Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 976369, 13 pages
doi:10.1155/2009/976369
Research Article
Total Stability in Nonlinear Discrete Volterra
Equations with Unbounded Delay
Sung Kyu Choi,1 Yoon Hoe Goo,2 Dong Man Im,3
and Namjip Koo1
1
Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea
Department of Mathematics, Hanseo University, Seosan 352-820, South Korea
3
Department of Mathematics Education, Cheongju University, Cheongju 360-764, South Korea
2
Correspondence should be addressed to Namjip Koo, njkoo@math.cnu.ac.kr
Received 15 December 2008; Revised 20 February 2009; Accepted 24 March 2009
Recommended by Elena Litsyn
We study the total stability in nonlinear discrete Volterra equations with unbounded delay, as a
discrete analogue of the results for integrodifferential equations by Y. Hamaya 1990.
Copyright q 2009 Sung Kyu Choi et al. 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 concepts of stability and asymptotic stability introduced by Lyapunov could be called
stabilities under sudden perturbations. The perturbation suddenly moves the systems from
its equilibrium state but then immediately disappears. Stability says that the effect of this
will not be great if the sudden perturbation is not too great. Asymptotic stability states, in
addition, that if the sudden perturbation is not great, the effect of the perturbation will tend
to disappear. In practice, however, the perturbations are not simply impulses, and this led
Duboshin 1940 and Malkin 1944 to consider what they called stability under constantly
acting perturbations, today known as total stability. This says that if the perturbation is not
too large and if the system is not too far from the origin initially it will remain near the
origin. Total stability can be described roughly as the property that a bounded perturbation
has a bounded effect on the solution 1. Many results have been obtained concerning total
stability 1–9.
In 10, Hamaya discussed the relationship between total stability and stability under
disturbances from hull for the integrodifferential equation
xt x t ft,
0
−∞
Ft, s, xt s, xtds,
1.1
2
Abstract and Applied Analysis
where f : R × Rd → Rd is continuous and is almost periodic in t uniformly for x ∈ Rd ,
and F : R × −∞, 0 × Rd × Rd → Rd is continuous and is almost periodic in t uniformly for
s, x, y ∈ R∗ −∞, 0 × Rd × Rd . He showed that for a periodic integrodifferential equation,
uniform stability and stability under disturbances from hull are equivalent. Also, he showed
the existence of an almost periodic solution under the assumption of total stability in 11.
Song and Tian 12 studied periodic and almost periodic solutions of discrete Volterra
equations with unbounded delay of the form
0
B n, j, x n j , xn ,
xn 1 fn, xn n ∈ Z ,
1.2
j−∞
where f : Z × Rd → Rd is continuous in x ∈ Rd for every n ∈ Z, and for any j, n ∈ Z j ≤ n,
B : Z × Z × Rd × Rd → Rd is continuous for x, y ∈ Rd . They showed that under some suitable
conditions, if the bounded solution of 1.2 is totally stable, then it is an asymptotically almost
periodic solution of 1.2, and 1.2 has an almost periodic solution. Also, Song 13 proved
that if the bounded solution of 1.2 is uniformly asymptotically stable, then 1.2 has an
almost periodic solution.
Choi and Koo 9 investigated the existence of an almost periodic solution of 1.2 as
a discretization of the results in 10. The purpose of this paper is to study the total stability
for the discrete Volterra equation of the form
0
B n, j, x n j , xn hn, xn .
xn 1 fn, xn 1.3
j−∞
To do this, we will employ to change Hamaya’s results in 2 for the integrodifferential
equation
xt x t ft,
0
−∞
xt ,
Ft, s, xt s, xtds ht,
1.4
into results for the discrete Volterra equation 1.3.
2. Preliminaries
We denote by Z, Z , Z− , respectively, the set of integers, the set of nonnegative integers, and
the set of nonpositive integers. Let Rd denote d-dimensional Euclidean space.
