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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Existence of positive periodic solutions for a nonautonomous neutral delay n-species competitive model with impulsesI Zhenguo Luo a , Binxiang Dai a,∗ , Qi Wang b a

School of Mathematical Sciences and Computer Technology, Central South University, Changsha, Hunan 410075, China

b

School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China

article

abstract

info

Article history: Received 23 July 2009 Accepted 10 March 2010

In this paper, by using some analysis techniques and a new existence theorem, which is different from Gaines and Mawhin’s continuation theorem and abstract continuation theory for k-set contraction, we obtain some sufficient and realistic conditions for the existence of positive periodic solution of a general neutral delay n-species competitive model with impulses. As an application, we also examine some special cases which have been studied extensively in the literature. Some known results are improved and generalized. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Positive periodic solution Neutral Competitive model Impulse

1. Introduction In this paper, we investigate the existence of the positive periodic solution of the following n-species competition system with general periodic neutral delay and impulse

" Z t n n X X 0 N ( t ) = N ( t ) r ( t ) − a ( t ) N ( t ) − b ( t ) kij (t − s)Nj (s)ds i i ij j ij i −∞ j =1 j =1 # n n X X − cij (t )Nj (t − τij (t )) − dij (t )Nj0 (t − γij (t )) , i = 1, 2, . . . , n, t 6= tk , j =1 j=1 1Ni (tk ) = Ni (tk+ ) − Ni (tk ) = θik Ni (tk ), i = 1, 2, . . . , n, k = 1, 2, . . . ,

(1.1)

1 2 where aij , bij , cij ∈ C (R, [0, +∞)), dij ∈ C 1 (R, [0, +∞)), R ωτij ∈ C (R, R), γij ∈ C (R, R) are continuous ω-periodic functions, ri ∈ C (R, R) are continuous ω-periodic functions with 0 ri (t )dt > 0. The growth functions ri are not necessarily positive, since the environment fluctuates randomly, in some conditions ri may be negative. τ = max{maxt ∈[0,ω] {τij (t ), γij (t ), i, j =

R∞

R +∞

1, 2, . . . , n}}. 0 kij (s)ds = 1, 0 skij (s)ds < +∞, i, j = 1, 2, . . . , n. For the ecological background of (1.1), we refer to [1–6] and the references cited therein. In 1993, Kuang in [1] proposed an open problem (Open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation dN dt

= N (t )[a(t ) − β(t )N (t ) − b(t )N (t − τ (t )) − c (t )N 0 (t − τ (t ))].

(1.2)

I Research was supported by the National Natural Science Foundation of China (10971229) and the National Basic Research Program of China (2007CB714107). ∗ Corresponding author. E-mail addresses: [email protected] (Z. Luo), [email protected] (B. Dai).

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.03.003

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Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

In [2], Fang and Li studied Eq. (1.2) and gave an answer to the open problem 9.2 of [1]. But paper [2] required that c (t ) b(t ) ≥ 0, c (t ) ≥ 0 and c00 (t ) < b(t ), β(t ) ≥ 0 or c00 (t ) ≤ b(t ), β(t ) > 0 for t ∈ [0, ω], where c0 (t ) = 1−τ 0 (t ) . For multiple species system, Li [3] considered the following n-species neutral Lotka–Voterra system with constant delay of the form:

" Ni (t ) = Ni (t ) αi (t ) − 0

n X

βij (t )Nj (t − τij ) −

j =1

n X

# cij (t )Nj (t − γij ) ,

i = 1, 2, . . . , n.

0

(1.3)

j=1

Author used the continuation theorem of the coincidence degree theory to study the existence of a positive periodic solution of the system (1.3). In [4], Fang investigated the following n-species competition system with general periodic neutral delay:

" Ni (t ) = Ni (t ) ai (t ) − 0

n X

βij (t )Nj (t ) −

j =1

n X

bij (t )Nj (t − τij (t )) −

j =1

n X

# cij (t )Nj (t − τij (t )) , 0

i = 1 , 2 , . . . , n.

(1.4)

j =1

Author established some criteria to guarantee the existence of positive periodic solutions of system (1.4). Yang and Cao [5] also applied the theory of coincidence degree to obtain verifiable sufficient conditions of the existence of positive periodic solutions of system (1.4). Recently, Liu and Chen [6] further discussed the above system (1.4). They introduced a new existence theorem to obtain a set of sufficient conditions for the existence of positive periodic solutions for system (1.4), improved and generalized some known results. On the other hand, there are some other perturbations in the real world such as fires and floods, that are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations involving impulsive differential equations have attracted the interest of many researchers in the past twenty years [7–10], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, optimal control, etc. For details, see [11–13]. Recently, the corresponding theory for impulsive functional differential equations has been studied by many authors [14–20]. In [19], Wang and Dai investigated the following periodic neutral population model with delays and impulse:

dN dt

" = N (t ) a(t ) − e(t )N (t ) −

N (tk+ ) = (1 + θk )N (tk ),

n X

bj (t )N (t − σj (t )) −

j=1

m X

# ci (t )N (t − τi (t )) , 0

t 6= tk ,

i =1

k = 1, 2, . . .

