Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 85, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A CLASS OF
LINEAR NON-AUTONOMOUS NEUTRAL DELAY
DIFFERENTIAL EQUATIONS
GUILING CHEN
Abstract. We study a class of linear non-autonomous neutral delay differential equations, and establish a criterion for the asymptotic behavior of their
solutions, by using the corresponding characteristic equation.
1. Introduction
Let C be the complex numbers with norm | · |. For r ≥ 0, let C = C([−r, 0], C)
be the space of continuous functions taking [−r, 0] into C with norm defined by
kϕk = max−r≤θ≤0 |ϕ|. A functional differential equation of neutral type, or shortly
a neutral equation, is a system of the form
d
M xt = L(t)xt , t > t0 ∈ R,
(1.1)
dt
where xt ∈ C is defined by xt (θ) = x(t + θ), −r ≤ θ ≤ 0, M : C → C is continuous,
linear and atomic at zero, (see [5, page 255] for the concept of atomic at zero),
Z 0
M ϕ = ϕ(0) −
ϕ(θ) dµ(θ),
(1.2)
−r
where Var[s,0] µ → 0, as s → 0.
For (1.1), L(t) denote a family of bounded linear functionals on C. By the Riesz
representation theorem, for each t > t0 , there exists a complex valued function of
bounded variation η(t, ·) on [−r, 0], normalized so that η(t, 0) = 0 and η(t, ·) is
continuous from the left in (−r, 0) such that
Z 0
L(t)ϕ =
ϕ(θ) dθ η(t, θ).
(1.3)
−r
For any ϕ ∈ C, σ ∈ [t0 , ∞), a function x = x(σ, ϕ) defined on [σ − r, σ + A) is
said to be a solution of (1.1) on (σ, σ + A) with initial ϕ at σ if x is continuous on
[σ − r, σ + A), xσ = ϕ, M xt is continuously differentiable on (σ, σ + A), and relation
(1.1) is satisfied on (σ, σ + A). For more information on this type of equations, see
[5].
2000 Mathematics Subject Classification. 34K11, 34K40, 34K25.
Key words and phrases. Neutral delay differential equation; characteristic equation;
asymptotic behavior.
c
2011
Texas State University - San Marcos.
Submitted May 24, 2011. Published June 29, 2011.
1
2
G. CHEN
EJDE-2011/85
The initial-value problem (IVP) is stated as
d
M xt = L(t)xt
dt
xσ = ϕ.
t > σ,
(1.4)
For µ = 0 in (1.2), M ϕ = ϕ(0) and equation (1.1) becomes the retarded functional differential equation
x0 (t) = L(t)xt .
(1.5)
Consider the characteristic equation associated with (1.5),
Z r
Z t
λ(t) =
exp −
λ(s)ds dθ η(t, θ)
(1.6)
0
t−θ
which is obtained by looking for solutions to (1.5) of the form
hZ t
i
λ(s) ds .
x(t) = exp
(1.7)
0
The solutions of (1.6) are continuous functions λ(·) defined in [t0 − r, ∞) which
satisfy (1.5).
Cuevas and Frasson [1] studied the asymptotic behavior of solutions of (1.5) with
initial condition xσ = ϕ, and obtained the following result.
Theorem 1.1. Assume that λ(t) is a solution of (1.6) such that
Z r
Rt
lim sup
θ|e− t−θ λ(s)ds |dθ |η|(t, θ) < 1.
t→∞
0
Then for each solution x of (1.5), we have that the limit
−
lim x(t)e
Rt
t0
λ(s)ds
t→∞
exists, and
i0
h
R
− t λ(s)ds
= 0.
lim x(t)e t0
t→∞
Furthermore,
−
lim x0 (t)e
t→∞
R
− tt λ(s)ds
if limt→∞ λ(t)x(t)e
0
Rt
t0
λ(s)ds
−
= lim λ(t)x(t)e
t→∞
Rt
t0
λ(s)ds
,
exists.
Motivated by the work in [1], we provide a generalization of [1], and consider the
asymptotic behavior of solutions to (1.4). The method for the proving our main
result is similar to the one in [1, 2]. In Section 2, we state the main results. In
Section 3, some examples will be shown as applications of the main results of this
paper.
