IJMMS 2004:27, 1447–1453
PII. S016117120430801X
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ESTIMATES FOR THE NORMS OF SOLUTIONS OF DIFFERENCE
SYSTEMS WITH SEVERAL DELAYS
RIGOBERTO MEDINA
Received 1 August 2003
We derive explicit stability conditions for time-dependent difference equations with several
delays in C n (the set of n complex vectors) and estimates for the size of the solutions. The
growth rates obtained here are not necessarily decay rates.
2000 Mathematics Subject Classification: 39A10.
1. Introduction. Stability of systems of difference equations with delays has been
discussed by many authors, for example, see Gil’ and Cheng [6], Zhang [11], Elaydi and
Zhang [5], Pituk [10], Agarwal [1], and the references therein.
In the stability literature, one can find two major trends; stability using the first
approximation Lyapunov method and the direct Lyapunov functional method. For this
latter trend, see Zhang and Chen [12], Crisci et al. [4], Lakshmikantham and Trigiante
[7], and Agarwal and Wong [2]. By this method many very strong results are obtained.
But finding Lyapunov functionals is usually a difficult task.
In this note, we consider a class of perturbed difference equations with several delays
and, by means of a Gronwall inequality and the recent estimates for the powers Ak of a
constant matrix A established in Corduneanu [3], we derive explicit stability conditions.
Further, we suppose that the unperturbed linear difference equations have a bounded
growth. Actually, this work is an extension of Medina [8] to time-dependent difference
equations with several delays.
2. Preliminary facts. Let C n be the set of n complex vectors endowed with a norm
· . Let Ak (k = 1, 2, . . .) be n × n-complex matrices.
Consider in C n the equation
uj+1 = Aj uj + fj uj−σ1 , . . . , uj−σp ,
j = 0, 1, . . . ,
(2.1)
where p ≥ 1, σ1 , σ2 , . . . , σp are nonnegative integers such that 0 = σ1 < σ2 < · · · <
σp , σi ∈ Z + , and Z + is the set of nonnegative integers, fj , j = 0, 1, 2, . . . , maps C np
into C n .
We will consider (2.1) subject to the initial conditions
uj = τj ,
j = −σp , −σp + 1, . . . , 0.
(2.2)
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RIGOBERTO MEDINA
It is assumed that there are nonnegative sequences ql (l = 1, 2, 3, . . . , p) such that
p
m
fj uj−σ , . . . , uj−σ ≤
ql (j)uj−σl ,
p
1
j = 0, 1, . . . ,
(2.3)
l=1
and m is a fixed positive real number.
Unlike differential equations, discrete equations with the given initial conditions always have a solution.
In order to establish our main result, we will use the following discrete Gronwall-type
inequality.
Theorem 2.1 [9]. Assume that
z(k) ≤ C +
p
k−1
m
aj (i)z i − σj ,
k ∈ Z +,
(2.4)
i=0 j=1
where m > 0, 0 = σ1 < σ2 < · · · < σp , p ≥ 1, C > 0, aj (k) ≥ 0 for j = 1, 2, . . . , p and
k ∈ Z + , and z(k) ≤ C for k = −σp , −σp + 1, . . . , 0.
(a) If 0 < m < 1 and C ≤ 1, then
z(k) ≤ C
mk
k−1
p
1+
i=0
k ∈ Z +.
aj (i) ,
(2.5)
j=1
(b) If m = 1, then
z(k) ≤ C
k−1
1+
i=0
p
aj (i) ,
k ∈ Z +.
(2.6)
j=1
(c) If m > 1, then
z(k) ≤ 1 − (m − 1)C m−1 ·
C
k−1 p
1/(m−1) ,
j=1 aj (i)
i=0
k ∈ Z +,
(2.7)
provided that
1 − (m − 1)C m−1
p
k−1
aj (i) > 0,
k ∈ Z +.
