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Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form
Applied Mathematics Letters 24 (2011) 1218–1224
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form Kaizhong Guan a,∗ , Jianhua Shen b a
Research Institute of Mathematics, University of South China, Hengyang, Hunan 421001, PR China
b
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, PR China
article
abstract
info
Article history: Received 9 October 2009 Received in revised form 13 February 2011 Accepted 14 February 2011
This paper is concerned with an impulsive neutral differential equation of Euler form with unbounded delays
d P (t ) x(β t ) = 0, t ≥ t0 > 0, t ̸= tk , [x(t ) − C (t )x(α t )] + dt t∫ t k P (s/β) x(t ) = b x(t − ) + (1 − b ) x(s)ds, k = 1, 2, . . . . k k k k
Keywords: Asymptotic behavior Neutral differential equation Lyapunov functional Impulse Euler form
β tk
(∗)
s
Sufficient conditions are obtained for every solution of (∗) to tend to a constant as t → ∞. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive differential equations are now recognized as an excellent source of models to simulate processes and phenomena observed in control theory, physics, chemistry, population dynamics, industrial robotics, economics, etc. [1,2]. In recent years, there has been increasing interest in the oscillation and stability theory of impulsive delay differential equations and many results have been obtained (see [3–6] and the references cited therein). In particular, stability of some impulsive neutral differential equations with constant delays has also been studied by several authors [7,8]. However, to the best of our knowledge, there is very little in the way of results for the asymptotic behavior of solutions of impulsive neutral differential equations with variable delays though there are many results on the qualitative properties of delay differential equations with variable delays [9,10]. In this paper, we consider the asymptotic behavior of solutions of the following neutral differential equation in Euler form with impulsive perturbations d dt
[x(t ) − C (t )x(α t )] +
P (t ) t
x(tk ) = bk x(tk− ) + (1 − bk )
∫
x(β t ) = 0, tk
β tk
P (s/β) t
t ≥ t0 > 0, t ̸= tk ,
x(s)ds,
k = 1, 2, . . . ,
(1.1) (1.2)
where 0 < α, β < 1, C (t ), P (t ) ∈ PC ([t0 , ∞), R) and P (t ) ≥ 0, 0 < t0 < tk < tk+1 , with limt →∞ tk = ∞, and bk , k = 1, 2, . . . , are constants. PC ([t0 , ∞), R) denotes the set of all functions g : [t0 , ∞) → R such that g is continuous for tk ≤ t < tk+1 and limt →t − g (t ) = g (tk− ). k
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (K. Guan).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.02.012
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
1219
In system (1.1) and (1.2) the impulsive term is also delayed, that is, it contains an integral term. When all bk = 1, k = 1, 2, . . . , the system (1.1) and (1.2) reduces to the first-order neutral differential equation of Euler form d dt
[x(t ) − C (t )x(α t )] +
P (t ) t
x(β t ) = 0,
t ≥ t0 > 0.
(1.3)
The oscillatory behavior of solutions of (1.3) and its corresponding differential equation with certain impulsive perturbations was investigated in [11,12], respectively. However, the asymptotic behavior of solutions of such equations is not still studied. Thus, there is strong interest in investigating such problems. The main purpose of this paper is to investigate the asymptotic behavior of solutions of the system (1.1) and (1.2). As a consequence, some sufficient conditions are obtained for the asymptotic stability of solutions of (1.3). With the system (1.1) and (1.2), one associates an initial condition of the form xt0 = φ(η),
η ∈ [ρ, 1],
(1.4)
where ρ = min{α, β}, xt0 = x(ηt0 ) for ρ ≤ η ≤ 1 and φ ∈ PC ([ρ, 1], R) = {φ : [ρ, 1] → R|φ is continuous everywhere except at a finite number of points η¯ , and φ(η¯ − ) and φ(η¯ + ) = limη→η¯ + φ(η) exist with φ(η¯ + ) = φ(η)} ¯ . A function x(t ) is said to be a solution of (1.1) and (1.2) satisfying the initial value condition (1.4) if (i) x(t ) = φ(t /t0 ) for ρ t0 ≤ t ≤ t0 , x(t ) is continuous for t ≥ t0 and t ̸= tk , k = 1, 2, . . .; (ii) x(t ) − C (t )x(α t ) is continuously differentiable for t > t0 , t ̸= tk , k = 1, 2, . . . , and satisfies (1.1); (iii) x(tk+ ) and x(tk− ) exist with x(tk+ ) = x(tk ) and satisfy (1.2). Using the method of steps as in the case without impulses, one can show the global existence and uniqueness of the solution of the initial problem (1.1), (1.2) and (1.4). A solution of (1.1) and (1.2) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, the solution is said to be oscillatory. 2. Main results Theorem 2.1. Assume that the following conditions hold: ∞ − (1 − bk ) < ∞;
α tk is not an impulsive point , 0 < bk ≤ 1(k = 1, 2, . . .) and
(2.1)
k=1
C (tk ) = bk C (tk− );
(2.2)
lim |C (t )| = µ < 1;
(2.3)
[ ∫ t /β ] P (t /αβ) P (s/β) lim sup µ 1 + + ds < 2. P (t /β) s t →∞ βt
(2.4)
t →∞
Then every solution of (1.1) and (1.2) tends to a constant as t → ∞. Proof. Let x(t ) be any solution of (1.1) and (1.2). We shall prove that limt →∞ x(t ) exists and is finite. For this purpose, we rewrite the system (1.1) and (1.2) in the form
[
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
x(tk ) = bk x(tk− ) + (1 − bk )
P (s/β)
tk
∫
x(s)ds
]′
s
β tk
+
P (t /β) t
x(s)ds,
x(t ) = 0,
t ≥ t0 , t ̸= tk ,
k = 1, 2, . . . .