Definition 2.1 see 12. A continuous function f : Z × Rd → Rd is said to be almost periodic
in n ∈ Z uniformly for x ∈ Rd if for every ε > 0 and every compact set K ⊂ Rd , there
corresponds an integer N Nε, K > 0 such that among N consecutive integers there is
one, here denoted by p, such that
f n p, x − fn, x < ε
for all n ∈ Z, uniformly for x ∈ Rd .
2.1
Abstract and Applied Analysis
3
Definition 2.2 see 12. Let B : Z×Z∗ → Rd be continuous for x, y ∈ Rd , for any n ∈ Z, j ∈ Z− ,
where Z∗ Z− × Rd × Rd . Bn, j, x, y is said to be almost periodic in n uniformly for j, x, y ∈ Z∗
if for any ε > 0 and any compact set K ∗ ⊂ Z∗ , there exists a number l lε, K ∗ > 0 such that
any discrete interval of length l contains a τ for which
B n τ, j, x, y − B n, j, x, y ≤ ε
2.2
for all n ∈ Z and all j, x, y ∈ K ∗ .
For the basic results of almost periodic functions, see 8, 14, 15.
Let l− Rd denote the space of all Rd -valued bounded functions on Z− with
φ supφ j < ∞
2.3
j∈Z−
for any φ ∈ l− Rd .
Consider the discrete Volterra equation with unbounded delay of the form
xn 1 fn, xn 0
B n, j, x n j , xn hn, xn ,
n ∈ Z ,
2.4
j−∞
under certain conditions for f, B, and h see below. We assume that, given φ ∈ l− Rd , there
is a solution x of 1.3 such that xn φn for n ∈ Z− , passing through 0, φ0 , φ0 ∈ l− Rd .
Let K be any compact subset of Rd such that φj ∈ K for all j ≤ 0 and xn ∈ K for all
n ≥ 1.
For any φ, ψ ∈ l− Rd , we set
ρq φ, ψ
,
q
q0 2 1 ρq φ, ψ
∞
ρ φ, ψ 2.5
where ρq φ, ϕ max−q≤j≤0 |φj − ψj|, q ≥ 0. Then ρ defines a metric on the space l− Rd .
Note that the induced topology by ρ is the same as the topology of convergence on any finite
subset of Z− 12.
In view of almost periodicity, for any sequence nk ⊂ Z with nk → ∞ as k → ∞,
there exists a subsequence nk ⊂ nk such that
fn nk , x −→ gn, x
2.6
uniformly on Z × S for any compact set S ⊂ Rd ,
B n nk , n l nk , x, y −→ D n, n l, x, y
2.7
4
Abstract and Applied Analysis
uniformly on Z × S∗ for any compact set S∗ ⊂ Z∗ . We define the hull
H f, B g, D : 2.6 and 2.7 hold for some sequence
nk ⊂ Z with nk → ∞ as k → ∞}.
2.8
Note that f, B ∈ Hf, B and for any g, D ∈ Hf, B, we can assume the almost periodicity
of g and D, respectively 12.
3. Main Results
We deal with the discrete Volterra equation with unbounded delay of the form
0
B n, j, x n j , xn hn, xn ,
xn 1 fn, xn n ∈ Z .
3.1
j−∞
Throughout this paper we assume the following.
H1 f : Z × Rd → Rd is continuous in x ∈ Rd for every n ∈ Z and is almost periodic in
n ∈ Z uniformly for x ∈ Rd .
H2 B : Z × Z∗ → Rd is continuous in x, y ∈ Rd for any n ∈ Z, j ∈ Z− and is almost
periodic in n ∈ Z uniformly for j, x, y ∈ Z∗ Z− × Rd × Rd . Moreover, for any ε > 0
and any τ > 0 there exists a number M Mε, τ > 0 such that
−M B n, j, x n j , xn ≤ ε
3.2
j−∞
for all n ∈ Z whenever |xj| ≤ τ for all j ∈ Z− .
H3 h : Z × l− Rd → Rd is continuous in φ ∈ l− Rd for every n ∈ Z. xn ∈ l− Rd is
defined as xn j xn j for j ∈ Z− . Furthermore, for any r > 0, there exists a
function αr : Z → Rd with the property that αr n → 0 as n → ∞ and
h n, φ ≤ αr n
3.3
whenever |φj| ≤ r for all j ∈ Z− .
