Advances in Difference Equations Volume 2010, Article ID 540365,18pages doi:10.1155/2010/540365
Research Article
The Existence and Exponential Stability for
Random Impulsive Integrodifferential Equations of
Neutral Type
Huabin Chen, Xiaozhi Zhang, and Yang Zhao
Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China
Correspondence should be addressed to Huabin Chen,chb [email protected]
Received 24 March 2010; Revised 9 July 2010; Accepted 28 July 2010
Academic Editor: Claudio Cuevas
Copyrightq2010 Huabin Chen 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.
By applying the Banach fixed point theorem and using an inequality technique, we investigate a kind of random impulsive integrodifferential equations of neutral type. Some sufficient conditions, which can guarantee the existence, uniqueness, and exponential stability in mean square for such systems, are obtained. Compared with the previous works, our method is new and our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the proposed results.
1. Introduction
and uniqueness of the solutions to random impulsive differential equations, and in 18
Zhao and Zhang discussed the exponential stability of random impulsive integro-differential equations by employing the comparison theorem. Very recently, the existence, uniqueness and stability results of random impulsive semilinear differential equations, the existence and uniqueness for neutral functional differential equations with random impulses are discussed by using the Banach fixed point theorem in19,20, respectively.
It is well known that the nonlinear impulsive delay differential equations of neutral type arises widely in scientific fields, such as control theory, bioscience, physics, etc. This class of equations play an important role in modeling phenomena of the real world. So it is valuable to discuss the properties of the solutions of these equations. For example, Xu et al. in21, have considered the exponential stability of nonlinear impulsive neutral differential equations with delays by establishing singular impulsive delay differential inequality and transforming the n-dimensional impulsive neutral delay differential equation into a 2n -dimensional singular impulsive delay differential equations; and the results about the global exponential stability for neutral-type impulsive neural networks are obtained by using the linear matrix inequalityLMIin9,10, respectively.
However, most of these studies are in connection with deterministic impulses and finite delay. And, to the best of author’s knowledge, there is no paper which investigates the existence, uniqueness and exponential stability in mean square of random impulsive integrodifferential equation of neutral type. One of the main reason is that the methods to discuss the exponential stability of deterministic impulsive differential equations of neutral type and the exponential stability for random differential equations can not be directly adapted to the case of random impulsive differential equations of neutral type, especially, random impulsive integrodifferential equations of neutral type. That is, the methods proposed in 15, 16 are ineffective for the exponential stability in mean square for such systems. Although the exponential stability of nonlinear impulsive neutral integrodifferential equations can be derived in22, the method used in22is only suitable for the deterministic impulses. Besides, the methods introduced to deal with the exponential stability of random impulsive integrodifferential equations in18and study the exponential stability in mean square of random impulsive differential equations in 19, can not be applied to deal with our problem since the neutral item arises. So, the technique and the method dealt with the exponential stability in mean square of random impulsive integrodifferential equations of neutral type are in need of being developed and explored. Thus, with these aims, we will make the first attempt to study such problems to close this gap in this paper.
The format of this work is organized as follows. In Section 2, some necessary definitions, notations and lemmas used in this paper will be introduced. In Section 3, The existence and uniqueness of random impulsive integrodifferential equations of neutral type are obtained by using the Banach fixed point theorem. Some sufficient conditions about the exponential stability in mean square for the solution of such systems are given inSection 4. Finally, an illustrative example is provided to show the obtained results.
2. Preliminaries
Let| · |denote the Euclidean norm inRn. IfAis a vector or a matrix, its transpose is denoted
byAT; and ifAis a matrix, its Frobenius norm is also represented by| · | traceATA.
Assumed thatΩis a nonempty set andτkis a random variable defined fromΩtoDk0, dk
for allk 1,2, . . ., where 0 < dk ≤ ∞. Moreover, assumed thatτiand τj are independent
Let BCX, Ybe the space of bounded and continuous mappings from the topological space X into Y, and BC1X, Y be the space of bounded and continuously differentiable mappings from the topological spaceXintoY. In particular, Let BCBC−∞,0, Rnand
BC1 BC1−∞,0, Rn. PCJ, Rn {φ : J → Rn|φs is bounded and almost surely
continuous for all but at most countable pointss ∈ J and at these points s ∈ J,φsand
φs−exist,φs φs}, whereJ ⊂Ris an interval,φsandφs−denote the right-hand and left-hand limits of the functionφs, respectively. Especially, let PC PC−∞,0, Rn.
