In this article, we study the controllability of impulsive functional differential equations with nonlocal conditions

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

CONTROLLABILITY OF IMPULSIVE FUNCTIONAL DIFFERENTIAL SYSTEMS WITH NONLOCAL CONDITIONS

YANSHENG LIU, DONAL O’REGAN

Abstract. In this article, we study the controllability of impulsive functional differential equations with nonlocal conditions. We establish sufficient condi- tions for controllability, via the measure of noncompactness and M¨onch fixed point theorem.

1. Introduction Consider the impulsive functional differential equation

x⁰(t) =A(t)x(t) +f(t, x(t), x_t) +Bu(t), a.e. t∈[0, a];

∆x _t=t

i =Ii(x(ti)), i= 1,2, . . . k;

x(t) =φ(t), t∈[−τ,0);

x(0) +M(x) =x0,

(1.1)

where ∆x|t=t_i =x(ti+ 0)−x(ti−0), A(t) is a family of linear operators which generates an evolution operator

U: ∆ ={(t, s)∈J×J: 0≤s≤t≤a} →L(X),

X is a Banach space,J = [0, a],L(X) is the space of all bounded linear operators in X, M : P C(J, X)→ X, B is a bounded linear operator from a Banach space V to X and the control function u(·) is given in L²(J, V), 0 = t₀ < t₁ < t₂ <

· · · < t_k < t_k+1 = a, I_i : X → X, i = 1, . . . , k are impulsive functions, f : J ×X ×L([−τ,0], X)→X is a given function satisfying some assumptions that will be specified later,φ∈L([−τ,0], X) andL([−τ,0], X) is the space of X-valued Bochner integrable functions on [−τ,0] with the norm kφk_L[−τ,0]=R0

−τkφ(t)kdt.

Abstract differential systems in infinite-dimensional spaces appear in many bran- ches of science and engineering, such as heat flow in materials with memory, vis- coelasticity, and other physical phenomena. Systems with short-term perturbations are often naturally described by impulsive differential equations [18, 25]. Impulsive interruptions are observed in mechanics, radio engineering, communication security, control theory, optimal control, biology, medicine, bio-technologies, electronics, neu- ral networks and economics (see for example [4, 5, 8, 19, 26, 27]). We also refer the

2000Mathematics Subject Classification. 34K10, 34K21, 34K35.

Key words and phrases. Controllability; fixed point theorem; nonlocal conditions;

impulsive functional differential equations.

c

2013 Texas State University - San Marcos.

Submitted April 12, 2012. Published August 30, 2013.

1

reader to recent results in impulse theory [6, 7, 24, 28]. The semilinear nonlocal initial problem was first discussed by Byszewski [2, 3]. It was studied extensively under various conditions on A(or A(t)) and f by several authors (see [4, 15] and the references therein). Recently, Ji et al [15] studied the impulsive differential equation

x⁰(t) =A(t)x(t) +f(t, x(t)) +Bu(t), a.e. t∈[0, a];

∆x _t=t

i =I_i(x(t_i)), i= 1,2, . . . k;

x(0) +M(x) =x0.

(1.2)

Time delays are often encountered unavoidably in many practical systems such as automatic control systems, population models, inferred grinding models, the AIDS epidemic, and neural networks; see [9, 10, 17, 11, 22] and the references therein.

They describe phenomenon present in real systems where the rate of change of the state depends on not only the current state of the system but also its state at some time in history. Therefore, it is natural and necessary to study (1.2) with time delay, i.e. the (1.1).

To the best of our knowledge there is no paper studying such systems. The purpose of the present paper is to fill this gap. In this paper some sufficient condi- tions for controllability are established by using the measure of noncompactness and M¨onch’s fixed point theorem. The main features in the present paper are as follows.

First, the (1.1) considers the effect of time delay. Also we relax the assumptions on the functionsf,M, and Ii in [15].

The organization of this article is as follows. We shall introduce some prelimi- naries and some lemmas in Section 2. The main results and their proof are given in Section 3.

2. Preliminaries

For the sake of simplicity, we putJ₀= [0, t₁] andJ_i= (t_i, t_i+1],i= 1, . . . , k. Let P C(J, X) ={x:xis a map fromJ intoX such that x(t) is continuous att6=t_i, and left continuous att =ti, and the right limit x(t⁺_i ) exists for i= 1,2, . . . , k}.

