Global attractivity of the almost periodic solution of a delay logistic population model with impulses

Nonlinear Analysis 73 (2010) 3688–3697

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Global attractivity of the almost periodic solution of a delay logistic population model with impulsesI Qi Wang ∗ , Hongyan Zhang, Minmin Ding, Zhijie Wang School of Mathematical Sciences, Anhui University, Hefei, 230039, PR China

article

info

Article history: Received 26 February 2010 Accepted 7 July 2010 MSC: 34C25 34A39

abstract In this paper, we study the existence of almost periodic solutions of a delay logistic model with fixed moments of impulsive perturbations. By using a comparison theorem and constructing a suitable Lyapunov functional, a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution is obtained. As applications, some special models are studied; our new results improve and generalize former results. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Impulse Delay Persistence Lyapunov functional Almost periodic solution Global attractivity

1. Introduction In [1], Yan and Feng studied the global attractivity and oscillation of the nonlinear delay equation z 0 (t ) = z (t )[a(t ) + b(t )z p (t − mω) − c (t )z q (t − mω)],

(1.1)

where a, b, c ∈ C ([0, +∞), R) with a common period w > 0 (a > 0, c > 0), m is a positive integer, and p and q are positive constants with q > p. Eq. (1.1) can be used to describe the evolution of a single species (see [2,3] and the references therein). In [4], Chen considered the delayed periodic logistic equation N 0 (t ) = N (t )[a(t ) − b(t )N p (t − σ (t )) − c (t )N q (t − τ (t ))],

(1.2)

which describes the evolution of a single species. The existence of a positive periodic solution is established by using the method of coincidence degree [5]. On the other hand, if at certain moments of time some biotic factors act momentarily on the population, then the population number varies by jumps, and a number of models in ecology can be formulated as systems of impulsive differential equations. In recent years, impulsive differential equations have been intensively investigated (see [6–19] for details). In [6], the authors considered the following model:



x0 (t ) = x(t )[r (t ) − a(t )x(t ) − b(t )xp (t − σ (t )) − c (t )xq (t − τ (t ))], 1x(tk ) = dk x(tk ), k ∈ Z ,

t 6= tk ,

(1.3)

I Research supported by the Anhui Provincial Natural Science Foundation (090416237), NNSF of China (10971229, 10771001) and Foundation of Person with Ability of Anhui University (02303129). Financed by the 211 Project of Anhui University (KJTD002B) and the Foundation of Anhui Education Bureau (KJ2009A49, KJ2009A005Z) of China. ∗ Corresponding author. Fax: +86 0551 5107241. E-mail address: [email protected] (Q. Wang).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.07.016

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

3689

where x(t ) is the density of the population at time t, r (t ) is called the intrinsic growth rate, and a(t ), b(t ) and c (t ) are the self-inhibition coefficients. σ (t ) and τ (t ) are delays. The authors obtained the permanence and the existence, uniqueness, and global attractivity of the positive periodic solution of this system. Motivated by the above works, we discuss the permanence and the existence, uniqueness, and global attractivity of the positive almost periodic solution of system (1.3). The organization of this paper is as follows. In Section 2, we present some notations and lemmas. In Section 3, we study the existence of bounded solutions. In Section 4, we study the existence of a unique almost periodic solution of system (1.3). 2. Preliminaries Let R be the one-dimensional Euclidean space with elements u and norm kuk = supt ∈R |u(t )| and R+ = [0, +∞), R+ = (0, +∞), Ω ⊂ R, Ω 6= ∅. Let t0 ∈ R. We introduce the following notations. PC (t0 ) is the space of all functions φ : [t0 − h, t0 ] → Ω having points at θ1 , θ2 , . . . , θs ∈ [t0 − h, t0 ] of the first kind being left continuous at these points. For J = [t0 − h, t0 ], PC [J , R] = {ϕ | J → R, ϕ is continuously differentiable everywhere except at the points tk , {tk } ∈ B at which ϕ(tk− ) and ϕ(tk+ ) exist, and ϕ(tk− ) = ϕ(tk )}. Let φ0 ∈ PC (t0 ); denote by x(t ) = x(t ; t0 , φ0 ), x ∈ Ω , the solution of system (1.3) satisfying the initial conditions x(s; t0 , φ0 ) = φ0 (t ),

s ∈ [−h, t0 ],

x(t0 + 0, t0 , φ0 ) = φ0 (t0 ).

