# Positive periodic solutions of a periodic neutral delay logistic equation with impulses

## Positive periodic solutions of a periodic neutral delay logistic equation with impulses

Computers and Mathematics with Applications 56 (2008) 2189–2196 Contents lists available at ScienceDirect Computers and Mathematics with Application...

Computers and Mathematics with Applications 56 (2008) 2189–2196

Contents lists available at ScienceDirect

Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

Positive periodic solutions of a periodic neutral delay logistic equation with impulsesI Yongkun Li Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China

article

info

a b s t r a c t

Article history: Received 4 April 2006 Received in revised form 1 March 2008 Accepted 25 March 2008

By using a fixed point theorem of strict-set-contraction, some criteria are established for the existence of positive periodic solutions for a periodic neutral functional differential equation with impulses of the form

   dx(t )

Keywords: Positive periodic solution Neutral functional differential equation Impulse Strict-set-contraction

dt

 

" = x(t ) a(t ) −

m X

bi (t )x(t − τi (t )) −

i=1

x(tk ) − x(tk ) = −Ik (x(tk )),

k = 1, 2, . . . .

+

n X

# cj (t )x (t − σj (t )) , 0

j=1

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction Recently, by using the continuation theory for k-set-contractions, Lu  studied the existence of positive periodic solutions of the following neutral logistic equation dN (t ) dt

" = N (t ) a(t ) − β(t )N (t ) −

m X

bi (t )N (t − τi (t )) −

n X

# cj (t )N (t − σj (t )) , 0

(1.1)

j=1

i=1

where a, β, bi , cj ∈ C (R, [0, +∞)), τi , σj ∈ C (R, [0, +∞)) (i = 1, 2, . . . , m, j = 1, 2, . . . , n) are ω-periodic functions. Also, Yang and Cao , by using Mawhin’s continuation theorem , investigated the existence of positive periodic solutions of (1.1) in the case of m = n. However, the main results obtained in [1,2] all required that cj ∈ C 1 , σj ∈ C 2 and σj0 < 1 (j = 1, 2, . . . , n). In this paper, we are concerned with the following neutral delay logistic equation with impulses

   dx(t ) dt

 

" = x(t ) a(t ) −

m X i=1

bi (t )x(t − τi (t )) −

x(tk ) − x(tk ) = −Ik (x(tk )), +

k = 1, 2, . . . ,

n X j=1

# cj (t )x (t − σj (t )) , 0

(1.2)

where a, bi , cj ∈ C (R, R ) and τi , σj ∈ C (R, R), i = 1, 2, . . . , m, j = 1, 2, . . . , n are ω-periodic functions, Ik ∈ C (R+ , R+ ), there exists a positive integer p such that tk+p = tk + ω, Ik+p = Ik , k ∈ Z, [0, ω) ∩ {tk : k ∈ Z} = {t1 , t2 , . . . , tp }. For the ecological justification of (1.1), one can refer to [4–7]. The main purpose of this paper is to establish criteria to guarantee the existence of positive periodic solutions of (1.2) by using a fixed point theorem of strict-set-contraction [Theorems 3, 8]. +

I This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10361006 and the Natural Sciences Foundation of Yunnan Province under Grant 2003A0001M. E-mail address: [email protected]

0898-1221/\$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.03.044

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Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

For convenience, we introduce the notation R − 0ω a(s)ds

Θ := e

,

ω

Z

∆=

" m X

0

Π=

ω

Z

" m X

0

bi (s) +

i=1

n X

Θ bi (s) −

i=1

n X

# cj (s) ds,

j=1

# cj (s) ds,

f M = max {f (t )},

j=1

t ∈[0,ω]

f m = min {f (t )}, t ∈[0,ω]

where f is a continuous ω-periodic function. Throughout this paper, we assume that: Rω

− 0 a(s)ds (H1 ) Θ < 1. P := e P n (H2 ) m Θ bi (t ) − j=1 cj (t ) ≥ 0. i=1 Pm Pn 2 (H3 ) (1 + am ) 1Θ−Θ∆ ≥ maxt ∈[0,ω] i=1 bi (t ) + j=1 cj (t ) . Pm Pn M (H4 ) ΠΘ((a1−−Θ1)) ≤ mint ∈[0,ω] i=1 Θ bi (t ) − j=1 cj (t ) . P (H5 ) 1Θ−2Θ∆ nj=1 cjM < 1.

