IJMMS 2003:48, 3059–3066
PII. S016117120321022X
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ACCURATE SOLUTION ESTIMATES FOR VECTOR
DIFFERENCE EQUATIONS
RIGOBERTO MEDINA
Received 10 October 2002
Accurate estimates for the norms of the solutions of a vector difference equation
are derived. They give us stability conditions and bounds for the region of attraction of the stationary solution. Our approach is based on estimates for the powers
of a constant matrix. We also discuss applications of our main results to partial
reaction-diffusion difference equations and to a Volterra difference equation.
2000 Mathematics Subject Classification: 39A10.
1. Introduction. Let Cn be the set of n-complex vectors endowed with the
Euclidean norm · . Consider the equation
xk+1 = Axk + fk xk ,
k = 0, 1, 2, . . . ,
(1.1)
where A is an n × n complex matrix and fk : Cn → Cn are given functions.
A well-known result of Perron which dates back to 1929 (see [8], [12, page
270], and [10, Theorem 9.14]) states that (1.1) is asymptotically stable provided
that A is stable (i.e., the spectral radius ρ(A) of A is less than 1) and fk (x) =
f(x) = o(x). Clearly, this kind of results for the perturbed equation (1.1) is
purely local. It gives no information about the size of the region of asymptotic
stability and the norms of solutions.
In this paper, we derive accurate estimates for the norms of solutions. They
give us stability conditions for (1.1) and bounds for the region of attraction
of the stationary solution. Our approach is based on recent estimates for the
powers of a constant matrix, namely, Corduneanu [3] established that for any
constant matrix A, there exists a constant N ≥ 1, independent of the integers
k = 0, 1, 2, . . . , such that
k
A ≤ Nρ k (A) for k = 0, 1, 2, . . . .
(1.2)
In particular, if A = (aij ) is a triangular constant matrix, then N = 1. On the
other hand, Gil’ [5, Theorem 1.2.1] established a very sharp estimate for the
powers Ak of a constant matrix A. In order to establish Theorems 2.1 and 2.3,
we use a method introduced by Gil’ and Cheng [7] which deals with perturbed
linear discrete dynamical systems.
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RIGOBERTO MEDINA
We also discuss the applications of our main results to a partial reactiondiffusion difference equation and to the discrete analogous of an integrodifferential equation, respectively.
2. Main results. For a positive number r ≤ ∞, denote the ball Br = {x ∈ Cn :
x ≤ r } and assume that there are constants q, µ ≥ 0 such that
fk (x) ≤ qx + µ
x ∈ Br , k = 0, 1, 2, . . . .
(2.1)
Assume that
0 < ρ(A) < 1
(2.2)
and put θ = N/(1 − ρ(A)).
Now, we are in a position to formulate the main result of this paper.
Theorem 2.1. Under conditions (2.1) and (2.2), let qθ < 1. Then any solution
{xk }∞
k=0 of (1.1) satisfies the inequality
N x0 + µθ
sup xk ≤
1 − qθ
k=1,2,...
(2.3)
N x0 + µθ (1 − qθ)−1 ≤ r .
(2.4)
provided that
Proof. By inductive arguments, it is easy to check that the unique solution
{xk }∞
k=0 of (1.1) under the initial condition x0 is given by
xk = A k x0 +
k−1
Ak−j−1 fj xj ,
k = 0, 1, 2, . . . .
(2.5)
j=0
There are two cases to consider: r = ∞ and r < ∞.
Assuming first that r = ∞, hence
k k−1
Ak−j−1 qxj + µ ,
xk ≤ A x0 +
k = 0, 1, 2, . . . .
(2.6)
j=0
Let Γ = supk≥0 Ak . It follows that
xk ≤ Γ x0 + q
+µ
k−1
j=0
max
i=0,1,...,k−1
k−j−1 A
,
k−1
xi Ak−j−1 j=0
k = 0, 1, 2, . . . .
(2.7)
ACCURATE SOLUTION ESTIMATES FOR VECTOR . . .
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By (1.2) and (2.2), we have
Γ = sup Am ≤ sup Nρ m (A) = N sup ρ m (A) = N.
m≥0
m≥0
(2.8)
m≥0
Furthermore, since 0 < ρ(A) < 1, we obtain
∞
∞
m
A ≤ N
ρ m (A) =
m=0
m=0
N
= θ.
1 − ρ(A)
(2.9)
In view of (2.7) and the condition qθ < 1, it follows that
xk ≤ N x 0 + q
max
i=0,1,...,k−1
∞
∞
j
j
xi A + µ
A ,
j=0
k = 0, 1, 2, . . . .
(2.10)
j=0
Consequently,
max xi ≤ N x0 + qθ max xi + µθ.
i=0,1,...,k
i=0,1,...,k
(2.11)
Hence, it follows that
sup xi (1 − qθ) ≤ N x0 + µθ,
i≥0
N x0 + µθ
.
sup xi ≤
1 − qθ
i≥0
(2.12)
Next, we will consider the case r < ∞.
Define the function