(2.1)

Note that the solution x(t ) of system (1.3) is a piecewise continuous functions with points of discontinuity of the first kind at tk , k ∈ Z , where it is left continuous, i.e. the following relations are satisfied: x(tk− ) = x(tk ), x(tk+ ) = (1 + dk )x(tk ), k ∈ Z . Now, we introduce some definitions. j

Definition 2.1 ([9]). The set of sequences {tk } = {tk+j − tk }, k, j ∈ Z , is said to be uniformly almost periodic if for any ε > 0 there exists a relatively dense set in R of ε -almost periods common for all of the sequences. j

Definition 2.2 ([9]). The set of sequences {tk }, k, j ∈ Z , is said to be uniformly almost periodic if and only if for each infinite sequence of shift {tk − αn }, k ∈ Z , n = 1, 2, . . . , αn ∈ R, we can choose a subsequence, convergent in B. Definition 2.3 ([9]). The function ϕ ∈ PC [J , Rn ] is said to be almost periodic if the followings hold. j

(a) The set of sequences {tk }, k, j ∈ Z is uniformly almost periodic. (b) For any ε > 0 there exists a real number δ > 0 such that, if the points t 0 and t 00 belong to one and the same interval of continuity of ϕ(t ) and satisfy the inequality |t 0 − t 00 | < δ , then |ϕ(t 0 ) − ϕ(t 00 )| < ε . (c) For any ε > 0 there exists a relatively dense set T such that, if τ ∈ T , then |ϕ(t + τ ) − ϕ(t )| < ε for all t ∈ R satisfying the condition |t − τk | > ε , k ∈ Z . The elements of T are called ε -almost periods. The sequences {φn }, φn = (ϕn (t ), Tn ) ∈ (PC [J , Rn ] × B) if and only if for any ε > 0 there exists n0 > 0 such that for n > n0 it follows that

ρ(T , Tn ) < ε,

kϕn (t ) − ϕ(t )k < ε

holds uniformly for t ∈ R \ Fε (s(Tn ∪ T )). See [9] for more details about almost periodicity of impulsive differential systems. Throughout this paper, we shall use the following notations. (N1) If f (t ), t ∈ R, is an almost periodic function, we set f u = supt ∈R f (t ), f l = inft ∈R f (t ).

RT

(N2) Denote the mean value m(f ) = limT →+∞ T1 0 f (t )dt. When f (t ) is an ω-periodic function, then m(f ) = ω1 Rω f (t )dt. Obviously, when f (t ) is an ω-periodic function, m(f ) > 0 ⇔ 0 f (t )dt > 0. We denote the hull of f (t ) by 0 H (f ), where H (f ) is the set of real function g (t ) such that there exists a sequence tn such that limn→+∞ f (t + tn ) = g (t ) uniformly on R. We also introduce the following assumptions.

Rω

(A1) r (t ), a(t ), b(t ), c (t ) ∈ C (R, [0, +∞)) are all continuous almost periodic functions, which are bounded above and below by positive constants. (A2) σ (t ) and τ (t ) are nonnegative, continuously differentiable and almost periodic functions in t ∈ R, and µ1 (t ) = t −σ (t ) and µ2 (t ) = t − τ (t ) are invertible. Moreover, σ 0 (t ), τ 0 (t ) are all uniformly continuous almost periodic functions on R with σ 0 (t ) < 1, τ 0 (t ) < 1. (A3) m(r ) > 0. Q (A4) The sequences {dk }, −1 < dk ≤ 0, 0
3690

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697 j

(A5) The set of sequences {tk }, k, j ∈ Z , is uniformly almost periodic and there exists κ > 0 such that infk∈Z tk1 = κ > 0. Consider the following model: y0 (t ) = y(t )[r (t ) − A(t )y(t ) − B(t )yp (t − σ (t )) − C (t )yq (t − τ (t ))], where A(t ) = a(t )

Q

(1 + dk ), B(t ) = b(t )

0
Q

0
(2.2)

(1 + dk ), C (t ) = c (t )

Q

0 < tk < t

( 1 + dk ) .

Lemma 2.1. For systems (1.3) and (2.2), the following results hold. (1) If y(t ) is a solution of (2.2), then x(t ) = y(t )

Q (1 + dk ) is a solution of (1.3). Q0
0 < tk < t

= (1 + dk )y(tk )

Y

0
(1 + dk ) = (1 + dk )x(tk ).

0
So the last equation of (1.3) also holds. This proves the conclusion of (1). (2) We first show that y(t ) is continuous. Since y(t ) is continuous on each Q interval (tk , tk+1 ], it is sufficient to check the continuity of y(t ) at the impulse points tk , k = 1, 2, . . . . Since y(t ) = x(t ) 0
Y

y(tk+ ) = x(tk+ )

(1 + dk )−1 = x(tk )

Y

y(tk ) = x(tk ) −

(1 + dj )−1 = y(tk ),

0
0 < tj ≤ tk

−

Y

( 1 + dk )

−1

= x(tk )

Y

(1 + dj )−1 = y(tk ).

