- Email: [email protected]

Positive periodic solutions for a neutral delay Lotka–Volterra system Guirong Liu a,b,∗ , Jurang Yan a , Fengqin Zhang b a School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China b Department of Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, PR China

Received 6 June 2006; accepted 11 September 2006

Abstract Applying a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions for a class of neutral delay Lotka–Volterra systems: n n n X X X 0 0 ai j (t)x j (t) − xi (t) = xi (t) ri (t) − bi j (t)x j (t − τi j (t)) − ci j (t)x j (t − σi j (t)) , i = 1, 2, . . . , n. j=1

j=1

j=1

Compared to known results, our main generalization is that delays of the derivatives are not assumed to be constants, and our results are more easily verifiable. c 2006 Elsevier Ltd. All rights reserved.

MSC: 34K13 Keywords: Positive periodic solution; Neutral delay differential equation; Lotka–Volterra; Fixed point theorem; Strict-set-contraction

1. Introduction We can observe that populations in the real world tend to fluctuate. Therefore, the theory of existence for periodic solutions occupies an important place in mathematical biology. In recent years, the existence of periodic solutions for neutral delay differential equations in population dynamics has been studied extensively [2–4,7–16]. However, most of the studies have dealt with one-dimensional problems. In the real world, species interact and coexist in somewhat closed environments. It is thus more realistic to model population growths in system settings; that is, modeling the growths of interacting species by systems of equations. Recently, by using Mawhin’s continuation theorem, Li [10] studied the existence of positive periodic solutions for the Lotka–Volterra system # " n n X X 0 0 xi (t) = xi (t) ri (t) − bi j (t)x j (t − τi j ) − ci j (t)x j (t − σi j ) , i = 1, 2, . . . , n, (1) j=1

j=1

∗ Corresponding author at: School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China.

E-mail addresses: [email protected] (G. Liu), [email protected] (J. Yan), [email protected] (F. Zhang). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.029

G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

2643

where ri , bi j , ci j ∈ C(R, (0, ∞)) are ω-periodic functions, and τi j , σi j are constants, i, j = 1, 2, . . . , n. In addition, [10] required that either ci j ∈ C 1 (R, (0, ∞)), ci j 0 (t) < bi j (t), τi j = σi j (i, j = 1, 2, . . . , n), or ci j (t) ≡ ci j (i, j = 1, 2, . . . , n) are constants. Later on, applying the same method as that in [10], Liu [12] established the sufficient conditions for the existence of positive periodic solutions for the neutral delay multi-species ecological competitive system " # n n n X X X 0 0 xi (t) = xi (t) ri (t) − ai j (t)x j (t) − bi j (t)x j (t − τi j (t)) − ci j x j (t − σi j ) , i = 1, 2, . . . , n,(2) j=1

j=1

j=1

where ri , ai j , bi j ∈ C(R, (0, ∞)), τi j ∈ C(R, R) are ω-periodic functions, and ci j ≥ 0, σi j are constants, i, j = 1, 2, . . . , n. In this paper, we improve the sufficient conditions given in [10,12] by weakening the assumptions on the functions ci j , σi j and by obtaining more manageable criteria. Moreover, we in fact study a more general neutral delay Lotka–Volterra system of the form " # n n n X X X 0 0 xi (t) = xi (t) ri (t) − ai j (t)x j (t) − bi j (t)x j (t − τi j (t)) − ci j (t)x j (t − σi j (t)) , j=1

j=1

j=1

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

(3)

and thus cover many systems considered in [1,10,12] as special cases. By a positive solution of system (3) we understand a solution (x1 , x2 , . . . , xn )T satisfying that xi (t) ≥ 0(i = 1, 2, . . . , n) and there exists some 1 ≤ i 0 ≤ n such that xi0 (t) > 0, t ∈ R. In addition, our approach is based on a fixed point theorem for strict-set-contractions which has been proved in [13]. For convenience, we introduce the notation: p¯ = max { p(t)}, t∈[0,ω]

p = min { p(t)}, t∈[0,ω]

where p is a continuous ω-periodic function. In system (3), we shall use the following hypotheses: (H1 ) ri , ai j , bi j , ci j ∈ C(R, (0, ∞)), τi j , σi j ∈ C(R, [0, ∞)) (i, j = 1, 2, . . . , n) are ω-periodic functions. Also, for i, j = 1, 2, . . . , n, let Rω

qi = e− 0 ri (u)du , Z ω Mi j = [q j ai j (s) + q j bi j (s) − ci j (s)]ds

(4)

0

and ω

Z Ni j =

[ai j (s) + bi j (s) + ci j (s)]ds.