fk (x), x ≤ r ,
k (x) =
f
0,
x > r .
(2.13)
k (x) ≤ qx + µ (k = 0, 1, 2, . . . ) and x ∈ B∞ , then the sequence
Since f
∞
k }k=0 defined by
{x
0 = x0 ,
x
k x
k + f
k , k = 0, 1, . . . ,
x
k+1 = Ax
(2.14)
satisfies the inequality
N x0 + µθ
k ≤
≤r
sup x
1 − qθ
k≥0
(2.15)
provided that (Nx0 + µθ)(1 − qθ)−1 ≤ r .
k (x) coincide on Br . From this, we infer that xk = x
k for
But fk (x) and f
k = 0, 1, 2, . . . , and therefore (2.3) is satisfied, concluding the proof.
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RIGOBERTO MEDINA
Remarks. (a) Under (2.1) with µ = 0, fk (0) = 0 so that {0} is a solution
of (1.1). Under condition qθ < 1, Theorem 2.1 asserts that the trivial solution
is stable and that any initial vector x0 ∈ Br , satisfying the condition x0 ≤
(1 − qθ)/N, belongs to the region of attraction.
(b) If q = 0, then, by (2.4), every solution of (1.1) with the initial vector x0
satisfying Nx0 + µθ ≤ r is bounded.
Corollary 2.2. If A is a triangular matrix and q < 1 − ρ(A), then any
solution {xk }∞
k=0 of (1.1) satisfies
1 − ρ(A) x0 + µ
sup xk ≤
1 − q + ρ(A)
k≥0
(2.16)
−1
≤ r.
1 − ρ(A) x0 + µ 1 − q + ρ(A)
(2.17)
provided that
Theorem 2.3. Suppose that fk : Cn → Cn satisfies the condition
fk (x) ≤ qk x + µ
such that
satisfies
∞
j=0 qj ρ
j
x ∈ Br , k = 0, 1, 2, . . .
(2.18)
(A) = β < ∞. If βN < 1, then any solution {xk }∞
k=0 of (1.1)
N x0 + µθ
sup xk ≤
1 − βN
k=1,2,...
(2.19)
provided that (Nx0 + µθ)(1 − βN)−1 ≤ r .
By using arguments similar to those in Theorem 2.1, the result follows, thus
we will omit the proof.
3. Applications. In this section, we will illustrate our main results by considering a partial difference equation and the discrete analogous of an integrodifferential equation, respectively.
We consider a simple three-level discrete reaction-diffusion equation of the
form
(j+1)
ui
(j)
(j)
(j)
(j)
(j)
= aui−1 + bui + cui+1 + gi + fj ui ,
(3.1)
(j)
defined on Ω = {(i, j) : i = 0, 1, 2, . . . , n + 1; j = 0, 1, 2, . . .}, where g = {gi }
is a complex function defined on Ω and fj : C → C (j = 0, 1, 2, . . .) are given
functions.
ACCURATE SOLUTION ESTIMATES FOR VECTOR . . .
3063
Assume that the side conditions
(j)
u0 = δj ∈ C,
j = 0, 1, 2, . . . ,
(j)
un+1
= γj ∈ C,
j = 0, 1, 2, . . . ,
(0)
= τi ∈ C,
i = 1, 2, . . . ,
ui
(3.2)
are imposed, where τ = col(τ1 , τ2 , . . . , τn ) ∈ Cn . A solution of problem (3.1),
(j)
(3.2) is a discrete function u = {ui }(i,j)∈Ω which satisfies the relations (3.1)
and (3.2).
The existence and uniqueness of solutions of that problem is obvious provided the functions fj are well defined.
With the notation
(j) (j)
(j) u(j) = col u1 , u2 , . . . , un ,
(3.3)
(j)
the sequence {ui }∞
j=0 satisfies the vector equation
u(j+1) = Au(j) + Gj + Fj u(j) ,
j = 0, 1, 2, . . . ,
(3.4)
and the initial condition u(0) = τ, where

··· ··· 0

0
· · · 0

c
· · · 0
,

..
..
.. 
.
.
.

0
a
b
(j)
(j)
Gj = col g1 , . . . , gn + col aδj , 0, . . . , 0, cνj ,
Fj (x) = col fj x1 , . . . , fj xn , x = x1 , . . . , xn .

b

a

0
A=

 ..
.