(2.5)
(2.6)
From (2.3) and (2.4), we can select λ > 0 and δ > 0 sufficiently small and T > t0 sufficiently large such that µ + λ < 1,
[
(µ + λ) 1 +
P (t /αβ) P (t /β)
∫
t /β
+ βt
P (s/β) s
]
ds < 2 − δ,
for t ≥ T ,
(2.7)
and
|C (t )| ≤ µ + λ,
for t ≥ T .
(2.8)
In what follows, for the sake of convenience, when we write a functional inequality without specifying its domain of validity, we mean that it holds for all sufficiently large t. Now we introduce two functionals:
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
[
V1 (t ) = x(t ) − C (t )x(α t ) − V2 ( t ) =
P (s/β 2 )
t
∫
s
βt
dt
u
dV1 dt
and
dV2 dt
= −2 x(t ) − C (t )x(α t ) − =−
≤−
P (t /β) t P (t /β)
]2
,
x (u)duds + (µ + λ) 2
P (s/αβ)
t
∫
s
αt
∫
t
P (s/β)
]
x(s)ds
s
βt
2x (t ) − 2C (t )x(t )x(α t ) −
t P (s/β)
t
∫
2
P (t /β)
s
βt
x(t )
2x(s)x(t )ds
2x (t ) − |C (t )|x (α t ) − |C (t )|x (t ) − x (t ) 2
t
x2 (s)ds.
along the solution of (1.1), we have
[
[
x(s)ds
s
βt
P (u/β)
s
At t ̸= tk , calculating dV1
t
∫
P (s/β)
t
∫
2
2
2
∫
t
]
P (s/β) s
βt
∫
t
ds −
P (s/β)
βt
s
x (s)ds , 2
and dV2 dt
=−
P (t /β)
∫
t
t
P (u/β) u
βt
+ (µ + λ)
P (t /αβ) t
P (t /β)
x (u)du + 2
t
x2 (t ) − (µ + λ)
x (t ) 2
P (t /β) t
∫
t
P (s/β 2 ) s
βt
ds
x2 (α t ).
Let V (t ) = V1 (t ) + V2 (t ). From the above two inequalities and (2.7) and (2.8), it follows that dV dt
=
dV1
≤−
dV2
+
dt
dt
P (t /β)
− x (t )
≤−
2x2 (t ) − |C (t )|x2 (t ) − x2 (t )
t 2
=−
t P (t /β)
≤ −δ
ds + (µ + λ)
[
x (t ) 2 − |C (t )| − 2
[
x (t ) 2 −
P (t /β) t
s
βt
P (t /β)
t
P (s/β 2 )
t
∫
2
∫
t /β
∫
t /β
s
βt
t P (s/β) s
s
P (s/β)
t
P (t /αβ)
βt
P (s/β)
βt
∫
ds
x2 ( t )
ds − (µ + λ)
ds − (µ + λ) 1 +
P (t /αβ)
]
P (t /β)
P (t /αβ)
]
P (t /β)
x2 (t ).
(2.9)
At t = tk , from (1.2) and (2.1), it follows that
[
V (tk ) =
x(tk ) − C (tk )x(α tk ) −
∫
tk
+ β tk
P (s/β 2 )
tk
∫
s
tk
∫
P (s/β)
P (u/β) u
s
s
β tk
x2 (u)duds + (µ + λ)
[ ∫ = b2k x(tk− ) − C (tk− )x(α tk ) −
tk
β tk
∫
tk
+ β tk
P (s/β ) 2
tk
∫
s
P (u/β) u
s
≤ V (tk− ).
x(s)ds
]2
P (s/β) s
x(s)ds
tk
α tk
P (s/αβ) s
x2 (s)ds
]2
x (u)duds + (µ + λ) 2
∫
∫
tk
α tk
P (s/αβ) s
x2 (s)ds (2.10)
This and (2.9) show that V (t ) is decreasing. In view of the fact that V (t ) ≥ 0, limt →∞ V (t ) = γ exists and γ ≥ 0. By using (2.9) and (2.10) again, we can easily get ∞
∫ T
P (t /β) t
x2 (t )dt ≤
V (T )
δ
.
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
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This implies that P (t /β) t
x2 (t ) ∈ L1 (t0 , ∞), P (s/β) 2 x s
t
and hence for any 0 < η < 1 we have limt →∞ ηt t
∫
P (s/β 2 ) s
βt
P (u/β)
t
∫
u
s
x2 (u)duds ≤
(s)ds = 0. Thus, it follows from (2.4) that ∫ t P (s/β 2 ) P (u/β) 2 ds x (u)du
t
∫
s
βt
t /β
∫
P (s/β)
=
s
t t /β
∫
s
βt
u
βt
ds
P (u/β) u
t
∫ ds
P (u/β) u
βt
P (u/β)
t
∫ ≤2
t
βt
P (s/β)
≤
u
βt
∫
x2 (u)du → 0,
x2 (u)du x2 (u)du as t → ∞,
and
(µ + λ)
∫
t
P (s/αβ) s
αt
x2 (s)ds = (µ + λ)
P (s/β)
αt
P (s/β)
t
∫
P (s/αβ)
t
∫
≤2
s
αt
·
P (s/β) s
x2 (s)ds → 0,
x2 (s)ds
as t → ∞.
It follows that limt →∞ V2 (t ) = 0. Thus, limt →∞ V1 (t ) = limt →∞ V (t ) = γ , that is,
[ lim
t →∞
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
]2
x(s)ds
= γ.