H4 Equation 3.1 has a bounded solution un un, φ0 defined on Z , through
0, φ0 , φ0 ∈ l− Rd such that for some 0 ≤ M < ∞,
u n, φ0 ≤ M.
3.4
Abstract and Applied Analysis
5
Note that for any g, D ∈ Hf, B, Dn, j, x, y satisfies H2 with B D 12. The
limiting equation of 3.1 is defined as
xn 1 gn, xn 0
D n, j, x n j , xn ,
n ∈ Z ,
3.5
j−∞
where g, D ∈ Hf, B. We assume that for any solution vn of 3.5, vn ∈ K for all n ≥ 1,
where K is the above-mentioned compact set in Rd .
Theorem 3.1 see 12. Under the assumptions (H1)–(H4), if un un, φ0 , φ0 ∈ l− Rd is a
bounded solution of 3.1, passing through 0, φ0 and g, D ∈ Hf, B, then the limiting equation
3.5 of 3.1 has a bounded solution on Z.
x is “stable” not only with
Total stability requires that the solution of x t ft,
respect to the small perturbations of the initial conditions, but also with respect to the
perturbations, small in a suitable sense, of the right-hand side of the equation.
Definition 3.2. The bounded solution un of 3.1 is said to be totally stable if for any ε > 0
there exists a δ δε > 0 such that if n0 ≥ 0, ρun0 , xn0 < δ, and pn is a function such that
|pn| < δ for all n ≥ n0 , then ρun , xn < ε for all n ≥ n0 , where xn is any solution of
0
B n, j, x n j , xn hn, xn pn
xn 1 fn, xn 3.6
j−∞
such that xn0 j ∈ K for all j ∈ Z− .
Definition 3.3. A function φ : Z → Rd is called asymptotically almost periodic if it is a sum of an
almost periodic function φ1 and a function φ2 defined on Z which tends to zero as n → ∞,
that is, φn φ1 n φ2 n for all n ∈ Z.
It is known 15 that the decomposition φ φ1 φ2 in Definition 3.3 is unique.
Moreover, φ is asymptotically almost periodic if and only if for any integer sequence τk with τk → ∞ as k → ∞, there exists a subsequence τk ⊂ τk for which φn τk converges
uniformly for n ∈ Z as k → ∞ 15.
Theorem 3.4. Under the assumptions (H1)–(H4), if un is a bounded and totally stable solution of
3.1, then it is asymptotically almost periodic.
Proof. Let nk be an integer sequence with nk → ∞ as k → ∞. Set uk n un nk , k 1, 2, . . . . Then uk n is a solution of
xn 1 fn nk , xn 0
B n nk , j, x n j , xn hn nk , xn 3.7
j−∞
and uk n ∈ K for k 1, 2, . . . . Clearly, uk n is totally stable with the same number ε, δ for
the total stability of un. We can assume that uk n converges uniformly on any finite set
6
Abstract and Applied Analysis
in Z− as k → ∞ by taking a subsequence if necessary. Then there exists a number k1 ε > 0
such that ρuk0 , um
0 < δε whenever k, m ≥ k1 ε. Put
pn fn nm , um n 0
B n nm , j, um n j , um n
j−∞
m
hn nm , um
n − fn nk , u n
−
3.8
0
B n nk , j, um n j , um n − hn nk , um
n .
j−∞
Then um n un nm is a solution of
xn 1 fn nk , xn 0
B n nk , j, x n j , xn hn nk , xn pn
3.9
j−∞
and um n ∈ K for all n ∈ Z. We will show that there exists a number k2 ε > 0 such that
|pn| < δε for all n ≥ 0 whenever k, m ≥ k2 ε.
Note that for all x ∈ K, there exists a number c > 0 such that |x| ≤ c. It is clear that
|uk n| ≤ c and |um n| ≤ c for all n ∈ Z.
In view of H2, there exists a number M Mc, ε > 0 such that
−M B n nm , j, um n j , um n ≤ 1 δε,
5
j−∞
n ∈ Z,
−M B n nk , j, um n j , um n ≤ 1 δε,
5
j−∞
n ∈ Z.