Evidently,P C(J, X) is a Banach space with the norm kxk_{P C} = sup

t∈J

{kx(t)k}, ∀x∈P C(J, X).

Notice that the interaction of time delay and impulse give rise to discontinuity.

Therefore, we introduce the special complete space L([−τ,0], X) to overcome the difficulty arising from time delay. For any function y ∈P C(J, X) and anyt ∈J, we denote a functionyt by

yt(θ) =

(y(t+θ), t+θ≥0;

φ(t+θ), t+θ <0 (2.1) for θ ∈ [−τ,0], where φ(t) is the same as in (1.1). Then it is easy to see y_t ∈ L([−τ,0], X). Moreover, we have the following Lemma.

Lemma 2.1. Supposeyn, y0∈P C(J, X)withkyn−y0kP C→0asn→+∞. Then for eacht∈J, we have

kynt−y0tk_L[−τ,0]→0, as n→+∞, whereynt(θ) andy0t(θ) are defined by (2.1).

Proof. From (2.1), it follows that kynt−y0tk_L[−τ,0] =

(Rt

0|yn(s)−y₀(s)|ds, t≤τ;

Rt

t−τ|yn(s)−y0(s)|ds, t≥τ.

The conclusion follows.

The basic space to study (1.1) in this paper isP C(J, X). For a bounded subset Ω of Banach space X, let β(Ω) be the Hausdorff noncompactness measure of Ω, which is defined by β(Ω) = inf{ε > 0 : Ω has a finiteε-net inX} (see [1, 16]).

In this paper, the Hausdorff measure of noncompactness of a bounded set in X, P C(J, X), and L([−τ,0], X) are denoted by β(·), βP C(·), and βτ(·), respectively.

As in [13], we have the following result on the Hausdorff noncompactness measure.

Lemma 2.2. SupposeE is a Banach space. Let H be a countable set of strongly measurable functionx:J →E such that there exists a µ∈L[J, R⁺]with kx(t)k ≤ µ(t)a.e. t∈J for allx∈H. Then β(H(t))∈L[J, R⁺]and

β Z

J

x(t)dt:x∈Ho

≤2 Z

J

β(H(t))dt,

whereβ(·) denotes the Hausdorff noncompactness measure,J = [0, a].

Lemma 2.3(M¨onch fixed point theorem [20]).SupposeEis a Banach space. LetD be a closed and convex subset ofEandu∈D. Assume that the continuous operator A : D → D has the following property: C ⊂ D countable, C ⊂ co({u} ∪A(C)) impliesC is relatively compact. Then Ahas a fixed point inD.

Definition 2.4. A functionx∈P C(J;X) is said to be a mild solution of (1.1) if x(0) +M(x) =x₀and

x(t) =U(t,0)x(0) + Z t

0

U(t, s) (f(s, x(s), x_s) +Bu(s)

ds+ X

0<ti<t

U(t, t_i)I_i(x(t_i)), for allt∈J, wherexsis defined by (2.1).

Definition 2.5. Equation (1.1) is said to be nonlocally controllable on J if, for everyx0, x1∈X, there exists a control u∈L²(J, V) such that the mild solutionx of (1.1) satisfiesx(b) +M(x) =x1.

A two parameter family of bounded linear operatorsU(t, s), 0≤s≤t ≤aon X is called an evolution system if the following two conditions are satisfied:

(i) U(s, s) =I,U(t, r)U(r, s) =U(t, s) for 0≤s≤t≤a;

(ii) (t, s)→U(t, s) is strongly continuous for 0≤s≤t≤a.

Since the evolution systemU(t, s) is strongly continuous on the compact setJ×J, then there exists L_U >0 such that kU(t, s)k ≤L_U for any (t, s)∈ J ×J. More details about evolution systems can be found in [23].

3. Main results We will use the following hypotheses:

(S1) A(t) is a family of linear operators,A(t) :D(A)→X,D(A) not depending ontis a dense subset ofX, generating an equicontinuous evolution system {U(t, s) : (t, s)∈J×J}, i.e., (t, s)→ {U(t, s)x:x∈Ω} is equicontinuous fort >0 and for all bounded subsets Ω.