PC1J, Rn {φ : J → Rn|φsis bounded and almost surely continuously differentiable for all but at most countable points s ∈ J and at these points s ∈ J, φs and φs−,
φs φs, φs φs}, where φs denote the derivative of φs. Especially, let PC1PC1−∞,0, Rn.
Forφ∈PC1, we introduce the following norm:
φ∞max
sup
−∞<θ≤0
φθ, sup
−∞<θ≤0
φθ
. 2.1
In this paper, we consider the following random impulsive integrodifferential equations of neutral type:
xt Axt Dxt−r f1t, xt−ht
0
−∞f2θ, xtθdθ, t /ξk, t≥0,
2.2
xξk bkτkxξk−, k1,2, . . . , 2.3
xt0ϕ∈PC
1, 2.4
whereA, Dare two matrices of dimensionn×n;f1 :0,∞×Rn → Rnandf2 :−∞,0×
Rn → Rn are two appropriate functions;b
k : Dk → Rn×n is a matrix valued functions for
eachk1,2, . . .; assume thatt0∈0,∞is an arbitrary real number,ξ0t0andξkξk−1τk
for k 1,2, . . .; obviously, t0 ξ0 < ξ1 < ξ2 < · · · < ξk < · · ·; xξ−k limt→ξk−0xt; h : 0,∞ → 0, ρ ρ > 0is a bounded and continuous function andτ max{r, ρ}r > 0.
xt : xts xtsfor all s ∈ −∞,0. Let us denote by{Bt, t ≥ 0}the simple counting
process generated by{ξn}, that is,{Bt≥n}{ξn≤t}, and presentIttheσ-algebra generated
by{Bt, t≥0}. Then,Ω,{It}, Pis a probability space.
Firstly, define the spaceBconsisting of PC1−∞, T, Rn T > t0-valued stochastic
processϕ:−∞, T → Rnwith the norm
ϕ 2
E sup
−∞<θ≤T
ϕθ2
. 2.5
It is easily shown that the spaceB, · is a completed space.
Definition 2.1. A functionx∈ Bis said to be a solution of2.2–2.4ifxsatisfies2.2and conditions2.3and2.4.
Definition 2.2. The fundamental solution matrix{Φt expAt, t ≥ 0}of the equation
xt Axtis said to be exponentially stable if there exist two positive numbersM≥1 and
Definition 2.3. The solution of system 2.2 with conditions 2.3 and 2.4 is said to be exponentially stable in mean square, if there exist two positive constantsC1 > 0 andλ >0
such that
E|xt|2≤C1e−λt, t≥0. 2.6
Lemma 2.4see23. For any two real positive numbersa, b >0, then
ab2≤ν−1a2 1−ν−1b2, 2.7
whereν∈0,1.
Lemma 2.5 see 23. Let u, ψ, and χ be three real continuous functions defined on a, b and
χt≥0, fort∈a, b, and assumed that ona, b, one has the inequality
ut≤ψt
t
a
χsusds. 2.8
Ifψis differentiable, then
ut≤ψaexp
t
a
χsds
t
a
exp
t
s
χrdr
ψsds, 2.9
for allt∈a, b.
In order to obtain our main results, we need the following hypotheses.
H1The function f1 satisfies the Lipschitz condition: there exists a positive constant
L1>0 such that
f1t, x−f1
t, y≤L1x−y, 2.10
forx, y∈Rn,t∈0, T, andf
1t,0 0.
H2The functionf2satisfies the following condition: there also exist a positive constant
L2 > 0 and a function k : −∞,0 → 0,∞ with two important properties,
0
−∞ktdt1 and
0
−∞kte−ltdt <∞l >0, such that
f2t, x−f2
t, y≤L2ktx−y, 2.11
forx, y∈Rn,t∈0, T, andf
2t,0 0.
H3Emaxi,k{kji|bjτj|2} is uniformly bounded. That is, there exists a positive
constantL >0 such that
E
⎛
⎝max
i,k
⎧ ⎨ ⎩
k
ji
bj
τj2
⎫ ⎬ ⎭
⎞
⎠≤L, 2.12
for allτj∈Djandj 1,2, . . .. H4κ
3. Existence and Uniqueness
In this section, to make this paper self-contained, we study the existence and uniqueness for the solution to system2.2with conditions2.3and2.4by using the Picard iterative method under conditionsH1–H4. In order to prove our main results, we firstly need the
following auxiliary result.