(1.5)

they obtained some sufficient conditions for the existence of positive periodic solutions of the model (1.5) by using the theory of abstract continuous theorem of k-set contractive operator and some analysis techniques. In [20], Huo studied the following n-species neutral impulsive delay Lotka–Volterra system:

" Ni (t ) = Ni (t ) αi (t ) − 0

n X

βij (t )Nj (t − τij (t )) −

j =1

1Ni (tk ) = Ni (tk ) − Ni (tk ) = bik Ni (tk ), +

n X

# cij (t )Nj (t − γij (t )) , 0

i = 1, 2, . . . , n, t 6= tk ,

j =1

(1.6)

i = 1, 2, . . . , n, k = 1, 2, . . . ,

by using some techniques of Mawhin coincidence degree theory, author obtained sufficient conditions for the existence of periodic positive solutions of the system (1.6). However, to this day, no scholars had done works on the existence of positive periodic solution of (1.1). One could easily see that system (1.2)–(1.6) are all special cases of system (1.1). Therefore, we propose and study the system (1.1) in this paper. For the sake of generality and convenience, we make the following notation: Cω = {x|x ∈ C (R, R), x(t + ω) = x(t )}, with the norm defined by |x|0 = maxt ∈[0,ω] |x(t )|; Cω1 = {x|x ∈ C 1 (R, R), x(t + ω) = x(t )}, with the norm defined by kxk = maxt ∈[0,ω] {|x|0 , |x0 |0 }; PC = {x|x : R → R+ , lims→t x(s) = x(t ), if t 6= tk ; limt →t − x(t ) = x(tk ), limt →t + x(t ) exists, k ∈ Z + }; k

k

PC 1 = {x|x : R → R+ , x0 ∈ PC }; PCω = {x|x ∈ PC , x(t + ω) = x(t )}, with the norm defined by |x|0 = maxt ∈[0,ω] |x(t )|; PCω1 = {x|x ∈ PC 1 , x(t + ω) = x(t )}, with the norm defined by kxk = maxt ∈[0,ω] {|x|0 , |x0 |0 }. Then those spaces are all Banach spaces. We also denote f =

1

ω

ω

Z 0

f (t )dt ,

f L = min f (t ), t ∈[0,ω]

for any f ∈ PCω ,

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

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and make the following assumptions:

(H1 ) 0 < t1 < t2 < · · · < tk < · · · and limk→∞ tk = +∞ Q; (H2 ) {θik } are real sequences such that θik + 1 > 0, and 0

" yi (t ) = yi (t ) ri (t ) − 0

n X

n X

Aij (t )yj (t ) −

j =1 n X

−

Bij (t )

Cij (t )yj (t − τij (t )) −

j =1

kij (t − s)yj (s)ds −∞

j =1 n X

t

Z

# Dij (t )yj (t − γij (t )) ,

i = 1 , 2 , . . . , n,

0

(1.7)

j=1

where

Y

Aij (t ) = aij (t )

(1 + θik ),

0

(1 + θik ),

0 < tk < t

Y

Cij (t ) = cij (t )

Y

Bij (t ) = bij (t )

(1 + θik ),

Y

Dij (t ) = dij (t )

0

(1 + θik ),

i , j = 1 , 2 , . . . , n.

0

The following lemma will be used in the proof of our results. The proof is similar to that of Theorem 1 in [7]. Lemma 1.1. Suppose that (H1 ), (H2 ) hold, then T T (1) if Q (y1 (t ), y2 (t ), . . . , yn (t )) is a solution of (1.7), then (N1 (t ), N2 (t ), . . . , Nn (t )) is a solution of (1.1), where Ni (t ) = 0

Proof. (1) It is easy to see that Ni (t ) = 0

Q

" Ni (t ) − Ni (t ) ri (t ) − 0

n X

aij (t )Nj (t ) −

j=1

−

n X

n X

bij (t )

cij (t )Nj (t − τij (t )) −

j =1

kij (t − s)Nj (s)ds −∞

j =1 n X

t

Z

# dij (t )Nj (t − γij (t )) 0

j =1

=

Y

(1 + θik )yi (t ) −

n X

bij (t )

Y

(1 + θik )

n X

t

kij (t − s)yj (s)ds − −∞

Y

dij (t )

Y

(1 + θik )yj (t − τij (t ))

0

(1 + θik )yj (t − γij (t )) "

(1 + θik ) yi (t ) − yi (t ) ri (t ) −

n X

Aij (t )yj (t ) −

j =1

Cij (t )yj (t − τij (t )) −

j =1

n X

n X j =1

Bij (t )

Z

t

kij (t − s)yj (s)ds −∞

#) Dij (t )yj (t − γij (t )) 0

j =1

= 0.

(1.8)

On the other hand, for any t = tk , k = 1, 2, . . . , Ni (tk+ ) = lim

+

Y

(1 + θij )yi (t ) =

t →tk 0

and Ni (tk ) =

cij (t )

0

0

n X

(1 + θik )yj (t )

0 < tk < t

j =1

0

−

n X

Y

0

( =

aij (t )

j =1

Y

n X j=1

Z

0

j =1

−

(1 + θik )yi (t ) ri (t ) −

0 < tk < t

0

−

Y

0

Y 0

(1 + θij )yi (tk ),

Y 0

(1 + θij )yi (tk ),

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Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

thus Ni (tk+ ) = (1 + θik )Ni (tk ),

(1.9)

T which implies that (N Q1 (t ), N2 (t ), . . . , Nn (t )) is a solution of (1.1). (2) Since Ni (t ) = 0