(2.11)
Next, we shall prove that the limit
[ lim
t →∞
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
]
x(s)ds
t
exists and is finite. Set y(t ) = x(t ) − C (t )x(α t ) − β t y(tk ) = x(tk ) − C (tk )x(α tk ) −
[
∫
tk
P (s/β)
= bk x(tk ) − C (tk )x(α tk ) − −
−
s
β tk
tk
∫
P (s/β) x s
x(s)ds
P (s/β) s
β tk
(s)ds, from (1.2) and (2.1), we then have
x(s)ds
]
= bk y(tk− ).
(2.12)
From (2.11), it follows that lim y2 (t ) = γ .
(2.13)
t →∞
Moreover, in view of (2.5) and (2.12), system (2.5) and (2.6) can be rewritten as
y′ (t ) +
P (t /β)
x(t ) = 0, t ≥ t0 > 0, t ̸= tk , t y(tk ) = bk y(tk− ), k = 1, 2, . . . .
(2.14)
If γ = 0, then limt →∞ y(t ) = 0. If γ > 0, then there exists a sufficiently large T1 such that y(t ) ̸= 0 for t ≥ T1 . Otherwise, there is a sequence {τk } with limk→∞ τk = ∞ such that y(τk ) = 0, and so y2 (τk ) → 0 as k → ∞. This contradicts γ > 0. Therefore, for any tk > T1 , t ∈ [tk , tk+1 ), we have y(t ) > 0 or y(t ) < 0 because y(t ) is continuous on [tk , tk+1 ). Without loss of generality, we assume that y(t ) > 0 on [tk , tk+1 ). It follows that y(tk+1 ) = bk y(tk−+1 ) > 0, and thus y(t ) > 0 on [tk+1 , tk+2 ). By induction, we can conclude that y(t ) > 0 on [tk , ∞). This and (2.11) imply that lim y(t ) = lim
t →∞
t →∞
x(t ) − C (t )x(α t ) −
∫
t
βt
P (s/β) s
x(s)ds
= ν,
(2.15)
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
√ γ and is finite. In view of (2.14), we have
where ν =
∫
t
P (s/β) s
βt
x(s)ds = y(β t ) − y(t ) −
t
lim
t →∞
(y(tk ) − y(tk− )) = y(β t ) − y(t ) −
β t
Letting t → ∞ and noticing that
∫
−
P (s/β)
k=1
(1 − bk )y(tk− ).
β t < tk < t
(1 − bk ) < ∞, we have
x(s)ds = 0.
s
βt
∑∞
−
(2.16)
It follows from (2.15) and (2.16) that lim (x(t ) − C (t )x(α t )) = ν.
(2.17)
t →∞
Finally, we shall prove that lim x(t ) exists and is finite.
(2.18)
t →∞
To this end, we need to show that |x(t )| is bounded. As a matter of fact, if |x(t )| is unbounded, then there exists a sequence {sn } such that sn → ∞, |x(s− n )| → ∞ as n → ∞, and
|x(s− n )| = sup |x(sn )|, t0 ≤t ≤sn
where, if sn is not an impulsive point, then x(s− n ) = x(sn ). Thus, we have − − − − |x(s− n ) − C (sn )x(α sn )| ≥ |x(sn )| − |C (sn )||x(α sn )| ≥ |x(sn )|(1 − µ − λ) → ∞,
as n → ∞,
which contradicts (2.17) and so |x(t )| is bounded. If µ = 0, clearly limt →∞ x(t ) = ν , which shows that (2.18) holds. If 0 < µ < 1, one can see that C (t ) is eventually positive or negative. Otherwise, there is a sequence {τk } with limk→∞ τk = ∞ such that C (τk ) = 0, and so C (τk ) → 0 as k → ∞. It is a contradiction with µ > 0. By condition (2.3), one can find a sufficiently large T2 such that for t > T2 , |C (t )| < 1. Set
ξ = lim inf x(t ), t →∞
ϱ = lim sup x(t ). t →∞
Then we can select two sequences {un } and {vn } such that un → ∞, vn → ∞ as n → ∞, and lim x(un ) = ξ ,
lim x(vn ) = ϱ.
n→∞
n→∞
For t > T2 , we consider the following two possible cases.
Case 1. When −1 < C (t ) < 0 for t > T2 , we have
ν = lim [x(un ) − C (un )x(α un )] ≤ ξ + µϱ, n→∞
and
ν = lim [x(vn ) − C (vn )x(αvn )] ≥ ϱ + µξ . n→∞
Thus, we obtain
ξ + µϱ ≥ ϱ + µξ , that is,
(1 − µ)ξ ≥ (1 − µ)ϱ. Since 0 < µ < 1 and ϱ ≥ ξ , it follows that ϱ = ξ . It follows from (2.17) that
ϱ=ξ =
ν 1+µ
,
which shows that (2.18) holds.
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
1223
Case 2. When 0 < C (t ) < 1 for t > T2 , we have
ξ = lim x(un ) = lim [(x(un ) − C (un )x(α un )) + C (un )x(α un )] ≥ ν + µξ . n→∞
n→∞
It follows that
ν
ξ≥
1−µ
.
Similarly,
ϱ = lim x(vn ) = lim [(x(vn ) − C (vn )x(αvn )) + C (vn )x(αvn )] ≤ ν + µϱ, n→∞
n→∞
ν ν which implies ϱ ≤ 1−µ . Thus, ξ = ϱ = 1−µ and so (2.18) holds. Summarizing the above discussion, we conclude that (2.18) holds and so the proof is completed. By Theorem 2.1, we have the following asymptotic behavior result immediately.