0
0 < tj < tk

Thus y(t ) is continuous on [0, +∞). It is easy to check that y(t ) satisfies (2.2). Therefore, it is a solution of (2.2). This completes the proof of Lemma 2.1.  Lemma 2.2. Let y(t ) be any solution of (2.2) such that y(0) > 0; then y(t ) > 0 for all t ≥ 0. Proof. From (2.2), we have y(t ) = y(0) exp

t

Z



(r (s) − A(s)y(s) − B(s)y (s − σ (s)) − C (t )y (s − τ (s)))ds > 0. p

q

0

This completes the proof of Lemma 2.2.



We easily get the following. (A0 1). r (t ), A(t ), B(t ), C (t ) ∈ C (R, [0, +∞)) are all continuous almost periodic functions, which are bounded above and below by positive constants. In the following sections, we only discuss initial value problems (2.1)–(2.2). 3. Existence of bounded solutions Definition 3.1. The initial value problems (2.1)–(2.2) are said to be persistent if for any solution of them there exist positive constants m and M such that for all solutions y(t ) there exists T > 0 such that m ≤ y(t ) ≤ M, for all t ≥ T . The solution of the initial value problem is also called ultimately bounded above and below. Definition 3.2. The initial value problems (2.1)–(2.2) are said to be globally attractive if any two positive solutions x(t ) and y(t ) of the initial value problems (2.1)–(2.2) satisfy limt →+∞ |x(t ) − y(t )| = 0. In the following, we will prove a preliminary result which will be used in the proof of our main results. Now we consider the generalized almost periodic logistic equation x0 (t ) = x(t )[r (t ) − a(t )xα (t )],

t ∈ R,

(3.1)

where r (t ) and a(t ) are continuous almost periodic functions with α > 0, a > 0 and m(r ) > 0. We now state the following lemma. l

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

3691 1

Lemma 3.1. System (3.1) has a unique positive globally attractive almost periodic solution x˜ (t ) with x˜ (t ) ≤ lim supt →+∞ x˜ (t ) ≤

 u  α1 r al

 l α r au

≤ lim inft →+∞

. Let x˜ i (t ) (i = 1, 2) be the unique solution of (3.1) when replacing r (t ) by ri (t ) and a(t ) by

ai (t ) (i = 1, 2), respectively. If r2 (t ) ≥ r1 (t ) and a2 (t ) ≥ a1 (t ), then x˜ 2 (t ) ≥ x˜ 1 (t ). Proof. Similarly to the proof of Lemma 2.1 of [20], we can easily get Lemma 3.1. If (A0 1) and (A3) hold, it follows from Lemma 3.1 that the following system (α = 1) y0 (t ) = y(t )[r (t ) − A(t )y(t )],

t ∈ R,

(3.2)

has a unique globally attractive positive almost periodic solution, denoted by Y (t ).



Lemma 3.2. If (A0 1), (A2)–(A5) hold, then

" m[r (t ) − B(t )Y (t − σ (t )) − C (t )Y (t − τ (t ))] = m r (t ) − p

1 B(µ− 1 (t ))

q

1 1 − σ (µ− 1 (t ))

Y (t ) − p

1 C (µ− 2 (t )) 1 1 − τ (µ− 2 (t ))

Proof. From (A0 1), (A2) and the properties of almost periodic functions, we note that σ (t ), τ (t ), −1

C (µ2 (t )) are all almost periodic functions. −1 1−τ (µ2 (t )) −1 −1 B(µ1 (t )) T −σ T −τ C (µ2 (t )) Y p t dt T Yq t −1 −1 T 1−σ (µ1 (t )) 1−τ (µ2 (t ))

() ,

R

# Y (t ) . q

−1 B(µ1 (t )) −1 1−σ (µ1 (t ))

and

By the boundedness of the almost periodic functions, we can verify that

( )dt are bounded. Then, we have

R

m(B(t )Y (t − σ (t ))) = p

lim T

−1

T →+∞

T −σ (T )

Z

T →+∞

T →+∞

1 B(µ− 1 (t ))

T

Z

B(µ1−1 (t )) 1 1 − σ (µ− 1 (t ))

−σ (0)

= lim T −1

1 1 − σ 0 (µ− 1 (t ))

0

1 B(µ− 1 (t ))

0

+ −σ (0)

= lim T

1 − σ 0 (µ1 (t )) −1

−1

0

Y p (t )dt

Y p (t )dt

Y p (t )dt +

T −σ (T )

Z T

1 B(µ− 1 (t ))

T

Z

T →+∞

=m

B(t )Y p (t − σ (t ))dt 0

= lim T −1

Z

T

Z

1 1 − σ 0 (µ− 1 (t ))

1 B(µ− 1 (t )) 1 1 − σ 0 (µ− 1 (t ))

Y p (t )dt

Y p (t )dt

!