yi (tk+ ) =

Y

Y

(1 + θik )−1 Ni (tk+ ) =

0

(1 + θik )−1 Ni (tk ) = yi (tk ),

0

and

Y

yi (tk− ) =

(1 + θij )−1 Ni (tk− ) = yi (tk ),

i = 1, 2, . . . , n,

0

which implies that yi (t ), i = 1, 2, . . . , n, are continuous. It is easy to prove that yi (t ), i = 1, 2, . . . , n, are absolutely continuous. Similar to the proof of (1), we can check that (y1 (t ), y2 (t ), . . . , yn (t ))T is a solution of (1.7). The proof of Lemma 1.1 is completed. From Lemma 1.1, if we want to discuss the existence and global asymptotic stability of positive periodic solutions of systems (1.1), we only discuss the existence and global asymptotic stability of positive periodic solutions of systems (1.7). The present paper is organized as follows: In the next section we introduce some lemmas and an important existence theorem developed in [21,22]. In Section 3, we derive some sufficient conditions which ensure the existences of positive periodic solution of system (1.1) by applying this theorem and some other techniques. Finally as an application, we study some special cases of system (1.1) which have been studied extensively in the literature. These examples show that our sufficient conditions are new, and some known results can be improved and generalized. 2. Preliminaries In this section, in order to obtain the existence of a periodic solution for system (1.7), we shall give some concepts and results from [22], state an existence theorem and some lemmas. For a fixed τ > 0, let C =: C ([−τ , 0]; Rn ). If x ∈ C ([−τ , 0]; Rn ) for some δ > 0 and η ∈ R, then xt ∈ C for t ∈ [η, η + δ] is defined by xt (θ ) = x(t + θ ) for θ ∈ [−τ , 0]. The supremum Pn norm in C is denoted by k.k, nthat is, kφk = maxt ∈[−τ ,0] |φ(θ )| for φ ∈ C , where |.| denotes the norm in Rn , and |u| = j=1 |ui | for u = (u1 , . . . , un ) ∈ R . Consider the following neutral functional differential equation: d dt

[x(t ) − b(t , xt )] = f (t , xt ),

(2.1)

where f : R × C → Rn is completely continuous and b : R × C → Rn is continuous. Moreover, we assume: (1) there exists ω > 0 such that for every (t , φ) ∈ R × C , we have b(t + ω, φ) = b(t , φ) and f (t + ω, φ) = f (t , φ); (2) there exists a constant k < 1 such that |b(t , φ) − b(t , ϕ)| ≤ kkφ − ϕk, for t ∈ R and φ, ϕ ∈ C . By using the continuation theorem for composite coincidence degree, Erbe et al. [21] proved the following existence theorem (see also Theorem 4.7.1 in [22]). Theorem A ([22]). Assume that there exists a constant M > 0 such that: (i) for any λ ∈ (0, 1) and any ω-periodic solution x of the system d dt

[x(t ) − λb(t , xt )] = λf (t , xt ),

(2.2)

we have |Rx(t )| < M for t ∈ R; ω (ii) g (u) =: 0 f (s, uˆ )ds 6= 0 for u ∈ ∂ BM (Rn ), where BM (Rn ) = {u ∈ Rn : |u| < M }, and uˆ denotes the constant mapping from [−τ , 0] to Rn with the value u ∈ Rn ; (iii) deg(g , BM (Rn )) 6= 0. Then there exists at least one ω-periodic solution of the system (2.1) that satisfies supt ∈R |x(t )| < M. The following remark is introduced by Fang (see Remark 1 in [3]). Remark 2.1. Theorem A still remains valid if the assumption (2) is replaced by (20 ) there exists a constant k < 1 such that |b(t , φ) − b(t , ϕ)| ≤ kkφ − ϕk for t ∈ R and φ, ϕ ∈ {φ ∈ C : kφk < M } with M as given in condition (i) of Theorem A.

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

3959

We will also need the following lemma. Lemma 2.1 ([4,19]). Suppose σ ∈ Cω1 and σ 0 (t ) < 1, t ∈ [0, ω]. Then the function t − σ (t ) has a unique inverse µ(t ) satisfying µ ∈ C (R, R) with µ(a + ω) = µ(a) + ω, ∀a ∈ R; Furthermore, if g ∈ PCω , then g (µ(t )) ∈ PCω . Lemma 2.2 ([23,24]). Suppose x(t ) is a differently continuous ω-periodic function on R. Then for any t∗ ∈ R, maxt∗ ≤t ≤t∗ +ω |x(t )| Rω ≤ |x(t∗ )| + 12 0 |x0 (t )|dt . Lemma 2.3. The region Rn+ = {Ni (t ) : Ni (0) > 0, i = 1, 2, . . . , n} is the positive invariable region of the system (1.1). Proof. In view of biological population, we obtain Ni (0) > 0. By the system (1.1), we have

Z t

Ni (t ) = Ni (0) exp

ri (ξ ) −

0

−

n X

n X

aij (ξ )Nj (ξ ) −

n X

j =1

bij (ξ )

n X

j =1

kij (ξ − s)Nj (s)ds

−∞

j =1

cij (ξ )Nj (ξ − τij (ξ )) −

ξ

Z

dij (ξ )Nj0 (ξ − γij (ξ )) dξ ,

t ∈ [0, t1 ], i = 1, 2, . . . , n,

j =1

and

(Z "

n X

0

j =1

t

Ni (t ) = Ni (tk ) exp

−

n X

ri (ξ ) −

aij (ξ )Nj (ξ ) −

cij (ξ )Nj (ξ − τij (ξ )) −

n X

bij (ξ )

j =1

kij (ξ − s)Nj (s)ds

−∞

j =1 n X

ξ

Z #

)

dij (ξ )Nj0 (ξ − γij (ξ )) dξ

,

t ∈ (tk , tk+1 ], i = 1, 2, . . . , n, k ≥ 1,

j =1

Ni (tk+ ) = (1 + θik )Ni (tk ) > 0,

k ∈ N , i = 1, 2, . . . , n.

Then the solution of (1.1) is positive.