3.10
From the almost periodicity of fn, x and Bn, j, x, y, respectively, there exists a number
k2 ε ≥ k1 ε for which
B n nm , j, um n j , um n − B n nk , j, um n j , um n <
δε
,
5M
3.11
n ∈ Z, j ∈ −M 1, 0,
fn nm , um n − fn nk , um n < 1 δε,
5
n ∈ Z,
3.12
whenever k, m ≥ k2 ε. Since hn, φ → 0 as n → ∞, we obtain that if k, m ≥ k2 ε, then
m
|hn nm , um
n − hn nk , un | <
1
δε,
5
n ∈ Z .
3.13
Abstract and Applied Analysis
7
Then, by 3.10, and 3.11, we have
0 B n nm , j, um n j , um n − B n nk , j, um n j , um n j−∞
≤
−M B n nm , j, um n j , um n j−∞
−M B n nk , j, um n j , um n j−∞
3.14
0
B n nm , j, um n j , um n − B n nk , j, um n j , um n j−M1
<
1
δε
1
δε δε M
5
5
5M
3
δε.
5
Therefore we have
pn ≤ fn nm , um n − fn nk , um n
0 B n nm , j, um n j , um n − B n nk , j, um n j , um n j−∞
m
|hn nm , um
n − hn nk , un |
<
3.15
3
1
1
δε δε δε
5
5
5
δε,
k, m ≥ k2 ε, by 3.12, 3.13, and 3.14. Since uk n is totally stable, we obtain that
ρukn , um
n < ε for all n ≥ 0 if k, m ≥ k2 ε. This implies that for all k, m ≥ k2 ε,
|un nk − un nm | ≤ sup |un nk s − un nm s| < 4ε
−M≤s≤0
3.16
for all ε ≤ 1/4 and all n ≥ 0. It follows that for any nk ⊂ Z with nk → ∞ as k → ∞ there
exists a subsequence nkj ⊂ nk such that un nkj converges uniformly on Z as j → ∞,
that is, un is asymptotically almost periodic. This completes the proof.
Remark 3.5. Hino et al. 5 showed that for the functional differential equation
xt ,
x t ft,
3.17
8
Abstract and Applied Analysis
the solution vt of the limiting equation
x t g t, xt ,
v, g ∈ H u, f
3.18
means
of 3.17 is asymptotically almost periodic if vt is totally stable. Here v, g ∈ Hu, f
that there exists a sequence tk , tk → ∞ as k → ∞, such that ft tk , φ → g t, φ ∈ Hf
uniformly on any compact set in B and ut tk → vt uniformly on any compact interval in
the set of nonnegative real numbers, where the space B is the fading memory space by Hale
and Kato 16.
Theorem 3.6. Assume that (H1)–(H4). If the solution vn of 3.5 satisfying v, g, D ∈
Hu, f, B is totally stable, then the bounded solution un of 3.1 is also totally stable.
Proof. From v, g, D ∈ Hu, f, B, there exists a sequence nk ⊂ Z, nk → ∞ as k → ∞, such
that fn nk , x → gn, x uniformly on Z × K, Bn nk , j, x, y → Dn, j, x, y uniformly on
Z × S∗ × K × K for any compact set S∗ ⊂ Z− , and un nk → vn uniformly on any compact
set in R as k → ∞. Set uk n un nk , k 1, 2, . . . . Then it is clear that uk n is a solution
of
xn 1 fn nk , xn 0
B n nk , j, x n j , xn hn nk , xn ,
3.19
j−∞
such that uk0 j ∈ K for all j ≤ 0, where uk0 j uk 0 j uj nk . Note that for all
x ∈ K, |x| ≤ c for some c > 0. Let xτ be a function such that xτ ∈ K for all τ ≤ n. By H2,
there exists a number M Mc, ε > 0 such that
−M B n nk , j, x n j , xn ≤ 1 δ 1 δ ε ,
5
2 2
j−∞
3.20
where δ· is the number for the total stability of vn. Also, we have
−M D n, j, x n j , xn ≤ 1 δ 1 δ ε
5
2 2
j−∞
3.21
for the same M since Bn nk , j, x, y → Dn, j, x, y. Hence, by the same argument as in the
proof of Theorem 3.4, there exists a positive integer k0 ε such that if k ≥ k0 ε, then
0
fn nk , xn B n nk , j, x n j , xn hn nk , xn j−∞
0
1 ε
−gn, xn −
δ
,
D n, j, x n j , xn < δ
2 2
j−∞
ρ
uk0 , v0
1 ε
δ
.
