Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 172, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
STABILITY FOR LINEAR NEUTRAL INTEGRO-DIFFERENTIAL
EQUATIONS WITH VARIABLE DELAYS
ABDELOUAHEB ARDJOUNI, AHCENE DJOUDI, IMENE SOUALHIA
Abstract. In this article we study a linear neutral integro-differential equation with variable delays and give suitable conditions to obtain asymptotic
stability of the zero solution, by means of fixed point technique. An asymptotic stability theorem with a necessary and sufficient condition is proved,
which improves and generalizes previous results due to Burton [5], Becker and
Burton [4] and Jin and Luo [15]. We provide an example that illustrates our
results.
1. Introduction
Without doubt, Lyapunov’s direct method has been, for more than 100 years,
the main tool for investigating the stability properties of a wide variety of ordinary, functional, partial differential and integro-differential equations. Nevertheless,
the application of this method to problems of stability in differential and integrodifferential equations with delays has encountered serious obstacles if the delays are
unbounded or if the equation has unbounded terms [6]–[8]. In recent years, several
investigators have tried stability by using a new technique. Particularly, Burton,
Furumochi, Becker and others began a study in which they noticed that some of
these difficulties vanish or might be overcome by means of fixed point theory (see
[1]–[18], [20]). The fixed point theory does not only solve the problem on stability
but has other significant advantage over Lyapunov’s. The conditions of the former
are often averages but those of the latter are usually pointwise (see [6]).
In this article we consider the linear neutral integro-differential equation with
variable delays
N
N Z t
X
X
cj (t)x0 (t − τj (t)),
(1.1)
x0 (t) = −
aj (t, s)x(s)ds +
j=1
t−τj (t)
j=1
with the initial condition
x(t) = ψ(t)
for t ∈ [m(0), 0],
where ψ ∈ C([m(0), 0], R) and
mj (0) = inf{t − τj (t), t ≥ 0},
m(0) = min{mj (0), 1 ≤ j ≤ N }.
2000 Mathematics Subject Classification. 34K20, 34K30, 34K40.
Key words and phrases. Fixed point; stability; integro-differential equation; variable delay.
c
2012
Texas State University - San Marcos.
Submitted June 20, 2012. Published October 12, 2012.
1
2
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
Here C(S1 , S2 ) denotes the set of all continuous functions ϕ : S1 → S2 with
the supremum norm k · k. Throughout this paper we assume that aj ∈ C(R+ ×
[m(0), ∞), R), cj ∈ C 1 (R+ , R), and τj ∈ C(R+ , R+ ) with t − τj (t) → ∞ as t → ∞
for j = 1, 2, . . . , N .
Special cases of equation (1.1) have been investigated by many authors. For
example, Burton in [5], Becker and Burton in [4], Jin and Luo in [15] studied the
equation
Z t
x0 (t) = −
a1 (t, s)x(s)ds,
(1.2)
t−τ1 (t)
and proved the following theorems, respectively,
Theorem 1.1 ([5]). Suppose that τ1 (t) = r and there exists a constant α < 1 such
that
Z t
2
|A(t, s)|ds ≤ α for all t ≥ 0,
(1.3)
t−r
t
Z
A(s, s)ds → ∞
as t → ∞,
(1.4)
0
where
Z
r
Z
A(t, s) =
a1 (u + s, s)du
r
with A(t, t) =
a1 (u + t, t)du.
t−s
0
Then the zero solution of (1.2) is asymptotically stable.
Theorem 1.2 ([4]). Suppose that τ1 is differentiable, t−τ1 (t) is strictly increasing,
and there exist constants k ≥ 0, α ∈ (0, 1) such that for t ≥ 0,
Z t
−
G(s, s)ds ≤ k,
(1.5)
0
Z t
Z t R
Z s
t
|G(t, s)|ds +
e− s G(u,u)du |G(s, s)|
|G(s, u)|du ds ≤ α, (1.6)
t−τ1 (t)
0
s−τ1 (s)
with
Z
f (s)
G(t, s) =
Z
a1 (u, s)du,
t
G(t, t) =
f (t)
a1 (u, t)du,
t
where f is the inverse function of t−τ1 (t). Then for each continuous initial function
ψ : [m1 (0), 0] → R, there is a unique continuous function x : [m1 (0), ∞) → R with
x(t) = ψ(t) on [m1 (0), 0] that satisfies (1.2) on [0, ∞). Moreover, x is bounded
on [m1 (0), ∞). Furthermore, the zero solution of (1.2) is stable at t = 0. If, in
addition,
Z t
G(s, s)ds → ∞ as t → ∞,
(1.7)
0
then x(t) → 0 as t → ∞.