3. The main result If τij0 (t ) < 1, γij0 (t ) < 1, t ∈ [0, ω], then τij (t ), γij (t ) all have their inverse function. Throughout the following part, we set ζij (t ), ξij (t ) represent the inverse function of t − τij (t ), t − γij (t ), respectively. We denote

Γij (t ) = Aij (t ) + Bij (t ) +

Cij (ζij (t )) 1 − τij0 (ζij (t ))

−

Cij (ζij (t ))

Γij∗ (t ) = Aij (t ) + Bij (t ) +

1 − τij0 (ζij (t ))

+

D00,ij (ξij (t )) 1 − γij0 (ξij (t ))

;

Cij (ξij (t )) 1 − γij0 (ξij (t ))

+

|Cij (ξij (t )) − D00,ij (ξij (t ))| 1 − γij0 (ξij (t ))

;

∗ Γij (t ) , i, j = 1, 2, . . . , n Γ = max Γ (t ) ij

D (t )

where D0,ij (t ) = 1−γij 0 (t ) , i, j = 1, 2, . . . , n. ij Remark 3.1. From Lemma 2.1, we get ζij (ω) = ζij (0) + ω, ξij (ω) = ξij (0) + ω, i, j = 1, 2, . . . , n, then ω

Z

Cij (ζij (s)) 1 − τij0 (ζij (s))

0

ζij (ω)

Z ds =

Cij (t )(1 − τij0 (t )) 1 − τij0 (t )

ζij (0) ζij (0)+ω

Z =

ζij (0)

dt

Cij (t )dt = Cij ω,

i, j = 1, 2, . . . , n.

Similarly, ω

Z 0

D00,ij (ξij (s)) 1 − γij0 (ξij (s))

Z

ξij (ω)

ds =

D00,ij (t )(1 − γij0 (t ))

ξij (0)

Z

ξij (0)+ω

= ξij (0)

1 − γij0 (t ) D00,ij (t )dt = 0,

dt i, j = 1, 2, . . . , n.

(3.1)

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Thus,

Γij ω =

ω

Z

Γij (t )dt = (Aij + Bij + Cij )ω.

(3.2)

0

Theorem 3.1. Suppose that the following conditions hold: (1) The system of algebraic equations n X (Aij + Bij + Cij )µj = ri ,

i = 1, 2, . . . , n,

j =1

has a unique positive solution µ∗ = (µ∗1 , . . . , µ∗n ); (2) Aij + Bij + Cij > 0, ri >

Pn

j=1,j6=i

(Aij + Bij + Cij ) A

(3) K0 =: L0 eM0 < 1.

rj

jj +Bjj +Cjj

, τij0 (t ) < 1, γij0 (t ) < 1 and Γij (t ) > 0;

Then (1.7) has at least one positive ω-periodic solution. Here we have

ΓijL (1 − γij0 )L

αij =

1

Z

ω

(

, Ri = |ri (t )|dt , L0 = max (1 − γij0 )L + |D0,ij |0 ω 0 ( ) n n X X ∗ ∗ M0 = max | ln µi |0 , H , ω∆ + ∆i , H = max {Hi }, i=1

P ∆ = ∗

n

|ri |0 +

i=1

n

PP

n

|Aij |0 eHj +

i=1 j=1

n

PP

n

|Bij |0 eHj +

i =1 j =1

1−

n P n P

|D0,ij |0 ,

n X n X

Hi = ln

) |Dij |0 ,

i =1 j =1

i=1 j=1

i∈[1,n]

i=1

n

n X n X

ri

αii

+

n X ri + (Ri + Γ ri )ω, αij j =1

n

PP

|Cij |0 eHj

i=1 j=1

,

|Dij |0 eHj

i=1 j=1

n P r ri − (Aij + Bij + Cij ) A +Bj +C jj jj jj ri j=1,j6=i , ln ∆i = max ln Aii + Bii + Cii Aii + Bii + Cii

,

where Γij (t ), Γij∗ (t ), Γ are defined by (3.1). To prove the above theorem, we make the change of variables yi (t ) = exi (t ) ,

i = 1, 2, . . . , n.

(3.3)

Then the system (1.7) can be rewritten in the following form xi ( t ) = r i ( t ) − 0

n X

Aij (t )e

xj (t )

−

j =1

−

n X

n X

Bij (t )

j =1

t

kij (t − s)exj (s) ds

−∞

j =1

Cij (t )exj (t −τij (t )) −

Z

n X

Dij (t )x0j (t − γij (t ))exj (t −γij (t )) ,

i = 1 , 2 , . . . , n.

(3.4)

j =1

Let X denote the linear space of real value continuous Pn ω-periodic functions on R. The linear space X is a Banach space with the usual norm kxk0 = maxt ∈R |x(t )| = maxt ∈R j=1 |xi (t )| for a given x = (x1 , . . . , xn ) ∈ X . We define the following maps: b : R × C → Rn , bi (t , φ) = −

n X

b(t , φ) = (b1 (t , φ), b2 (t , φ), . . . , bn (t , φ)), D0,ij (t )eφj (−γij (t )) ;

j =1

f : R × C → Rn ,

f (t , φ) = (f1 (t , φ), f2 (t , φ), . . . , fn (t , φ)), Z 0 n n X X fi (t , φ) = ri (t ) − Aij (t )eφj (0) − Bij (t ) kij (t − s)eφj (s) ds j =1

−

n X j =1

Cij (t )eφj (−τij (t )) +

j =1 n X j=1

−∞

D00,ij (t )eφj (−γij (t )) ,

i = 1, 2, . . . , n, φ = (φ1 , φ2 , . . . , φn ) ∈ C , t ∈ R.

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

3961

Clearly, b : R × C → Rn and f : R × C → Rn are complete continuation functions and system (3.4) takes the form d

[x(t ) − b(t , xt )] = f (t , xt ).

dt

(3.5)

In the proof of our main result below, we will use the following two important lemmas. Lemma 3.1. If the assumptions of Theorem 3.1 are satisfied and if Ω = {φ ∈ C : kφk < M }, where M > M0 is such that k = L0 eM < 1, then

|b(t , φ) − b(t , ϕ)| ≤ kkφ − ϕk,

for t ∈ R and φ, ϕ ∈ Ω .

