞−hn.
H∞ j−Hn, j zj
jN1
jN1
5.31
I. Győri and L. Horváth
15
But under the hypotheses of Theorem 3.1 we find
N
∞ H∞ jzj
lim supgn, N ≤
lim H∞ j − Hn, jzj n→∞
j0
n→∞
jN1
n
Hn, jcz lim h∞ − hn
lim supV −
n→∞
n→∞
jN1
∞ n
H∞ jzj lim supV −
Hn, jcz , N ≥ 0.
n→∞
jN1
jN1
5.32
Now, by Lemma 5.2
∞ n
H∞ jzj lim lim supV −
lim sup lim supgn, N ≤ lim
Hn, jcz 0,
N→∞
N→∞
n→∞
n→∞
N→∞
jN1
jN1
5.33
and hence 5.30 holds. On the other hand, it can be easily seen that by the above definition of
gn, N the relation 5.29 also holds. The proof is complete.
Now, we prove Theorem 3.1.
Proof. Let n > N ≥ 0 be arbitrarily fixed. Then, 3.1 can be written in the form
zn 1 N
Hn, izi i0
n
Hn, izi hn.
5.34
iN1
Subtracting 5.29 from the above equation, we get
n
zn 1 − cz Hn, i zi − cz − gn, N,
n > N ≥ 0.
5.35
iN1
On the other hand, by Lemma 5.1, z : znn≥0 is bounded and hence
η : lim supzn − cz 5.36
n→∞
is finite. Let > 0 be arbitrarily fixed and e : 1, . . . , 1T . Then there is an N0 ≥ 0 such that
zn − cz ≤ η e, n ≥ N0 .
5.37
Thus, 5.35 yields
n zn 1 − cz ≤
Hn, iη e gn, N,
n > N ≥ N0 .
5.38
iN1
From this it follows:
n Hn, iη e lim supgn, N,
η lim supzn − cz ≤ lim sup
n→∞
n→∞
iN1
n→∞
N ≥ N0 . 5.39
16
Advances in Difference Equations
Thus,
η ≤ lim
N→∞
n Hn, i η e lim lim sup gn, N ,
lim sup
n→∞
N→∞
iN1
5.40
n→∞
and hence Lemma 5.4 implies that
η ≤ Wη e.
5.41
Since W is a nonnegative matrix with ρW < 1, we have that I − W−1 ≥ O. Thus,
η ≤ I − W−1 We −→ 0,
−→ 0,
5.42
and hence the proof of Theorem 3.1 is complete.
5.2. Proof of Theorem 3.3
Theorem 3.3 will be proved after some preparatory lemmas.
In the next lemma, we show that 3.7 can be transformed into an equation of the form
3.1 by using the transformation
zn λ−n
0 xϕn,
n ∈ Z .
5.43
Lemma 5.5. Under the conditions of Theorem 3.3, the sequence z : Z →R defined by 5.43 satisfies
3.1, where the sequences H : {n, i | 0 ≤ i ≤ n}→R and h : Z →R are defined by
Hn, i : −λ−1
0
hn :
λ−1
0
n
λ−i
0
i0
∞
−j
0 ≤ i ≤ n,
Kjλ0 ,
jn−i1
−1
5.44
Ki − jϕj gi
ϕ0,
n ∈ Z .
5.45
j−∞
Proof. Let z : znn≥0 be defined by 5.43. Then,
xϕn λn0 zn,
n ∈ Z .
5.46
Thus,
n
n1
− λn0 zn,
Δxϕn λn1
zn 1 − zn λn1
0 zn 1 − λ0 zn λ0
0
n ∈ Z . 5.47
On the other hand, from 3.7 it follows:
Δxϕn Axϕn n
j0
Kn − jxϕj −1
j−∞
Kn − jϕj gn,
n ∈ Z .
5.48
I. Győri and L. Horváth
17
Thus,
−n1
Δxϕn λ−1
0 − 1 zn
−n1
n
−n1
Axϕn λ−1
Kn − jxϕj
0 − 1 zn λ0
Δzn λ0
λ0
j0
−n1
λ0
−1
−n1
Kn − jϕj λ0
gn,
5.49
n ∈ Z ,
j−∞
and hence
n
j
−n1
−1
Δzn λ−1
A
λ
−
1
zn
λ
Kn − jλ0 zj
0
0
0
j0
−1
−n1
λ0
Kn − jϕj 5.50
−n1
λ0
gn,
n∈Z .
j−∞
But
n
Δzi zn 1 − z0 zn 1 − ϕ0,
n ∈ Z .