Theorem 1.3 ([15]). Let τ1 be differentiable. Suppose that there exist constants
k ≥ 0, α ∈ (0, 1) and a function h1 ∈ C(R+ , R) such that for t ≥ 0,
Z t
−
h1 (s)ds ≤ k,
(1.8)
0
EJDE-2012/172
Z
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
3
t
|h1 (s) + B1 (t, s)|ds
t−τ1 (t)
Z t R
Z s
t
+
e− s h1 (u)du |h1 (s)|
0
s−τ1 (s)
Z t R
t
+
e− s h1 (u)du |h1 (s − τ1 (s)) +
0
|h1 (u) + B1 (s, u)|du ds
(1.9)
B1 (s, s − τ1 (s))||1 − τ10 (s)| ≤ α,
where
Z
B1 (t, s) =
s
Z
with B1 (t, t − τ1 (t)) =
a1 (u, s)du
t
t−τ1 (t)
a1 (u, t − τ1 (t))du.
t
Then for each continuous initial function ψ : [m1 (0), 0] → R, there is an unique
continuous function x : [m1 (0), ∞) → R with x(t) = ψ(t) on [m1 (0), 0] that satisfies
(1.2) on [0, ∞). Moreover, x is bounded on [m1 (0), ∞). Furthermore, the zero
solution of (1.2) is stable at t = 0. If, in addition,
Z t
h1 (s)ds → ∞ as t → ∞,
(1.10)
0
then x(t) → 0 as t → ∞.
Remark 1.4. The result by Becker and Burton in Theorem 1.2 requires that
t − τ1 (t) be strictly increasing. In Theorem 1.3, this condition is clearly removed.
Also, the conditions of stability in Theorem 1.3 are less restrictive than Theorem
1.2. Thus, Theorem 1.3 improves Theorems 1.1 and 1.2.
Our objective here is to improve Theorem 1.3 and extend it to investigate a
wide class of linear integro-differential equation with variable delays of neutral type
presented in (1.1). To do this we define a suitable continuous function H (see
Theorem 2.2 below) and find conditions for H, with no need of further assumptions
on the inverse of delays t − τj (t), so that for a given continuous initial function ψ a
mapping P for (1.1) is constructed in such a way to map a complete metric space
Sψ in itself and in which P possesses a fixed point. This procedure will enable us to
establish and prove an asymptotic stability theorem for the zero solution of (1.1)
with a necessary and sufficient condition and with less restrictive conditions. The
results obtained in this paper improve and generalize the main results in [4, 5, 15].
We provide an example to illustrate our claim.
2. Main results
For each ψ ∈ C([m(0), 0], R), a solution of (1.1) through (0, ψ) is a continuous
function x : [m(0), σ) → R for some positive constant σ > 0 such that x satisfies
(1.1) on [0, σ) and x = ψ on [m(0), 0]. We denote such a solution by x(t) = x(t, 0, ψ).
From the existence theory we can conclude that for each ψ ∈ C([m(0), 0], R), there
exists a unique solution x(t) = x(t, 0, ψ) of (1.1) defined on [0, ∞). We define
kψk = max{|ψ(t)| : m(0) ≤ t ≤ 0}. Stability definitions may be found in [6], for
example.
Our aim here is to generalize Theorem 1.3 to equation (1.1) by giving a necessary
and sufficient condition for asymptotic stability of the zero solution.
It is known that studying the stability of an equation using a fixed point technic
involves the construction of a suitable fixed point mapping. This can be an arduous
task. So, to construct our mapping, we begin by transforming (1.1) to a more
4
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
tractable, but equivalent, equation, which we then invert to obtain an equivalent
integral equation from which we derive a fixed point mapping. After that, we define
a suitable complete space, depending on the initial condition, so that the mapping is
a contraction. Using Banach’s contraction mapping principle, we obtain a solution
for this mapping, and hence a solution for (1.1), which is asymptotically stable.
First, we have to transform (1.1) into an equivalent equation that possesses the
same basic structure and properties to which we apply the variation of parameters
to define a fixed point mapping.