Theorem 2.2. The conditions of Theorem 2.1 imply that every oscillatory solution of (1.1) and (1.2) tends to zero as t → ∞. In Theorem 2.1, taking bk ≡ 1, k = 1, 2, . . . , we obtain the following Corollary 2.3. Assume that (2.3) and (2.4) hold. Then every solution of (1.3) tends to a constant as t → ∞. Theorem 2.4. The conditions of Theorem 2.1 together with ∞
∫
P (s/β) s
t0
ds = ∞
(2.19)
imply that every solution of (1.1) and (1.2) tends to zero as t → ∞. Proof. By Theorem 2.2, we only have to prove that every nonoscillatory solution of (1.1) and (1.2) tends to zero as t → ∞. Without loss of generality, let x(t ) be an eventually positive solution of (1.1) and (1.2), we shall prove that limt →∞ x(t ) = 0. As in the proof of Theorem 2.1, we can rewrite (1.1) and (1.2) in the form (2.14). Integrating from t0 to t both sides of (2.14) yields t
∫
P (s/β) s
t0
∑∞
k =1
P (s/β)
∞
s
t0
− t0 < tk < t
Using (2.15) and
∫
x(s)ds = y(t0 ) − y(t ) −
(1 − bk )y(tk− ).
(1 − bk ) < ∞, we obtain
x(s)ds < ∞.
This, together with (2.19), implies that lim inft →∞ x(t ) = 0. By Theorem 2.1, limt →∞ x(t ) = 0 and so the proof is completed. The following corollary follows from Corollary 2.3 and Theorem 2.4. Corollary 2.5. Assume that (2.3), (2.4) and (2.19) hold. Then every solution of (1.3) tends to zero as t → ∞. 3. Examples In this section, we give two examples to illustrate the usefulness of our main results. Example 3.1. Consider the neutral differential equation d dt
[
x( t ) −
1 4
]
x(t /2) +
1 8t
x(t /4) = 0,
t ≥ 1.
(3.1)
One can easily see that the conditions (2.3), (2.4) and (2.19) hold. By Corollary 2.5, every solution of (3.1) tends to zero as t → ∞. Indeed, x(t ) = 1/t (t ≥ 1) is such a solution. Example 3.2. Consider the impulsive neutral differential equation
d [x(t ) − C (t )x(t /e)] +
1 x(t /e) = 0, t ≥ t0 = 3, t ̸= k, 4t∫(ln t − 1) k 2 k −1 1 1 x(k− ) + 2 x(s)ds, k = 4, 5, . . . , x(k) = k2 k k/e 4s ln s dt
where P (t ) = 4(ln1t −1) , bk =
k2 −1 , k2
and C (t ) = 4(k−k 1)2 [t ], t ∈ [k − 1, k), k = 4, 5, . . . .
(3.2)
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
One can easily find that lim |C (t )| =
t →∞
1 4
< 1,
k2 − 1
C (k) =
k2
∫
C (k− ),
∞
P (s/β)
t0
s
∞
∫ ds = 3
1
ds = ∞,
4s ln s
and
[
lim sup µ 1 + t →∞
P (t /αβ) P (t /β)
∫
t /β
+ βt
P (s/β) s
] ds
[ 1
= lim sup
4
t →∞
= lim sup t →∞
=
1 2
1 4
1+
1+
ln t 1 + ln t ln t
1 + ln t
∫
et
+ t /e
+ ln
]
1 4s ln s
ln t + 1
ds
ln t − 1
< 2.
By Theorem 2.4, every solution of (3.2) tends to zero as t → ∞. Acknowledgement The authors are very grateful to the referee for his (her) suggestions for the improvement of this paper. References [1] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [2] A.M. Samoilenko, A.V. Perestynk, Differential Equations with Impulsive Effect, Visca Skola, Kive, 1987. [3] D.D. Bainov, M.B. Dinitrova, A.B. Dishliev, Oscillation of the solutions of impulsive differential equations and inequalities with a retarded argument, Rocky Mountain J. Math. 28 (1998) 25–40. [4] Zh.G. Luo, J.H. Shen, Stability and boundedness for impulsive differential equations with infinite delays, Nonlinear Anal. 46 (2001) 475–493. [5] J.R. Graef, J.H. Shen, I.P. Stavroulakis, Oscillation of impulsive neutral delay differential equations, J. Math. Anal. Appl. 268 (2002) 310–333. [6] J. Yan, A. Zhao, Oscillation and stability of linear impulsive differential equations, J. Math. Anal. Appl. 227 (1998) 187–194. [7] X.Z. Liu, J.H. Shen, Asymptotic behavior of solutions of impulsive neutral differential equations, Appl. Math. Lett. 12 (1999) 51–58. [8] J.H. Shen, Y.J. Liu, J.L. Li, Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses, J. Math. Anal. Appl. 332 (2007) 179–189. [9] Q.R. Wang, Oscillation criteria for first-order neutral differential equations, Appl. Math. Lett. 8 (2002) 1025–1033. [10] T. Yoneyama, The 3/2 stability theorem for one-dimensional delay-differential equations with unbounded delay, J. Math. Anal. Appl. 165 (1992) 133–143. [11] K.Z. Guan, J.H. Shen, On first-order neutral differential equations of Euler form with unbounded delay, Appl. Math. Comput. 189 (2007) 1419–1427. [12] K.Z. Guan, J.H. Shen, Oscillation criteria for a first-order impulsive neutral differential equations of Euler form, Comput. Math. Appl. 58 (2009) 670–677.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form Kaizhong Guan a,∗ , Jianhua Shen b a
Research Institute of Mathematics, University of South China, Hengyang, Hunan 421001, PR China
b
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, PR China
article
abstract
info
Article history: Received 9 October 2009 Received in revised form 13 February 2011 Accepted 14 February 2011
This paper is concerned with an impulsive neutral differential equation of Euler form with unbounded delays