(5)

0

(H2 ) q j ai j (t) + q j bi j (t) − ci j (t) ≥ 0, t ∈ [0, ω], i, j = 1, 2, . . . , n. q2

i (H3 ) Mi j (1 + r i ) 1−q ≥ maxt∈[0,ω] {ai j (t) + bi j (t) + ci j (t)}, i, j = 1, 2, . . . , n. i nP o n i × max c ¯ (H4 ) max1≤i≤n 1−q 1≤i≤n 2 j=1 i j < 1.

qi Mii

2. Preliminaries In order to prove our results, the following definitions and lemmas are needed. Let X be a Banach space and P be a cone in X . θ denotes the zero element in X . The semi-order induced by the cone P is denoted by “≤”; that is, x ≤ y if and only if y − x ∈ P. In addition, for a bounded subset A ⊂ X , let α X (A) denote the (Kuratowski) measure of

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

non-compactness defined by (

)

α X (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 . Definition 2.1 ([5,6]). Let X and Y be Banach spaces and D ⊂ X . A continuous and bounded map T : D → Y is called k-set-contractive if, for any bounded set S ⊂ D, αY (T (S)) ≤ kα X (S). Definition 2.2 ([5,6]). T is called strict-set-contractive if it is k-set-contractive for some 0 ≤ k < 1. Lemma 2.1 ([13]). Let X be a Banach space and P a cone in X . For R > r > 0, set Ω R = {x ∈ X : kxk < R}

and

Ωr = {x ∈ X : kxk < r }.

Assume that T : P ∩ Ω R → P is strict-set-contractive and satisfies T x 6≥ x

for any x ∈ (P ∩ Ω r ) \ {θ}

(6)

T x 6≤ x

for any x ∈ P ∩ ∂Ω R .

(7)

and

Then T has at least one fixed point in P ∩ (Ω R \ Ω r ). In order to apply Lemma 2.1 to system (3), set Z = {x = (x1 , x2 , . . . , xn )T ∈ C(R, Rn ) : x(t + ω) = x(t), t ∈ R} with the norm defined by kxk0 = max {|xi |0 },

(8)

1≤i≤n

and X = {x = (x1 , x2 , . . . , xn )T ∈ C 1 (R, Rn ) : x(t + ω) = x(t), t ∈ R} with the norm defined by kxk1 = max {|xi |1 },

(9)

1≤i≤n

where, for i = 1, 2, . . . , n, |xi |0 = max {|xi (t)|},

(10)

|xi |1 = max{|xi |0 , |xi 0 |0 }.

(11)

t∈[0,ω]

and

Then Z and X are all Banach spaces. Also, for R, r ∈ (0, ∞), set Ω R = {x ∈ X : kxk1 < R}

and

Ωr = {x ∈ X : kxk1 < r }.

Define the cone P in X by P = {x = (x1 , x2 , . . . , xn )T ∈ X : xi (t) ≥ qi |xi |1 , t ∈ [0, ω], i = 1, 2, . . . , n},

(12)

where qi , |xi |1 are defined by (4) and (11), respectively. Let the map T be defined by (T x)(t) = ((T x)1 , (T x)2 , . . . , (T x)n )T ,

x ∈ P, t ∈ R,

(13)

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

where, for i = 1, 2, . . . , n, " # Z t+ω n n n X X X 0 (T x)i (t) = G i (t, s)xi (s) ai j (s)x j (s) + bi j (s)x j (s − τi j (s)) + ci j (s)x j (s − σi j (s)) ds, t