Lemma 3.1. Letf1:0,∞×Rn → Rnandf2:−∞,0×Rn → Rnbe two continuous functions. Then,xis the unique solution of the random impulsive integrodifferential equations of neutral type:
xt Axt Dxt−r f1t, xt−ht
0
−∞f2θ, xtθdθ, t /ξk, t≥0,
xξk bkτkxξ−k, k1,2, . . . ,
xt0ϕ∈PC
1,
3.1
if and only ifxis a solution of impulsive integrodifferential equations:
ixt0θ ϕθ,θ∈−∞,0,
ii
xt ∞
k0
⎡
⎣k
i1
biτiΦt−t0x0 k
i1 k
ji
bj
τj ξi
ξi−1
Φt−sDdxs−r
t
ξkΦ
t−sDdxs−r
k
i1 k
ji
bj
τj
× ξi
ξi−1
Φt−sf1s, xs−hsds
t
ξkΦ
t−sf1s, xs−hsds k
i1 k
ji
bj
τj ξi
ξi−1
Φt−s
×
0
−∞f2θ, xsθdθds
t
ξkΦ
t−s
0
−∞f2θ, xsθdθds
Iξk,ξk1t,
3.2
for all t∈t0, T, wherenjm· 1asm > n,
k
jibjτj bkτkbk−1τk−1· · ·biτi, andIΩ·denotes the index function, that is,
IΩt
⎧ ⎨ ⎩
1, if t∈Ω,
0, if t /∈Ω.
Proof. The approach of the proof is very similar to those in17,19,20. Here, we omit it.
Theorem 3.2. Provided that conditions (H1)–(H4) hold, then the system2.2with the conditions 2.3and2.4has a unique solution onB.
Thus, due toLemma 2.4, it follows that
From conditionH3, we have
In view ofLemma 2.5, it yields that
We will show3.16by mathematical induction. From3.12, it is easily seen that3.16holds asn0. Under the inductive assumption that3.16holds for somen≥1. We will prove that
For anym > n≥1, it follows that
xm−xn
E
sup
−∞<t≤T|x
mt−xnt|2
1/2
≤∞
kn
E
sup
−∞<t≤T
xk1t−xkt2
1/2
≤∞
kn
Λ5Λ1T −t0 k
k!
1/2
−→0,
3.20
asn → ∞. Thus,{xnt}
n≥0t∈−∞, Tis a Cauchy sequence in Banach spaceB. Denote
the limit byx∈B. Now, lettingn → ∞in both sides of3.4, we obtain the existence for the solution of system2.2with conditions2.3and2.4.
Uniqueness. Letx, y ∈ Bbe two solutions of system2.2with conditions2.3and
2.4. By the same ways as above, we can yield that
E
sup
−∞<t≤T
xs−ys2
E
sup
t0≤t≤T
xt−yt2
≤Λ1
T
t0
E
sup
−∞<s≤t
xs−ys2
dt.
3.21
Applying the Gronwall inequality into3.21, it follows that
E
sup
−∞<t≤T
xt−yt2
0. 3.22
That is,x−y0. So, the uniqueness is also proved. The proof of this theorem is completed.
4. Exponential Stability
In this section, the exponential stability in mean square for system2.2with initial conditions
2.3and2.4is shown by using an integral inequality.
Theorem 4.1. Supposed that the conditions of Theorem 3.2 holds, then the solution of the system 2.2with conditions2.3and2.4is exponential stable in mean square if the inequality
4M2max{1, L}|D|2|A|2L2 1L22
1−κ2a < a≤l
4.1
Now, in order to show our main result, we only claim that4.4implies
E|xt|2 ≤Mεe−μt−t0, t∈−∞,∞. 4.6
It is easily seen that4.6holds for anyt ∈ −∞, t0. Assume, for the sake of contradiction,
that there exists at1> t0and
E|xt|2< Mεe−μt−t0, t∈−∞, t1, E|xt1|2Mεe−μt1−t0. 4.7
Then, it, from4.4, implies that
E|xt1|2 ≤λ1e−at1−t0κMεeμτe−μt1−t0λ2Mε
t1
t0
e−at1−t0−s sup
θ∈−τ,0e
−μsθds
λ3Mε
t1
t0
e−at1−t0−s
0
−∞kθe
−μsθdθds
≤
λ1−Mε
λ2 e μτ
a−μ λ3
a−μ
0
−∞kθe −μθdθ
e−at1−t0
Mε
κeμrλ2e
μτ
a−μ λ3
a−μ
0
−∞kθe −μθdθ
e−μt1−t0.