d P (t ) x(β t ) = 0, t ≥ t0 > 0, t ̸= tk , [x(t ) − C (t )x(α t )] + dt t∫ t k P (s/β) x(t ) = b x(t − ) + (1 − b ) x(s)ds, k = 1, 2, . . . . k k k k
Keywords: Asymptotic behavior Neutral differential equation Lyapunov functional Impulse Euler form
β tk
(∗)
s
Sufficient conditions are obtained for every solution of (∗) to tend to a constant as t → ∞. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive differential equations are now recognized as an excellent source of models to simulate processes and phenomena observed in control theory, physics, chemistry, population dynamics, industrial robotics, economics, etc. [1,2]. In recent years, there has been increasing interest in the oscillation and stability theory of impulsive delay differential equations and many results have been obtained (see [3–6] and the references cited therein). In particular, stability of some impulsive neutral differential equations with constant delays has also been studied by several authors [7,8]. However, to the best of our knowledge, there is very little in the way of results for the asymptotic behavior of solutions of impulsive neutral differential equations with variable delays though there are many results on the qualitative properties of delay differential equations with variable delays [9,10]. In this paper, we consider the asymptotic behavior of solutions of the following neutral differential equation in Euler form with impulsive perturbations d dt
[x(t ) − C (t )x(α t )] +
P (t ) t
x(tk ) = bk x(tk− ) + (1 − bk )
∫
x(β t ) = 0, tk
β tk
P (s/β) t
t ≥ t0 > 0, t ̸= tk ,
x(s)ds,
k = 1, 2, . . . ,
(1.1) (1.2)
where 0 < α, β < 1, C (t ), P (t ) ∈ PC ([t0 , ∞), R) and P (t ) ≥ 0, 0 < t0 < tk < tk+1 , with limt →∞ tk = ∞, and bk , k = 1, 2, . . . , are constants. PC ([t0 , ∞), R) denotes the set of all functions g : [t0 , ∞) → R such that g is continuous for tk ≤ t < tk+1 and limt →t − g (t ) = g (tk− ). k
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (K. Guan).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.02.012
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
1219
In system (1.1) and (1.2) the impulsive term is also delayed, that is, it contains an integral term. When all bk = 1, k = 1, 2, . . . , the system (1.1) and (1.2) reduces to the first-order neutral differential equation of Euler form d dt
[x(t ) − C (t )x(α t )] +
P (t ) t
x(β t ) = 0,
t ≥ t0 > 0.
(1.3)
The oscillatory behavior of solutions of (1.3) and its corresponding differential equation with certain impulsive perturbations was investigated in [11,12], respectively. However, the asymptotic behavior of solutions of such equations is not still studied. Thus, there is strong interest in investigating such problems. The main purpose of this paper is to investigate the asymptotic behavior of solutions of the system (1.1) and (1.2). As a consequence, some sufficient conditions are obtained for the asymptotic stability of solutions of (1.3). With the system (1.1) and (1.2), one associates an initial condition of the form xt0 = φ(η),
η ∈ [ρ, 1],
(1.4)
where ρ = min{α, β}, xt0 = x(ηt0 ) for ρ ≤ η ≤ 1 and φ ∈ PC ([ρ, 1], R) = {φ : [ρ, 1] → R|φ is continuous everywhere except at a finite number of points η¯ , and φ(η¯ − ) and φ(η¯ + ) = limη→η¯ + φ(η) exist with φ(η¯ + ) = φ(η)} ¯ . A function x(t ) is said to be a solution of (1.1) and (1.2) satisfying the initial value condition (1.4) if (i) x(t ) = φ(t /t0 ) for ρ t0 ≤ t ≤ t0 , x(t ) is continuous for t ≥ t0 and t ̸= tk , k = 1, 2, . . .; (ii) x(t ) − C (t )x(α t ) is continuously differentiable for t > t0 , t ̸= tk , k = 1, 2, . . . , and satisfies (1.1); (iii) x(tk+ ) and x(tk− ) exist with x(tk+ ) = x(tk ) and satisfy (1.2). Using the method of steps as in the case without impulses, one can show the global existence and uniqueness of the solution of the initial problem (1.1), (1.2) and (1.4). A solution of (1.1) and (1.2) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, the solution is said to be oscillatory. 2. Main results Theorem 2.1. Assume that the following conditions hold: ∞ − (1 − bk ) < ∞;
α tk is not an impulsive point , 0 < bk ≤ 1(k = 1, 2, . . .) and
(2.1)
k=1
C (tk ) = bk C (tk− );
(2.2)
lim |C (t )| = µ < 1;
(2.3)
[ ∫ t /β ] P (t /αβ) P (s/β) lim sup µ 1 + + ds < 2. P (t /β) s t →∞ βt
(2.4)
t →∞
Then every solution of (1.1) and (1.2) tends to a constant as t → ∞. Proof. Let x(t ) be any solution of (1.1) and (1.2). We shall prove that limt →∞ x(t ) exists and is finite. For this purpose, we rewrite the system (1.1) and (1.2) in the form
[
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
x(tk ) = bk x(tk− ) + (1 − bk )
P (s/β)
tk
∫
x(s)ds
]′
s
β tk
+
P (t /β) t
x(s)ds,
x(t ) = 0,
t ≥ t0 , t ̸= tk ,
k = 1, 2, . . . .