1 B(µ− 1 (t ))

Y (t ) . p

1 1 − σ 0 (µ− 1 (t ))

Similarly, we have m(C (t )Y (t − τ (t ))) = m

1 C (µ− 2 (t ))

q

1 1 − τ (µ− 2 (t ))

! Y (t ) q

" m[r (t ) − B(t )Y (t − σ (t )) − C (t )Y (t − τ (t ))] = m r (t ) − p

Thus the lemma is proved.

q

1 B(µ− 1 (t )) 1 1 − σ (µ− 1 (t ))

Y (t ) − p

1 C (µ− 2 (t )) 1 1 − τ (µ− 2 (t ))

# Y (t ) . q



Theorem 3.1. to (A0 1), (A2)–(A5), we further assume the following.  In addition 

(A6) m r (t ) −

−1 B(µ1 (t )) Yp −1 1−σ (µ1 (t ))

(t ) +

−1 C (µ2 (t )) Yq −1 1−τ (µ2 (t ))

(t )

> 0, where Y (t ) is the unique globally attractive positive almost

periodic solution of system (3.2). Then the initial value problems (2.1)–(2.2) is persistent. Proof. First, we show that any positive solution of the initial value problems (2.1)–(2.2) is ultimately bounded above by some positive constant. Let y(t ) be any other solution of the initial value problems (2.1)–(2.2). It follows from (2.2) that y0 (t ) ≤ y(t )[r (t ) − A(t )y(t )],

3692

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

for all t ≥ t0 . By using the comparison theorem, we have y(t ) ≤ Y (t ),

for all t ≥ t0 ,

(3.3)

where Y (t ) is the unique globally attractive positive almost periodic solution of system (3.2) which satisfies the initial condition y(t0 ) ≤ Y (t0 ). From Lemma 3.1 and (3.3), it is not difficult to obtain that the following two inequalities hold: lim sup y(t ) ≤ t →+∞

ru Al

,

for all t ∈ R,

and there is a T0 ≥ t0 such that, for ∀ε > 0, y(t ) ≤

ru Al

+ ε = H,

for all t ≥ T0 .

(3.4)

Second, we shall show that any positive solution of system (2.2) is ultimately bounded below by some positive constant. To this end, we proceed with two steps. Step 1: We show that there exists δ0 > 0 such that lim supt →+∞ y(t ) ≥ δ0 . In fact, it follows from (3.3) that, for any constant  > 0, there exists T () ≥ t0 such that y(t ) ≤ Y (t ) + ,

for all t ≥ T ().

(3.5)

Denote K (t ) = r (t ) − B(t )Y p (t − σ (t )) − C (t )Y q (t − τ (t )), K (t , ) = r (t ) − B(t )(Y (t − σ (t )) + )p − C (t )(Y (t − τ (t )) + )q ,

for all t ≥ T ().

It follows from (A6) and Lemma 3.2 that lim T −1

T →∞

Z

"

T +s

K (t )dt = m r (t ) −

s

1 B(µ− 1 (t )) 1 1 − σ (µ− 1 (t ))

Y p (t ) −

1 C (µ− 2 (t )) 1 1 − τ (µ− 2 (t ))

# Y q (t )

> 0,

uniformly for s ∈ R. Therefore, there exist positive constants and such that t +λ

Z

K (u)du ≥ δ,

for all t ∈ R.

(3.6)

t

From (3.6), we can choose sufficiently small positive constants 0 , δ0 , and γ0 , such that t +λ

Z

[K (u, 0 ) − a(u)(δ0 + 0 )]du ≥ γ0 ,

for all t ∈ R.

(3.7)

t

Now we claim that the following inequality holds: lim sup y(t ) ≥ δ0 .

(3.8)

t →+∞

By way of contradiction, suppose that lim supt →+∞ y(t ) < δ0 ; then there exists T > T () such that y(t ) < δ0 + 0 , for all t ≥ T . This, together with (3.5), gives y0 (t ) ≥ y(t )[r (t ) − A(t )(δ0 + 0 ) − B(t )(Y (t − σ (t )) + 0 )p − C (t )(Y (t − τ (t )) + 0 )q ]

= y(t )[K (t , 0 ) − A(t )(δ0 + 0 )],

(3.9)

for all t ≥ T . An integration of (3.9) over [T , t ] leads to y(t ) ≥ y(T ) exp

t

Z



[K (s, 0 ) − A(s)(δ0 + 0 )]ds . T

Obviously, it follows from (3.7) that y(t ) → +∞, which contradicts y(t ) ≤ H, for all t ≥ T in (3.5). Hence, the inequality (3.8) is correct. Step 2: We show that, for any solution y(t ), there exists a positive constant L > 0 such that lim inf y(t ) ≥ L.

(3.10)

t →+∞

In fact, suppose the contrary, i.e., there exists a sequence of solutions {yn (t )}, n = 1, 2, . . . , such that lim inf yn (t ) < t →+∞

1 n

.