4.8
From the definitions ofμandMε, we have
κeμr λ2e
μτ
a−μ λ3
a−μ
0
−∞kθe
−μθdθ1,
λ1−Mε
λ2 e μτ
a−μ λ3
a−μ
0
−∞kθe −μθdθ
≤λ1−
λ2 e μτ
a−μ λ3
a−μ
0
−∞kθe −μθdθ
ελ1 a−μ
λ2eμτ λ3
0
−∞kθe−μθdθ
<0.
4.9
Thus,4.8yields that
E|xt1|2< Mεe−μt1−t0, 4.10
which contradicts4.7, that is,4.4holds.
Asε >0 is arbitrarily small, in view of4.6, it follows that
E|xt|2≤Me−μt−t0, t≥t
whereMmax{λ1a−μ/λ2eμτλ3
0
−∞kθe−μθdθ, λ1}>0.
That is,
E|xt|2≤Me−αt−t0, t≥t
0, 4.12
whereα∈0, aand
Mmax
⎧ ⎨ ⎩
8 max{L,1}M2|D|2
LEϕ2∞a−μ
1−κ
× ⎛ ⎜
⎝4 max{L,1}M
2|D|2|A|2L2 1
eμτ 1−κa
4 max{L,1}M2L2 2 1−κa
0
−∞kθe −μθdθ
⎞ ⎟ ⎠
−1
,
8 max{L,1}M2|D|2LEϕ2 ∞
a−μ
1−κ
⎫ ⎬ ⎭>0.
4.13
The proof is completed.
In particular, whenD≡0,τ ≡0,andf2 ≡0, system2.2is turned into the following
form:
xt Axt f1t, xt, t /ξk, t≥0,
xξk bkτkxξk−, k1,2, . . . ,
xt0x0.
4.14
Remark 4.2. Obviously, we can also give the existence, uniqueness, and exponential stability in mean square for the solution of system4.14by employing the Picard iterative method and a similar impulsive-integral inequality proposed in24. So, the following corollary can be given as follows.
Corollary 4.3. Under conditions (H1) and (H3), the existence, uniqueness, and exponential stability in mean square for the solution of system4.14can be obtained only if the inequality
max{1, L}M2L21< a2 4.15
holds.
Proof. The proofs of this corollary are very similar to those of Theorems3.2and4.1. So, we omit them.
5. An Illustrative Example
Letτkbe a random variable defined inDk ≡ 0, dkfor allk 1,2, . . ., where 0 < dk ≤ ∞.
Furthermore, assume thatτiandτjare independent of each other asi /jfori, j1,2, . . ..
Consider the random impulsive integrodifferential equations of neutral type:
dx1t
dt −0.2x1t
1 2√100
dx1t−r
dt −
1 2√100
dx2t−r
dt δ1costx1t−ht
δ3
0
−∞−θ
−1/2eπ2θ
x1sθds,
dx2t
dt −0.1x1t−0.1x2t
1 3√100
dx1t−r
dt
1 4√100
dx2t−r
dt δ2sintx2t−ht
δ4
0
−∞−θ
−1/2eπ2θ
x2sθds,
5.1
asξk−1≤t < ξk,k1,2, . . . ,and for allk1,2,3, . . .,
x1ξk ckτk·x1
ξk−,
x2ξk ckτk·x2
ξk−, 5.2
where ξ0 t0 and ξk ξk−1 τk for all k 1,2, . . ., ck is a function of k. Denote that
c maxk{ck} and there isρ : 0 ≤ ρ < 1 such thatEcτk2 ≤ ρ for allk 1,2, . . .. and
|δi| ≤L1 i1,2and|δi| ≤L2 i3,4. And|D|0.0846, |A|0.2449, κ0.0846.
The corresponding linear homogeneous equations:
⎛ ⎜ ⎜ ⎝
dx1t
dt dx2t
dt
⎞ ⎟ ⎟
⎠
+
−0.2 0
−0.1 −0.1
,+
x1t
x2t
,
. 5.3
And, the fundamental solution matrix of system5.3can be given by
Φt
+
e−0.2t 0
e−0.2t−e−0.1t e−0.1t
,
. 5.4
Then, it is easily obtain that
|Φt| ≤Me−0.1t, t≥0, 5.5
whereM√2>1 anda0.1>0, and it is easily seen that we can derive that the functionsf1
andf2satisfy conditionsH1andH2with the Lipschitz coefficientsL1andL2, respectively.