Lemma 2.1. Equation (1.1) is equivalent to
x0 (t) =
N
X
Bj (t, t − τj (t))(1 − τj0 (t))x(t − τj (t))
j=1
Z
N
N
X
X
d t
Bj (t, s)x(s)ds +
cj (t)x0 (t − τj (t)),
+
dt
t−τ
(t)
j
j=1
j=1
(2.1)
where
Z
Bj (t, s) =
s
Z
aj (u, s)du
and
t−τj (t)
Bj (t, t − τj (t)) =
t
aj (u, t − τj (t))du.
t
Proof. Differentiating the integral term in (2.1), we obtain
Z
d t
Bj (t, s)x(s)ds
dt t−τj (t)
Z t
0
= Bj (t, t)x(t) − Bj (t, t − τj (t))(1 − τj (t))x(t − τj (t)) +
t−τj (t)
∂
Bj (t, s)x(s)ds.
∂t
Substituting this into (2.1), it follows that (2.1) is equivalent to (1.1) provided Bj
satisfies the following conditions
Bj (t, t) = 0
and
∂
Bj (t, s) = −aj (t, s).
∂t
(2.2)
aj (u, s)du + φ(s),
(2.3)
This euqality implies
t
Z
Bj (t, s) = −
0
for some function φ, and Bj (t, s) must satisfy
Z t
Bj (t, t) = −
aj (u, t)du + φ(t) = 0.
0
Consequently,
Z
φ(t) =
t
aj (u, t)du.
0
Substituting this into (2.3), we obtain
Z t
Z
Bj (t, s) = −
aj (u, s)du +
0
0
s
Z
aj (u, s)du =
s
aj (u, s)du.
t
This definition of Bj satisfies (2.2). Consequently, (1.1) is equivalent to (2.1).
EJDE-2012/172
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
5
Theorem 2.2. Suppose that τj is twice differentiable and τj0 (t) 6= 1 for all t ∈ R+ ,
and there exist continuous functions hj : [mj (0), ∞) → R for j = 1, 2, . . . , N and a
constant α ∈ (0, 1) such that for t ≥ 0
Z t
lim inf
H(s)ds > −∞,
(2.4)
t→∞
and
N
X
j=1
+
|
N Z t
X
cj (t)
|
+
|hj (s) + Bj (t, s)|ds
1 − τj0 (t)
j=1 t−τj (t)
N Z
X
j=1
+
0
e−
Rt
−
Rt
s
H(u)du
|[hj (s − τj (s)) + B(s, s − τj (s))](1 − τj0 (s)) − rj (s)|ds (2.5)
H(u)du
Z
|H(s)|(
0
N Z
X
j=1
t
t
e
s
0
s
|hj (u) + Bj (s, u)|du)ds ≤ α,
s−τj (s)
where
H(t) =
N
X
hj (t),
rj (t) =
j=1
[cj (t)H(t) + c0j (t)](1 − τj0 (t)) + cj (t)τj00 (t)
,
(1 − τj0 (t))2
and
Z
Bj (t, s) =
s
Z
aj (u, s)du
with
t−τj (t)
Bj (t, t − τj (t)) =
t
aj (u, t − τj (t))du.
t
Then the zero solution of (1.1) is asymptotically stable if and only if
Z t
H(s)ds → ∞ as t → ∞.
(2.6)
0
Proof. First, suppose that (2.6) holds. We set
K = sup{e−
Rt
0
H(s)ds
}.
(2.7)
t≥0
Let ψ ∈ C([m(0), 0], R) be fixed and define
Sψ := {ϕ ∈ C([m(0), ∞), R) : ϕ(t) = ψ(t) for t ∈ [m(0), 0] and ϕ(t) → 0 as t → ∞}.
Endowed with the supremum norm k · k; that is, for φ ∈ Sψ ,
kφk := sup{|φ(t)| : t ∈ [m(0), ∞)}.