(2.8)
i=0 j=1
Assumption 2.2. It is assumed that the unperturbed linear difference equation
uj+1 = Aj uj ,
j = 0, 1, . . . ,
(2.9)
has a bounded growth, that is, there exist real constants γ ≥ 1 and α > 0 such that
Φ(j, i) ≤ γαj−i ,
where Φ(k, l) =
k−1
j=l
∀j ≥ i ≥ 0,
Aj , k > l, is the fundamental matrix solution of (2.9).
(2.10)
ESTIMATES FOR THE NORMS OF SOLUTIONS OF DIFFERENCE SYSTEMS . . .
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3. Main results. Now, we are in a position to establish our main results pertaining
to the bounded growth and the zero convergence properties of the solutions of (2.1)
subject to the conditions (2.2).
Theorem 3.1. Assume that conditions (2.3) and (2.10) hold. In addition, assume that
p
∞ α(m−1)i−mσl ql (i) < ∞.
(3.1)
i=0 l=1
Then,
(a) if 0 < m ≤ 1 and C = γτ0 ≤ 1, every solution uj of (2.1), (2.2), such that uj ≤
Cαj for j = −σp , −σp + 1, . . . , 0, satisfies
∞ p
j
uj ≤ αj γ τ0 m exp γ
βl (i) ,
j = 1, 2, . . . ,
(3.2)
i=0 l=1
where βl (i) = α(m−1)i−mσl −1 ql (i),
(b) if m > 1 and
τ0 ≤
η
(m − 1)λγ m
1/(m−1)
,
λ=
p
∞ βl (i),
(3.3)
i=0 l=1
for some η ∈ (0, 1) and C = γτ0 , every solution uj of (2.1), (2.2) such that uj ≤ Cαj
for j = −σp , −σp + 1, . . . , 0, satisfies
uj ≤
γαj
1/(m−1)
(1 − η)
τ0 ,
j = 0, 1, 2, . . . .
(3.4)
Proof. By inductive arguments, we can prove that the unique solution {uj }∞
j=−σp
of (2.1), subject to given initial values u0 = τ0 , u−1 , . . . , u−σp , satisfies
j−1
uj = Φ(j, 0)τ0 +
Φ(j, i + 1)fi u(i−σ1 ) , . . . , u(i−σp ) ,
j ∈ Z +.
(3.5)
i=0
Hence, by conditions (2.3) and (2.10),
j−1 p
m
uj ≤ γαj τ0 + γ
αj−i−1 ql (i)ui−σl .
(3.6)
i=0 l=1
This yields
j−1 p
m
α−j uj ≤ γ τ0 + γ
α−i−1 ql (i)ui−σl .
(3.7)
i=0 l=1
By setting z(j) = α−j uj and βl (i) = α(m−1)i−mσl −1 ql (i), it follows that
z(j) ≤ C + γ
p j−1
βl (i)zm i − σl ,
l=1 i=0
where C = γτ0 and z(j) ≤ C for j = −σp , −σp + 1, . . . , 0.
(3.8)
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RIGOBERTO MEDINA
Case 1. If 0 < m ≤ 1 and C ≤ 1, then by Theorem 2.1(a) it follows that
z(j) ≤ C m
j
j−1 p
1+γ
i=0
j
βl (i) ≤ C m exp γ
l=1
p
∞ βl (i) ,
(3.9)
i=0 l=1
and the proof of Case 1 is complete.
Case 2. If m > 1, proceeding in a similar way to Case 1, we arrive at the inequality
(3.8). Hence, by Theorem 2.1(b), it follows that
z(j) ≤ 1 − (m − 1)γC m−1
C
j−1 p
i=0
1/(m−1) ,
β
(i)
l
l=1
(3.10)
provided that
j−1 p
1 − (m − 1)γC m−1
βl (i) > 0.
(3.11)
i=0 l=1
Let η ∈ (0, 1) be an arbitrary number. We will prove that the condition (3.11) holds for
all τ0 satisfying
τ0 ≤
η
(m − 1)λγ m
1/(m−1)
=: R,
(3.12)
∞ p
where λ = i=0 l=1 βl (i) < ∞.