<δ
2 2
3.22
3.23
Abstract and Applied Analysis
9
Put
0
rn f n nk , uk n B n nk , j, uk n j , uk n
j−∞
0
h n nk , ukn − g n, uk n −
D n, j, uk n j , uk n .
3.24
j−∞
Then uk n is a solution of
0
xn 1 gn, xn D n, j, x n j , xn rn
3.25
j−∞
such that uk0 j ∈ K for j ≤ 0. Note that |rn| < δ1/2δε/2 for n ≥ 0 by 3.22. From
3.22 and the fact that vn is totally stable, we have
1 ε
ρ ukn , vn < δ
2 2
3.26
for all n ≥ 0.
Let m k0 ε. To show that un is totally stable we will show that if n0 ≥ 0,
ρun0 , yn0 < 1/2δε/2, and |pn| < 1/2δε/2 for n ≥ n0 , then ρun , yn < ε for all
n ≥ n0 , where yn is a solution of
xn 1 fn, xn 0
B n, j, x n j , xn hn, xn pn
3.27
j−∞
such that yn0 j ∈ K for all j ≤ 0. Suppose that this is not the case. Then there exists an integer
σ > n0 such that
ρ uσ , yσ ε
ρ un , yn < ε
for σ > n0 ,
for n0 ≤ n < σ.
3.28
We set zn yn nm . Then zn is a solution of
xn 1 fn nm , xn 0
B n nm , j, x n j , xn hn nm , xn pn nm j−∞
3.29
defined on n0 − nm , σ − nm such that zn0 −nm j yn0 j ∈ K for all j ≤ 0. Also, zn is a
solution of
xn 1 gn, xn 0
D n, j, x n j , xn qn,
j−∞
3.30
10
Abstract and Applied Analysis
where
qn fn nm , zn 0
B n nm , j, z n j , zn
j−∞
hn nm , zn pn nm − gn, zn −
0
D n, j, z n j , zn .
3.31
j−∞
Note that |zn| ≤ c for all n ≤ σ − nm , and |pn nm | < 1/2δε/2 for n ≥ n0 − nm . Thus
|qn| < δε/2 for n0 − nm ≤ n ≤ σ − nm . Also, we have
ρun0 , vn0 −nm <
1 ε
δ
,
2 2
1 ε
ρun0 , zn0 −nm ρ un0 , yn0 < δ
2 2
3.32
from 3.26. Thus we obtain
ρvn0 −nm , zn0 −nm ≤ ρvn0 −nm , un0 ρun0 , zn0 −nm < δ
ε
2
.
3.33
Since vn is totally stable, we have
ρvσ−nm , zσ−nm <
ε
.
2
3.34
On the other hand, 3.26 implies that
ρun , vn−nm <
1 ε
δ
,
2 2
n ≥ n0 .
3.35
ε
1
Hence, if n0 ≥ 0, ρun0 , yn0 < δ , and |pn| < 1/2δε/2 for n ≥ n0 , then we obtain
2
2
ρ uσ , yσ ≤ ρuσ , vσ−nm ρvσ−nm , zσ−nm < ε.
3.36
This contradicts 3.28. Therefore ρun , yn < ε for all n ≥ n0 when n0 ≥ 0, ρun0 , yn0 < δ∗ ε
and |pn| < δ∗ ε for all n ≥ n0 , where δ∗ ε 1/2δε/2. Consequently, un is totally
stable.
The following definitions are the discrete analogues of Hamaya’s definitions in 2.