In other words, we carry out our investigations in the complete metric space (Sψ , ρ)
where ρ is supremum metric
ρ(x, y) := sup |x(t) − y(t)| = kx − yk,
for x, y ∈ Sψ .
t≥m(0)
Rewrite (1.1) in the following equivalent form
x0 (t) =
N
X
Bj (t, t − τj (t))(1 − τj0 (t))x(t − τj (t))
j=1
Z
N
N
X
X
d t
Bj (t, s)x(s)ds +
cj (t)x0 (t − τj (t))
+
dt
t−τj (t)
j=1
j=1
(2.8)
6
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
Rt
Multiplying both sides of (2.8) by exp 0 H(u)du and integrating with respect to
s from 0 to t, we obtain
Z t R
N
R
X
− 0t H(u)du
− st H(u)du
x(t) = ψ(0)e
+
e
hj (s)x(s)ds
0
Z
t
e−
+
Rt
s
H(u)du
0
Z
j=1
t
+
e−
Rt
e−
Rt
s
H(u)du
N
X
H(u)du
N
X
0
Z
N
X
j=1
s
Z
d
ds
Bj (s, u)x(u)du
s−τj (s)
Bj (s, s − τj (s))(1 − τj0 (s))x(s − τj (s))ds
j=1
t
+
s
0
cj (s)x0 (s − τj (s))ds.
j=1
Thus,
x(t) = ψ(0)e−
Rt
H(u)du
0
+
N Z
X
N Z
X
N Z
X
Rt
s
H(u)du
t
e−
Rt
e−
Rt
s
Rt
s
H(u)du
N Z
X
d
ds
Z
s
Bj (s, u)x(u)du
s−τj (s)
H(u)du
Bj (s, s − τj (s))(1 − τj0 (s))x(s − τj (s))ds
t
s
H(u)du
cj (s)x0 (s − τj (s))ds.
0
j=1
Performing an integration by parts, we obtain
N Z t
Z
Rt
Rt
X
x(t) = ψ(0)e− 0 H(u)du +
e− s H(u)du d
j=1
+
N Z t
X
e−
Rt
s
H(u)du
0
× (1 −
τj0 (s))x(s
= ψ(0) −
N
X
j=1
+
N
X
j=1
+
[hj (u) + Bj (s, u)]x(u)du
s−τj (s)
[hj (s − τj (s)) + Bj (s, s − τj (s))]
− τj (s))ds +
N Z
X
j=1
×e
s
0
j=1
−
hj (s)x(s)ds
0
j=1
+
e−
0
j=1
+
t
e−
0
j=1
+
t
Rt
0
t
cj (s) − R t H(u)du
e s
dx(s − τj (s))
1 − τj0 (s)
N Z 0
X
cj (0)
ψ(−τ
(0))
−
[h
(s)
+
B
(0,
s)]ψ(s)ds
j
j
j
1 − τj0 (0)
j=1 −τj (0)
H(u)du
0
N Z t
X
cj (t)
x(t
−
τ
(t))
+
[hj (s) + Bj (t, s)]x(s)ds
j
1 − τj0 (t)
j=1 t−τj (t)
N Z
X
j=1
0
t
e−
Rt
s
H(u)du
{[hj (s − τj (s)) + Bj (s, s − τj (s))]
EJDE-2012/172
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
7
× (1 − τj0 (s)) − rj (s)}x(s − τj (s))ds
N Z t
Z s
R
X
− st H(u)du
−
e
H(s)
[hj (u) + Bj (s, u)]x(u)du ds.
j=1
0
s−τj (s)
Now use this equality to define the operator P : Sψ → Sψ by (P ϕ)(t) = ψ(t) if
t ∈ [m(0), 0] and for t ≥ 0 we let
N
X
(P ϕ)(t) = ψ(0) −
cj (0)
ψ(−τj (0))
1 − τj0 (0)
j=1
−
N Z
X
j=1
0
Rt
[hj (s) + Bj (0, s)]ψ(s)ds e− 0 H(u)du
−τj (0)
N
X
N Z t
X
cj (t)
ϕ(t
−
τ
(t))
+
[hj (s) + Bj (t, s)]ϕ(s)ds
j
1 − τj0 (t)
(2.9)
j=1
j=1 t−τj (t)
N Z t
Rt
X
+
e− s H(u)du {[hj (s − τj (s)) + Bj (s, s − τj (s))]
+
j=1
0
× (1 − τj0 (s)) − rj (s)}ϕ(s − τj (s))ds
N Z t
Z s
Rt
X
−
e− s H(u)du H(s)
[hj (u) + Bj (s, u)]ϕ(u)du ds.
j=1
0
s−τj (s)
It is clear that (P ϕ) ∈ C([m(0), ∞), R). We will show that (P ϕ)(t) → 0 as t → ∞.