2. Main results
For equation (1.1), the characteristic equation is
Z 0
Z t
Z 0
Z
λ(t) =
dµ(θ)λ(t + θ) exp −
λ(s)ds +
dθ η(t, θ) exp −
−r
t+θ
−r
t
λ(s)ds ,
t+θ
(2.1)
which is obtained by looking for solutions of (1.1) of the form (1.7) and the solutions of (2.1) are continuous functions defined in [σ − r, ∞) satisfying (2.1). For
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS
3
autonomous neutral functional differential equations (NFDEs), the constant solutions of (2.1) are the roots of the so called characteristic equation, for detailed
discussion of this type, refer to [3, 4, 5].
Theorem 2.1. Assume that λ(t) is a solution of (2.1) such that
lim sup χλ,t < 1,
(2.2)
t→∞
where
0
Z
|e−
χλ,t =
Rt
t+θ
λ(s) ds
| d|µ|(θ)
−r
Z
0
(−θ)|e−
+
Rt
t+θ
λ(s) ds
| |λ(t + θ)| d|µ|(θ) + dθ |η|(t, θ) .
−r
Then for each solution x of (1.4), we have that the limit
−
lim x(t)e
Rt
t0
λ(s) ds
(2.3)
t→∞
exists, and
i0
h
R
− t λ(s) ds
= 0.
lim x(t)e t0
(2.4)
t→∞
Furthermore,
−
lim x0 (t)e
Rt
−
λ(s) ds
t0
= lim λ(t)x(t)e
t→∞
Rt
t0
λ(s) ds
t→∞
(2.5)
if the limit in the right-hand side exists.
Proof. From (2.2), there exists t1 ≥ t0 , such that
sup χλ,t < 1.
t≥t1
Hence without loss of generality, we assume that t0 = 0 and define
Γλ := sup χλ,t < 1.
t≥0
For solutions x of (1.4), we set
y(t) = x(t)e−
Rt
0
λ(s) ds
t > −r.
,
Then (1.4) becomes
y 0 (t) + λ(t)y(t) −
Z
0
dµ(θ)y 0 (t + θ)e−
Rt
t+θ
λ(s) ds
−r
Z
0
=
−
y(t + θ)e
Rt
t+θ
(2.6)
λ(s) ds
λ(t + θ) dµ(θ) + dθ η(t, θ)
−r
and the initial condition is equivalent to
y(t) = ϕ(t)e−
Rt
0
λ(s) ds
,
−r ≤ t ≤ 0.
(2.7)
Combining (2.7) with (2.1), for t ≥ −r, we have
Z 0
Rt
y 0 (t) =
dµ(θ)y 0 (t + θ)e− t+θ λ(s) ds
Z
−r
0
−
−r
e−
Rt
t+θ
λ(s) ds
Z
0
−r
(2.8)
y 0 (s) ds λ(t + θ) dµ(θ) + dθ η(t, θ) .
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G. CHEN
EJDE-2011/85
From the definition of the solutions to (1.4), we know that y 0 (t) is continuous, Let
Mϕ,λ0 = max{|ϕ0 (t)e−
Rt
λ(s) ds
0
− λ(t)ϕ(t)e−
Rt
0
λ(s) ds
| : −r ≤ t ≤ 0}.
0
We shall show that Mϕ is also a bound of y on the whole interval [−r, ∞); i.e.,
|y 0 (t)| ≤ Mϕ,λ0 ,
t ≥ −r.
(2.9)
For this purpose, let us consider an arbitrary number ε > 0. Then
|y 0 (t)| < Mϕ,λ0 + ε
for t ≥ −r.
(2.10)
∗
Indeed, in the opposite case, we suppose there exists a point t > 0 such that
|y 0 (t)| < Mϕ,λ0 + ε
for − r ≤ t < t∗ ,
|y(t∗ )| = M (λ0 , µ0 ; φ) + ε.