5.51
i0
Therefore,
i
n
n j
−i1
−1
zn 1 λ−1
zi λ0
Ki − jλ0 zj hn,
0 A λ0 − 1
i0
n ∈ Z ,
5.52
i0 j0
where h : Z →R is defined in 5.45. By interchanging the order of the summation we get
i
n −i1
λ0
j
Ki − jλ0 zj i0 j0
n−j
n
n n j
−i1
λ0
Ki − jλ0 zj λ−1
λ−i
0
0 Kizj
j0 ij
λ−1
0
n n−i
j0 i0
−j
λ0 Kjzi,
5.53
n ∈ Z .
i0 j0
This and 5.52 yield
zn 1 n
λ−1
0 A
λ−1
0
−1
λ−1
0
i0
n−i
−j
λ0 Kj
zi hn,
n ∈ Z .
5.54
j0
By using the definition λ0 , we have
λ−1
0
A 1 − λ0 n−i
j0
−j
λ0 Kj
−λ−1
0
∞
jn−i1
−j
λ0 Kj,
0 ≤ i ≤ n,
5.55
18
Advances in Difference Equations
and hence
zn 1 −
λ−1
0
n
∞
−j
λ0 Kj
zi hn,
n ∈ Z .
5.56
i0 jn−i1
But by using the definition of Hn, i in 5.44 the proof of the lemma is complete.
In the next lemma, we collect some properties of the function G defined in 3.13.
0 for some n0 ≥ 1 and
Lemma 5.6. Let K : Z →R be a sequence such that Kn0 /
1/n
r : lim supKn
< ∞.
5.57
n→∞
Then, the function G defined in 3.13 has the following properties.
a The series of functions
∞ iKiμ−i1 ,
μ>0
5.58
i1
is convergent on r, ∞ and it is divergent on 0, r.
b limμ→∞ Gμ 0
c limμ→r Gμ Gr
d G is strictly decreasing.
e If Gr ≥ 1, then the equation Gμ 1 has a unique solution.
Proof. a The root test can be applied. b The series of functions 5.58 is uniformly convergent
on a, ∞ for every a > r, and this, together with limμ→∞ μ−i1 0 i ≥ 1, implies the result.
c If Gr is finite, then the series of functions 5.58 is uniformly convergent on r, ∞, hence
G is continuous on r, ∞. Suppose now that Gr ∞ and r > 0. Let c > 0 be fixed and let
i0 ∈ Z such that
i0 iKir −i1 > c.
5.59
i1
Since
lim
μ→r
i0 i0 iKiμ−i1 iKir −i1 > c,
i1
5.60
i1
there is a δ > 0 such that
Gμ ≥
i0 iKiμ−i1 > c,
5.61
i1
whenever μ ∈ r, r δ, and this shows limμ→r Gμ ∞. Finally, we consider the case r 0. Then, limμ→r Gμ ∞ follows from the condition Kn0 /
0. d The series of functions
5.58 can be differentiated term-by-term within r, ∞, and therefore G μ < 0, μ ∈ r, ∞.
Together with c this gives the claim. e We have only to apply d, c, and b. The proof is
complete.
I. Győri and L. Horváth
19
We are now in a position to prove Theorem 3.3.
Proof. a Let Kn0 0 for all n ≥ 1. Then, it is easy to see that there is a λ0 ∈ R \ {0} such
that 3.10 holds if and only if 3.15 is true, and in this case 3.11 is also satisfied. Suppose
Kn0 /
0 for some n ≥ 1. Let r be finite. By the root test, the series
∞
∞
Kiμ−i ,
i0
−1i Kiμ−i ,
5.62
μ>0
i0
are convergent for all μ > r. Moreover, it can be easily verified that the series
∞
∞
Kir −i ,
i0
−1i Kir −i
5.63
i0
are absolutely convergent, whenever Gr is finite. Define the functions Fi : D→R i 1, 2 by
∞
F1 μ : μ − 1 − A −
Kiμ−i ,
i0
F2 μ : μ 1 A ∞
5.64
i
−i
−1 Kiμ ,
i0
where D : r, ∞ if Gr is finite and D : r, ∞, otherwise. The series of functions in 5.64
are uniformly convergent on a, ∞ for every a ∈ D, hence F1 and F2 are continuous. Further,
F1 μ 1 ∞
iKiμ−i1 ≥ 1 −
i1
F2 μ
1−
∞
∞ iKiμ−i1 1 − Gμ,
μ > r,
i1
i
−1 iKiμ
−i1
∞ ≥1−
iKiμ−i1 1 − Gμ,
i1
5.65
μ > r.