2. Preliminaries The following lemma is fundamental to our discussion. Since the proof is similar to those in the literature [9,10], we omit the proof. Lemma 2.1. x(t ) is an ω-periodic solution of (1.2) is equivalent to x(t ) is an ω-periodic solution of the following: x(t ) =

Z

t +ω

G(t , s)x(s)

" m X

t

bi (s)x(s − τi (s)) +

i=1

n X

# cj (s)x (s − σj (s)) + 0

j=1

X

G(t , tk )Ik (x(tk )),

(2.1)

k:tk ∈[t ,t +ω)

where G(t , s) =

e−

Rs

t a(θ)dθ

1 − e−

0 a(θ)dθ

,

s ∈ [t , t + ω].

(2.2)

In order to obtain the existence of a periodic solution of system (2.1), we first make the following preparations: Let E be a Banach space and K be a cone in E. The semi-order induced by the cone K is denoted by ‘‘≤’’. That is, x ≤ y if and only if y − x ∈ K . In addition, for a bounded subset A ⊂ E, let αE (A) denote the (Kuratowski) measure of non-compactness defined by

( αE (A) = inf δ > 0 : there is a finite number of subsets Ai ⊂ A ) such that A =

[

Ai and diam(Ai ) ≤ δ

,

i

where diam(Ai ) denotes the diameter of the set Ai . Let E, F be two Banach spaces and D ⊂ E, a continuous and bounded map Φ : D → F is called k-set contractive if for any bounded set S ⊂ D we have

αF (Φ (S )) ≤ kαE (S ). Φ is called strict-set-contractive if it is k-set-contractive for some 0 ≤ k < 1. The following lemma cited from Ref. [8,11] is useful for the proof of our main results of this paper. Lemma 2.2 ([8,11]). Let K be a cone of the real Banach space X and Kr ,R = {x ∈ K : r ≤ kxk ≤ R} with R > r > 0. Suppose that Φ : Kr ,R → K is strict-set-contractive such that one of the following two conditions is satisfied: (i) Φ x ≤ 6 x, ∀x ∈ K , kxk = r and Φ x ≥ 6 x, ∀x ∈ K , kxk = R. (ii) Φ x ≥ 6 x, ∀x ∈ K , kxk = r and Φ x ≤ 6 x, ∀x ∈ K , kxk = R. Then Φ has at least one fixed point in Kr ,R .

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

2191

Define PC (R) = x : R → R, x|(tk ,tk+1 ) ∈ C ((tk , tk+1 ), R), ∃x(tk− ) = x(tk ), x(tk+ ), k ∈ Z



and

 PC 1 (R) = x : R → R, x|(tk ,tk+1 ) ∈ C 1 ((tk , tk+1 ), R), ∃x0 (tk− ) = x0 (tk ), x(tk+ ), k ∈ Z . Set

X = {x : x ∈ PC (R), x(t + ω) = x(t ), t ∈ R} with the norm defined by |x|0 = supt ∈[0,ω] {|x(t )|}, and

Y = x : x ∈ PC 1 (R), x(t + ω) = x(t ), t ∈ R



with the norm defined by |x|1 = max{|x|0 , |x0 |0 }. Then X and Y are all Banach spaces. Define the cone K in Y by K = {x ∈ Y, x(t ) ≥ Θ |x|1 , t ∈ [0, ω]}.

(2.3)

Let the map Φ be defined by

(Φ x)(t ) =

Z

t +ω

G(t , s)x(s)

" m X

t

n X

bi (s)x(s − τi (s)) +

# G(t , tk )Ik (x(tk )),

(2.4)

k:tk ∈[t ,t +ω)

j=1

i=1

X

cj (s)x (s − σj (s)) + 0

where x ∈ K , t ∈ R, and G(t , s) is defined by (2.2). Definition 2.1 (). The set F is said to be quasi-equicontinuous in [0, ω] if for any  > 0 there exists a δ > 0 such that if x ∈ F , k ∈ Z, t 0 , t 00 ∈ (tk−1 , tk ] ∩ [0, ω], and |t 0 − t 00 | < d, then |x(t 0 ) − x(t 00 )| <  . Lemma 2.3 (Compactness criterion ). The set F ⊂ X is relatively compact if and only if: (a) F is bounded, that is, kxk ≤ M for each x ∈ F and some M > 0; (b) F is quasi-equicontinuous in [0, ω]. In what follows, we will give some lemmas concerning K and Φ defined by (2.3) and (2.4), respectively. Lemma 2.4. Assume that (H1 )–(H3 ) hold. (i) If aM ≤ 1, then Φ : K → K is well defined. (ii) If (H4 ) holds and aM > 1, then Φ : K → K is well defined. Proof. For any x ∈ K , it is clear that Φ x ∈ Y. In view of (H2 ), for x ∈ K , t ∈ [0, ω], we have m X