(2.11)
Then combining (2.8) and (2.11), we obtain
M (λ0 , µ0 ; φ) + ε
= y 0 (t∗ )
Z 0
R t∗
≤
y 0 (t∗ + θ)e− t∗ +θ λ(s) ds dµ(θ)
−r
Z
+
0
R t∗
−
e
Z
λ(s) ds
t∗ +θ
−r
≤ (Mϕ,λ0
0
nZ
+ ε)
−
|e
0
−r
R t∗
y 0 (s) ds λ(t∗ + θ) dµ(θ) + dθ η(t∗ , θ) (2.12)
λ(s) ds
t∗ +θ
| d|µ|(θ)
−r
Z
0
+
R t∗
(−θ)|e−
t∗ +θ
λ(s) ds
o
| |λ(t∗ + θ)| d|µ|(θ) + dθ |η|(t∗ , θ)
−r
= (Mϕ,λ0 + ε)Γλ
< (Mϕ,λ0 + ε),
which is a contradiction, so (2.10) holds. Since (2.10) holds for every ε > 0, it
follows that |y 0 (t)| ≤ Mϕ,λ0 , for all t ≥ −r. By using (2.8) and (2.9), for t ≥ 0 we
have
Z 0
Rt
|y 0 (t)| ≤ y 0 (t + θ)e− t+θ λ(s)ds dµ(θ)
−r
Z
+
0
−
e
Rt
t+θ
λ(s) ds
−r
≤ Mϕ,λ0
Z
0
−r
nZ
0
|e−
Rt
t+θ
y 0 (s) ds λ(t + θ) dµ(θ) + dθ η(t, θ) λ(s) ds
| d|µ|(θ)
(2.13)
−r
Z
0
+
(−θ)|e−
Rt
t+θ
λ(s) ds
o
| |λ(t + θ)| d|µ|(θ) + dθ |η|(t, θ)
−r
= Mϕ,λ0 Γλ ,
which means, for t ≥ 0,
|y 0 (t)| ≤ Mϕ,λ0 Γλ0 .
One can show by induction, that y 0 (t) satisfies
|y 0 (t)| ≤ Mϕ,λ0 (Γλ )n
for t ≥ nr − r,
(n = 0, 1, 2, 3, . . . ).
(2.14)
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5
Since 0 ≤ χλ,t < 1, it follows that y 0 (t) tends to zero as t → ∞. So we proved (2.4).
In the following, we will show (2.3) holds.
To prove that limt→∞ y(t) exists, we consider (2.14). For an arbitrary t ≥ 0,
we set n = [t/r] + 1 (the greatest integer less than or equal to t/r + 1), then from
n = [t/r] + 1 ≤ t/r + 1 ≤ [t/r] + 2 = n + 1, we have t/r ≤ n. From (2.14),
|y 0 (t)| ≤ Mϕ,λ0 (Γλ )n ≤ Mϕ,λ0 (Γλ )t/r
for t ≥ nr − r.
(2.15)
Now we use the Cauchy convergence criterion, for t > T ≥ 0, from (2.15), we have
Z t
Z t
|y(t) − y(T )| ≤
|y 0 (s)| ds ≤
Mϕ,λ0 (Γλ )s/r ds
T
T
is=t
r h
(Γλ )s/r
= Mϕ,λ0
ln Γλ
s=T
i
r h
t/r
(Γλ ) − (Γλ )T /r .
= Mϕ,λ0
ln Γλ
(2.16)
Let T → ∞, we have t → ∞, and by (2.16), we have
i
r h
Mϕ,λ
(Γλ )t/r − (Γλ )T /r → 0;
ln Γλ
and limT →∞ |y(t) − y(T )| = 0. The Cauchy convergence criterion implies the existence of limt→∞ y(t). We obtain (2.5) by a straight forward application of (2.4). Remark 2.2. Under the conditions of Theorem 2.1, a solution of (1.4) can not
grow faster than the exponential function; i.e., there exists a constant M > 0, such
that
Rt
(2.17)
|x(t)| ≤ M e 0 λ(s) ds , for t ≥ 0.
From (2.17), it is not difficult to show that:
Rt
• Every solution of (1.4) is bounded if and only if lim supt→∞ 0 λ(s) ds < ∞;
Rt
• Every solution of (1.4) tends to zero if and only if lim supt→∞ 0 λ(s) ds →
−∞.
Remark 2.3. If the characteristic equation (2.1) has a constant solution λ(t) = λ0 ,
then from Theorem 2.1, limt→∞ x(t)e−λ0 t exists.
3. Examples
Example 3.1. Consider the linear differential equation with distributed delay
Z 0
1 0
x(t + θ)
0
x (t) − x (t − 1) =
dθ, t > 1.