(S2) f :J×X×L([−τ,0], X)→X satisfies:

(i) t→f(t, x, y) is strongly measurable for eachx∈X, y∈L([−τ,0], X);

(x, y)→f(t, x, y) is continuous for almost allt∈J; (ii) there exist functions a₁, b₁, µ₁∈L(J;R⁺) such that kf(t, x, y)k ≤a1(t)kxk+b1(t)kyk_L[−τ,0]+µ1(t), for allt∈J,x∈X,y∈L([−τ,0], X);

(iii) there existl1, l2∈L¹(J;R⁺) such that for any bounded subsetsB1⊂ X, B2⊂L([−τ,0], X),

β(f(t, B1, B2))≤l1(t)β(B1) +l2(t)βτ(B2);

(S3) M : P C(J, X)→X is a continuous operator and there exist nonnegative numbersa2, b2, l3such that

kM(y)k ≤a2kyk+b2, ∀y∈P C(J, X);

β(M(B₁))≤l₃β_{P C}(B₁), for any boundedB₁⊂P C(J, X);

(S4) the linear operator W :L²(J, V)→X defined by W u=

Z a

0

U(a, s)Bu(s)ds is such that:

(i) Whas an invertible operatorW⁻¹which take values inL²(J, V)/kerW and there exist positive constants L_B and L_W such that kBk ≤ L_B andkW⁻¹k ≤L_W;

(ii) there isK_W ∈L¹(J, R⁺) such that, for any bounded set Q⊂X, βV((W⁻¹Q)(t))≤KW(t)β(Q).

(S5) Ii:X →X(i= 1, . . . , k) is a continuous operator and there exist nonneg- ative numbersci, di, ki (i= 1,2, . . . , k) such that:

kIi(x)k ≤cikxk+di, ∀x∈X, i= 1,2, . . . , k;

β(I_i(B₁))≤k_iβ(B₁), for any boundedB₁⊂X, i= 1,2, . . . , k.

Theorem 3.1. Assume that (S1)–(S5)are satisfied. In addition, assume that c:=L_Uh

(1 +L_BL_Wa^1/2) a₂+

Z a

0

a₁(s) +τ b₁(s) ds+

k

X

i=1

c_i

+LULBLWa2a^1/2i

<1,

(3.1)

d:=LU

h l3+ 2

Z a

0

(l1(s) +τ l2(s))ds+

k

X

i=1

ki

1 + 2LBLU

Z a

0

KW(s)ds + 2l3LB

Z a

0

KW(s)dsi

<1.

(3.2)

Then the impulsive functional differential system (1.1)is nonlocally controllable on J.

Proof. From (S4)(i), one can define the control:

ux(t) =W⁻¹[x1−M(x)−U(a,0)(x0−M(x))

− Z a

0

U(a, s)f(s, x(s), xs)ds−

k

X

i=1

U(a, ti)Ii(x(ti))](t), (3.3) for allx∈P C(J, X). Using this control, define the following operator onP C(J, X) by

(Gx)(t) =U(t,0)(x0−M(x)) + Z t

0

U(t, s) f(s, x(s), xs) +Bux(s) ds

+ X

0<t_i<t

U(t, t_i)I_i(x(t_i)), ∀x∈P C(J, X).

(3.4)

Obviously,Gx∈P C(J, X). We shall show thatGhas a fixed point, which is then a solution of (1.1). Clearly, ifxis a fixed point ofG, thenx1=M(x) +G(x)(a), which implies that the system (1.1) is controllable.

First we show thatGis continuous. To do this, supposex_n, x∈P C(J, X) and x_n→xas n→+∞. Then by (S3) and (S5) we know that

kGxn−GxkP C

≤L_U

kM(x_n)−M(x)k+ Z a

0

kf(s, x_n(s), x_ns)−f(s, x(s), x_s)kds

+L_B Z a

0

ku_xn(s)−u_x(s)kds+

k

X

i=1

kI_i(x_n(t_i))−I_i(x(t_i))k

≤LU

kM(xn)−M(x)k+ Z a

0

kf(s, xn(s), xns)−f(s, x(s), xs)kds

+LBa^1/2kux_n−uxkL²+

k

X

i=1

kIi(xn(ti)−Ii(x(ti))k .