bi (t )x(t − τi (t )) +

n X

cj (t )x0 (t − σj (t )) ≥

bi (t )x(t − τi (t )) −

m X

Θ bi (t )|x|1 −

i=1

=

n X

cj (t )|x0 (t − σj (t ))|

j=1

i=1

j=1

i=1

m X

n X

cj (t )|x|1

j=1

" m X

Θ bi (t ) −

i=1

n X

# cj (t ) |x|1 ≥ 0.

(2.5)

j=1

Therefore, for x ∈ K , t ∈ [0, ω], we find

|Φ x|0 ≤

1 1−Θ

ω

Z

x(s)

" m X

0

bi (s)x(s − τi (s)) +

i=1

#

n X

cj (s)x (s − σj (s)) ds +

1

0

j=1

p X

1 − Θ k=1

Ik (x(tk ))

and

Θ (Φ x)(t ) ≥ 1−Θ

Z

Θ = 1−Θ

Z

t +ω

x(s)

t

≥ Θ |Φ x|0 .

0

" m X

bi (s)x(s − τi (s)) +

n X

i=1

ω

x(s)

" m X i=1

# cj (s)x (s − σj (s)) ds + 0

j=1

#

n

bi (s)x(s − τi (s)) +

X j=1

cj (s)x (s − σj (s)) ds + 0

p Θ X Ik (x(tk )) 1 − Θ k=1

p Θ X Ik (x(tk )) 1 − Θ k=1

(2.6)

2192

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

Now, we show that (Φ x)(t ) ≥ Θ |(Φ x)|1 , t ∈ [0, ω]. From (2.4), we have

(Φ x) (t ) = G(t , t + ω)x(t + ω) 0

" m X

bi (t + ω)x(t + ω − τi (t + ω)) +

i=1

− G(t , t )x(t )

" m X

n X

# cj (t + ω)x (t + ω − σj (t + ω)) 0

j=1

bi (t )x(t − τi (t )) +

n X

# cj (t )x (t − σj (t )) + a(t )(Φ x)(t ) 0

j=1

i=1

" m X

= a(t )(Φ x)(t ) − x(t )

bi (t )x(t − τi (t )) +

i=1

n X

# cj (t )x (t − σj (t )) . 0

(2.7)

j=1

It follows from (2.7) that if (Φ x)0 (t ) ≥ 0, then

(Φ x)0 (t ) ≤ a(t )(Φ x)(t ) ≤ aM (Φ x)(t ) ≤ (Φ x)(t ).

(2.8)

On the other hand, from (2.5), (2.7) and (H3 ), if (Φ x)0 (t ) < 0, then

−(Φ x) (t ) = x(t ) 0

" m X

ai (t )x(t − τi (t )) +

≤ |x|

" m X

# cj (t )x (t − σj (t )) − a(t )(Φ x)(t ) 0

j=1

i=1 2 1

n X

bi (t ) +

n X

# cj (t ) − am (Φ x)(t )

j=1

i=1

# Z ω "X n m X Θ2 2 cj (s) ds − am (Φ x)(t ) |x|1 Θ bi (s) − ≤ (1 + a ) 1−Θ 0 j=1 i=1 # " Z ω n m X X Θ m |x|1 cj (s) ds − am (Φ x)(t ) Θ |x|1 bi (s) − = (1 + a ) Θ |x|1 1−Θ 0 j=1 i=1 " # Z t +ω n m X X 0 m cj (s)|x (s − σj (s))| ds − am (Φ x)(t ) ≤ (1 + a ) bi (s)x(s − τi (s)) − G(t , s)x(s) m

t

≤ (1 + a ) m

Z

t +ω

G(t , s)x(s)

i=1

j=1

" m X

n X

t

bi (s)x(s − τi (s)) +

i=1

# cj (s)x (s − σj (s)) ds − am (Φ x)(t ) 0

j=1

= (1 + am )(Φ x)(t ) − am (Φ x)(t ) = (Φ x)(t ).