To this end, denote the five terms on the right hand side of (2.9) by I1 , I2 , . . . I5 ,
respectively. It is obvious that the first term I1 tends to zero as t → ∞, by condition
(2.6). Also, due to the facts that ϕ(t) → 0 and t − τj (t) → ∞ for j = 1, 2, . . . , N as
t → ∞, the second term I2 in (2.9) tends to zero as t → ∞. What is left to show
is that each of the remaining terms in (2.9) go to zero at infinity.
Let ϕ ∈ Sψ be fixed. For a given ε > 0, we choose T0 > 0 large enough such
that t − τj (t) ≥ T0 , j = 1, 2, . . . , N , implies |ϕ(s)| < ε if s ≥ t − τj (t). Therefore,
the third term I3 in (2.9) satisfies
N Z t
X
|I3 | = |
[hj (s) + Bj (t, s)]ϕ(s)ds|
t−τj (t)
j=1
≤
N Z
X
j=1
≤ε
t
|hj (s) + Bj (t, s)||ϕ(s)|ds
t−τj (t)
N Z
X
j=1
t
|hj (s) + Bj (t, s)|ds ≤ αε < ε.
t−τj (t)
Thus, I3 → 0 as t → ∞. Now consider I4 . For the given ε > 0, there exists a T1 > 0
such that s ≥ T1 implies |ϕ(s − τj (s))| < ε for j = 1, 2, . . . , N . Thus, for t ≥ T1 ,
the term I4 in (2.9) satisfies
N Z t
X
Rt
|I4 | = e− s H(u)du [hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)
j=1
0
8
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
× ϕ(s − τj (s))ds
N Z
X
≤
T1
e−
Rt
s
H(u)du [hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)
0
j=1
|ϕ(s − τj (s))|ds
N Z t
Rt
X
+
e− s H(u)du |[hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)
T1
j=1
|ϕ(s − τj (s))|ds
N Z
X
≤ sup |ϕ(σ)|
σ≥m(0)
T1
e−
Rt
H(u)du s
[hj (s − τj (s)) + Bj (s, s − τj (s))]
0
j=1
× (1 − τj0 (s)) − rj (s)ds
N Z t
Rt
X
+ε
e− s H(u)du |[hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)|ds.
T1
j=1
By (2.6), we can find T2 > T1 such that t ≥ T2 implies
sup |ϕ(σ)|
σ≥m(0)
N Z
X
T1
e−
Rt
H(u)du
s
|[hj (s − τj (s)) + Bj (s, s − τj (s))]
0
j=1
× (1 − τj0 (s)) − rj (s)|ds
−
= sup |ϕ(σ)|e
Rt
H(u)du
T2
σ≥m(0)
N Z
X
j=1
T1
e−
RT
s
2
H(u)du
|[hj (s − τj (s)) + Bj (s, s − τj (s))]
0
× (1 − τj0 (s)) − rj (s)|ds < ε.
Now, apply (2.5) to have |I4 | < ε + αε < 2ε. Thus, I4 → 0 as t → ∞. Similarly, by
using (2.5), then, if t ≥ T2 then term I5 in (2.9) satisfies
N
X
|I5 | = Z
t
−
e
Rt
s
H(u)du
Z
H(s)
0
j=1
≤ sup |ϕ(σ)|e
Rt
T2
H(u)du
σ≥m(0)
×
[hj (u) + Bj (s, u)]ϕ(u)du ds
s−τj (s)
−
Z
s
N Z
X
j=1
s
T1
e−
RT
s
2
H(u)du
|H(s)|
0
|hj (u) + Bj (s, u)|du ds
s−τj (s)
+ε
N Z t
X
j=1
e−
T1
Rt
s
H(u)du
Z
|H(s)|
s
|hj (u) + Bj (s, u)|du ds
s−τj (s)
< ε + αε < 2ε.