(3.5)

Notice that

kxns−xsk_L[−τ,0] ≤τkxn−xkP C. (3.6) From (3.3), we have

ku_xn−u_xk_L2

≤LWkM(xn)−M(x)k+LWLU

hkM(xn)−M(x)k

+ Z a

0

kf(s, xn(s), xns)−f(s, x(s), xs)kds+

k

X

i=1

kIi(xn(ti)−Ii(x(ti))ki .

(3.7)

Then by (3.5)–(3.7), (S2)–(S5), and the Lebesgue dominated convergence theorem, we obtain

kGxn−GxkP C→0 as n→+∞, soGis continuous.

Next, choose a positive numberrsatisfying r > LU

1−c h

(1 +LULBLWa^1/2)

kx0k+b2+ Z a

0

b1(s)ds· kφk_L[−τ,0]

+ Z a

0

µ1(s)ds+

k

X

i=1

di

+LBLWa^1/2(kx1k+b2)i .

(3.8)

We now show that

G:B(0, r)→B(0, r), (3.9)

where B(0, r) ={x∈P C(J, X) :kxkP C ≤r}. In fact, for eachx∈P C(J, X), by (3.3), we have

kuxk_L2=Z a 0

kux(s)k²ds^1/2

≤LW(kx1k+a2kxkP C+b2) +LWLU

hkx0k+a2kxkP C+b2

+ Z a

0

a1(s)kx(s)k+b1(s)kxsk_L[−τ,0]+µ1(s) ds+

k

X

i=1

(cikx(ti)k+di)i

≤LW(kx1k+a2kxkP C+b2) +LWLU

hkx0k+a2kxkP C+b2

+ Z a

0

a1(s)kxkP C+b1(s)(τkxkP C+kφk_L[−τ,0]) +µ1(s) ds

+

k

X

i=1

(cikxkP C+di)i .

This together with (3.4) guarantees that kGxkP C

≤L_Uh

kx₀k+kM(x)k+ Z a

0

kf(s, x(s), x_s) +Bu_x(s)kds+

k

X

i=1

kI_i(x(t_i))ki

≤L_Uh

kx0k+a₂kxkP C+b₂+ Z a

0

a₁(s)kx(s)k+b₁(s)kxsk_L[−τ,0]+µ₁(s) ds

+LB

Z a

0

kux(s)kds+

k

X

i=1

(cikx(ti)k+di)i

≤LU

hkx0k+a2kxkP C+b2+ Z a

0

a1(s)kxkP C+b1(s)(τkxkP C+kφk_L[−τ,0])

+µ1(s)

ds+LBa^1/2kuxk_L2+

k

X

i=1

(cikxkP C+di)i

≤ckxkP C+LU

h

(1 +LULBLWa^1/2)

kx0k+b2+ Z a

0

b1(s)ds· kφk_L[−τ,0]

+ Z a

0

µ1(s)ds+

k

X

i=1

di

+LBLWa^1/2(kx1k+b2)i .

From (3.8) we havekGxkP C≤rifkxkP C ≤r; that is, (3.9) holds.

Next we prove that ifD⊂B(0, r) is countable and

D⊂co({u0} ∪G(D)), (3.10)

where u₀ ∈ B(0, r), then D is relatively compact. Without loss of generality, suppose thatD={xn}^∞_n=1. First we show{Gxn}^∞_n=1is equicontinuous on eachJi, i= 0, . . . , k. If this is true thenco({u0} ∪G(D)) is also equicontinuous on eachJi. To this end, notice that for eachx∈D,t⁰, t⁰⁰∈Ji, we have

k(Gx)(t⁰⁰)−(Gx)(t⁰)k

=k[U(t⁰⁰,0)−U(t⁰,0)](x₀−M(x))k+k

i

X

j=1

U(t⁰⁰, t_j)−U(t⁰, t_j)

I_j(x(t_j))k

+k Z t⁰⁰

0

U(t⁰⁰, s) f(s, x(s), xs) +Bux(s) ds

− Z t⁰

0

U(t⁰, s) f(s, x(s), xs) +Bux(s) dsk

≤ k[U(t⁰⁰,0)−U(t⁰,0)](x₀−M(x))k+

i

X

j=1

k U(t⁰⁰, t_j)−U(t⁰, t_j)