(2.5)
(2.6)
From (2.3) and (2.4), we can select λ > 0 and δ > 0 sufficiently small and T > t0 sufficiently large such that µ + λ < 1,
[
(µ + λ) 1 +
P (t /αβ) P (t /β)
∫
t /β
+ βt
P (s/β) s
]
ds < 2 − δ,
for t ≥ T ,
(2.7)
and
|C (t )| ≤ µ + λ,
for t ≥ T .
(2.8)
In what follows, for the sake of convenience, when we write a functional inequality without specifying its domain of validity, we mean that it holds for all sufficiently large t. Now we introduce two functionals:
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
[
V1 (t ) = x(t ) − C (t )x(α t ) − V2 ( t ) =
P (s/β 2 )
t
∫
s
βt
dt
u
dV1 dt
and
dV2 dt
= −2 x(t ) − C (t )x(α t ) − =−
≤−
P (t /β) t P (t /β)
]2
,
x (u)duds + (µ + λ) 2
P (s/αβ)
t
∫
s
αt
∫
t
P (s/β)
]
x(s)ds
s
βt
2x (t ) − 2C (t )x(t )x(α t ) −
t P (s/β)
t
∫
2
P (t /β)
s
βt
x(t )
2x(s)x(t )ds
2x (t ) − |C (t )|x (α t ) − |C (t )|x (t ) − x (t ) 2
t
x2 (s)ds.
along the solution of (1.1), we have
[
[
x(s)ds
s
βt
P (u/β)
s
At t ̸= tk , calculating dV1
t
∫
P (s/β)
t
∫
2
2
2
∫
t
]
P (s/β) s
βt
∫
t
ds −
P (s/β)
βt
s
x (s)ds , 2
and dV2 dt
=−
P (t /β)
∫
t
t
P (u/β) u
βt
+ (µ + λ)
P (t /αβ) t
P (t /β)
x (u)du + 2
t
x2 (t ) − (µ + λ)
x (t ) 2
P (t /β) t
∫
t
P (s/β 2 ) s
βt
ds
x2 (α t ).
Let V (t ) = V1 (t ) + V2 (t ). From the above two inequalities and (2.7) and (2.8), it follows that dV dt
=
dV1
≤−
dV2
+
dt
dt
P (t /β)
− x (t )
≤−
2x2 (t ) − |C (t )|x2 (t ) − x2 (t )
t 2
=−
t P (t /β)
≤ −δ
ds + (µ + λ)
[
x (t ) 2 − |C (t )| − 2
[
x (t ) 2 −
P (t /β) t
s
βt
P (t /β)
t
P (s/β 2 )
t
∫
2
∫
t /β
∫
t /β
s
βt
t P (s/β) s
s
P (s/β)
t
P (t /αβ)
βt
P (s/β)
βt
∫
ds
x2 ( t )
ds − (µ + λ)
ds − (µ + λ) 1 +
P (t /αβ)
]
P (t /β)
P (t /αβ)
]
P (t /β)
x2 (t ).
(2.9)
At t = tk , from (1.2) and (2.1), it follows that
[
V (tk ) =
x(tk ) − C (tk )x(α tk ) −
∫
tk
+ β tk
P (s/β 2 )
tk
∫
s
tk
∫
P (s/β)
P (u/β) u
s
s
β tk
x2 (u)duds + (µ + λ)
[ ∫ = b2k x(tk− ) − C (tk− )x(α tk ) −
tk
β tk
∫
tk
+ β tk
P (s/β ) 2
tk
∫
s
P (u/β) u
s
≤ V (tk− ).
x(s)ds
]2
P (s/β) s
x(s)ds
tk
α tk
P (s/αβ) s
x2 (s)ds
]2
x (u)duds + (µ + λ) 2
∫
∫
tk
α tk
P (s/αβ) s
x2 (s)ds (2.10)
This and (2.9) show that V (t ) is decreasing. In view of the fact that V (t ) ≥ 0, limt →∞ V (t ) = γ exists and γ ≥ 0. By using (2.9) and (2.10) again, we can easily get ∞
∫ T
P (t /β) t
x2 (t )dt ≤
V (T )
δ
.
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
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This implies that P (t /β) t
x2 (t ) ∈ L1 (t0 , ∞), P (s/β) 2 x s
t
and hence for any 0 < η < 1 we have limt →∞ ηt t
∫
P (s/β 2 ) s
βt
P (u/β)
t
∫
u
s
x2 (u)duds ≤
(s)ds = 0. Thus, it follows from (2.4) that ∫ t P (s/β 2 ) P (u/β) 2 ds x (u)du
t
∫
s
βt
t /β
∫
P (s/β)
=
s
t t /β
∫
s
βt
u
βt
ds
P (u/β) u
t
∫ ds
P (u/β) u
βt
P (u/β)
t
∫ ≤2
t
βt
P (s/β)
≤
u
βt
∫
x2 (u)du → 0,
x2 (u)du x2 (u)du as t → ∞,
and
(µ + λ)
∫
t
P (s/αβ) s
αt
x2 (s)ds = (µ + λ)
P (s/β)
αt
P (s/β)
t
∫
P (s/αβ)
t
∫
≤2
s
αt
·
P (s/β) s
x2 (s)ds → 0,
x2 (s)ds
as t → ∞.
It follows that limt →∞ V2 (t ) = 0. Thus, limt →∞ V1 (t ) = limt →∞ V (t ) = γ , that is,
[ lim
t →∞
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
]2
x(s)ds
= γ.