Indeed, for all such a τ0 , we have
p
p
∞ m−1 j−1
m−1 βl (i) ≤ (m − 1)γ m τ0 βl (i) ≤ η.
(m − 1)γ m τ0 i=0 l=1
(3.13)
i=0 l=1
Thus,
p
m−1 j−1
βl (i) ≥ 1 − η > 0.
1 − (m − 1)γ m τ0 (3.14)
i=0 l=1
Consequently, for all τ0 such that τ0 ≤ R, we have
Cαj τ0 uj ≤ 1/(m−1)
j−1 p
1 − (m − 1)γC m−1 i=0 l=1 βl (i)
γαj
τ0 ,
≤
1/(m−1)
(1 − η)
(3.15)
+
j∈Z .
Hence the proof of Case 2 is complete.
Remark 3.2. We want to point out the explicit dependence of the growth constants
of the perturbed equation (2.1) upon the growth constants of the unperturbed equation
(2.9) and the estimate for the perturbation f . Further, the growth rates obtained here
are not necessarily decay rates.
ESTIMATES FOR THE NORMS OF SOLUTIONS OF DIFFERENCE SYSTEMS . . .
1451
Corollary 3.3. Under the assumptions of Theorem 3.1, with α in the open interval
(0, 1), we have
(i) if 0 < m ≤ 1 and C = γτ0 ≤ 1, every solution uj with sufficiently small initial data
tends to zero as j → ∞,
(ii) if m > 1 and τ0 ≤ R, then the zero solution uj of (2.1), (2.2) is asymptotically
stable.
Indeed, the inequality uj ≤ γαj /(1−η)1/(m−1) τ0 := Kαj τ0 , j ∈ Z + , shows that
for any ε > 0, we can choose a suitable number 0 < δ < min{R, ε/K} and a number N > 0
such that for all k > N and τ0 < δ, we have uj < ε.
Remark 3.4. If Ak ≡ A is a constant matrix, whose spectral radius is less than 1,
then the zero solution of (2.9) is uniformly asymptotically stable. However, this result
cannot be extended to nonautonomous equations (see [7, Theorem 4.4.1]).
4. Special cases. If the system (2.9),
uj+1 = Aj uj ,
(4.1)
has slowly varying coefficients, then the condition (2.10) concerning growth of the
solutions can be avoided in the case
Ak − Aj ≤ qk−j
qk = q−k = const > 0, q0 = 0; j, k = 1, 2, . . . .
(4.2)
On the other hand, Corduneanu [3] established that for any constant matrix A there
exists a constant Γ ≥ 1, independent of the integers j = 0, 1, 2, . . . such that
j
A ≤ Γ ρ j (A),
j = 0, 1, . . . ,
(4.3)
where ρ(A) is the spectral radius of A.
In particular, if A = (aij ) is a triangular constant matrix, then Γ = 1.
Consider in C n the equation
uj+1 = Aj uj + gj ,
j ∈ Z +,
(4.4)
where Aj (j = 0, 1, . . .) are n × n-complex matrices and gj , uj are vectors in C n .
Theorem 4.1. Under condition (4.2), assume that
ρ0 = sup
l=0,1,...
S0 (A) =
Γl ρ Al < 1,
∞
qk ρ0k < 1,
k=0
S1 (A; g) =
∞
k=0
ρ0k gk < ∞,
(4.5)
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RIGOBERTO MEDINA
where Γl and ρ(Al ) have the same meaning as the quantities in (4.3) referring to A. Then
any solution {uj }∞
j=0 of (4.4) satisfies the inequality
ρ0 u0 + S1 (A; g)
.
sup uj ≤
1 − S0 (A)
j=1,2,...
(4.6)
uj+1 − Al uj = Aj − Al uj + gj
(4.7)
Proof. Rewrite (4.4) as
with a fixed integer l. The variation of parameters formula yields
ul+1 = Al+1
l u0 +
l
l−j A j − A l uj + g j .