i1
Let μ2 : μ1 if Gr ≥ 1, and let μ2 : r if Gr < 1. It now follows from the previous inequalities
and Lemma 5.6d that
Fi μ > 0,
μ > μ2 ,
i 1, 2,
5.66
and hence Fi i 1, 2 is strictly increasing on μ2 , ∞. It is immediate that limμ→∞ Fi μ ∞ i −i
1, 2. If 3.10 is hold for some λ0 ∈ R \ {0}, then the convergence of the series ∞
i0 Kiλ0
implies r < ∞. Suppose r < ∞. It is simple to see that there is a λ0 ∈ R \ {0} satisfying 3.10
if and only if either F1 λ0 0 in case λ0 > 0 or F2 −λ0 0 in case λ0 < 0. Moreover, the
existence of a λ0 ∈ R \ {0} satisfying 3.11 is equivalent to either |λ0 | > μ1 in case Gr ≥ 1
or |λ0 | ≥ r in case Gr < 1. Now, the result follows from the properties of the functions
Fi i 1, 2. The proof of a is complete. b In virtue of Lemma 5.5 the proof of the theorem
will be complete if we show that the sequences H and h satisfy the conditions H2 –H5 in
−i
Section 3. Since the series ∞
i0 Kiλ0 is convergent,
H∞ i : lim Hn, i lim
n→∞
n→∞
−
λ−1
0
∞
jn−i1
−j
λ0 Kj
0,
i ∈ Z ,
5.67
20
and hence
In fact
Advances in Difference Equations
n
i0 |H∞ i| 0. Thus H2 holds. Now, let n ≥ N ≥ 0 and consider
iN |Hn, i|.
∞
n n ∞
−j
Hn, i λ0 −1 λ0 Kj
jn−i1
iN
iN
n ∞
−j
λ0 −1
λ0 |Kj|
≤
5.68
jn−i1
iN
n−N1
∞ λ0 −1 λ0 −j Kj.
l1
jl
Thus,
lim sup
n→∞
n ∞ ∞ Hn, i ≤ λ0 −1
λ0 −j Kj
iN
l1 jl
j
∞ −1 −j λ0 Kj
λ0 −1
λ0 5.69
j0 l1
∞ −j j λ0 Kj < 1.
j1
Thus,
W : lim
N→∞
n lim sup Hn, i
n→∞
5.70
iN
is finite and ρW < 1. In a similar way, one can easily prove that
V λ−1
0
lim
N→∞
lim
n→∞
n
Hn, i
λ−1
0
∞
−j
jλ0 Kj
5.71
j1
iN
is also finite. It is also clear that
h∞ lim hn
n→∞
5.72
is finite. Thus, by Theorem 3.1 we get that the limit
cϕ lim zn lim λ−n
0 xϕn
n→∞
n→∞
5.73
is finite and satisfies the required relation 3.22. The proof is now complete.
Lemma 5.7. Let K : Z →R be a sequence such that Kn0 / 0 for some n0 ≥ 1. Then, there is at most
one λ0 ∈ C \ {0} satisfying 3.10 and 3.11.
I. Győri and L. Horváth
21
Proof. Suppose on the contrary that there exist two different numbers from C \ {0}, denoted by
λ0 and λ1 , such that 3.10 and 3.11 hold. Then,
λj 1 A ∞
i0
Ki
1
λj
i
∞ −i1
iKiλj < 1,
,
j 0, 1,
j 0, 1.
5.74
5.75
i1
It follows from 5.74, the mean value inequality and 5.75 that
i−1 ∞ 1
1
1
− 1 λ0 − λ1 ≤
iKi max
,
|λ0 | |λ1 |
λ0
λ1 i1
−2 i1 ∞ 1
1
1
1
1
1 max , i Ki max , λ − λ λ0 λ1 λ0 λ1 0
1
i1
−2 1
1
1
− 1 ≤ λ0 − λ1 ,
< max , λ0 λ1 λ0 λ1 5.76
and this is a contradiction.