(2.9)

It follows from (2.8) and (2.9) that |(Φ x)0 |0 ≤ |Φ x|0 . So |Φ x|1 = |Φ x|0 . By (2.6) we have

(Φ x)(t ) ≥ Θ |Φ x|1 . Hence, Φ x ∈ K . The proof of (i) is complete. (ii) In view of the proof of (i), we only need to prove that (Φ x)0 (t ) ≥ 0 implies

(Φ x)0 (t ) ≤ (Φ x)(t ). From (2.5), (2.7), (H2 ) and (H4 ), we obtain

(Φ x) (t ) ≤ a(t )(Φ x)(t ) − Θ |x|1 0

" m X

bi (t )x(t − τi (t )) −

i=1

" ≤ a(t )(Φ x)(t ) − Θ |x|

2 1

Θ

m X

# cj (t )|x (t − σj (t ))| 0

j=1

bi (t ) −

i=1

aM − 1

n X

n X

# cj (t )

j=1

Z

ω

" m X

n X

#

cj (s) ds bi (s) + Θ (1 − Θ ) 0 j=1 i=1 " # Z t +ω m n X X 1 M M ≤ a (Φ x)(t ) − (a − 1) |x|1 bi (s)|x|1 + cj (s)|x|1 ds 1−Θ t i=1 j=1

≤ a (Φ x)(t ) − Θ |x| M

2 1

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

≤ aM (Φ x)(t ) − (aM − 1)

Z

t +ω

G(t , s)x(s)

" m X

t

≤ a (Φ x)(t ) − (a − 1) M

M

Z

bi (s)x(s − τj (s)) +

i=1 t +ω

G(t , s)x(s)

" m X

t

n X

2193

# cj (s)|x0 (s − σj (s))| ds

j=1

bi (s)x(s − τj (s, x(s))) +

i=1

n X

# cj (s)x (s − σj (s)) ds 0

j=1

= aM (Φ x)(t ) − (aM − 1)(Φ x)(t ) = (Φ x)(t ). The proof of (ii) is complete.



Lemma 2.5. Assume that (H1 )–(H3 ) hold and R

Pn

j=1

cjM < 1.

¯ R → K is strict-set-contractive, (i) If aM ≤ 1, then Φ : K Ω T ¯ R → K is strict-set-contractive, (ii) If (H4 ) holds and aM > 1, then Φ : K Ω T

where ΩR = {x ∈ Y : |x|1 < R}. Proof. We only need to prove P (i), sincethe proof of (ii) is similar. It is easy to see that Φ is continuous and bounded. Now n ¯ R . Let η = αY (S ). Then, for any positive number we prove that αY (Φ (S )) ≤ R j=1 cjM αY (S ) for any bounded set S ⊂ Ω

ε< R

Pn

cjM η, there is a finite family of subsets {Si } satisfying S =



j=1

S

i

Si with diam(Si ) ≤ η + ε . Therefore

|x − y|1 ≤ η + ε for any x, y ∈ Si .

(2.10)

As S and Si are precompact in X, it follows that there is a finite family of subsets {Sij } of Si such that Si =

S

j

Sij and

|x − y|0 ≤ ε for any x, y ∈ Sij .

(2.11)

In addition, for any x ∈ S and t ∈ [0, ω], we have

|(Φ x)(t )| =

t +ω

Z

G(t , s)x(s)

t

" m X

bi (s)x(s − τi (s)) +

1−Θ

Z 0

ω

" m X

# cj (s)x (s − σj (s)) ds 0

j=1

i=1

R2

n X

bi (s) +

i=1

n X

# cj (s) ds := H

j=1

and

# " n m X X 0 cj (t )x (t − σj (t )) bi (t )x(t − τi (t )) + |(Φ x) (t )| = a(t )(Φ x)(t ) − x(t ) j=1 j=1 ! n m X X ≤ aM H + R2 cjM . bM + i 0

i=1

j=1

Applying Lemma 2.3, we know that Φ (S ) is precompact in X. Then, there is a finite family of subsets {Sijk } of Sij such that S Sij = k Sijk and

|Φ x − Φ y|0 ≤ ε for any x, y ∈ Sijk . From (2.7), (2.10)–(2.12) and (H2 ), for any x, y ∈ Sijk , we obtain

