Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 804509, 10 pages
doi:10.1155/2012/804509
Research Article
Asymptotic Stability of Differential Equations
with Infinite Delay
D. Piriadarshani1 and T. Sengadir2
1
Department of Mathematics, Hindustan Institute of Technology and Science,
Rajiv Gandhi Salai, Kelambakkam, Chennai 603 103, India
2
Department of Mathematics, Central University of Tamil Nadu, Thiruvarur 610 004, India
Correspondence should be addressed to D. Piriadarshani, piriadarshani@gmail.com
Received 3 January 2012; Accepted 5 April 2012
Academic Editor: Mehmet Sezer
Copyright q 2012 D. Piriadarshani and T. Sengadir. 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.
A theorem on asymptotic stability is obtained for a differential equation with an infinite delay in a
function space which is suitable for the numerical computation of the solution to the infinite delay
equation.
1. Introduction and Preliminaries
In this paper, we study the asymptotic stability of the solutions to the infinite delay differential equation given below:
x t axt ∞
bi xt − τi ,
t ≥ 0,
i1
xθ φθ,
θ ∈ −∞, 0,
under the following assumptions.
i There exists p > 0 with |bi | ≤ pγ −i for all i ∈ N.
ii τi ≤ iτ1 for all i ∈ N.
1.1
2
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The asymptotic stability of a linear infinite delay equation is studied in 1–5 in the
context of abstract phase spaces which includes the space:
φ ∈ C−∞, 0 : sup e
γθ
φθ < ∞, lim e φθ exists .
γθ θ→∞
θ∈−∞,0
1.2
The asymptotic constancy neutral equations are studied in 6. Linear time-invariant systems
with constant point delays are studied in 7 and in 8; a Razumikhin approach is used to
study exponential stability of delay equations. Asymptotic stability and stabilization of linear
delay-differential equations are studied in 9.
In this paper, the phase space Cσ −∞, 0 for the initial function is chosen as follows.
Let mi iτ1 > 0 and βi pγ −i . The space Cσ −∞, 0 is defined as
φ ∈ C−∞, 0 :
∞
i1
βi
sup φθ < ∞ .
1.3
θ∈−mi ,0
Here C−∞, 0 is the set of continuous complex valued functions defined on −∞, 0.
The motivation to consider the above type of phase space is that for numerical
computation of solutions it is enough to know the values of the initial data over a finite
domain at every stage of computation. See 10, 11.
The following definitions and results are well known, see for example 5 or 12.
Definition 1.1. The Kuratowski measure of noncompactness αV of the subset V of a Banach
space X is defined by
αV inf d > 0 : there exists a finite number of sets V1 , V2 , . . . , Vn ,
with diam Vj ≤ d such that V ∪nj1 Vj .
1.4
For a bounded linear operator L : X → Y , |L|α is defined as
|L|α inf{k > 0 : αLV ≤ kαV for all bounded sets V }.
1.5
Proposition 1.2. Let X, Y, Z be Banach spaces and M : X → Y , L : Y → Z be bounded linear
operators. Then, |M ◦ L|α ≤ |M|α |L|α . Further, if M : X → Y is compact, then |M|α 0.
Theorem 1.3. Let X be a Banach space and let A : DA → X be the infinitesimal generator of a
semigroup of operators St : X → X. Then, the growth bound of the semigroup ω0 defined as
1
ln
St inf ω : ∃M ≥ 1 such that St ≤ Meωt ,
t→∞ t
ω0 lim
1.6
is given by
ω0 max{sA, ωess },
1.7
Journal of Applied Mathematics
3
where sA sup{λ : λ ∈ spec A} and
ωess lim
1
t→∞ t
ln|St |α .
1.8
In Theorem 1.3, specA is the compliment of the resolvent set ρA which is the set of
all λ ∈ C such that the operator λI − A is one-one and onto and λI − A−1 is a bounded linear
map.
For a real number r, r max{n ∈ Z : n ≤ r} and r min{n ∈ Z : n ≥ r}. We will
make use of the observation r ≤ r ≤ r 1 for r ∈ R.
2. Asymptotic Stability of a PDE
Consider the following simple initial boundary value problem for a PDE:
∂u ∂u
,
∂t
∂θ
t ≥ 0, θ ≤ 0,
ut, 0 0,
t ≥ 0,
u0, θ u0 θ,
2.1
θ ≤ 0,
where u0 ∈ Cσ,0 −∞, 0 {u ∈ Cσ −∞, 0 : u0 0}.
Its mild solution is given by the semigroup Tt : Cσ,0 −∞, 0 → Cσ,0 −∞, 0 defined as
Tt u0 θ u0 t θ,
tθ <0
0, t θ ≥ 0.