j=1

j=1

j=1

(14) and G i (t, s) =

e−

Rs

ri (u)du Rω , − 0 ri (u)du t

s ∈ [t, t + ω]. 1−e It is easy to see that, for s ∈ [t, t + ω], t ∈ R, i = 1, 2, . . . , n, G i (t + ω, s + ω) = G i (t, s) and 1 qi ≤ G i (t, s) ≤ . 1 − qi 1 − qi In the following, we will give some lemmas concerning P and T defined by (12) and (13), respectively. Lemma 2.2. In addition to (H1 )–(H3 ), assume further that, for i = 1, 2, . . . , n, one of the following two conditions is satisfied: (i) r¯i ≤ 1; N (¯ri −1) ≤ mint∈[0,ω] {q j ai j (t) + q j bi j (t) − ci j (t)}, j = 1, 2, . . . , n, where Ni j is defined by (5). (ii) r¯i > 1, qii j(1−q i) Then T : P → P. Proof. For any x ∈ P, it is easy to see that T x ∈ C 1 (R, Rn ). From (14), for t ∈ R, i = 1, 2, . . . , n, we obtain " Z t+2ω n X G i (t + ω, s)xi (s) ai j (s)x j (s) (T x)i (t + ω) = t+ω

+

j=1

n X

bi j (s)x j (s − τi j (s)) +

n X

j=1 t+ω

Z =

# ci j (s)x j (s − σi j (s)) ds 0

j=1

" G i (t + ω, u + ω)xi (u + ω)

t

+

n X

ai j (u + ω)x j (u + ω)

j=1 n X

bi j (u + ω)x j (u + ω − τi j (u + ω)) +

j=1 t+ω

Z =

G i (t, u)xi (u)

t

+

# ci j (u + ω)x j (u + ω − σi j (u + ω)) du

n X

ai j (u)x j (u) +

n X

bi j (u)x j (u − τi j (u))

j=1

j=1

# ci j (u)x j (u − σi j (u)) du = (T x)i (t). 0

j=1

So T x ∈ X . From (H2 ) and (12), for x ∈ P, t ∈ [0, ω], i = 1, 2, . . . , n, we have n n n X X X bi j (t)x j (t − τi j (t)) + ci j (t)x j 0 (t − σi j (t)) ai j (t)x j (t) + j=1

≥

j=1 n X j=1

≥

n X j=1

ai j (t)x j (t) +

0

j=1

"

n X

n X

j=1 n X

bi j (t)x j (t − τi j (t)) −

j=1 n X

q j ai j (t)|x j |1 +

j=1

n X

ci j (t)|x j 0 (t − σi j (t))|

j=1

q j bi j (t)|x j |1 −

n X j=1

ci j (t)|x j |1

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

=

n X q j ai j (t) + q j bi j (t) − ci j (t) |x j |1 j=1

≥ 0.

(15)

From (10), (14) and (15), for i = 1, 2, . . . , n, we obtain ( |(T x)i |0 = max {|(T x)i (t)|} ≤ max t∈[0,ω]

+

t∈[0,ω]

n X

1 1 − qi

bi j (s)x j (s − τi j (s)) +

j=1

t∈[0,ω]

+

n X

" xi (s)

t

n X

ai j (s)x j (s)

j=1

#

)

ci j (s)x j (s − σi j (s)) ds 0

j=1

( = max

n X

t+ω

Z

1 1 − qi

ω

Z

" xi (s)

n X

0

n X

ai j (s)x j (s) +

j=1

#

bi j (s)x j (s − τi j (s))

j=1

)

ci j (s)x j 0 (s − σi j (s)) ds

j=1

1 = 1 − qi

ω

Z

" xi (s)

n X

0

ai j (s)x j (s) +

j=1

n X

bi j (s)x j (s − τi j (s)) +

j=1

#

n X

ci j (s)x j 0 (s − σi j (s)) ds.

j=1

Therefore, for t ∈ [0, ω], i = 1, 2, . . . , n, we have qi (T x)i (t) ≥ 1 − qi

Z

qi = 1 − qi

Z

t+ω

" xi (s)

n X

t

ai j (s)x j (s) +

j=1 ω

" xi (s)

0

n X

n X

bi j (s)x j (s − τi j (s)) +

j=1

ai j (s)x j (s) +

j=1

n X

n X

# ci j (s)x j (s − σi j (s)) ds 0

j=1

bi j (s)x j (s − τi j (s)) +

j=1

n X

# ci j (s)x j (s − σi j (s)) ds 0

j=1

≥ qi |(T x)i |0 .

(16)

Now, we prove that, for t ∈ [0, ω], i = 1, 2, . . . , n, −(T x)i0 (t) ≤ (T x)i (t).