(3.11)

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

3693

Then there exist two time sequences {sn } and {tn } such that 0 < s1 < t1 < s2 < t2 < · · · < sn < tn < · · · , sn → ∞, tn → ∞,

y n ( sn ) =

as n → ∞,

for all n = 1, 2, . . . , 1 n2

,

yn (tn ) =

1 n

(3.12)

,

and 1 n2

< yn (t ) <

1 n

,

for all t ∈ (sn , tn ).

(3.13)

It follows from (3.5) that there exists Tn > T (0 ) such that yn (t ) ≤ Y (t ) + 0 ,

t ≥ Tn .

Clearly, there exists an integer N such that sn > Tn for n ≥ N. Hence, for any t ∈ [sn , tn ] and n ≥ N, we have y0 (t ) = yn (t )[r (t ) − A(t )yn (t ) − B(t )ypn (t − σ (t )) − C (t )yqn (t − τ (t ))]

≥ yn (t )[r (t ) − A(t )(Yn (t ) + 0 ) − B(t )(Yn (t − σ (t )) + 0 )p − C (t )(Yn (t − τ (t )) + 0 )q ] ≥ −kyn (t ),

(3.14)

where k = sup A(t )(Yn (t ) + 0 ) + B(t )(Yn (t − σ (t )) + 0 )p + C (t )(Yn (t − τ (t )) + 0 )q .





t ∈R

An integration of (3.14) over [sn , tn ] leads to 1 n2

= yn (tn ) ≥ yn (sn ) exp(−k(tn − sn )) =

1 n

exp(−k(tn − sn )),

for n ≥ N, which implies that tn − sn ≥

ln n k

,

for all n ≥ N .

(3.15)

It follows from (3.15) that there exists a sufficiently large integer n0 such that

δ0 >

1 n

,

tn − sn ≥ λ,

for all n ≥ n0 ,

(3.16)

Therefore, for any n ≥ n0 and t ∈ [sn , tn ], it follows from (3.12) and (3.13) that y0 (t ) = yn (t )[r (t ) − A(t )yn (t ) − B(t )ypn (t − σ (t )) − C (t )yqn (t − τ (t ))]

  1 ≥ yn (t ) r (t ) − A(t ) − B(t )(Yn (t − σ (t )) + 0 )p − C (t )(Yn (t − τ (t )) + 0 )q n

≥ yn (t )[r (t ) − A(t )δ0 − B(t )(Yn (t − σ (t )) + 0 )p − C (t )(Yn (t − τ (t )) + 0 )q ] = yn (t )[K (t , 0 ) − A(t )δ0 ].

(3.17)

Together with (3.7), (3.12) and (3.13), an integration of (3.17) over [tn − λ, tn ] leads to 1 n2

= yn (tn ) ≥ yn (tn − λ) exp

Z

tn

tn −λ

[K (t , 0 ) − A(t )δ0 ]dt



>

eγ 0 n2

>

1 n2

.

This is a contradiction; thus the inequality (3.10) is correct. That is to say, any positive solution y(t ) of the initial value problems (2.1)–(2.2) is ultimately bounded below by a positive constant m. From Definition 2.1, the proof of Theorem 3.1 is complete.  Denote K = {y(t ) : m ≤ y(t ) ≤ M , t ∈ R}. Theorem 3.2. If (A0 1), (A2)–(A6) hold, then the initial value problems (2.1)–(2.2) have at least one positive solution on all of R belonging to K .

3694

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

Proof. Lemma 3.2 implies that there exists T0 ≥ T0 such that the initial value problems (2.1)–(2.2) have at least one positive solution y(t ) satisfying 0 < m ≤ y(t ) ≤ M for t ≥ T0 . In what follows, we will prove that the initial value problems (2.1)–(2.2) have at least one positive solution v(t ) defined on R such that m ≤ v(t ) ≤ M, for all t ∈ R. Since r (t ), A(t ), B(t ), C (t ), τ (t ), σ (t ) ∈ C (R, [0, +∞)) are all continuous almost periodic functions, there is a sequence {tn }, n → ∞, tn → ∞ such that r (t + tn ) → r (t ), A(t + tn ) → A(t ), B(t + tn ) → B(t ), C (t + tn ) → C (t ), τ (t + tn ) → τ (t ), σ (t + tn ) → σ (t ) uniformly for all t ∈ R as n → ∞. We claim that the sequence {y(t + tn )} is uniformly bounded and equi-continuous on any bounded interval in R. In fact, for any bounded interval [t− , t+ ] ⊂ R, if n is large enough, tn + h ≥ T0 . So 0 < λ ≤ y(t + tn ) ≤ β for t ∈ [t− , t+ ], which implies that the sequence {y(t + tn )} is uniformly bounded. On the other hand, ∀t1 , t2 ∈ [t− , t+ ], from the elementary mean value theorem of differential calculus, we have