(2.11)
Next, we shall prove that the limit
[ lim
t →∞
x(t ) − C (t )x(α t ) −
∫
t
P (s/β) s
βt
]
x(s)ds
t
exists and is finite. Set y(t ) = x(t ) − C (t )x(α t ) − β t y(tk ) = x(tk ) − C (tk )x(α tk ) −
[
∫
tk
P (s/β)
= bk x(tk ) − C (tk )x(α tk ) − −
−
s
β tk
tk
∫
P (s/β) x s
x(s)ds
P (s/β) s
β tk
(s)ds, from (1.2) and (2.1), we then have
x(s)ds
]
= bk y(tk− ).
(2.12)
From (2.11), it follows that lim y2 (t ) = γ .
(2.13)
t →∞
Moreover, in view of (2.5) and (2.12), system (2.5) and (2.6) can be rewritten as
y′ (t ) +
P (t /β)
x(t ) = 0, t ≥ t0 > 0, t ̸= tk , t y(tk ) = bk y(tk− ), k = 1, 2, . . . .
(2.14)
If γ = 0, then limt →∞ y(t ) = 0. If γ > 0, then there exists a sufficiently large T1 such that y(t ) ̸= 0 for t ≥ T1 . Otherwise, there is a sequence {τk } with limk→∞ τk = ∞ such that y(τk ) = 0, and so y2 (τk ) → 0 as k → ∞. This contradicts γ > 0. Therefore, for any tk > T1 , t ∈ [tk , tk+1 ), we have y(t ) > 0 or y(t ) < 0 because y(t ) is continuous on [tk , tk+1 ). Without loss of generality, we assume that y(t ) > 0 on [tk , tk+1 ). It follows that y(tk+1 ) = bk y(tk−+1 ) > 0, and thus y(t ) > 0 on [tk+1 , tk+2 ). By induction, we can conclude that y(t ) > 0 on [tk , ∞). This and (2.11) imply that lim y(t ) = lim
t →∞
t →∞
x(t ) − C (t )x(α t ) −
∫
t
βt
P (s/β) s
x(s)ds
= ν,
(2.15)
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
√ γ and is finite. In view of (2.14), we have
where ν =
∫
t
P (s/β) s
βt
x(s)ds = y(β t ) − y(t ) −
t
lim
t →∞
(y(tk ) − y(tk− )) = y(β t ) − y(t ) −
β t
Letting t → ∞ and noticing that
∫
−
P (s/β)
k=1
(1 − bk )y(tk− ).
β t < tk < t
(1 − bk ) < ∞, we have
x(s)ds = 0.
s
βt
∑∞
−
(2.16)
It follows from (2.15) and (2.16) that lim (x(t ) − C (t )x(α t )) = ν.
(2.17)
t →∞
Finally, we shall prove that lim x(t ) exists and is finite.
(2.18)
t →∞
To this end, we need to show that |x(t )| is bounded. As a matter of fact, if |x(t )| is unbounded, then there exists a sequence {sn } such that sn → ∞, |x(s− n )| → ∞ as n → ∞, and
|x(s− n )| = sup |x(sn )|, t0 ≤t ≤sn
where, if sn is not an impulsive point, then x(s− n ) = x(sn ). Thus, we have − − − − |x(s− n ) − C (sn )x(α sn )| ≥ |x(sn )| − |C (sn )||x(α sn )| ≥ |x(sn )|(1 − µ − λ) → ∞,
as n → ∞,
which contradicts (2.17) and so |x(t )| is bounded. If µ = 0, clearly limt →∞ x(t ) = ν , which shows that (2.18) holds. If 0 < µ < 1, one can see that C (t ) is eventually positive or negative. Otherwise, there is a sequence {τk } with limk→∞ τk = ∞ such that C (τk ) = 0, and so C (τk ) → 0 as k → ∞. It is a contradiction with µ > 0. By condition (2.3), one can find a sufficiently large T2 such that for t > T2 , |C (t )| < 1. Set
ξ = lim inf x(t ), t →∞
ϱ = lim sup x(t ). t →∞
Then we can select two sequences {un } and {vn } such that un → ∞, vn → ∞ as n → ∞, and lim x(un ) = ξ ,
lim x(vn ) = ϱ.
n→∞
n→∞
For t > T2 , we consider the following two possible cases.
Case 1. When −1 < C (t ) < 0 for t > T2 , we have
ν = lim [x(un ) − C (un )x(α un )] ≤ ξ + µϱ, n→∞
and
ν = lim [x(vn ) − C (vn )x(αvn )] ≥ ϱ + µξ . n→∞
Thus, we obtain
ξ + µϱ ≥ ϱ + µξ , that is,
(1 − µ)ξ ≥ (1 − µ)ϱ. Since 0 < µ < 1 and ϱ ≥ ξ , it follows that ϱ = ξ . It follows from (2.17) that
ϱ=ξ =
ν 1+µ
,
which shows that (2.18) holds.
K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
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Case 2. When 0 < C (t ) < 1 for t > T2 , we have
ξ = lim x(un ) = lim [(x(un ) − C (un )x(α un )) + C (un )x(α un )] ≥ ν + µξ . n→∞
n→∞
It follows that
ν
ξ≥
1−µ
.
Similarly,
ϱ = lim x(vn ) = lim [(x(vn ) − C (vn )x(αvn )) + C (vn )x(αvn )] ≤ ν + µϱ, n→∞
n→∞
ν ν which implies ϱ ≤ 1−µ . Thus, ξ = ϱ = 1−µ and so (2.18) holds. Summarizing the above discussion, we conclude that (2.18) holds and so the proof is completed. By Theorem 2.1, we have the following asymptotic behavior result immediately.