(17)

From (14), for t ∈ [0, ω], i = 1, 2, . . . , n, we find " (T x)i0 (t)

= ri (t)(T x)i (t) + G i (t, t + ω)xi (t + ω)

n X

ai j (t + ω)x j (t + ω)

j=1

+

n X

bi j (t + ω)x j (t + ω − τi j (t + ω)) +

j=1

# ci j (t + ω)x j (t + ω − σi j (t + ω)) 0

j=1

" − G i (t, t)xi (t)

n X

n X

ai j (t)x j (t) +

bi j (t)x j (t − τi j (t)) +

j=1

j=1

" = ri (t)(T x)i (t) − xi (t)

n X

n X j=1

ai j (t)x j (t) +

n X

# ci j (t)x j (t − σi j (t)) 0

j=1 n X j=1

From (H3 ) and (18), for t ∈ [0, ω], i = 1, 2, . . . , n, we have

bi j (t)x j (t − τi j (t)) +

n X j=1

# ci j (t)x j 0 (t − σi j (t)) .(18)

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

" −(T x)i0 (t)

= −ri (t)(T x)i (t) + xi (t) ≤ −r i (t)(T x)i (t) + |xi |1

n X

ai j (t)x j (t) +

j=1 n X

n X

bi j (t)x j (t − τi j (t)) +

j=1

n X

# ci j (t)x j (t − σi j (t)) 0

j=1

|x j |1 ai j (t) + bi j (t) + ci j (t)

j=1

Z ω n X qi2 ≤ −r i (t)(T x)i (t) + (1 + r i ) q j ai j (s) + q j bi j (s) − ci j (s) ds |xi |1 |x j |1 1 − qi 0 j=1 # " Z n ω X qi2 = −r i (T x)i (t) + (1 + r i ) |xi |1 |x j |1 q j ai j (s) + q j bi j (s) − ci j (s) ds 1 − qi 0 j=1 " Z t+ω n X qi = −r i (T x)i (t) + (1 + r i ) qi |xi |1 q j ai j (s)|x j |1 1 − qi t j=1 # n n X X + q j bi j (s)|x j |1 − ci j (s)|x j |1 ds j=1

j=1

≤ −r i (T x)i (t) + (1 + r i ) +

n X

t+ω

Z

" G i (t, s)xi (s)

t

n X

ai j (s)x j (s)

j=1

bi j (s)x j (s − τi j (s)) +

j=1

n X

# ci j (s)x j (s − σi j (s)) ds 0

j=1

= −r i (T x)i (t) + (1 + r i )(T x)i (t) = (T x)i (t). Hence, (17) holds. Next, we prove that, for t ∈ [0, ω], i = 1, 2, . . . , n, (T x)i0 (t) ≤ (T x)i (t).

(19)

Suppose (i) holds. From (15) and (18), we have (T x)i0 (t) ≤ ri (t)(T x)i (t) ≤ (T x)i (t),

t ∈ [0, ω].

Suppose (ii) holds. From (15) and (18), for t ∈ [0, ω], we have # " n n n X X X 0 bi j (t)x j (t − τi j (t)) − ci j (t)|x j |1 ai j (t)x j (t) + (T x)i (t) ≤ ri (t)(T x)i (t) − qi |xi |1 j=1

≤ r¯i (T x)i (t) − qi |xi |1 = r¯i (T x)i (t) − qi |xi |1

" n X j=1 n X

ai j (t)q j |x j |1 +

j=1

j=1 n X

bi j (t)q j |x j |1 −

# ci j (t)|x j |1

j=1

j=1

|x j |1 q j ai j (t) + q j bi j (t) − ci j (t)

n X

j=1

Z ωX n r¯i − 1 |xi |1 ai j (s) + bi j (s) + ci j (s) |x j |1 ds qi (1 − qi ) 0 j=1 " # Z t+ω n n n X X X 1 = r¯i (T x)i (t) − (¯ri − 1) |xi |1 ai j (s)|x j |1 + bi j (s)|x j |1 + ci j (s)|x j |1 ds 1 − qi t j=1 j=1 j=1 " Z t+ω n X ≤ r¯i (T x)i (t) − (¯ri − 1) G i (t, s)xi (s) ai j (s)x j (s) ≤ r¯i (T x)i (t) − qi

t

j=1

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

+

n X

bi j (s)x j (s − τi j (s)) +

n X

j=1

# ci j (s)x j (s − σi j (s)) ds 0

j=1

= r¯i (T x)i (t) − (¯ri − 1)(T x)i (t) = (T x)i (t). Hence, (19) holds. From (17) and (19), we have |(T x)i0 |0 ≤ |(T x)i |0 ,

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

which, together with (11) and (16), reduce to (T x)i (t) ≥ qi |(T x)i |1 ,

t ∈ [0, ω], i = 1, 2, . . . , n.