|y(t1 + tn ) − y(t2 + tn )| ≤ β[r u + au β + bu β p + c u β q ]|t1 − t2 |. The above inequality shows that the sequence {y(t + tn )} is equi-continuous on [t− , t+ ], and the claim follows. By the Ascoli–Arzela theorem, there exist a subsequence of {tn } (we still denote it as {tn } and a continuous function v(t ) such that v(t + tn ) → v(t ) as n → ∞ uniformly in t on any bounded interval in R. Let θ ∈ R be given. We may assume that tn + θ ≥ 0, for all n. For t ≥ t0 , an integration of (2.2) over [tn + θ , t + tn + θ] leads to y(t + tn + θ ) − y(tn + θ ) =

t +tn +θ

Z

y(s)[r (s) − A(s)y(s) − B(s)yp (s − σ (s)) − C (s)yq (s − τ (s))]ds

tn +θ t +θ

Z

y(s + tn )[r (s + tn ) − A(s + tn )y(s + tn ) − B(s + tn )yp (s + tn − σ (s + tn ))

= θ

− C (s + tn )yq (s + tn − τ (s + tn ))]ds. Using the Lebesgue dominated convergence theorem, one has

v(t + θ ) − v(θ ) =

t +θ

Z

v(s)[r (s) − A(s)v(s) − B(s)v p (s − σ (s)) − C (s)yq (s − τ (s))]ds.

θ

This means that v(t ) is a solution of (2.2), and by the arbitrariness of θ , v(t ) is a solution of (2.2) on R with 0 < m ≤ v(t ) ≤ M. Therefore, Theorem 3.2 is valid.  4. Main theorem Lemma 4.1 (See [21]). Let f be a nonnegative function defined on [0, +∞) such that f is integrable on [0, +∞) and is uniformly continuous on [0, +∞). Then limt →+∞ f (t ) = 0. Lemma 4.2 (See [22, Theorem 10.1], [23, Theorem 3.2]). Consider the system x0 = f (t , x), and suppose that f (t , x) is almost periodic in t uniformly in x ∈ K , and that K is compact in R. If each equation x0 = g (t , x), g ∈ H (f ) (where H (f ) is the hull of f ) has a unique solution on R belonging to K , then these solutions are almost periodic. Assume that α(t ) is an almost periodic function and that α(t ) is a continuously differentiable almost periodic function. Furthermore, α 0 (t ) is uniformly continuous on R with inft ∈R {1 − α 0 (t )} > 0. Obviously, α 0 (t ) is also a continuously almost periodic function. Lemma 4.3 (See [23,24]). Suppose that (α ∗ (t ), β ∗ (t )) ∈ H (α(t ), β(t )) and that σ −1 (t ), σ ∗−1 (t ) is the inverse of σ (t ) = t − β(t ), σ ∗ (t ) = t − β ∗ (t ), respectively. Then we have the following. (a) If there exists a sequence {tn } such that α(t + tn ) → α ∗ (t ), β(t + tn ) → β ∗ (t ) uniformly on R as n → ∞, then α(σ −1 (t + tn )) → α ∗ (σ ∗−1 (t )) uniformly on R. (b) α(σ −1 (t )) and α ∗ (σ ∗−1 (t )) are all almost periodic and α ∗ (σ ∗−1 (t )) ∈ H (α(σ −1 (t ))). For the convenience, we set

η(t ) = a(t ) − pρ(t )p−1

1 B(µ− 1 (t ))

1 − τ 0 (µ1 (t )) −1

− qρ(t )q−1

1 C (µ− 2 (t )) 1 1 − σ 0 (µ− 2 (t ))

,

where ρ(t ) is between x(t ) and y(t ) (see Lemma 4.4 for details). Lemma 4.4. In addition to (A0 1), (A2)–(A6), if the initial value problems (2.1)–(2.2) also satisfy the following conditions: R +∞ (A7) η(t ) > 0 and 0 η(t )dt = ∞, then the initial value problems (2.1)–(2.2) have a unique positive bounded solution y(t ) which is globally attractive.

(4.1)

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

3695

Proof. In order to show the global attractivity of the bounded solution x(t ) of the initial value problems (2.1)–(2.2), we shall show that the bounded solution y(t ) of the initial value problems (2.1)–(2.2) is globally attractive. Consider the following Lyapunov functional: V (t ) = |lnx(t ) − ln y(t )| +

Z

1 B(µ− 1 (s))

t

Z

t −τ (t )

1 C (µ− 2 (s))

t

+ t −σ (t )

1 1 − σ 0 (µ− 2 (s))

1 1 − τ 0 (µ− 1 (s))

|xp (s) − yp (s)|ds

|xq (s) − yq (s)|ds,

t ≥ 0.