(3.1)
2
−1 2(t + θ)
This equation can be written in the form (1.1) by setting µ(θ) = −1/2 for θ ≤ −1,
µ(θ) = 0 for θ > −1, η(t, θ) = ln t + 12 ln(t + θ) for t > 1 and θ ∈ [−1, 0]. Since both
θ 7→ η(t, θ) and θ 7→ µ(θ) are increasing functions, |µ| = µ, |η| = η.
The characteristic equation associated with (3.1) is
i Z 0
i
h Z t
h Z t
1
λ(t − 1)
λ(t) =
exp −
λ(s) ds +
exp −
λ(s) ds dθ, (3.2)
2
t−1
−1 2(t + θ)
t+θ
which has a solution
λ(t) = 1/t.
(3.3)
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G. CHEN
EJDE-2011/85
For this λ(t) and for t > 1, using the expression of χλ,t , we have
Z 0
h Z t ds i
1
1
1
−θ
1
dθ = < 1 as t → ∞.
1−
+
+
exp −
2
t
4t
2
−1 2(t + θ)
t+θ s
Hence the hypothesis (2.2) of Theorem 2.1 is fulfilled. So we obtain that
h x(t) i0
x(t)
x0 (t)
lim
= 0 and lim
exists, lim
= 0.
t→∞ t
t→∞
t→∞
t
t
Example 3.2. Consider the equation with variable delay
x(t − τ (t))
2
, t > t0 .
x0 (t) − x0 (t − 1) =
3
3(t + c − τ (t))
(3.4)
(3.5)
where c ∈ R and τ : [0, ∞) → [−1, 0] is a continuous function such that t+c−τ (t) >
0 for t > t0 . Equation (3.5) can be written in the form (1.1) by letting µ(θ) = −2/3
for θ ≤ −1, µ(θ) = 0 for θ > −1, η(t, θ) = 0 for θ < τ (t), η(t, θ) = (t + c − τ (t))/3
for θ > τ (t). Since both θ 7→ η(t, θ) and θ 7→ µ(θ) are increasing functions, we have
that |µ| = µ, |η| = η.
The characteristic equation associated with (3.5) is
h Z t
i
h Z t
i
2λ(t − 1)
1
λ(t) =
exp −
λ(s)ds +
exp −
λ(s)ds (3.6)
3
3(t + c − τ (t))
t−1
t−τ (t)
and we have that a solution of (3.6) is
1
.
(3.7)
t+c
For (3.7), the left hand side of (2.2) reads as
Z 0
h2
i
Rt
1 1
lim sup
1−
+
+
(−θ)|e− t−θ λ(s)ds |dθ |η|(t, θ)
3
t+c
6(t + c)
t→∞
−1
i
h2
τ (t)
2
−
= < 1.
= lim sup
3 3(t + c)
3
t→∞
λ(t) =
and hence hypothesis (2.2) of Theorem 2.1 is fulfilled and therefore, for all solutions
x(t) of (3.5), we have that
h x(t) i0
x(t)
= 0.
(3.8)
lim
exists, and lim
t→∞ t + c
t→∞ t + c
Manipulating further the limits in (3.5), we are able to establish that x(t) = O(t)
and x0 (t) = o(t) as t → ∞.
Acknowledgements. I express my thanks to my supervisors Sjoerd Verduyn
Lunel and Onno van Gaans who have provided me with valuable guidance in every
stage of my research. Also, I would like to show my deepest gratitude to Chinese
Scholarship Council.
References
[1] Cuevas, Claudio; Frasson, Miguel V. S. Asymptotic properties of solutions to linear nonautonomous delay differential equations through generalized characteristic equations. Electron.
J. Differential Equations 2010, (2010), No. 95, pp. 1-5.
[2] Dix, J. G., Philos, C. G., and Purnaras, I. K. Asymptotic properties of solutions to linear
non-autonomous neutral differential equations. J. Math. Anal. Appl. 318, 1 (2006), 296–304.
[3] Frasson, M. On the dominance of roots of characteristic equations for neutral functional differential equations. Journal of Mathematical Analysis and Applications 360 (2009), 27–292.
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[4] Frasson, M. V. S., and Verduyn Lunel, S. M. Large time behaviour of linear functional differential equations. Integral Equations Operator Theory 47, 1 (2003), 91–121.
[5] Hale, J. K., and Verduyn Lunel, S. M. Introduction to functional-differential equations, vol. 99
of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.
Guiling Chen
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA, Leiden, The Netherlands
E-mail address: guiling@math.leidenuniv.nl