Hence, T x ∈ P. The proof of Lemma 2.2 is complete.

Lemma 2.3. In addition to (H1 )–(H3 ), let R > 0 be a constant such that ( ) n X R max c¯i j < 1. 1≤i≤n

j=1

Assume further that, for i = 1, 2, . . . , n, one of the following two conditions is satisfied: (i) r¯i ≤ 1; N (¯ri −1) ≤ mint∈[0,ω] {q j ai j (t) + q j bi j (t) − ci j (t)}, j = 1, 2, . . . , n. (ii) r¯i > 1, qii j(1−q T i) Then T : P Ω R → P is strict-set-contractive. T Proof. From (13) and (14) and LemmaT2.2, we know that T : P Ω R → P is continuous and bounded. Now we prove that, for any bounded set S ⊂ P Ω R , ) ( n X (20) α X (T (S)) ≤ R max c¯i j α X (S). 1≤i≤n

j=1

Let η = α X (S). Then, for any positive number ε < R max1≤i≤n S satisfying S = p S p with diam(S p ) ≤ η + ε. Therefore

nP

n j=1 c¯i j

o

η, there is a finite family of subsets {S p }

kx − yk1 ≤ η + ε, for any x, y ∈ S p . (21) T Note that S p ⊂ S ⊂ P Ω R and S is bounded. Set kxk1 ≤ K p , for any x ∈ S p . From (8)–(11), we find that, for any x ∈ Sp, kxk0 ≤ K p

and

kx 0 k0 ≤ K p .

Using the Arzela–Ascoli theorem, we know that S S p are precompact in Z . Therefore, it follows that there is a finite family of subsets {S pq } of S p such that S p = q S pq and kx − yk0 ≤ ε,

for any x, y ∈ S pq .

(22)

In addition, for any x ∈ S and t ∈ [0, ω], i = 1, 2, . . . , n, we have " # Z t+ω n n n X X X 0 |(T x)i (t)| = G i (t, s)xi (s) ai j (s)x j (s) + bi j (s)x j (s − τi j (s)) + ci j (s)x j (s − σi j (s)) ds t

≤

j=1

R2 1 − qi

j=1

j=1

n ωX

Z

ai j (s) + bi j (s) + ci j (s) ds =: Ai

0

j=1

and " # n n n X X X 0 0 |(T x)i (t)| = ri (t)(T x)i (t) − xi (t) ai j (t)x j (t) + bi j (t)x j (t − τi j (t)) + ci j (t)x j (t − σi j (t)) j=1 j=1 j=1

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

≤ r¯i Ai + R 2

n X

(a¯ i j + b¯i j + c¯i j ) =: Bi .

j=1

Hence, k(T x)0 k0 ≤ max {Bi }.

kT xk0 ≤ max {Ai }, 1≤i≤n

1≤i≤n

Applying the Arzela–Ascoli theorem, we know that T (S) is precompact in Z . Then, there is a finite family of subsets S {S pqr } of S pq such that S pq = r S pqr and kT x − T yk0 ≤ ε,

for any x, y ∈ S pqr .

(23)

From (21)–(23), for any x, y ∈ S pqr , i = 1, 2, . . . , n, we obtain |(T x)i0

− (T y)i0 |0

= max

t∈[0,ω]

{|(T x)i0 (t) − (T y)i0 (t)|} "

− xi (t)

n X

ai j (t)x j (t) +

j=1

"