2.2
Proposition 2.1. Let mi iτ1 and βi pγ −i . The infinitesimal generator of the semigroup defined by
2.2 is given by B : DB → Cσ,0 −∞, 0, Bφ φ , where
DB φ ∈ Cσ,0 −∞, 0 : φ ∈ Cσ,0 −∞, 0 .
2.3
Further, ρB {λ : λ > − ln γ/τ1 }.
Besides, if λ > − ln γ/τ1 , then eλ ∈ Cσ −∞, 0 and for every f ∈ Cσ −∞, 0, h
θ
defined as hθ 0 eλθ−ξ fξdξ and eλ defined as eλ θ eλθ are elements of Cσ −∞, 0.
Finally, for the semigroup Tt defined in 2.2, ω0 − ln γ/τ1 .
Proof. Since θ ∈ −iτ1 , 0 ⇒ t θ ∈ −iτ1 , t,
sup Tt φθ ≤
θ∈−iτ1 ,0
sup φθ,
θ∈−iτ1 ,0
2.4
and hence Tt σ ≤ 1, Tts Tt Ts is obvious, then
limTt φ − φσ 0
t→0
2.5
4
Journal of Applied Mathematics
can be proved using Proposition 1.9 of 10. The proof that B is the infinitesimal generator of
Tt is also easy.
λ ∈ ρB. Define φ, as φθ Note that λ 0 trivially satisfies λ > − ln γ/τ1 . Let 0 /
−i
pγ
<
∞,
φ
∈
C
−∞,
0
and
hence
there
is
a
unique ψ ∈ DB, such that
θ. Since ∞
σ,0
i1
λψ − ψ φ. Indeed, ψ λI0 − B−1 φ. Here, I0 is the identity on Cσ,0 −∞, 0. Let us note
that ψ0 0. Now, we find that ψ1 , defined as ψ1 θ θ/λ 1/λ2 1 − eλθ is the unique
continuously differentiable function such that λψ1 − ψ1 φ and ψ1 0 0. From this we
infer that ψ1 λI0 − B−1 φ and hence ψ1 ∈ Cσ,0 −∞, 0. Now, since φ ∈ Cσ,0 −∞, 0, we
obtain 1 − eλ ∈ Cσ,0 −∞, 0 ⊆ Cσ −∞, 0. Since the constant function 1 is an element of
Cσ −∞, 0, eλ ∈ Cσ −∞, 0. Noting that − ln γ/τ1 inf{λ : eλ ∈ Cσ −∞, 0}, we obtain
λ > − ln γ/τ1 .
Let t ≥ τ1 . It is clear that for all i ≤ t/τ1 , and θ ∈ −iτ1 , 0, Tt φθ 0. For i > t/τ1 ,
and θ ∈ −iτ1 , 0, we have t θ ≥ t − iτ1 ≥ −i − t/τ1 τ1 . Thus,
sup Tt φθ ≤
θ∈−iτ1 ,0
sup φt θ
θ∈−iτ1 ,0
≤
sup
φθ.
2.6
θ∈−i−t/τ1 τ1 ,0
Hence
∞ Tt φ ≤
βi sup Tt φθ
σ
i1
≤
∞
θ∈−iτ1 ,0
βi it/τ1 1
sup
φθ
θ∈−i−t/τ1 τ1 ,0
βi φθ
≤
βi−t/τ1 sup
βi−t/τ θ∈−i−t/τ τ ,0
1
1 1
it/τ1 1
∞
βi βi−t/τ φθ
≤ sup sup
1
i>t/τ1 βi−t/τ1 it/τ1 1 θ∈−i−t/τ1 τ1 ,0
βi φσ
≤ sup i>t/τ1 βi−t/τ1 ∞
≤ sup
i>t/τ1 γ −i
γ −it/τ1 2.7
φ
σ
≤ γ −t/τ1 φσ .
Hence, the operator norm Tt σ ≤ γ −t/τ1 .
To prove the equality, we construct a function η ∈ Cσ,0 −∞, 0 such that Tt η
σ γ −t/τ1 η
σ and the result follows.
Let δ t/τ1 1τ1 − t τ1 t/τ1 1 − t/τ1 . We have, δ < τ1 . Define,
−θ
, −δ ≤ θ ≤ 0
δ
1, θ < −δ.
ηθ 2.8
Journal of Applied Mathematics
It is clear that η
σ ∞
i1
5
pγ −i , Now,
θt
, −δ − t ≤ θ ≤ −t
δ
1, θ < −δ − t.
Tt ηθ −
2.9
∞
−i
it/τ1 1 γ .