Acknowledgment
This work was supported by Hungarian National Foundation for Scientific Research Grant no.
K73274.
References
1 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA,
2nd edition, 2000.
2 S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New
York, NY, USA, 3rd edition, 2005.
3 W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Application, Academic Press,
Boston, Mass, USA, 1991.
4 V. B. Kolmanovskii, E. Castellanos-Velasco, and J. A. Torres-Muñoz, “A survey: stability and
boundedness of Volterra difference equations,” Nonlinear Analysis: Theory, Methods & Applications,
vol. 53, no. 7-8, pp. 861–928, 2003.
5 R. Medina, “Asymptotic behavior of Volterra difference equations,” Computers & Mathematics with
Applications, vol. 41, no. 5-6, pp. 679–687, 2001.
6 R. Medina and M. I. Gil, “The freezing method for abstract nonlinear difference equations,” Journal of
Mathematical Analysis and Applications, vol. 330, no. 1, pp. 195–206, 2007.
7 Y. Song and C. T. H. Baker, “Admissibility for discrete Volterra equations,” Journal of Difference
Equations and Applications, vol. 12, no. 5, pp. 433–457, 2006.
8 R. P. Agarwal and M. Pituk, “Asymptotic expansions for higher-order scalar difference equations,”
Advances in Difference Equations, vol. 2007, Article ID 67492, 12 pages, 2007.
9 L. Berezansky and E. Braverman, “On exponential dichotomy, Bohl-Perron type theorems and
stability of difference equations,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp.
511–530, 2005.
22
Advances in Difference Equations
10 S. Bodine and D. A. Lutz, “Asymptotic solutions and error estimates for linear systems of difference
and differential equations,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 343–
362, 2004.
11 S. Bodine and D. A. Lutz, “On asymptotic equivalence of perturbed linear systems of differential and
difference equations,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1174–1189,
2007.
12 R. D. Driver, G. Ladas, and P. N. Vlahos, “Asymptotic behavior of a linear delay difference equation,”
Proceedings of the American Mathematical Society, vol. 115, no. 1, pp. 105–112, 1992.
13 S. Elaydi, S. Murakami, and E. Kamiyama, “Asymptotic equivalence for difference equations with
infinite delay,” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 1–23, 1999.
14 S. Elaydi, “Asymptotics for linear difference equations. II. Applications,” in New Trends in Difference
Equations, pp. 111–133, Taylor & Francis, London, UK, 2002.
15 J. R. Graef and C. Qian, “Asymptotic behavior of a forced difference equation,” Journal of Mathematical
Analysis and Applications, vol. 203, no. 2, pp. 388–400, 1996.
16 I. Győri, “Sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of
difference equations,” Applied Mathematics and Computation, vol. 36, no. 2, pp. 89–111, 1990.
17 V. Kolmanovskii and L. Shaikhet, “Some conditions for boundedness of solutions of difference
Volterra equations,” Applied Mathematics Letters, vol. 16, no. 6, pp. 857–862, 2003.
18 Z. H. Li, “The asymptotic estimates of solutions of difference equations,” Journal of Mathematical
Analysis and Applications, vol. 94, no. 1, pp. 181–192, 1983.
19 W. F. Trench, “Asymptotic behavior of solutions of Poincaré recurrence systems,” Computers &
Mathematics with Applications, vol. 28, no. 1–3, pp. 317–324, 1994.
20 J. A. D. Applelby, I. Győri, and D. W. Reynolds, “On exact convergence rates for solutions of linear
systems of Volterra difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 12,
pp. 1257–1275, 2006.
21 Ch. G. Philos and I. K. Purnaras, “The behavior of solutions of linear Volterra difference equations with
infinite delay,” Computers & Mathematics with Applications, vol. 47, no. 10-11, pp. 1555–1563, 2004.
22 Ch. G. Philos and I. K. Purnaras, “On linear Volterra difference equations with infinite delay,” Advances
in Difference Equations, vol. 2006, Article ID 78470, 28 pages, 2006.
23 I. Győri and L. Horváth, “Limit theorems for discrete sumsand convolutions,” submitted.