( = max ri (t)(T x)i (t) − ri (t)(T y)i (t) t∈[0,ω]

n X

bi j (t)x j (t − τi j (t)) +

j=1

n X

# ci j (t)x j (t − σi j (t)) 0

j=1

# ) + yi (t) ai j (t)y j (t) + bi j (t)y j (t − τi j (t)) + ci j (t)y j (t − σi j (t)) j=1 j=1 j=1 n X

n X

n X

0

≤ max {|ri (t)[(T x)i (t) − (T y)i (t)]|} t∈[0,ω] ( " # n n n X X X ai j (t)x j (t) + bi j (t)x j (t − τi j (t)) + ci j (t)x j 0 (t − σi j (t)) + max xi (t) t∈[0,ω] j=1 j=1 j=1 " # ) n n n X X X − yi (t) ai j (t)y j (t) + bi j (t)y j (t − τi j (t)) + ci j (t)y j 0 (t − σi j (t)) j=1 j=1 j=1 ( " n X ≤ r¯i |(T x)i − (T y)i |0 + max xi (t) ai j (t)(x j (t) − y j (t)) t∈[0,ω] j=1 +

n X

bi j (t)(x j (t − τi j (t)) − y j (t − τi j (t)))

j=1

# ) + ci j (t)(x j (t − σi j (t)) − y j (t − σi j (t))) j=1 ( " n n X X ai j (t)y j (t) + bi j (t)y j (t − τi j (t)) + max (xi (t) − yi (t)) t∈[0,ω] j=1 j=1 # ) n X ci j (t)y j 0 (t − σi j (t)) + j=1 ! ! ! n n n n n n X X X X X X ≤ r¯i ε + R a¯ i j + b¯i j ε + R c¯i j (η + ε) + R a¯ i j + b¯i j + c¯i j ε n X

0

j=1

=

R

n X j=1

0

j=1

j=1

j=1

j=1

j=1

! c¯i j η + Wi ε,

(24)

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

Pn

where Wi = r¯i + 2R

¯i j j=1 a

kT x − T yk1 ≤

R max

+ 2R ( n X

1≤i≤n

Pn

¯ + 2R

j=1 bi j

Pn

j=1 c¯i j .

From (23) and (24), we have

)! c¯i j

η + max {Wi }ε,

for any x, y ∈ S pqr .

1≤i≤n

j=1

As ε is arbitrary small, it follows that (20) holds. Therefore, T : P Lemma 2.3 is complete.

T

Ω R → P is strict-set-contractive. The proof of

From (18), it is easy to see that the following lemma holds. Lemma 2.4. In addition to (H1 )–(H3 ), assume further that, for i = 1, 2, . . . , n, one of the following two conditions is satisfied: (i) r¯i ≤ 1; (ii) r¯i > 1,

Ni j (¯ri −1) qi (1−qi )

≤ mint∈[0,ω] {q j ai j (t) + q j bi j (t) − ci j (t)}, j = 1, 2, . . . , n.

If x is a nonzero fixed point of the operator T on P, then x is a positive ω-periodic solution for system (3). 3. Main results In this section, we shall study the existence of positive ω-periodic solutions for system (3). Theorem 3.1. In addition to (H1 )–(H4 ), assume further that, for i = 1, 2, . . . , n, one of the following two conditions is satisfied: (i) r¯i ≤ 1; (ii) r¯i > 1,

Ni j (¯ri −1) qi (1−qi )

≤ mint∈[0,ω] {q j ai j (t) + q j bi j (t) − ci j (t)}, j = 1, 2, . . . , n.

Then system (3) has at least one positive ω-periodic solution. 1−qi q (1−q ) i i Proof. Let R = max1≤i≤n 2 and 0 < r < min1≤i≤n Pn N . Note that qi < 1 and Mii ≤ Nii , i = qi Mii j=1 i j T 1, 2, . . . , n. We then have r < R. From (HT 4 ) and Lemma 2.3, Twe know that T : P Ω R → P is strict-set-contractive. From Lemma 2.4, if there exists x˜ ∈ P ∂Ω R or x˜ ∈ P ∂Ωr such that T x˜ = x, ˜ then x˜ is one positive ω-periodic solution for system (3). T T Now, we shall prove that (6) and (7) hold if T x 6= x for any x ∈ P ∂Ω R and x ∈ P ∂Ωr . First, we prove that (6) holds. Otherwise, there exists x ∈ (P ∩ Ω r ) \ {θ} such that T x ≥ x. So kxk1 > 0 and T x − x ∈ P, which implies that, for i = 1, 2, . . . , n, t ∈ [0, ω], (T x)i (t) − xi (t) ≥ qi |(T x)i − xi |1 ≥ 0.