(4.2)

Calculating the upper right derivative D+ V (t ) of V (t ) along the solution of (2.2) D+ V (t ) ≤ −a(t )|x(t ) − y(t )| +

1 B(µ− 1 (t ))

1 − τ 0 (µ1 (t )) −1

≤ −a(t )|x(t ) − y(t )| + pρ(t )p−1

|xp (t ) − yp (t )| +

1 B(µ− 1 (t )) 1 1 − τ 0 (µ− 1 (t ))

1 C (µ− 2 (t )) 1 1 − σ 0 (µ− 2 (t ))

|x(t ) − y(t )| + qρ(t )q−1

|xq (t ) − yq (t )| 1 C (µ− 2 (t )) 1 1 − σ 0 (µ− 2 (t ))

|x(t ) − y(t )|

≤ −η(t )|x(t ) − y(t )| < −ηl |x(t ) − y(t )| < 0,

(4.3)

where ρ(t ) is between x(t ) and y(t ). On integrating (4.3) over [T0 , t ], we obtain that

ηl

t

Z

|x(s) − y(s)|ds < V (T0 ) − V (t ),

for t ≥ t0 .

T0

Therefore, we have

Z t →+∞

V (T0 )

t

|x(s) − y(s)|ds <

lim sup

ηl

T0

< +∞,

for t ≥ t0 .

(4.4)

By Lemma 4.1, from (4.4), one can easily deduce that lim |x(t ) − y(t )| = 0,

t →+∞

which implies the global attractivity of system (2.2). By using the equivalence between (2.2) and (1.3), it follows that the bounded solution x(t ) of system (1.3) is also globally attractive. This completes the proof of Lemma 4.4.  Now consider the hull system y0 (t ) = y(t )[r ∗ (t ) − A∗ (t )y(t ) − B∗ (t )yp (t − σ ∗ (t )) − C ∗ (t )yq (t − τ ∗ (t ))],

(4.5)

where for some sequence {tn } with tn → ∞ as n → ∞, r (t + tn ) → r (t ), A(t + tn ) → A (t ), B(t + tn ) → B (t ), C (t + tn ) → C ∗ (t ), τ (t + tn ) → τ ∗ (t ), σ (t + tn ) → σ ∗ (t ) uniformly for all t ∈ R as n → ∞. From Lemma 4.3, it follows that ∗

∗

∗

lim [r (t + tn ) − B(t + tn )Y p (t + tn − σ (t + tn )) − C (t + tn )Y q (t + tn − τ (t + tn ))]

n→∞

= r ∗ (t ) −

1 B∗ (µ∗− (t )) 1 1 1 − σ ∗ (µ∗− (t )) 1

Y ∗p ( t ) −

1 C (µ∗− (t )) 2

1 − τ ∗ (µ2∗−1 (t ))

Y ∗q (t ). ∗−1 B∗ (µ1 (t ))Y ∗p (t ) ∗−1 1−σ ∗ (µ1 (t ))

Notice that r ∗ (t ), A∗ (t ), B∗ (t ), C ∗ (t ), σ ∗ (t ), τ ∗ (t ), Y ∗ (t ), r ∗ (t ) − in t. We then have the following.

(4.6)

−

∗−1 C (µ2 (t ))Y ∗q (t ) ∗−1 1−τ ∗ (µ2 (t ))

are all almost periodic

Lemma 4.5. Suppose that the conditions in Lemma 4.4 hold; then the hull system (4.5) has a unique bounded solution y¯ (t ) ∈ K on R, which is globally attractive. Proof. By the definition of mean value and the assumptions (A0 1), (A2), together with (4.6), it is easy to prove that m(r ∗ (t )) = m(r (t )) > 0,