I_j(x(t_j))k

+ Z t⁰

0

kU(t⁰⁰, s)−U(t⁰, s) f(s, x(s), xs) +Bux(s) kds +

Z t⁰⁰

t⁰

kU(t⁰⁰, s)k · kf(s, x(s), xs) +Bux(s)kds

(3.11) From the equicontinuity property of U(·, s) and the absolute continuity of the Lebesgue integral, we see that the right-hand side of the inequality (3.11) tends to zero independent of x ∈ D as |t⁰⁰−t⁰| → 0, t⁰⁰, t⁰ ∈ Ji. Therefore, G(D) is equicontinuous on everyJi.

Next notice that

kxns−xmsk_L[−τ,0] ≤τkxn−xmkP C, kxn(s)−xm(s)k ≤ kxn−xmkP C, s∈J, which implies

β_τ({x_ns}^∞_n=1)≤τ β_{P C}({x_n}^∞_n=1), β({x_n(s)}^∞_n=1)≤β_{P C}({x_n}^∞_n=1), s∈J.

Then from (S2), (S3), (S4) and (S5), for eacht∈J, we have βV({ux_n(t)}^∞_n=1)

≤KW(t)β

{M(xn) +U(a,0)(x0−M(xn)) + Z a

0

U(a, s)f(s, xn(s), xns)ds

+

k

X

i=1

U(a, ti)Ii(xn(ti))}^∞_n=1

≤K_W(t)

l₃(1 +L_U)β_{P C}({x_n}^∞_n=1) + 2L_U Z a

0

l₁(s)β({x_n(s)}^∞_n=1)

+l₂(s)β_τ({x_ns}^∞_n=1)

ds+L_U

k

X

i=1

k_iβ({x_n(t_i)}^∞_n=1)

≤KW(t)

l3(1 +LU) + 2LU

Z a

0

l1(s) +τ l2(s)

ds+LU k

X

i=1

ki

βP C({xn}^∞_n=1), and

β({(Gxn)(t)}^∞_n=1)

≤β

{U(t,0)(x₀−M(x_n))}^∞_n=1 +β

{ Z t

0

U(t, s) f(s, x_n(s), x_ns) +Bu_xn(s)

ds}^∞_n=1

+β { X

0<ti<t

U(t, ti)Ii(xn(ti))}^∞_n=1

≤LUl3βP C({xn}^∞_n=1) + 2LU

Z a

0

l1(s)β({xn(s)}^∞_n=1) +l2(s)βτ({xns}^∞_n=1) ds

+ 2LULB

Z a

0

βV({uxn(s)}^∞_n=1)ds+LU k

X

i=1

kiβ({xn(ti)}^∞_n=1)

≤LU

h l3+ 2

Z a

0

l1(s) +τ l2(s)

ds+ 2LB

l3(1 +LU) + 2LU

Z a

0

l1(s) +τ l2(s) ds

+LU k

X

i=1

ki

Z a

0

KW(s)ds+

k

X

i=1

ki

i

βP C({xn}^∞_n=1)

≤L_Uh l₃+ 2

Z a

0

(l₁(s) +τ l₂(s))ds+

k

X

i=1

k_i

1 + 2L_BL_U Z a

0

K_W(s)ds

+ 2l3LB

Z a

0

KW(s)dsi

βP C({xn}^∞_n=1)

=d·β_{P C}({xn}^∞_n=1).

(3.12) Note since{Gx_n}^∞_n=1 is equicontinuous on eachJ_i,i= 0, . . . , kwe have (from a well known result on measures of noncompactness)

βP C({Gxn}^∞_n=1) = sup

0≤i≤k

sup

t∈Ji

β({(Gxn)(t)}^∞_n=1).

This together with (3.2), (3.10) and (3.12) guarantees that

βP C({xn}^∞_n=1)≤βP C({Gxn}^∞_n=1)≤d·βP C({xn}^∞_n=1), which implies thatD={x_n}^∞_n=1is relatively compact.