(25)

In addition, there exists some 1 ≤ i 0 ≤ n, such that |xi0 |0 > 0. Further, for t ∈ [0, ω], we obtain " Z t+ω n n X X (T x)i0 (t) = G i0 (t, s)xi0 (s) ai0 j (s)x j (s) + bi0 j (s)x j (s − τi0 j (s)) t

+

j=1 n X

j=1

# ci0 j (s)x j 0 (s − σi0 j (s)) ds

j=1

r |xi |0 ≤ 1 − qi0 0 =

ω

Z 0

"

n X j=1

ai0 j (s) +

n X j=1

bi0 j (s) +

n X

# ci0 j (s) ds

j=1

n X r |xi0 |0 Ni 0 j 1 − qi0 j=1

≤ qi0 |xi0 |0 .

(26)

G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

2651

From (25) and (26), we find |xi0 |0 ≤ |(T x)i0 |0 ≤ qi0 |xi0 |0 < |xi0 |0 , which is a contradiction. Next, we prove that (7) also holds. For (7), we only need to prove that T x 6< x

for any x ∈ P ∩ ∂Ω R .

T Assume, for the sake of contradiction, that there exists x ∈ P ∂Ω R such that T x < x. Hence x − T x ∈ P \ {θ}. In addition, there exists some 1 ≤ i 1 ≤ n, such that |xi1 |1 = kxk1 = R and xi1 (t) − (T x)i1 (t) ≥ qi1 |xi1 − (T x)i1 |1 ≥ 0,

t ∈ [0, ω].

(27)

Further, for any t ∈ [0, ω], we find " Z t+ω n n X X G i1 (t, s)xi1 (s) ai1 j (s)x j (s) + (T x)i1 (t) = bi1 j (s)x j (s − τi1 j (s)) t

+

j=1 n X

j=1

# ci1 j (s)x 0j (s − σi1 j (s)) ds

j=1

Z ω qi1 qi1 |xi1 |1 ai1 i1 (s)xi1 (s) + bi1 i1 (s)xi1 (s − τi1 i1 (s)) + ci1 i1 (s)xi01 (s − σi1 i1 (s)) ds > 1 − qi1 0 Z ω qi1 ≥ qi1 |xi1 |1 qi1 ai1 i1 (s)|xi1 |1 + qi1 bi1 i1 (s)|xi1 |1 − ci1 i1 (s)|xi1 |1 ds 1 − qi1 0 (qi1 )2 Mi i R 2 1 − qi1 1 1 ≥ R, =

which, together with (27), implies that xi1 (t) ≥ (T x)i1 (t) > R,

t ∈ [0, ω],

which contradicts |xi1 |1 = R. Therefore, (6) and (7) hold. By Lemma 2.1, we see that T has at least one fixed point in P ∩ (Ω R \ Ω r ). Applying Lemma 2.4, system (3) has at least one positive ω-periodic solution. The proof of Theorem 3.1 is complete. Remark 3.1. The deviating arguments τi j , σi j and the period ω have no effect on the existence of positive periodic solutions for system (3). But, in [10,12], a few restrictions were imposed on the period ω. Remark 3.2. Theorem 2.1 [10] required that ci j ∈ C 1 (R, (0, ∞)), ci j 0 (t) < bi j (t), τi j = σi j , i, j = 1, 2, . . . , n; Theorem 2.2 [10] required that ci j (t) ≡ ci j (i, j = 1, 2, . . . , n) are constants; Theorem [12] required that ci j (t) ≡ ci j , σi j (t) ≡ σi j (i, j = 1, 2, . . . , n) are constants. But in our paper, we only require that ci j ∈ C(R, (0, ∞)), and τi j (t) is independent of σi j (t), i, j = 1, 2, . . . , n. Therefore, our main result generalizes and improves the corresponding results in [10] and [12]. This advantage is illustrated by the following example. Example 3.1. Consider the following neutral delay Lotka–Volterra system: 3 + sin t 3 + cos t 2 + sin t 0 − (x1 (t) + x2 (t)) − (x1 (t − 2 − sin t) + x2 (t − 2 − cos t))− x (t) = x (t) 1 1 128 96 96 4 + sin t 0 4 + cos t 0 x1 (t − 3 − sin t) − x2 (t − 3 − cos t) , 120 120 (28) 2 + cos t 1 2 + cos t 0 x (t) = x (t) − (x (t) + x (t)) − (x (t − 4 − sin t) + x (t − 4 − cos t))− 2 2 1 2 1 2 128 24 72 3 + cos t 0 3 + sin t 0 x1 (t − 5 − sin t) − x2 (t − 5 − cos t) . 120 120