Proof. For t ∈ R and φ, ϕ ∈ Ω ,we have n X

|bi (t , φ) − bi (t , ϕ)| ≤

D0,ij (t )|eφj (−γij (t )) − eϕj (−γij (t )) |

j =1 n X

≤

D0,ij (t )eσij φj (−γij (t ))+(1−σij )ϕj (−γij (t )) |φj (−γij (t )) − ϕj (−γij (t ))|

j =1

for some σij ∈ (0, 1). Then we get

|bi (t , φ) − bi (t , ϕ)| ≤

n X

|D0,ij |0 eM kφ − ϕk.

j =1

Hence,

|b(t , φ) − b(t , ϕ)| ≤

n X n X

|D0,ij |0 eM kφ − ϕk ≤ L0 eM kφ − ϕk = kkφ − ϕk.

i=1 j=1

The proof is thus complete.

Lemma 3.2. If the assumptions of Theorem 3.1 are satisfied. Then every solution x ∈ X of the system d

[x(t ) − λb(t , xt )] = λf (t , xt ),

dt

λ ∈ (0, 1),

(3.6)

satisfies kxk0 ≤ M0 .

[x(t ) − λb(t , xt )] = λf (t , xt ) for x ∈ X , that is, #0 " Z n n n X X X xj (−γij (t )) xi (t ) + λ D0,ij (t )e = λ ri (t ) − Aij (t )exj (t ) − Bij (t ) d dt

Proof. Let

"

j =1 n

−

X

j =1

Cij (t )exj (t −τij (t )) +

j =1

t

kij (t − s)exj (s) ds

−∞

j =1

#

n

X

D00,ij (t )exj (t −γij (t )) ,

i = 1, 2, . . . , n; λ ∈ (0, 1),

(3.7)

j =1

which yields, after integrating from 0 to ω, that ω

Z 0

n X

" Aij (t )e

xj (t )

+ Bij (t )

Z

t

kij (t − s)e

xj (s)

−∞

j =1

ds + Cij (t )e

xj (t −τij (t ))

−

n X j =1

# D0,ij (t )e 0

xj (t −γij (t ))

ω

Z dt =

ri (t )dt = ri ω,

0

i = 1, 2, . . . , n. From Remark 3.1, we can get ω

Z 0

n X

Γij (t )exj (t ) dt = ri ω,

i = 1, 2, . . . , n,

j =1

where Γij (t ) is defined by (3.1). From (3.7) we derive ω

Z 0

" # 0 Z ω " Z t n n n X X X xj (−γij (t )) D0,ij (t )e kij (t − s)exj (s) ds Aij (t )exj (t ) − Bij (t ) xi (t ) + λ dt = λ ri (t ) − 0 −∞ j =1 j =1 j =1

(3.8)

3962

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967 n X

−

Cij (t )e

xj (t −τij (t ))

n X

+

j =1

ω

Z

ω

Z 0

0

# dt Z t

xj (t −γij (t ))

j =1

|ri (t )|dt +

≤

D0,ij (t )e 0

n X A (t )exj (t ) + Bij (t ) j=1 ij

kij (t − s)e

xj (s)

ds + Cij (t )e

xj (t −τij (t ))

−∞

− D0,ij (t )e 0

xj (t −γij (t ))

dt . (3.9)

Note that

Z

Z t n X xj (s) xj (t −τij (t )) 0 xj (t −γij (t )) xj (t ) kij (t − s)e ds + Cij (t )e − D0,ij (t )e Aij (t )e + Bij (t ) dt 0 j =1 −∞ Z ω X Z t n kij (t − s)exj (s) ds + Cij (t )exj (t −τij (t )) − Cij (t )exj (t −γij (t )) = Aij (t )exj (t ) + Bij (t ) 0 j=1 −∞ xj (t −γij (t )) 0 xj (t −γij (t )) − D0,ij (t )e ) dt + (Cij (t )e ω

ω

Z

n X

≤ 0

Aij (t )exj (t ) + Bij (t )

t

Z

kij (t − s)exj (s) ds + Cij (t )exj (t −τij (t )) + Cij (t )exj (t −γij (t ))

−∞

j=1

+ |Cij (t ) − D0,ij (t )|e 0

xj (t −γij (t ))

dt .

In view of Remark 3.1, and by a similar analysis, we obtain ω

Z

n X

0

Aij (t )exj (t ) + Bij (t )

+ Cij (t )e ω

n X

= 0

Γij∗ (s)exj (s) ds =

1 − τij0 (ζij (s)) ω

Z

+

n X Γij∗ (s)

0

j =1

xj (t −γij (t ))

Cij (ζij (s))

j =1

ω

Γij (s)

j =1

dt

Cij (ξij (s)) 1 − γij0 (ξij (s))

+

|Cij (ξij (s)) − D00,ij (ξij (s))| 1 − γij0 (ξij (s))

exj (s) ds

Γij (s)exj (s) ds

n X Γij∗ (s) xj (s) Γ (s) Γij (s)e ds

ω

≤ 0

≤Γ

+ |Cij (t ) − D0,ij (t )|e 0

Aij (s) + Bij (s) +

0

Z

xj (t −γij (t ))

n X

= Z

kij (t − s)exj (s) ds + Cij (t )exj (t −τij (t ))

−∞

j =1

Z

t

Z

ij

j=1

Z

ω

0

n X

Γij (s)exj (s) ds,

(3.10)

j =1

where Γij∗ (s), Γ is defined by (3.1). It follows from (3.9) and (3.10) that ω

Z 0

" # 0 n X D0,ij (t )exj (−γij (t )) dt ≤ (Ri + Γ ri )ω, xi ( t ) + λ j =1

i = 1, 2, . . . , n.