" m r (t ) − ∗

1 B∗ (µ∗− (t )) 1 1 1 − σ ∗ (µ∗− (t )) 1

" = m r (t ) −

Y

1 B(µ− 1 (t )) 1 1 − σ (µ− 1 (t ))

∗p

(t ) −

1 C (µ− 2 (t )) 1 1 − τ (µ− 2 (t ))

Y (t ) ∗q

1 1 − τ ∗ (µ∗− (t )) 2

Y (t ) − p

#

1 C (µ∗− (t )) 2

# Y (t ) q

> 0,

3696

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

and mint ∈R η∗ (t ) > η∗l > 0. These imply that, corresponding to (4.5), all the requirements in Lemma 4.4 are satisfied. Then applying Lemma 4.4 to the hull system (4.5), we obtain that system (4.5) has a unique positive bounded solution y∗ (t ) ∈ K on R, which is globally attractive. This completes the proof Lemma 4.5.  By Lemma 4.5, it follows that, for each g (t , X ) ∈ H (f (t , X )), the hull equation x0 = g (t , X ) has a unique bounded solution on R with value in K . Hence, from Lemma 4.2, this unique solution is almost periodic. By the global attractivity, y(t ) is the unique almost periodic solution of (2.2) contained in K . Thus, our main results follows. Theorem 4.1. Suppose the conditions in Lemma 4.4 hold; then the initial value problems (2.1)–(2.2) have a unique positive almost periodic solution y(t ) ∈ K on R, which is globally attractive. Similarly to the proofs of Lemma 31 and Theorem 79 in [9], we have the following. Theorem 4.2. Suppose that the conditions in Lemma 4.4 hold; then the initial value problems (1.3)–(2.1) have a unique positive almost periodic solution x(t ) ∈ K on R, which is globally attractive. When we consider (2.2) in a periodic environment, i.e., r (t ), a(t ), b(t ), c (t ), σ (t ), τ (t ), dk and impulses are all ωperiodic, we have the following. Theorem 4.3. Suppose that the coefficients of (2.2) are all ω-periodic and that all the conditions in Theorem 4.1 hold; then the initial value problems (2.1)–(2.2) have a unique positive ω-periodic solution y(t ) ∈ K on R, which is globally attractive. Proof. Let xQ (t ) be the unique positive almost periodic solution of (2.1)–(2.2), but in the periodic case, r (t ), a(t ), b(t ), c (t ), σ (T ), τ (t ), 0
J. Yan, Q. Feng, Global attractivity and oscillation in a nonlinear delay equation, Nonlinear Anal. 43 (2001) 101–108. K. Gopalsamy, G. Ladss, On the oscillation and asymptotic behavior of N˙ (t ) = N (t )[a + bN (t − τ ) − cN 2 (t − τ )], Quart. Appl. Math. 48 (1990) 433–440. B.S. Lalli, B.G. Zhang, On a periodic delay population model, Quart. Appl. Math. 52 (1994) 35–42. Yuming Chen, Periodic solutions of a delayed periodic logistic equation, Appl. Math. Lett. 16 (2003) 1047–1051. R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. Pei Yongzhen, Li Changguo, Chen Lansun, Global attractivity of the positive periodic solution of a delay logistic population model with impulses, J. Math. (PRC) 30 (2010) 10–14. M.X. He, Z. Li, F.D. Chen, Permanence, extinction and global attractivity of the periodic Gilpin–Ayala competition system with impulses, Nonlinear Anal. RWA 11 (3) (2010) 1537–1551. D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, 1993. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. Z. Jin, M.A. Han, G.H. Li, The persistence in a Lotka–Volterra competition systems with impulsive, Chaos Solitons Fractals 24 (2005) 1105–1117. X.Q. Li, J.W. Dou, G.H. Song, Asymptotic behavior of a periodic impulsive ordinary equations model of plankton allelopathy, J. Biomath. 21 (2006) 216–224 (in Chinese). Shair Ahmad, Gani Tr. Stamov, Almost periodic solutions of N-dimensional impulsive competitive systems, Nonlinear Anal. RWA 10 (3) (2009) 1846–1853. Z.J. Liu, L.S. Chen, Periodic solution of a two-species competitive system with toxicant and birth pulse, Chaos Solitons Fractals 32 (5) (2007) 1703–1712. M.X. He, F.D. Chen, Dynamic behaviors of the impulsive periodic multi-species predator–prey system, Comput. Math. Appl. 57 (2) (2009) 248–265.

Q. Wang et al. / Nonlinear Analysis 73 (2010) 3688–3697

3697

[16] Shair Ahmad, Gani Tr. Stamov, On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations, Nonlinear Anal. RWA 10 (2009) 2857–2863. [17] S. Ahmad, Ivanka M. Stamova, Asymptotic stability of competitive systems with delays and impulsive perturbations, J. Math. Anal. Appl. 334 (2007) 686–700. [18] S. Ahmad, I.M. Stamova, Asymptotic stability of an N-dimensional impulsive competitive system, Nonlinear Anal. RWA 8 (2007) 654–663. [19] Shair Ahmad, Ivanka M. Stamova, Global exponential stability for impulsive cellular neural networks with time-varying delays, Nonlinear Anal. TMA 69 (2008) 786–795. [20] Y.H. Xia, M.A. Han, Zh.K. Huang, Global attractivity of an almost periodic N-species nonlinear ecological competitive model, J. Math. Anal. Appl. 337 (2008) 144–168. [21] I. Barbălat, Systems d’quations differentielle d’scillations nonlineaires, Rev. Roumaine Math. Pures Appl. 4 (1959) 267–270. [22] A.M. Fink, Almost Periodic Differential Equations, in: Lecture Notes in Math., vol. 3777, Springer-Verlag, Berlin, 1974. [23] C. He, Almost Periodic Differential Equations, Higher Education Press, 1992 (in Chinese). [24] Z. Teng, Permanence and stability of Lotka–Volterra type n-species competitive systems, Acta Math. Sinica 45 (5) (2002) 905–918.