Definition 3.7. The bounded solution un of 3.1 is said to be attracting in Hf, B if there
exists a δ0 > 0 such that for any n0 ≥ 0 and any v, g, D ∈ Hu, f, B, ρvn0 , xn0 < δ0 implies
ρvn , xn → 0 as n → ∞, where xn is a solution of 3.5 such that xn0 j ∈ K for all j ≤ 0.
Definition 3.8. The bounded solution un of 3.1 is said to be totally asymptotically stable if it is
totally stable and there exists a δ0 > 0 and for any ε > 0 there exists an ηε > 0 and a T ε > 0
Abstract and Applied Analysis
11
such that if n0 ≥ 0, ρun0 , xn0 < δ0 and pn is any function which satisfies |pn| < ηε for
n ≥ n0 , then ρun , xn < ε for all n ≥ n0 T ε, where xn is a solution of
xn 1 fn, xn 0
B n, j, x n j , xn hx, xn pn
3.37
j−∞
such that xn0 j ∈ K for all j ≤ 0.
Note that the total asymptotic stability is equivalent to the uniform asymptotic stability
whenever pn ≡ 0.
Theorem 3.9. Under the assumptions (H1)–(H4), if the bounded solution un of 3.1 is attracting
in Hf, B and totally stable, then it is totally asymptotically stable.
Proof. Let δ0 be the number for the attracting of un in Hf, B and let δ0∗ δδ0 /2, where
δ· is the number for the total stability of un. Suppose that un is not totally asymptotically
stable. Then there exists a number ε > 0 with ε ≤ δ0 /4 and exist sequences jk , nk , pk ,
and xk such that jk ≥ 0, nk ≥ jk 2k, ρujk , xjkk < δ0∗ and ρunk , xnkk ≥ ε for all k 1, 2, . . .,
where xk n is a solution of
xn 1 fn, xn 0
B n, j, x n j , xn hx, xn pk n.
3.38
j−∞
such that xjkk j ∈ K for all j ≤ 0 and pk : Z → Rd with |pk n| < min{1/k, δ0∗ } for n ≥ jk . Note
that ρujk , xjkk < δ0∗ and |pk n| < δ0∗ for n ≥ jk . Then we have
1
ρ un , xnk < δ0
2
3.39
for all n ≥ jk and k 1, 2, . . . , since un is totally stable. Also, there exists an integer number
k0 ε > 0 such that if k ≥ k0 ε, then |pk n| < 1/k < δε for all n ≥ jk .
We claim that ρun , xnk ≥ δε on jk k, jk 2k if n ≥ n0 . If we assume that ρun , xnk <
δε on jk k, jk 2k, then ρun , xnk < ε for n ≥ jk 2k since un is totally stable. This
contradicts ρunk , xnkk ≥ ε, k 1, 2, . . ., because nk ≥ jk 2k.
Now, for the sequence jk k, taking a subsequence if necessary, there exists a
v, g, D ∈ Hu, f, B.
If we set yk n xk n jk k, then yk n is the defined on −k, k. There exists a
subsequence of yk n, which we denote by yk n again, and a function yn such that
yk n → yn uniformly on any compact set in R as k → ∞ such that y0 j ∈ K for all j ≤ 0.
Moreover, we can show that yn is a solution of
xn 1 gn, xn 0
D n, j, x n j , xn
j−∞
3.40
12
Abstract and Applied Analysis
such that y0 j ∈ K for all j ≤ 0, by the same method as in 13, Theorem 3.1 . We have
δ
0
δε < ρ unjk k , ynk < ,
2
0 ≤ n ≤ k, k ≥ k0 .
3.41
Then, by letting k → ∞, we obtain
δ0
δε < ρ vn , yn ≤ ,
2
n ≥ 0.
3.42
Since un is attracting in Hf, B, we have ρvn , yn → 0 as n → ∞. This contradicts
ρvn , yn ≥ δε. Hence un is totally asymptotically stable. This completes the proof of the
theorem.
Acknowledgment
The authors are thankful to the anonymous referees for their valuable comments and
corrections to improve this paper.
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