From (3.8) we see that ω

Z 0

n X

Γij (t )exj (t ) dt =

j =1

ω

n Z X j=1

Γij (t )exj (t ) dt = ri ω,

i = 1, 2, . . . , n.

0

So we have ri ω =

ω

n Z X j =1

0

Γij (t )e

xj ( t )

dt =

ω

n Z X j =1

[Γij (t )exj (t ) − (αij exj (t ) + αij D0,ij (t )exj (t −γij (t )) )

0

+ (αij exj (t ) + αij D0,ij (t )exj (t −γij (t )) )]dt

(3.11)

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

ω

n Z X

=

[Γij (t )exj (t ) − (αij exj (t ) + αij D0,ij (t )exj (t −γij (t )) )]dt +

0

j=1

ω

n Z X

3963

[αij exj (t ) + (αij D0,ij (t )exj (t −γij (t )) )]dt . (3.12)

0

j =1

In view of Remark 3.1, and by a similar analysis, we obtain ω

n Z X j=1

As αij =

xj (t )

[Γij (t )e

− (αij e

xj (t )

xj (t −γij (t ))

+ αij D0,ij (t )e

)]dt =

0

ri ω ≥

ω

n Z X

D

" Γij (s) − αij − αij

0

j =1

ΓijL (1−γij0 )L (1−γij0 )L +|D0,ij |0

ω

n Z X

D0,ij (ξij (s)) 1 − γij0 (ξij (s))

# exj (s) dt . (3.13)

(ξ (s))

0,ij ij , it follows that Γij (s) − αij − αij 1−γ 0 (ξ (s)) ≥ 0. So we find from (3.12) that ij

ij

[αij exj (t ) + (αij D0,ij (t )exj (t −γij (t )) )]dt .

0

j =1

That is ri ω ≥

ω

Z

n X [αij exj (t ) + (αij D0,ij (t )exj (t −γij (t )) )]dt .

0

(3.14)

j=1

By the mean value theorem, we see that there exist points βi ∈ [0, ω], (i = 1, . . . , n) such that ri ≥

n X

αij exj (βi ) +

j=1

n X

αij D0,ij (βi )exj (βi −γij (βi )) ,

i = 1, . . . , n,

j =1

which implies that ri

xi (βi ) ≤ ln

αii

D0,ij (βi )exj (βi −γij (βi )) ≤

,

ri

αij

,

i = 1 , . . . , n.

(3.15)

By (3.11) and (3.15), we can see xi (t ) + λ

n X

D0,ij (t )e

xj (t −γij (t ))

≤ xi (βi ) + λ

n X

Dij (βi )

j=1

1 − γij0 (βi )

j =1

≤ ln For λ

Pn

j =1

e

xj (βi −γij (βi ))

ω

Z + 0

" #0 n X xj (t −γij (t )) D0,ij (t )e xi ( t ) + λ dt j =1

n

ri

+

αii

X ri + (Ri + Γ ri )ω = Hi , αij j=1

i = 1, 2, . . . , n.

D0,ij (t )exj (t −γij (t )) ≥ 0, one can find that

xi (t ) ≤ Hi ,

i = 1, . . . , n.

(3.16)

Besides, from (3.7) we have n X

xi (t ) = λ ri (t ) − 0

Aij (t )e

xj (t )

−

j =1

−

n X

n X

Bij (t )

Z

Dij (t )xj (t − γij (t ))e

kij (t − s)exj (s) ds −

−∞

j =1

0

t

xj (t −γij (t ))

,

n X

Cij (t )exj (t −τij (t ))

j =1

i = 1, 2, . . . , n.

j =1

Then by (3.16) we get

Z n n X X |x0i |0 ≤ λ ri (t ) + Aij (t )exj (t ) + Bij (t ) j =1

+

n X

kij (t − s)exj (s) ds +

−∞

j =1

Dij (t )x0j (t − γij (t ))exj (t −γij (t ))

t

n X

Cij (t )exj (t −τij (t ))

j =1

j=1

≤ |ri |0 +

n X j =1

|Aij |0 eHj +

n X j=1

|Bij |0 eHj +

n X j =1

|Cij |0 eHj +

n X j =1

|Dij |0 |x0j |0 eHj ,

i = 1, 2, . . . , n.

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Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

Furthermore, we have

kx 0 k0 =

n X

|x0i |0

i=1

≤

n X

n X n X

|ri |0 +

i=1

n X n X

|Aij |0 eHj +

|Bij |0 eHj +

|Cij |0 eHj +

n X n X

|Dij |0 kx0 k0 eHj ,

i = 1, 2, . . . , n.

i =1 j =1

i=1 j=1

i =1 j =1

i =1 j =1

n X n X

By the assumption (3) of Theorem 3.1, we see n X n X

|Dij |0 eHj ≤

n X n X

n X n X

|Dij |0 eH ≤

i =1 j =1

i=1 j=1

|Dij |0 eM0 < 1.

i=1 j=1

Then, we have n P

kx 0 k0 ≤

|ri |0 +

i=1

n P n P

|Aij |0 eHj +

n P n P

|Bij |0 eHj +

n n P P

1−

|Cij |0 eHj

i=1 j=1

i=1 j=1

i=1 j=1

n P n P

|Dij |0 e

=: ∆∗ .

(3.17)

Hj

i=1 j=1

By using the extended integral mean value theorem, from (3.8), we can find points δj , (j = 1, . . . , n) such that ri ω =

ω

n Z X j=1

Γij (t )e

xj (t )

dt =

0

n X

e

xj (δj )

Z

ω

Γij (t )dt ,

i = 1, . . . , n.