Thus, I5 → 0 as t → ∞. In conclusion (P ϕ)(t) → 0 as t → ∞, as required. Hence
P maps Sψ into Sψ . Also, by condition (2.5), P is a contraction mapping with
contraction constant α. Indeed, for φ, η ∈ Sψ and t > 0
|(P ϕ)(t) − (P η)(t)|
EJDE-2012/172
≤
N
X
j=1
+
|
N Z
X
N Z
X
j=1
9
cj (t)
||ϕ(t − τj (t)) − η(t − τj (t))|
1 − τj0 (t)
t
|hj (s) + Bj (t, s)||ϕ(s) − η(s)|ds
t−τj (t)
j=1
+
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
t
e−
Rt
s
H(u)du
|[hj (s − τj (s)) + Bj (s, s − τj (s))]
0
× (1 − τj0 (s)) − rj (s)||ϕ(s − τj (s)) − η(s − τj (s))|ds
N Z t
Z s
Rt
X
+
e− s H(u)du H(s)
|hj (u) + Bj (s, u)||ϕ(u) − η(u)|du ds
0
j=1
≤
X
N
|
j=1
+
cj (t)
|+
1 − τj0 (t)
j=1
N Z
X
j=1
+
s−τj (s)
N Z t
X
t
Rt
−
Rt
s
H(u)du
|[hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)|ds
H(u)du
Z
H(s)
0
N Z t
X
j=1
e−
|hj (s) + Bj (t, s)|ds
t−τj (t)
e
s
0
s
|hj (u) + Bj (s, u)|du ds kϕ − ηk.
s−τj (s)
By condition (2.5), P is a contraction mapping with constant α. By the contraction mapping principle (Smart [19, p. 2]), P has a unique fixed point x in Sψ
which is a solution of (1.1) with x(t) = ψ(t) on [m(0), 0] and x(t) = x(t, 0, ψ) → 0
as t → ∞.
To obtain the asymptotic stability, we need to show that the zero solution of (1.1)
is stable. Let ε > 0 be given and choose δ > 0 (δ < ε) satisfying 2δK + αε < ε. If
x(t) = x(t, 0, ψ) is a solution of (1.1) with kψk < δ, then x(t) = (P x)(t) defined in
((2.9). We claim that |x(t)| < ε for all t ≥ t0 . Notice that |x(s)| < ε on [m(0), 0].
If there exists t∗ > 0 such that |x(t∗ )| = ε and |x(s)| < ε for m(0) ≤ s < t∗ , then
it follows from (2.9) that
|x(t∗ )|
N Z 0
N
R t∗
X
X
cj (0)
|
+
|h
(s)
+
B
(0,
s)|ds
e− 0 H(u)du
≤ kψk 1 +
|
j
j
0 (0)
1
−
τ
−τ
(0)
j
j
j=1
j=1
N Z t∗
X
cj (t∗ )
+ε
|
|hj (s) + Bj (t∗ , s)|ds
0 (t∗ ) | + ε
1
−
τ
∗ −τ (t∗ )
t
j
j
j=1
j=1
Z t∗ R ∗
t
+ε
e− s H(u)du |[hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)|ds
N
X
t0
+ε
N Z
X
j=1
t∗
−
e
R t∗
t0
≤ 2δK + αε < ε,
s
H(u)du
Z
|H(s)|(
s
s−τj (s)
|hj (u) + Bj (s, u)|du)ds
10
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
which contradicts the definition of t∗ . Thus, |x(t)| < ε for all t ≥ 0, and the zero solution of (1.1) is stable. This shows that the zero solution of (1.1) is asymptotically
stable if (2.6) holds.
Conversely, suppose (2.6) fails. Then, by (2.4) there exists a sequence {tn },
Rt
tn → ∞ as n → ∞ such that lim 0 n H(u)du = l for some l ∈ R. We may also
n→∞
choose a positive constant J satisfying
Z tn
−J ≤
H(u)du ≤ J,
0
for all n ≥ 1. To simplify our expressions, we define
ω(s) =
N
X
|[hj (s − τj (s)) + Bj (s, s − τj (s))](1 − τj0 (s)) − rj (s)|
j=1
+
N
X
Z
|H(s)|(
s
|hj (u) + B(s, u)|du),
s−τj (s)
j=1
for all s ≥ 0. By (2.5), we have
Z tn
e−
R tn
s
H(u)du
ω(s)ds ≤ α.
0
This yields
tn
Z
Rs
e
0
H(u)du
ω(s)ds ≤ αe
R tn
0
H(u)du
≤ J.
0
Rt Rs
The sequence { 0 n e 0 H(u)du ω(s)ds} is bounded, so there exists a convergent subsequence. For brevity of notation, we may assume that
Z tn R
s
lim
e 0 H(u)du ω(s)ds = γ,
n→∞
0
for some γ ∈ R+ and choose a positive integer m so large that
Z tn R
s
e 0 H(u)du ω(s)ds < δ0 /4K,
tm
for all n ≥ m, where δ0 > 0 satisfies 2δ0 KeJ + α ≤ 1.