( " # m n X X 0 |(Φ x) − (Φ y) |0 = max a(t )(Φ x)(t ) − a(t )(Φ y)(t ) − x(t ) bi (t )x(t − τi (t )) + cj (t )x (t − σj (t )) t ∈[0,ω] j=1 j=1 " # ) m n X X 0 + y(t ) bi (t )y(t − τi (t )) + cj (t )y (t − σj (t )) i=1 j=1 ( " # m n X X 0 ≤ max {|a(t )[(Φ x)(t ) − (Φ y)(t )]|} + max x(t ) bi (t )x(t − τi (t )) + cj (t )x (t − σj (t )) t ∈[0,ω] t ∈[0,ω] i=1 j=1 " # ) m n X X 0 − y(t ) bi (t )y(t − τj (t )) + cj (t )y (t − σj (t )) j=1 j=1 0

0

(2.12)

2194

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

( " ! m n X X 0 ≤ a |(Φ x) − (Φ y)|0 + max x(t ) bi (t )x(t − τi (t )) + cj (t )x (t − σj (t )) t ∈[0,ω] j=1 j=1 !# ) n m X X 0 bi (t )y(t − τi (t )) + cj (t )y (t − σj (t )) − i=1 j=1 ) ( " # n m X X 0 + max y(t ) cj (t )y (t − σj (t )) [x(t ) − y(t )] bi (t )y(t − τi (t )) + t ∈[0,ω] j=1 i=1 ( ) q m X X M 0 0 ≤ a ε + R max bi (t )|x(t − τi (t )) − y(t − τi (t ))| + cj (t )|x (t − σj (t )) − y (t − σj (t ))| M

t ∈[0,ω]

( + ε max

t ∈[0,ω]

i=1

j=1

n

q

X

X

bi (t )y(t − τi (t , y(t ))) +

i=1

! + R(η + ε)

BM i

n X

i=1

= Rη

n X

cj (t )|y (t − σj (t ))|

j=1

m X

≤ aM ε + Rε

) 0

! cjM

m X

+ Rε

j=1

bM i +

i=1

n X

! cjM

j=1

! + Hˆ ε,

cjM

(2.13)

j=1

Pm

ˆ = aM + 2R i=1 bM where H i + 2R From (2.12) and (2.13) we have |Φ x − Φ y|1 ≤

R

n X

Pn

j=1

cjM .

! η + Hˆ ε for any x, y ∈ Sijk .

cjM

j=1

As ε is arbitrary small, it follows that

αY (Φ (S )) ≤ R

n X

! αY (S ).

cjM

j=1

Therefore, Φ is strict-set-contractive. The proof of Lemma 2.5 is complete.



3. Main result Our main result of this paper is as follows: Theorem 3.1. Assume that (H1 )–(H3 ), (H5 ) hold. (i) If aM ≤ 1, then system (1.2) has at least one positive ω-periodic solution. (ii) If (H4 ) holds and aM > 1, then system (1.2) has at least one positive ω-periodic solution. Θ (1−Θ )

Proof. We only need to prove (i), since the proof of (ii) is similar. Let R = 1Θ−2Θ and 0 < r < . Then we have 0 < r < R. Π ∆ From Lemmas 2.4 and 2.5, we know that Φ is strict-set-contractive on Kr ,R . In view of (2.7), we see that if there exists x∗ ∈ K such that Φ x∗ = x∗ , then x∗ is one positive ω-periodic solution of system (1.2). Now, we shall prove that condition (ii) of Lemma 2.2 holds. First, we prove that Φ x 6≥ x, ∀x ∈ K , |x|1 = r. Otherwise, there exists x ∈ K , |x|1 = r such that Φ x ≥ x. So |x| > 0 and Φ x − x ∈ K , which imply that

(Φ x)(t ) − x(t ) ≥ Θ |Φ x − x|1 ≥ 0 for any t ∈ [0, ω].

(3.1)

Moreover, for t ∈ [0, ω], we have

(Φ x)(t ) =

Z

t +ω

G(t , s)x(s)

t

" m X

bi (s)x(s − τi (s)) +

j=1

i=1

1 1−Θ

Z r |x|0 0

ω

" m X i=1

n X

bi (s) +

n X j=1

# cj (s) ds

# cj (s)x (s − σj (s)) ds 0

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

1

=

1−Θ

2195

Π |x|0

< Θ |x|0 .

(3.2)

In view of (3.1) and (3.2), we have

|x|0 ≤ |Φ x| < Θ |x|0 < |x|0 , which is a contradiction. Finally, we prove that Φ x 6≤ x, ∀x ∈ K , |x|1 = R also hold. For this case, we only need to prove that

Φ x 6< x,

x ∈ K , |x|1 = R.