(3.18)

0

j =1

It follows from (3.2) and (3.18) that ri =

n X

exj (δj ) (Aij + Bij + Cij ),

i = 1, . . . , n.

(3.19)

j =1

From (3.19), we obtain exi (δi ) (Aii + Bii + Cii ) ≤ ri ,

i = 1 , . . . , n.

As Aii + Bii + Cii > 0, it follows from the above formula that xi (δi ) ≤ ln

ri Aii + Bii + Cii

,

i = 1, . . . , n.

(3.20)

On the other hand, from (3.19) and (3.20) we get ri = (Aii + Bii + Cii )exi (δi ) +

n X

(Aij + Bij + Cij )exj (δj )

j=1,j6=i

≤ (Aii + Bii + Cii )exi (δi ) +

n X

(Aij + Bij + Cij )

j=1,j6=i

rj Ajj + Bjj + Cjj

,

i = 1, . . . , n.

Therefore, by the assumption (2) of Theorem 3.1, we have ri − xi (δi ) ≥ ln

n P

(Aij + Bij + Cij ) A

j=1,j6=i

rj jj +Bjj +Cjj

Aii + Bii + Cii

,

i = 1 , . . . , n.

(3.21)

Therefore, (3.20) and (3.21) imply

n P r (Aij + Bij + Cij ) A +Bj +C ri − jj jj jj ri j=1,j6=i , ln |xi (δi )| ≤ max ln Aii + Bii + Cii Aii + Bii + Cii

=: ∆i ,

i = 1 , . . . , n.

(3.22)

From (3.17), (3.22) and Lemma 2.2, we have

|xi | ≤ |xi (δi )| +

1 2

ω

Z 0

|x0i (t )|dt ≤ ∆i +

1 2

ω

Z 0

|x0i |dt ,

i = 1, . . . , n.

(3.23)

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

3965

Then

kx k0 ≤

n X

|xi | ≤

n X

i =1

∆i +

i =1

1 2

ω

Z

kx0 k0 dt <

0

n X

∆i +

i=1

Obviously, M0 is independent of λ, the proof is complete.

1 2

∆∗ ω ≤ M 0 .

(3.24)

Based on the above results, we can now apply Theorem A and Remark 2.1 to (3.4) and obtain a proof of Theorem 3.1. Proof of Theorem 3.1. Obviously, for M as given in Lemma 3.1, condition (i) in Theorem A is satisfied. Let g (µ) = (g1 (µ), . . . , gn (µ)). Since gi (µ) =

ω

Z

fi (s, µ) ˆ ds =

ω

Z

ri (t )dt −

0

0

ω

n Z X i=1

Aij (t )dteµj −

0

ω

n Z X i =1

Bij (t )dteµj −

0

n Z X i=1

ω

Cij (t )dteµj

0

"

# n X µj = ri − (Aij + Bij + Cij )e ω, j=1 n and M > i=1 | ln µi |, we have g (µ) 6= 0 for any µ ∈ ∂ BM (R ). That is, condition (ii) in Theorem A holds. At last, we verify that condition (iii) of Theorem A also holds. By assumption (1) of Theorem 3.1 and the formula for the Brouwer degree (see Theorem 2.2.3 in [4]), a straightforward calculation shows that

Pn

∗

deg(g , BM (Rn )) =

X µ∈g −1 (0)

T

sign det Dg (µ) BM (Rn )

n P µ∗j

= sign (−1)n det[Aij + Bij + Cij ]ej=1

6= 0.

By now all the assumptions required in Theorem A hold. It follows from Theorem A and Remark 2.1 that system (3.4) has an ω-periodic solution. Returning to yi (t ) = exi (t ) , we infer that system (1.7) has at least one positive ω-periodic solution. By Lemma 1.1, we conclude that system (1.1) has at least one positive ω-periodic solution. The proof of Theorem 3.1 is complete. Similarly, we can get the following result. Theorem 3.2. Assume that conditions of Theorem 3.1 hold. Then the conclusion of Theorem 3.1 holds for the following system:

Z t n n X X 0 N ( t ) = − N ( t )[ r ( t ) − a ( t ) N ( t ) − b ( t ) kij (t − s)Nj (s)ds i i ij j ij i −∞ j=1 j =1 n n X X − c ( t ) N ( t − τ ( t )) − dij (t )Nj0 (t − γij (t ))], i = 1, 2, . . . , n, t 6= tk , ij j ij j =1 j =1 1Ni (tk ) = Ni (tk+ ) − Ni (tk ) = θik Ni (tk ), i = 1, 2, . . . , n, k = 1, 2, . . . . The proof is similar as that of Theorem 3.1. Here we omit it. 4. Applications In order to illustrate some features of our main result, in the following, we will apply Theorem 3.1 to some special cases which have been studied extensively in the literature. Application 4.1. Consider the single species neutral delay logistic equation with impulse:

(

dN (t )

= N (t )[r (t ) − a(t )N (t ) − b(t )N (t − τ (t )) − c (t )N 0 (t − σ (t ))] dt 1N (tk ) = N (tk+ ) − N (tk ) = θk N (tk ), k = 1, 2, . . . ,

(4.1)

where a, b ∈ C (R, [0, +∞)), c ∈ C 1 (R, [0, +∞)), τ R∈ C 1 (R, R), σ ∈ C 2 (R, R) are continuous ω-periodic functions. ω r ∈ C (R, R) are continuous ω-periodic functions with 0 r (t )dt > 0. Applying Theorem 3.1 to Eq. (4.1), we can obtain

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Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

the following theorem. Theorem 4.1. Assume that the following conditions are satisfied: (1) A + B > 0, τ 0 (t ) < 1, γ 0 (t ) < 1, Γ1 (t ) > 0; ∗ (2) K ∗ =: L∗ eM < 1. Then system (4.1) has at least one positive ω-periodic solution, where A(t ) = a(t )