By (2.4), K in (2.7) is well defined. We now consider the solution x(t) =
x(t, tm , ψ) of (1.1) with ψ(tm ) = δ0 and |ψ(s)| ≤ δ0 for s ≤ tm . We may choose ψ
so that |x(t)| ≤ 1 for t ≥ tm and
ψ(tm ) −
N
X
j=1
Z
[
cj (tm )
ψ(tm − τj (tm ))
1 − τj0 (tm )
tm
+
[hj (s) + Bj (tm , s)]ψ(s)ds]
tm −τj (tm )
≥
1
δ0 .
2
EJDE-2012/172
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
11
It follows from (2.9) with x(t) = (P x)(t) that for n ≥ m
Z tn
N h
i
X
cj (tn )
x(t
−
τ
(t
))
+
[h
(s)
+
B
(t
,
s)]x(s)ds
x(tn ) −
n
j
n
j
j
n
0
1 − τj (tn )
tn −τj (tn )
j=1
Z tn R
R tn
tn
1
e− s H(u)du ω(s)ds
≥ δ0 e− tm H(u)du −
2
tm
Z tn R
R tn
R
s
1 − t H(u)du
− 0tn H(u)du
m
−e
e 0 H(u)du ω(s)ds
= δ0 e
2
tm
Z tn R
1
R tn
R tm
s
e 0 H(u)du ω(s)ds
= e− tm H(u)du δ0 − e− 0 H(u)du
2
tm
Z tn R
1
R tn
s
≥ e− tm H(u)du δ0 − K
e 0 H(u)du ω(s)ds
2
tm
1 −2J
1 − Rttn H(u)du
≥ δ0 e
> 0.
≥ δ0 e m
4
4
(2.10)
On the other hand, if the zero solution of (1.1) is asymptotically stable, then
x(t) = x(t, tm , ψ) → 0 as t → ∞. Since tn − τj (tn ) → ∞ as n → ∞ and (2.5) holds,
we have
Z tn
N h
i
X
cj (tn )
x(t
−
τ
(t
))
+
[h
(s)
+
B
(t
,
s)]x(s)ds
→0
x(tn ) −
n
j
n
j
j
n
1 − τj0 (tn )
tn −τj (tn )
j=1
as n → ∞, which contradicts (2.10). Hence condition (2.6) is necessary for the
asymptotic stability of the zero solution of (1.1). The proof is complete.
Remark 2.3. It follows from the first part of the proof of Theorem 2.2 that the
zero solution of (1.1) is stable under (2.4) and (2.5). Moreover, Theorem 2.2 still
holds if (2.5) is satisfied for t ≥ tσ for some tσ ∈ R+ .
For the special case N = 1 and c1 = 0, we have the following result.
Corollary 2.4. Suppose that τ1 is differentiable and there exist continuous function
h1 : [m1 (0), ∞) → R and a constant α ∈ (0, 1) such that for t ≥ 0
Z t
lim inf
h1 (s)ds > −∞,
(2.11)
t→∞
0
and
Z
t
|h1 (s) + B1 (t, s)|ds
t−τ1 (t)
Z t R
t
+
e− s h1 (u)du |h1 (s − τ1 (s)) + B1 (s, s − τ1 (s))||1 − τ10 (s)|ds
0
Z t R
Z s
t
e− s h1 (u)du |h1 (s)|(
|h1 (u) + B1 (s, u)|du)ds ≤ α.
+
0
s−τ1 (s)
Then the zero solution of (1.2) is asymptotically stable if and only if
Z t
h1 (s)ds → ∞ as t → ∞.
0
(2.12)
(2.13)
12
A. ARDJOUNI, A. DJOUDI, I. SOUALHIA
EJDE-2012/172
Obviously, Corollary 2.4 extends Theorem 1.3. Thus, Theorem 2.2 generalizes
Theorem 1.3.
3. An Example
In this section, we give an example to illustrate the applications of Theorem 2.2.
Example 3.1. Consider the linear neutral integro-differential equation with variable delays
2 Z t
2
X
X
0
x (t) = −
aj (t, s)x(s)ds +
cj (t)x0 (t − τj (t)),
(3.1)
t−τj (t)
j=1
j=1
where τ1 (t) = 0.489t, τ2 (t) = 0.478t, a1 (t, s) = 0.48/(s2 +1), a2 (t, s) = 0.52/(s2 +1),
c1 (t) = 0.015, c2 (t) = 0.017. Then the zero solution of (3.1) is asymptotically
stable.