Theorem 2.2. The conditions of Theorem 2.1 imply that every oscillatory solution of (1.1) and (1.2) tends to zero as t → ∞. In Theorem 2.1, taking bk ≡ 1, k = 1, 2, . . . , we obtain the following Corollary 2.3. Assume that (2.3) and (2.4) hold. Then every solution of (1.3) tends to a constant as t → ∞. Theorem 2.4. The conditions of Theorem 2.1 together with ∞
∫
P (s/β) s
t0
ds = ∞
(2.19)
imply that every solution of (1.1) and (1.2) tends to zero as t → ∞. Proof. By Theorem 2.2, we only have to prove that every nonoscillatory solution of (1.1) and (1.2) tends to zero as t → ∞. Without loss of generality, let x(t ) be an eventually positive solution of (1.1) and (1.2), we shall prove that limt →∞ x(t ) = 0. As in the proof of Theorem 2.1, we can rewrite (1.1) and (1.2) in the form (2.14). Integrating from t0 to t both sides of (2.14) yields t
∫
P (s/β) s
t0
∑∞
k =1
P (s/β)
∞
s
t0
− t0 < tk < t
Using (2.15) and
∫
x(s)ds = y(t0 ) − y(t ) −
(1 − bk )y(tk− ).
(1 − bk ) < ∞, we obtain
x(s)ds < ∞.
This, together with (2.19), implies that lim inft →∞ x(t ) = 0. By Theorem 2.1, limt →∞ x(t ) = 0 and so the proof is completed. The following corollary follows from Corollary 2.3 and Theorem 2.4. Corollary 2.5. Assume that (2.3), (2.4) and (2.19) hold. Then every solution of (1.3) tends to zero as t → ∞. 3. Examples In this section, we give two examples to illustrate the usefulness of our main results. Example 3.1. Consider the neutral differential equation d dt
[
x( t ) −
1 4
]
x(t /2) +
1 8t
x(t /4) = 0,
t ≥ 1.
(3.1)
One can easily see that the conditions (2.3), (2.4) and (2.19) hold. By Corollary 2.5, every solution of (3.1) tends to zero as t → ∞. Indeed, x(t ) = 1/t (t ≥ 1) is such a solution. Example 3.2. Consider the impulsive neutral differential equation
d [x(t ) − C (t )x(t /e)] +
1 x(t /e) = 0, t ≥ t0 = 3, t ̸= k, 4t∫(ln t − 1) k 2 k −1 1 1 x(k− ) + 2 x(s)ds, k = 4, 5, . . . , x(k) = k2 k k/e 4s ln s dt
where P (t ) = 4(ln1t −1) , bk =
k2 −1 , k2
and C (t ) = 4(k−k 1)2 [t ], t ∈ [k − 1, k), k = 4, 5, . . . .
(3.2)
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K. Guan, J. Shen / Applied Mathematics Letters 24 (2011) 1218–1224
One can easily find that lim |C (t )| =
t →∞
1 4
< 1,
k2 − 1
C (k) =
k2
∫
C (k− ),
∞
P (s/β)
t0
s
∞
∫ ds = 3
1
ds = ∞,
4s ln s
and
[
lim sup µ 1 + t →∞
P (t /αβ) P (t /β)
∫
t /β
+ βt
P (s/β) s
] ds
[ 1
= lim sup
4
t →∞
= lim sup t →∞
=
1 2
1 4
1+
1+
ln t 1 + ln t ln t
1 + ln t
∫
et
+ t /e
+ ln
]
1 4s ln s
ln t + 1
ds
ln t − 1
< 2.
By Theorem 2.4, every solution of (3.2) tends to zero as t → ∞. Acknowledgement The authors are very grateful to the referee for his (her) suggestions for the improvement of this paper. References [1] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [2] A.M. Samoilenko, A.V. Perestynk, Differential Equations with Impulsive Effect, Visca Skola, Kive, 1987. [3] D.D. Bainov, M.B. Dinitrova, A.B. Dishliev, Oscillation of the solutions of impulsive differential equations and inequalities with a retarded argument, Rocky Mountain J. Math. 28 (1998) 25–40. [4] Zh.G. Luo, J.H. Shen, Stability and boundedness for impulsive differential equations with infinite delays, Nonlinear Anal. 46 (2001) 475–493. [5] J.R. Graef, J.H. Shen, I.P. Stavroulakis, Oscillation of impulsive neutral delay differential equations, J. Math. Anal. Appl. 268 (2002) 310–333. [6] J. Yan, A. Zhao, Oscillation and stability of linear impulsive differential equations, J. Math. Anal. Appl. 227 (1998) 187–194. [7] X.Z. Liu, J.H. Shen, Asymptotic behavior of solutions of impulsive neutral differential equations, Appl. Math. Lett. 12 (1999) 51–58. [8] J.H. Shen, Y.J. Liu, J.L. Li, Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses, J. Math. Anal. Appl. 332 (2007) 179–189. [9] Q.R. Wang, Oscillation criteria for first-order neutral differential equations, Appl. Math. Lett. 8 (2002) 1025–1033. [10] T. Yoneyama, The 3/2 stability theorem for one-dimensional delay-differential equations with unbounded delay, J. Math. Anal. Appl. 165 (1992) 133–143. [11] K.Z. Guan, J.H. Shen, On first-order neutral differential equations of Euler form with unbounded delay, Appl. Math. Comput. 189 (2007) 1419–1427. [12] K.Z. Guan, J.H. Shen, Oscillation criteria for a first-order impulsive neutral differential equations of Euler form, Comput. Math. Appl. 58 (2009) 670–677.