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G. Liu et al. / Nonlinear Analysis 67 (2007) 2642–2653

t 2+cos t 3+sin t 1 Clearly, r1 (t) = 2+sin 128 , r2 (t) = 128 , a11 (t) = a12 (t) = 96 , a21 (t) = a22 (t) = 24 , b11 (t) = b12 (t) 2+cos t 4+sin t 4+cos t 3+sin t 3+cos t 3+cos t 96 , b21 (t) = b22 (t) = 72 , c11 (t) = 120 , c12 (t) = 120 , c21 (t) = 120 , c22 (t) = 120 , τ11 (t) 2 + sin t, τ12 (t) = 2 + cos t, σ11 (t) = 3 + sin t, σ12 (t) = 3 + cos t, τ21 (t) = 4 + sin t, τ22 (t) = 4 + cos t, σ21 (t)

= = =

5 + sin t, σ22 (t) = 5 + cos t. All of the above functions are 2π -periodic functions. Obviously, (H1 ) holds. Further, we have r1 = r2 = R 2π

1 , 128

r¯1 = r¯2 =

3 , 128

c¯11 = c¯12 =

π

r1 (u)du

1 , 24

R 2π

c¯21 = c¯22 =

1 , 30

π

= e− 32 ≈ 0.9065, q2 = e− 0 r2 (u)du = e− 32 ≈ 0.9065, 1 5q1 1 q1 M21 = M22 = π − π ≈ 0.2385, M11 = M12 = π − π ≈ 0.1465, 8 15 36 20 5q1 − 3 q1 a11 (t) + q1 b11 (t) − c11 (t) > > 0, 120 1 q1 − > 0, q1 a21 (t) + q1 b21 (t) − c21 (t) > 18 30 5q2 − 3 q2 a12 (t) + q2 b12 (t) − c12 (t) > > 0, 120 q2 1 q2 a22 (t) + q2 b22 (t) − c22 (t) > − > 0, 18 30 q2 M11 (1 + r 1 ) 1 ≈ 1.2978 > 0.125 > max {a11 (t) + b11 (t) + c11 (t)}, t∈[0,2π] 1 − q1 q1 = e−

0

M12 (1 + r 1 )

q12 ≈ 1.2978 > 0.125 > max {a12 (t) + b12 (t) + c12 (t)}, t∈[0,2π] 1 − q1

M21 (1 + r 2 )

q22 ≈ 2.1118 > 0.1167 > max {a21 (t) + b21 (t) + c21 (t)}, t∈[0,2π] 1 − q2

M22 (1 + r 2 )

q22 ≈ 2.1118 > 0.1167 > max {a22 (t) + b22 (t) + c22 (t)}, t∈[0,2π] 1 − q2

and ( max

1≤i≤2

1 − qi qi2 Mii

)

( × max

1≤i≤2

2 X j=1

) c¯i j

≈ 0.7766 ×

1 < 1. 12

Hence, (H1 )–(H4 ) hold and r¯i ≤ 1 (i = 1, 2). By means of Theorem 3.1, system (28) has at least one positive 2π-periodic solution. Remark 3.3. As σi j (i, j = 1, 2) are not constants, Theorems 2.1 and 2.2 in [10] and Theorem in [12] cannot be used to obtain the existence of a positive 2π-periodic solution of system (28). Acknowledgements The authors would like to thank the reviewers and the editor for their valuable suggestions and comments. This work was supported by the Fund for Doctoral Program Research of the Education Ministry of China (20040108003) and the Natural Science Foundation of China (10471040). References [1] M. Fan, K. Wang, D. Jiang, Existence and global attractivity of positive periodic solutions of periodic n-species Lotka–Volterra competition systems with several deviating arguments, Math. Biosci. 160 (1999) 47–61. [2] H. Fang, J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl. 259 (2001) 8–17. [3] K. Gopalsamy, X. He, L. Wen, On a periodic neutral logistic equation, Glasg. Math. J. 33 (1991) 281–286.

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