4.1, the existence, uniqueness, and exponential stability in mean square of system5.1with
5.2are obtained if the constantsL1andL2satisfy the following inequality:
L2
1L22<0.02771. 5.6
Acknowledgment
The authors would like to thank the referee and the editor for their careful comments and valuable suggestions on this work.
References
1 A. Anokhin, L. Berezansky, and E. Braverman, “Exponential stability of linear delay impulsive differential equations,”Journal of Mathematical Analysis and Applications, vol. 193, no. 3, pp. 923–941, 1995.
2 Y. Hino, S. Murakani, and T. Naito,Functional Differential Equations with Infinite Delay, vol. 1473 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991.
3 J. Hale and S. M. Verguyn Lunel,Introduction to Functional Differential Equations, vol. 99 ofApplied Mathematical Science, Springer, New York, NY, USA, 1993.
4 J. Henderson and A. Ouadhab, “Local and global existence and uniqueness results for second and higher order impulsive functional differential equations with infinite delay,”The Australian Journal of Mathematical Analysis and Applications, vol. 4, pp. 1–26, 2007.
5 A. Korzeniowski and G. S. Ladde, “Modeling hybrid network dynamics under random perturba-tions,”Nonlinear Analysis: Hybrid Systems, vol. 3, no. 2, pp. 143–149, 2009.
6 X. Liu, X. Shen, and Y. Zhang, “A comparison principle and stability for large-scale impulsive delay differential systems,”The ANZIAM Journal, vol. 47, no. 2, pp. 203–235, 2005.
7 X. Liu, “Stability results for impulsive differential systems with applications to population growth models,”Dynamics & Stability of Systems. Series A, vol. 9, no. 2, pp. 163–174, 1994.
8 X. Liu and X. Liao, “Comparison method and robust stability of large-scale dynamic systems,”
Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 11, no. 2-3, pp. 413–430, 2004.
9 R. Rakkiyappan, P. Balasubramaniam, and J. Cao, “Global exponential stability results for neutral-type impulsive neural networks,”Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 122– 130, 2010.
10 R. Samidurai, S. M. Anthoni, and K. Balachandran, “Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays,”Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 103–112, 2010.
11 V. Lakshmikantham, D. D. Ba˘ınov, and P. S. Simeonov,Theory of Impulsive Differential Equations, vol. 6 ofSeries in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
12 A. M. Samo˘ılenko and N. A. Perestyuk,Impulsive Differential Equations, vol. 14 ofWorld Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
13 S. Wu and X. Meng, “Boundedness of nonlinear differential systems with impulsive effect on random moments,”Acta Mathematicae Applicatae Sinica, vol. 20, no. 1, pp. 147–154, 2004.
14 S. Wu, X. Guo, and Y. Zhou, “p-moment stability of functional differential equations with random impulses,”Computers & Mathematics with Applications, vol. 52, no. 12, pp. 1683–1694, 2006.
15 S. Wu, X. Guo, and R. Zhai, “Almost sure stability of functional differential equations with random impulses,”Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 3, pp. 403–415, 2008.
16 S. Wu and Y. Duan, “Oscillation, stability, and boundedness of second-order differential systems with random impulses,”Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1375–1386, 2005.
17 S. Wu, X. Guo, and S. Lin, “Existence and uniqueness of solutions to random impulses,” Acta Mathematicae Applicatae Sinica, vol. 22, pp. 595–600, 2004.
18 D. Zhao and L. Zhang, “Exponential asymptotic stability of nonlinear Volterra equations with random impulses,”Applied Mathematics and Computation, vol. 193, no. 1, pp. 18–25, 2007.
20 A. Anguraj and A. Vinodkumar, “Existence and uniqueness of neutral functional differential equations with random impulses,”International Journal of Nonlinear Science, vol. 8, no. 4, pp. 412–418, 2009.
21 D. Xu, Z. Yang, and Z. Yang, “Exponential stability of nonlinear impulsive neutral differential equations with delays,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1426– 1439, 2007.
22 L. Xu and D. Xu, “Exponential stability of nonlinear impulsive neutral integro-differential equations,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2910–2923, 2008.
23 W. Walter, Differential and Integral Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer, New York, NY, USA, 1970.