−t/τ1 Tt η
σ γ
η
σ .
Thus Tt η
σ p
Hence,
Now, ω0 limt → ∞ 1/t ln
Tt σ − lnγ/τ1 .
Let λ > − ln γ/τ1 . Since
∞
−1 −λt
e Tt gdt
λI0 − B g σ
σ
∞0
≤
e−λt Tt g σ dt
0
∞
∞
−λt ω0 t g σ dt ≤
e
e
eω0 −λt g σ dt
0
0∞
≤
e− lnγ/τ1 −λt g σ dt,
2.10
0
we have λ ∈ ρB.
Let f ∈ Cσ −∞, 0. Define gθ fθ − f0. Then g ∈ Cσ,0 −∞, 0.
Let ψ λI0 − B−1 g. We have, ψ0 0.
θ
Define ψ1 θ − 0 eλθ−ξ gξdξ. Now ψ1 0 0 and ψ1 0 0.
By the uniqueness of the solution to the initial value problem of the ODE:
λψ − ψ g,
2.11
ψ0 0,
it is now obvious that ψ1 ψ and hence ψ1 ∈ Cσ,0 −∞, 0.
Now,
θ
0
eλθ−ξ gξdξ θ
0
eλθ−ξ fξ − f0 dξ θ
eλθ−ξ fξdξ 0
1
1 − eλθ f0.
λ
2.12
Since 1 − e ∈ Cσ,0 −∞, 0, h ∈ Cσ,0 −∞, 0 ⊂ Cσ −∞, 0, where h is defined as hθ θ λθ−ξ λ
e
fξdξ.
0
3. Stability of the Infinite Delay Equation
The proof of the next theorem assuring the existence of a unique solution to 1.1 is similar to
the proof of Theorem 2.2 of 10.
6
Journal of Applied Mathematics
Theorem 3.1. Let a ∈ R and the sequences bi and βi be as in Section 1. Assume that τi ≤ iτ1 . Then
there exists a unique solution x : R → R to 1.1 such that its restriction to 0, ∞, denoted by y, is
in C1 0, ∞. Further, for any t ∈ 0, ∞, there is a constant ct > 0 such that
sup ys ≤ ctφσ .
3.1
s∈0,t
In addition, the family of operators {St : t ≥ 0} defined as
St φθ xt θ,
tθ ≥0
φt θ,
tθ <0
3.2
forms a semigroup. Also, the infinitesimal generator of St is given by A : DA → Cσ −∞, 0,
where
DA ∞
φ ∈ Cσ −∞, 0 : φ ∈ Cσ −∞, 0, φ 0 aφ0 bi φ−τi i1
3.3
Aφ φ .
Further, DA is dense and A is a closed operator.
Theorem 3.2. For the semigroup St defined by 3.2
|St |α ≤ γ −t/τ1 .
Further, assume that a by 3.3 and
spec A ∞
i1
3.4
bi /
0. Then for the generator of the semigroup St defined
∞
ln γ
ln γ
−λτi
∪ λ : λ > −
.
: λ a bi e
λ : λ ≤ −
τ1
τ1
i1
3.5
−λτi
Besides, suppose that for any λ ∈ C with λ a ∞
, we have λ < −μ1 for
i1 bi e
some μ1 > 0. Then, the semigroup St is asymptotically stable.
Proof. Let Tt be as in Proposition 2.1. Fix t > 0. Define Vt : Cσ −∞, 0 → Cσ −∞, 0 as
Vt φθ 0,
tθ ≥0
φt θ − φ0,
t θ < 0.
3.6
Journal of Applied Mathematics
7
Define Kt : C0, t → Cσ −∞, 0 as
Kt zθ zt θ − z0,
0,
tθ ≥0
3.7
t θ < 0.
It is easy to see that
∞ Kt z
σ ≤ 2 βi i1
sup |zs| .
3.8
s∈0,t
Thus, Kt is a bounded linear map.
Define K1 : Cσ −∞, 0 → Cσ −∞, 0 as K1 φθ φ0 for all θ ∈ −∞, 0. It is clear
that K1 is compact. Define Bt : Cσ −∞, 0 → C0, t as Bt φ z, where z is the restriction of y
to 0, t. From 3.1, Bt is a bounded linear map. Let St be as in 3.3. Then,
St Vt Kt Bt K1 .
3.9
Now, if I is the identity on Cσ −∞, 0 and J : Cσ,0 −∞, 0 → Cσ −∞, 0 is the inclusion
map, then Vt JTt I − K1 , and, finally,
St JTt I − K1 Kt Bt K1 .