From M¨onch’s fixed point theorem,Ghas a fixed point inB(0, r) and immedi- ately the system (1.1) is nonlocally controllable onJ. Remark 3.2. Note that (1.1) with no effect of time delay was considered in [15].

The assumptions on f, M, and Ii in [15] are relaxed in this paper. For example M is not necessarily compact here, and the the assumptions (S2), (S3), and (S5) in our paper are weaker than assumptions (H2), (H3), and (H5) in [15].

Acknowledgements. The authors wish to thank the anonymous referees for their valuable suggestions.

This research was supported by grants 11171192 from the NNSF of China, and BS2010SF025 from the Promotive Research Fund for Excellent Young and Middle- Aged Scientists of Shandong Province.

References

[1] J. Banas, K. Goebel; Measure of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.

[2] L. Byszewski; Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991) 494-505.

[3] L. Byszewski, H. Akca;Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal., 34 (1998) 65-72.

[4] M. Choisy, J. F. Guegan, P. Rohani;Dynamics of infectious diseases and pulse vaccination:

Teasing apart the embedded resonance effects, Physica D., 22 (2006) 26-35.

[5] A. d’Onofrio;On pulse vaccination strategy in the SIR epidemic model with vertical trans- mission, Appl. Math. Lett., 18 (2005) 729-32.

[6] X. Fu, J. Qi, Y. Liu; General comparison principle for impulsive variable time differential equations with application, Nonlinear Anal., 42 (2000) 1421-1429.

[7] X. Fu, B. Yan, Y. Liu;Introduction to Impulsive Differential System, China Science Publisher, Beijing, 2005 (in Chinese).

[8] S. Gao, L. Chen, J. J. Nieto, A. Torres; Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine., 24 (2006) 6037-6045.

[9] K. Gu, V. Kharitonov, J. Chen;Stability of Time-Delay Systems, Birkh¨auser, Boston, Mas- sachusetts, 2003.

[10] J. Hale, S. Verduyn Lunel; Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993.

[11] S. Haykin;Neural Networks, Prentice Hall, New Jersey, 1999.

[12] M. L. Heard; A quasilinear hyperbolic integrodifferential equation related to a nonlinear string, Trans. American Math. Soc., 285 (1984) 805-823.

[13] H. P. Heinz;On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983) 1351-1371

[14] E. Hernandez, D. O’Regan; Controllability of Volterra-Fredholm type systems in Banach space, J. Franklin Inst., 346 (2009) 95-101.

[15] S. Ji, G. Li, M. Wang;Controllability of impulsive differential systems with nonlocal condi- tions, Applied Mathematics and Computation, 217 (2011) 6981-6989.

[16] M. Kamenskii, P. Obukhovskii, P. Zecca;Condensing Multivalued Maps and Semilinear Dif- ferential Inclusions in Banach Spaces, De Gruyter, 2001.

[17] V. Kolmanovskii, A. Myshkis;Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Netherlands, 1992.

[18] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;Theory of Impulsive Differential Equa- tions, World Scientific, Singapore, 1989.

[19] S. K. Ntouyas, D. O’Regan;Some remarks on controllability of evolution equations in Banach spaces, Elect. J Diff. Eqns., Vol. 2009 (2009), No. 79, 1-6.

[20] H. M¨onch;Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980) 985-99.

[21] R. Narasimha;Nonlinear vibration of an elastic string, J. Sound Vibration, 8 (1968) 134-146.

[22] S. Niculescu;Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, New York, 2001.

[23] A. Pazy;Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[24] I. Rachunkov, M. Tvrdy;Non-ordered lower and upper functions in second-order impulsive periodic problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005) 397-415.

[25] A. M. Samoilenko, N.A. Perestyuk;Impulsive Differential Equations, World Scientific, Sin- gapore, 1995.

[26] S. Tang, L. Chen;Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44 (2002) 185-199.

[27] J. Yan, A. Zhao, J. J. Nieto;Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling., 40 (2004) 509-518.

[28] S. T. Zavalishchin, A. N. Sesekin; Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers Group, Dordrecht, 1997.

Yansheng Liu

Department of Mathematics, Shandong Normal University, Jinan, 250014, China E-mail address:[email protected]

Donal O’Regan

Department of Mathematics, National University of Ireland, Galway, Ireland E-mail address:[email protected]