Y

(1 + θk ),

Y

B(t ) = b(t )

0

(1 + θk ),

Y

C (t ) = c (t )

0

(1 + θk )

0

Z ω Γ1M (1 − σ 0 )M 1 , R = |r (t )|dt , L1 = max{|C0 |0 , |C |0 }, 1 − σ 0 (t ) (1 − σ 0 )M + |C0 |0 ω 0 Γ2 r r r ∗ , H ∗ , ∆∗ , r ω, M ∗ = max ln + + R + H = ln Γ α α A+B 1 0 ∗ ∗ H H | r | + | A | e r + | B | e C00 (ξ (t )) B(µ(t )) 0 0 0 , ∆∗ = − ; ω + ln Γ1 (t ) = A(t ) + ∗ H 0 1 − |C |0 e 1 − τ (µ(t )) 1 − σ 0 (ν(t )) A+B B(µ(t )) B(ν(t )) |B(ν(t )) − C00 (ν(t ))| Γ2 (t ) = A(t ) + + + , 1 − τ 0 (µ(t )) 1 − σ 0 (ν(t )) 1 − σ 0 (ν(t )) C0 ( t ) =

C (t )

,

α=

and µ(t ), ν(t ) represent the inverse function of t − τ (t ), t − σ (t ), respectively. Remark 4.1. Although the instinct growth rate r may not be positive, Theorem 4.1 is almost the same as Theorem 1 in [16]. When τ (t ) = σ (t ) and θk = 0 we can derive an immediate corollary of Theorem 4.1, which is also an answer to the open problem 9.2 due to Kuang [1]. Besides, we can see that Theorem 4.1 can hold without the assumption c00 (t ) < b(t ) or c00 (t ) ≤ b(t ). When c00 (t ) > b(t ) Fang’s main result (see Theorem 3.1 in [14]) cannot be applied. Therefore, comparing with [14], our result requires weaker condition. Application 4.2. Consider the single species neutral delay logistic equation with impulse:

dNi

" = Ni (t ) ai (t ) −

dt

n X

bij (t )Nj (t ) −

n X

cij (t )Nj (t − σij (t )) −

Ni (t ) = (1 + θik )Ni (tk ),

k = 1, 2, . . .

+

# dij (t )Nj (t − τij (t )) , 0

t 6= tk ,

j =1

j=1

j =1

m X

(4.2)

where bij , cij ∈ C (R, [0, +∞)), dij ∈ C 1 (R, [0, +∞)),Rσij ∈ C 1 (R, R), τij ∈ C 2 (R, R) are continuous ω-periodic functions. ω ai ∈ C (R, R) are continuous ω-periodic functions with 0 ai (t )dt > 0. Applying our Theorem 3.1 to Eq. (4.2), we can obtain the following theorem. Theorem 4.2. Assume that the following conditions are satisfied. (1) The system of algebraic equations n X (Bij + Cij )uj = ai ,

i = 1, 2, . . . , n,

j =1

has a unique positive solution u = (u∗1 , . . . , u∗n ); Pn aj (2) Bij + Cij > 0, ai > , σij0 (t ) < 1, τij0 (t ) < 1 and Θij (t ) > 0; j=1,j6=i (Bij + Cij ) Bjj +Cjj

(3) K2 =: L2 eM2 < 1.

Then (4.2) has at least one positive ω-periodic solution, where

ΘijL (1 − τij0 )L

αij =

1

ω

Z

( n X n X

, Ai = |ai (t )|dt , L3 = max (1 − τij0 )L + |D0,ij |0 ω 0 ( ) n n X X ∗ ∗ ∗ M3 = max | ln ui |0 , H , ω∆ + ∆i , H ∗ = max {Hi }, i=1 n P

∆ = ∗

i=1

| ai | 0 +

i=1 n P n P

|Bij |0 eHj +

i=1 j=1

1−

n P n P i =1 j =1

n P n P i =1 j =1 Hj

|Dij |0 e

|Cij |0 eHj ,

|D0,ij |0 ,

i =1 j =1

n X n X

) |Dij |0 ,

i=1 j=1

n X ai + (Ai + Θ ai )ω, i∈[1,n] αii αij j =1 n P aj (Bij + Cij ) B +C ai − jj jj ai j=1,j6=i , ln ∆i = max ln , Bii + Cii Bii + Cii

Hi = ln

ri

+

Z. Luo et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3955–3967

Θij (t ) = Bij (t ) + Θij0 (t ) = Bij (t ) + Θ = max

Bij (t ) = bij (t )

Cij (ρij (t )) 1 − σij (ρij (t )) 0

Cij (ρij (s)) 1 − σij (ρij (s))

Θij0 (s) Θij (s) Y 0

0

D0,ij (%ij (t ))

3967

0

− +

1 − τij0 (%ij (t )) Cij (%ij (s)) 1 − τij (%ij (s)) 0

, +

|Cij (%ij (s)) − D00,ij (%ij (s))| 1 − τij0 (%ij (s))

,

, i, j = 1, 2, . . . , n , 0

(1 + θik ),

Cij (t ) = cij (t )

Y

(1 + θik ),

Dij (t ) = dij (t )

0

Y

(1 + θik )

0

and ρij (t ), %ij (t ) represent the inverse function of t − σij (t ), t − τij (t ), respectively. Remark 4.2. When i = 1, or τij (t ) = σij (t ) and θik = 0 we can derive some immediate corollaries of Theorem 4.1. On the other hand, we can see that our Theorem 4.2 can hold without the assumption ai > 0. When a < 0 Wang’s main result (see Theorem 3.1 in [19]) cannot be applied. Therefore, comparing with [4,19], our result improves the results in [4,19]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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