3.10
Next, we show that Bt is, in fact, a compact map. Let x be the solution to 1.1 as in
Theorem 3.1:
zs eas φ0 eas
s
0
e−aη
∞
bi x η − τi dη,
s ∈ 0, t.
3.11
i1
Thus,
z s azs ∞
bi xs − τi .
3.12
i1
Consider n ∈ N such that t ∈ nτ1 , n 1τ1 . From 3.1 and 3.11, we obtain existence
of c1 t ≥ 0 such that
sup z s ≤ c1 tφσ .
s∈0,t
3.13
Hence by Arzela-Ascoli theorem, Bt is a compact operator.
It is easy to show that |J|α ≤ J
σ 1. By the compactness of K1 and Bt , |I − K1 |α 1
and |Kt Bt |α |K1 |α 0. Thus, from the relation
St JTt I − K1 Kt Bt K1 ,
3.14
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Journal of Applied Mathematics
and Propositions 1.2 and 2.1 of this paper, we obtain
|St |α ≤ |Tt |α ≤ Tt σ ≤ γ −t/τ1 .
3.15
So,
γ
1
ln|St |α ≤ − ln .
t→∞ t
τ1
ωess lim
3.16
Let 0 /
λ ∈ ρA.
There is a unique ψ ∈ DA such that
λψ − ψ −1,
∞
ψ 0 aψ0 bi ψ−τi .
3.17
i1
It is clear that there is c ∈ C such that ψθ c − 1/λeλθ − 1/λ. Now, we claim that
c/
1/λ. If c 1/λ, then ψθ −1/λ for all θ ∈ −∞, 0. Since ψ ∈ DA, we must have ψ 0 But this would imply that a ∞
aψ0 ∞
i1 bi ψ−τi .
i1 bi 0 which is a contradiction, to the
∞
0. Now, since c − 1/λ /
0, it is obvious that eλ ∈ Cσ −∞, 0. But
hypothesis that a i1 bi /
this implies that λ > − lnγ/τ1 . If 0 ∈ ρA, the condition λ > − lnγ/τ1 is obvious.
Thus,
ln γ
.
λ : λ > −
τ1
ρA ⊆
3.18
−λτi
We now infer that {λ : λ ≤ − lnγ/τ1 } ⊆ specA. Next, if λ a ∞
,
i1 bi e
and λ > − lnγ/τ1 , then eλ ∈ Cσ −∞, 0 and hence eλ ∈ DA with λeλ Aeλ . Thus,
λ ∈ specA. So,
∞
ln γ
−λτi
λ : λ > −
⊆ specA.
, λ a bi e
τ1
i1
3.19
−λτi
Let us assume that λ > − lnγ/τ1 and λ /
a ∞
.
i1 bi e
Then, by Proposition 2.1, we have eλ ∈ Cσ −∞, 0 and the function h defined as hθ θ λθ−ξ
e
fξdξ is in Cσ −∞, 0.
0
Defining Λ : Cσ −∞, 0 → C as Λφ aφ0 ∞
i1 bi φ−τi and taking c Λh −
θ λθ−ξ
f0/Λeλ − λ, we find that φ defined as φθ 0 e
fξdξ ceλθ is λI − A−1 f.
Thus,
∞
ln γ
−λτi
⊆ ρA.
, λ/
a bi e
λ : λ > −
τ1
i1
3.20
Journal of Applied Mathematics
9
From 3.18, 3.19, and 3.20, we finally conclude that
∞
ln γ
ln γ
−λτi
∪ λ : λ > −
,
, λ a bi e
λ : λ ≤ −
τ1
τ1
i1
specA 3.21
or
ρA ∞
ln γ
−λτi
λ : λ > −
.
, λ/
a bi e
τ1
i1
3.22
Since ω0 max{sA, ωess } ≤ max{−μ1 , − lnγ/τ1 }, the result follows.
Remark 3.3. Consider the PDE:
∂u ∂u
,
∂t
∂θ
u0, θ φθ.
3.23
Let B be as in Proposition 2.1 and A be as in Theorem 3.1. For φ ∈ DB, ut, θ Tt φ ∈
Cσ,0 −∞, 0 is the solution to the above PDE. For φ ∈ DA, ut, θ St φ ∈ Cσ −∞, 0 is the
solution to the above PDE. For the first solution ut θ 0, t θ ≥ 0 and for the second
solution ut θ xt θ, t θ ≥ 0. Here x is the solution to the delay equation.
Acknowledgment
D. Piriadarshani would like to thank Professor B. Praba, Department of Mathematics, SSN
College of Engineering, Kalavakkam, Tamil Nadu, India, for the support and advice received
from her as a Cosupervisor.
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