Suppose, for the sake of contradiction, that there exist x ∈ K and |x|1 = R such that Φ x < x. Thus x − Φ x ∈ K \ {0}. Furthermore, for any t ∈ [0, ω], we have x(t ) − (Φ x)(t ) ≥ Θ |x − Φ x|1 > 0.

(3.3)

In addition, for any t ∈ [0, ω], we find

(Φ x)(t ) =

t +ω

Z

G(t , s)x(s)

" m X

t

bi (s)x(s − τi (s)) +

i=1

Θ ≥ Θ |x|21 1−Θ

Z

ω

" m X

0

#

n X

cj (s)x (s − σj (s)) ds 0

j=1

Θ bi (s) −

n X

# cj (s) ds

j=1

i=1

Θ2 1R2 1−Θ = R.

=

(3.4)

From (3.3) and (3.4), we obtain

|x| > |Φ x|0 ≥ R, which is a contradiction. Therefore, conditions (i) and (ii) of Lemma 2.2 hold. By Lemma 2.2, we see that Φ has at least one non-zero fixed point in K . Thus, system (2.1) has at least one positive ω-periodic solution. Therefore, it follows from Lemma 2.1 that system (1.2) has a positive ω-periodic solution. The proof of Theorem 3.1 is complete.  Remark 3.1. Since we do not restrict σj to be non-negative, in fact, (1.2) is a neutral functional differential equation of mixed type. An example Consider the following non-impulsive equation x0 (t ) = x(t )



1 + cos t 4π

− (2 + sin t )x(t − τ1 (t )) −



1 − sin t 0 x (t − σ1 (t )) , 20

(3.5)

where τ , σ ∈ C (R, R) are 2π -periodic functions. Obviously, a(t ) =

1 + cos t 4π

,

b1 (t ) = 2 + sin t ,

c1 (t ) =

1 − sin t 20

.

Furthermore, we have 1

Θ = e− 2 , ∆=

Z

max {Θ b1 (t ) − c1 (t )} > 3 > 0,

0≤t ≤2π

1

[Θ b1 (s) − c1 (s)]ds = 16π e− 2 −

0

(1 + aM )

∆=

Θ2 ∆ > 59 and 1−Θ

1−Θ

Θ 2∆

1 10

99 10

1

π + 6π e− 2 ,

max {b1 (t ) + c1 (t )} < 4,

0≤t ≤2π

π,

c1M < 1.68 × 10−3 < 1.

Hence, (H1 )–(H3 ) and (H5 ) hold and aM ≤ 1. According to Theorem 3.1, system (3.5) has at least one positive 2π -periodic solution. Remark 3.2. Since we do not require σ10 (t ) < 1, the results in [1,2] cannot be used to obtain the existence of positive 2π -periodic solutions of (3.5).

2196

Y. Li / Computers and Mathematics with Applications 56 (2008) 2189–2196

Acknowledgment The author thanks the anonymous referee for his/her valuable comments and suggestions to improve the content of this article. References  S. Lu, On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments, J. Math. Anal. Appl. 280 (2003) 321–333.  Z. Yang, J. Cao, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models, Appl. Math. Comput. 142 (2003) 123–142.  R.E. Gaines, J.L. Mawhin, Coincidence Degree Theory and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.  Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.  Y. Kuang, A. Feldstein, Boundedness of solutions of a nonlinear nonautonomous neutral delay equation, J. Math. Anal. Appl. 156 (1991) 293–304.  K. Gopalsamy, X. He, L. Wen, On a periodic neutral logistic equation, Glasgow Math. J. 33 (1991) 281–286.  K. Gopalsamy, B.G. Zhang, On a neutral delay logistic equation, Dyn. Stab. Syst. 2 (1988) 183–195.  N.P. Các, J.A. Gatica, Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71 (1979) 547–557.  J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett. 15 (2002) 489–493.  X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal. 62 (2005) 683–701.  D. Guo, Positive solutions of nonlinear operator equations and its applications to nonlinear integral equations, Adv. Math. 13 (1984) 294–310 (in Chinese).  D. Bainov, P. Simeonov, Impulsive differential equations: Periodic solutions and applications, in: Pitman Monographs Surv. Pure Appl. Math., vol. 66, 1993.