0
c
b
a
..
.
···
0
c
b
..
.
···
(3.5)
Thus, we can write the considered problem as (1.1) with fj (x) = Fj (x) + Gj .
Assume that there are nonnegative constants q1 and µ1 such that
Fj (x) ≤ q1 x + µ1
x ∈ Br , j = 0, 1, 2, . . . .
(3.6)
In addition, assume that
µ2 =
∞
Gj < ∞.
j=0
Hence, condition (2.1) holds with µ = µ1 + µ2 .
(3.7)
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RIGOBERTO MEDINA
We want to point out that if ac > 0, then the spectral radius ρ(A) is equal
√
to |b| + 2 ac cos(π /(n + 1)) and when ac < 0, then
ρ(A) =
b2 − 4ac cos2
π
.
n+1
(3.8)
Then, as a direct consequence of Theorem 2.1, we get the following theorem.
Theorem 3.1. Assume that
√
(i) ac > 0 and |b| + 2 ac cos(π /(n + 1)) < 1, or
(ii) ac < 0 and b2 − 4ac cos2 (π /(n + 1)) < 1,
(iii) condition (2.1) holds with µ = µ1 + µ2 .
Then, for any x0 = τ such that (Nτ + µθ)/(1 − q1 θ) ≤ r , the unique solu(j)
tion xj = {ui }(i,j)∈Ω of problem (3.1), (3.2) satisfies
(j) Nτ + µθ
sup ui ≤
1 − q1 θ
j=1,2,...
for i = 0, 1, 2, . . . , n + 1.
(3.9)
Remarks. (1) An important class of admissible perturbation functions fj
for (3.1) is the polynomial functions. For example, if we consider
fj xi =
p
j
β
x l + ηi ,
2 +1 i
j
l=0
j = 0, 1, 2, . . . , i = 0, 1, 2, . . . , n,
(3.10)
where p is a nonnegative integer and ηi and βl , l = 0, 1, 2, . . . , p, are nonnegative
real numbers, then we infer that the requirements of Theorem 3.1 are satisfied
p
n
taking q1 = l=0 r βl −1 and µ1 = i=0 ηi .
Indeed, if u(j) ∈ Br , then
p
(j) (j) (j) βl −1
≤
fj u
u u + ηi
i
i
i
l=0
≤
p
(j) r βl −1 ui + ηi
l=0

≤
p

r
βl −1 
(3.11)
(j) ui + ηi .
l=0
Hence,
(j) (j) (j) = fj u
F j u
, . . . , fj un
1




p
p
βl −1  (j) βl −1  (j) 

r
r
≤
u1 + η 1 + · · · +
un + η n
l=0

β
−1
≤  r l u(j) + µ1 .

p
l=0
l=0
(3.12)
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ACCURATE SOLUTION ESTIMATES FOR VECTOR . . .
(2) We want to point out that Theorem 3.1 compares favorably with [11,
Theorems 1 and 2] and [2, Theorems 1–4]. On the other hand, Theorems 2.1
and 2.3 are comparable with the main results in [7]. However, our results,
though different in their approach in general, have very little overlapping with
those in [2, 7, 11]. Other important related results can be found in [1, pages
237–245], [6], [4, pages 196–204], and [9, pages 102–104].
We now consider the analogous of the integrodifferential equation
dx
= Bx(t) +
dt
t
f s, x(s) ds,
t ≥ 0.
0
(3.13)
Consider the discrete equation
xk+1 = Axk +
k
fj xj ,
k ≥ 0,
(3.14)
j=0
where A is an n × n complex matrix and fj : Cn → Cn (j = 0, 1, 2, . . .) are given
functions, satisfying
fj (x) ≤ qj x + µ
x ∈ Br , j = 0, 1, 2, . . . .
(3.15)
Define
S(A) =
∞
k
kρ (A),
ψ(A; f ) =
k=0
∞

k
ρ (A)
k
k=0

qi .
(3.16)
i=0
We are now in a position to establish the next theorem.
Theorem 3.2. Under condition (3.15), assume that
S(A) < ∞,
Nψ(A; f ) < 1.
(3.17)
Then any solution {xk }∞
k=0 of (3.14) satisfies the inequality
N x0 + µNS(A)
sup xk ≤
1 − Nψ(A; f )
k=1,2,...
(3.18)
−1
≤ r.
N x0 + µNS(A) 1 − Nψ(A; f )
(3.19)
provided that
Proof. The variation of parameters formula yields
xk = A k x0 +
k−1
j=0

Ak−j−1 
j

fi xi ,
k = 0, 1, 2, . . . .
i=0
There are two cases to consider: r = ∞ and r < ∞.
(3.20)
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RIGOBERTO MEDINA
Assuming first that (3.15) is valid for r = ∞, hence


j
k−1
k−j−1
A

xk ≤ A k x0 +
qi xi + µ ,
j=0
k = 0, 1, 2, . . . .
(3.21)
i=0
Thus, following the lines of the proof of Theorem 2.3, the result follows.
Acknowledgment. This research was supported by FONDECYT Chile under Grant 1.030.460.
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Rigoberto Medina: Departamento de Ciencias Exactas, Universidad de Los Lagos,
Casilla 933, Osorno, Chile
E-mail address: rmedina@ulagos.cl