Proof. We have
Z s
0.48
0.48(s − t)
B1 (t, s) =
du =
,
2+1
s
s2 + 1
t
Z
B2 (t, s) =
t
2
s
0.52
0.52(s − t)
du =
.
s2 + 1
s2 + 1
2
Choosing h1 (t) = 0.52t/(t + 1) and h2 (t) = 0.48t/(t + 1) in Theorem 2.2, we have
H(t) = t/(t2 + 1) and
2
X
|
j=1
2 Z
X
0.017
cj (t)
0.015
|+|
| < 0.062,
|=|
1 − τj0 (t)
1 − 0.489
1 − 0.478
t
|hj (s) + Bj (t, s)| ds
t−τj (t)
j=1
Z t
s − 0.48t
s − 0.52t
|ds
+
| 2
|ds
2+1
s
s +1
0.511t
0.522t
Z t
Z t
s − 0.52t
s − 0.48t
ds
+
ds
=
2+1
2
s
0.511t
0.522t s + 1
= t[0.48 arctan 0.511t + 0.52 arctan 0.522t − arctan t] + ln(t2 + 1)
1
− [ln(0.5112 t2 + 1) + ln(0.5222 t2 + 1)]
2
= ω(t).
Z
t
|
=
Since the function ω is increasing in [0, ∞) and
lim ω(t) = 1 − 0.48/0.511 − 0.52/0.522 − ln(0.511 × 0.522) ' 0.386,
t→∞
then
2 Z
X
j=1
2 Z
X
j=1
0
t
e−
Rt
s
H(u)du
t
|hj (s) + Bj (t, s)| ds < 0.386,
t−τj (t)
Z
|H(s)|
s
s−τj (s)
|hj (u) + Bj (s, u)|du ds < 0.386,
EJDE-2012/172
STABILITY FOR INTEGRO-DIFFERENTIAL EQUATIONS
13
and
2 Z
X
t
e−
Rt
s
H(u)du
|[hj (s − τj (s)) + B(s, s − τj (s))](1 − τj0 (s)) − rj (s)|ds
0
j=1
Z
t
=
−
e
0
t
Rt
u
du
s u2 +1
|0.511(
0.52 × 0.511s 0.48(0.511s − s)
0.015s
+
)−
|ds
0.5112 s2 + 1
0.5112 s2 + 1
0.511(s2 + 1)
0.48 × 0.522s 0.52(0.522s − s)
0.017s
+
)−
|ds
2
2
2
2
0.522 s + 1
0.522 s + 1
0.522(s2 + 1)
0
Z t R
Z
s
0.015 t − Rst u2u+1 du s
0.48
− t u du
ds
+
ds
)
≤ (1 −
e s u2 +1
e
0.511 0
s2 + 1/0.5112
0.511 0
s2 + 1
Z t R
Z
s
0.52
0.017 t − Rst u2u+1 du s
− t u du
+ (1 −
)
ds
+
ds
e s u2 +1
e
0.522 0
s2 + 1/0.5222
0.522 0
s2 + 1
0.015
0.52
0.017
0.48
+
+1−
+
< 0.127.
<1−
0.511 0.511
0.522 0.522
It is easy to see that all the conditions of Theorem 2.2 hold for α = 0.062 + 0.386 +
0.386 + 0.127 = 0.961 < 1. Thus, Theorem 2.2 implies that the zero solution of
(3.1) is asymptotically stable.
Z
+
−
e
Rt
u
du
s u2 +1
|0.522(
Acknowledgements. The authors would like to express their sincere thanks to
the anonymous referee for his/her careful reading of our manuscript and the helpful
report.
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Abdelouaheb Ardjouni
Applied Mathematics Lab., Facullty of Sciences, Department of Mathematics, Badji
Mokhtar University of Annaba, P.O. Box 12, Annaba 23000, Algeria
E-mail address: abd ardjouni@yahoo.fr
Ahcene Djoudi
Applied Mathematics Lab., Facullty of Sciences, Department of Mathematics, Badji
Mokhtar University of Annaba, P.O. Box 12, Annaba 23000, Algeria
E-mail address: adjoudi@yahoo.com
Imene Soualhia
Applied Mathematics Lab., Facullty of Sciences, Department of Mathematics, Badji
Mokhtar University of Annaba, P.O. Box 12, Annaba 23000, Algeria