Al
(4.8)
j=0
It follows from (4.2) and (4.3) that
l
l−j A ql−j uj + gj ul+1 ≤ ρ l+1 u0 +
0
l
j=0
≤ ρ0 u0 +
≤ ρ0 u0 +
max
j=0,1,2,...,l
max
j=0,1,2,...,l
l
j uj A qj + gl−j (4.9)
l
j=0
∞
∞
j uj ·
gj sup Aj .
qj sup Al +
l
j=0
l=0,1,2,...
j=0
l=0,1,2,...
Consequently,
max
j=0,1,2,...,l+1
uj ≤ ρ0 u0 + S0 (A) ·
max
j=0,1,...,l+1
uj + S1 (A; g),
(4.10)
and we infer that
ρ0 u0 + S1 (A; g)
,
sup uj ≤
1 − S0 (A)
j=1,2,...
(4.11)
concluding the proof.
Consider the equation
uj+1 = Aj uj + fj uj ,
j = 0, 1, 2, . . . .
(4.12)
Assume that there are constants ν, µ ≥ 0 such that
fj (u) ≤ νu + µ,
j = 0, 1, 2, . . . .
(4.13)
Γl ρ Al .
(4.14)
Denote
θ0 =
µ
,
1 − ρ0
ρ0 = sup
l=0,1,...
Now we are in a position to formulate the next result of this paper.
ESTIMATES FOR THE NORMS OF SOLUTIONS OF DIFFERENCE SYSTEMS . . .
1453
Theorem 4.2. Under conditions (4.2) and (4.13), assume that
ρ0 < 1,
ψ(A; f ) =
∞
j
qj + ν ρ0 < 1.
(4.15)
j=0
Then any solution {uj }∞
j=0 of (4.12) satisfies the inequality
ρ0 u0 + θ0
.
sup uj ≤
1 − ψ(A; f )
j=1,2,...
(4.16)
Proof. It can be proved in a similar way to Theorem 4.1, so we will omit the proof.
Acknowledgments. I thank the referees of this paper for their careful and insightful critique. This research was supported by Fondecyt Chile under Grant 1.030.460 and
by Dirección de Investigación, Universidad de Los Lagos.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
R. P. Agarwal, Difference Equations and Inequalities, Monographs and Textbooks in Pure
and Applied Mathematics, vol. 155, Marcel Dekker, New York, 1992.
R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Mathematics and
Its Applications, vol. 404, Kluwer Academic Publishers, Dordrecht, 1997.
A. Corduneanu, The asymptotic behaviour of the solution of some difference equations, Bul.
Inst. Politeh. Iaşi. Secţ. I. Mat. Mec. Teor. Fiz. 39 (1993), no. 1–4, 45–52.
M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, Stability of difference Volterra
equations: direct Liapunov method and numerical procedure, Comput. Math. Appl.
36 (1998), no. 10–12, 77–97.
S. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay,
Funkcial. Ekvac. 37 (1994), no. 3, 401–413.
M. I. Gil’ and S. S. Cheng, Stability of a time discrete perturbed dynamical system with delay,
Discrete Dyn. Nat. Soc. 3 (1999), 56–63.
V. Lakshmikantham and D. Trigiante, Theory of Difference Equations, Mathematics in Science and Engineering, vol. 181, Academic Press, Massachusetts, 1988.
R. Medina, Estimates for the norms of solutions of delay difference systems, Int. J. Math.
Math. Sci. 30 (2002), no. 11, 697–703.
V. N. Phat and J. Y. Park, On the Gronwall inequality and asymptotic stability of nonlinear
discrete systems with multiple delays, Dynam. Systems Appl. 9 (2000), no. 2, 309–
321.
M. Pituk, Convergence and uniform stability in a nonlinear delay difference system, Math.
Comput. Modelling 22 (1995), no. 2, 51–57.
S. Zhang, Estimate of total stability of delay difference systems, Comput. Math. Appl. 37
(1999), no. 9, 31–38.
S. Zhang and M. Chen, A new Razumikhin theorem for delay difference equations, Comput.
Math. Appl. 36 (1998), no. 10–12, 405–412.
Rigoberto Medina: Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933,
Chile
E-mail address: rmedina@ulagos.cl