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
Istv ´
an Gy ˝
ori and L ´
aszl ´
oHorv
´
ath
Department of Mathematics and Computing, University of Pannonia, 8200 Veszpr´
em,
Egyetem u. 10, Hungary
Correspondence should be addressed to Istv´
an Gy˝
ori, 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 q2008 I. Gy˝
ori and L. Horv´
ath. 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,Medina5,MedinaandGil6, and Song
and Baker 7; see 8–19for 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
zn1
n
i0
Hn, izihn,n∈Z,1.1
2 Advances in Difference Equations
with the initial condition
z0z0,1.2
where z0∈Rd,Hn, i0≤i≤nand hnn≥0are sequences with elements in Rd×dand 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,infactn≥2m2, the sum in
1.1can be split into three terms
m
i0
Hn, izi
n−m−1
im1
Hn, izi
m
j0
Hn, n −jzn−j,1.3
since
n
in−m
Hn, izi
m
j0
Hn, n −jzn−j.1.4
In 20, Theorem 3.1the middle sum in 1.3contributed nothing to the limit limn→∞zn,
since it was assumed that
lim
m→∞ lim sup
n→∞
n−m
im
Hn, i
O. 1.5
In our case, we split the sum in 1.1only into two terms, and the condition 1.5is not
assumed. In fact, we show an example in Section 4,where1.5does not hold and hence in 20,
Theorem 3.1is 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, 1.1has been proven, we may use it for
the scalar convolution Volterra difference equation with infinite delay,
ΔxnAxn
n
j−∞
Kn−jxjgn,n∈Z,1.6
with the initial condition,
xnϕn,n∈Z−,1.7
where A∈R,andKnn≥0,gnn≥0and ϕnn≤0are real sequences.
Here, Δdenotes the forward difference operator to be defined as usual, that is, Δxn:
xn1−xn,n∈Z.
If we look for a solution xnλn
0,n∈Zλ0∈R\{0}of the homogeneous equation
associated with 1.6, we see that λ0is a root of the characteristic equation
λ01A
∞
i0
Kiλ−i
0.1.8
I. Gy˝
ori and L. Horv´
ath 3
We immediately observe that λ0∈Ris a simple root if
λ0
−1∞
i1
i
Ki
λ0
−i<1.1.9
In the paper 21see also 22,itisshownthatifλ0>0satisfies1.8and 1.9,and
the initial sequence ϕnn≤0is suitable, then for the solution xof 1.6and 1.7the sequence
zn:λ−n
0xn,n∈Zis bounded. Furthermore, some extra conditions guarantee that the
limit z∞:limn→∞znis 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 1.8and 1.9are 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 nonconvolu-
tion 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 1.1in 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 Zbe the set of integers, Z:{n∈Z|n≥0}and Z−:{n∈Z|n≤0}.Rdstands
for the set of all d-dimensional column vectors with real components and Rd×dis the space
of all dby dreal matrices. The zero matrix in Rd×dis denoted by O, and the identity matrix
by I.LetEbe the matrix in Rd×dwhose elements are all 1. The absolute value of the vector
xx1,...,x
dT∈Rdand the matrix AAij 1≤i,j≤d∈Rd×dis defined by |x|:|x1|,...,|xd|T
and |A|:|Aij |1≤i,j≤d, respectively. The vector xand the matrix Ais nonnegative if xi≥0and
Aij ≥0, 1 ≤i, j ≤d, respectively. In this case, we write x≥0andA≥O.Rdcan be endowed
with any norms, but they are equivalent. A vector norm is denoted by · and the norm of
a matrix in Rd×dinduced by this vector norm is also denoted by ·. The spectral radius of
the matrix A∈Rd×dis given by ρA:limn→∞An1/n, which is independent of the norm
employed to calculate it.
A partial ordering is defined on RdRd×dby letting x≤yA≤Bifandonlyify−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,andρA≤ρBif
A, B ∈Rd×dand O≤A≤B.
3. The main results
First, consider the nonconvolutional linear Volterra difference equation
zn1
n
i0
Hn, izihn,n∈Z,3.1
4 Advances in Difference Equations
with initial condition
z0z0.3.2
Here, we assume
H1z0∈Rd,H:Hn, i0≤i≤nand h:hnn≥0are sequences with elements in Rd×d
and Rd, respectively;
H2for any fixed i≥0 the limit H∞i:limn→∞Hn, iis finite and ∞
i0|H∞i|<∞;
H3the matrix
V:lim
N→∞ lim
n→∞
n
jN
Hn, j3.3
is finite;
H4the matrix
W:lim
N→∞ lim sup
n→∞
n
jN
Hn, j
3.4
is finite and ρW<1;
H5the limit h∞:limn→∞hnis finite.
By a solution of 3.1, we mean a sequence z:znn≥0in Rdsatisfying 3.1for any
n∈Z. It is clear that 3.1with initial condition 3.2has 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∈Rdthe unique solution z:
znn≥0of 3.1and 3.2has a finite limit at ∞and it satisfies
lim
n→∞ znI−V−1∞
i0
H∞izih∞.3.5
Under conditions H3and H4
|V|lim
N→∞ lim
n→∞
n
jN
Hn, j
≤W, 3.6
and hence ρW<1 yields ρV<1, thus I−Vis invertible. On the other hand under our
conditions the unique solution z:znn≥0of 3.1and 3.2is a bounded sequence, therefore
∞
i0H∞iziis finite, and 3.5makes 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. Gy˝
ori and L. Horv´
ath 5
with the initial condition
xnϕn,n∈Z−,3.8
where A∈R,K:Z→R,g:Z→R,andϕ:Z−→Rare given.
By a solution of the Volterra difference equation 3.7we mean a sequence x:Z→R
satisfies 3.7for any n∈Z.
In what follows, by Swe will denote the set of all initial sequence ϕ:Z−→Rsuch that
for each n∈Z
−1
j−∞
Kn−jϕj3.9
exists.
It can be easily seen that for any initial sequence ϕ∈S,3.7has exactly one solution
satisfying 3.8.Thisuniquesolutionisdenotedbyxϕ:Z→Rand it is called the solution of
the initial value problem 3.7,3.8.
The asymptotic representation of the solutions of 3.7is given under the next condition.
AThere exists a λ0∈R\{0}for which
λ01A
∞
i0
Kiλ−i
0,3.10
λ0
−1∞
i1
i
Ki
λ0
−i<1.3.11
From the theory of the infinite series, one can easily see that condition Ayields
r:lim sup
n→∞
Kn
1/n 3.12
is finite. Moreover, the mapping G:r, ∞→0,∞,defined by
Gµ:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
i1
i
Ki
µ−i1if µ>r,
∞
i1
i
Ki
r−i1if µr>0,
∞if µr0,
3.13
is real valued on r, ∞. It is also clear see Section 5that if there is an n0≥1 such that
Kn0/
0, and if Gr≥1 then the equation
Gµ13.14
has a unique solution, say µ1.
Now we formulate the following more explicit condition:
Beither Kn0, n≥1, and
1AK0/
0,3.15