Electronic Journal of Differential Equations, Vol. 2007(2007), No. 13, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
POSITIVE PERIODIC SOLUTIONS OF NEUTRAL LOGISTIC
EQUATIONS WITH DISTRIBUTED DELAYS
YONGKUN LI, GUOQIAO WANG, HUIMEI WANG
Abstract. Using a fixed point theorem of strict-set-contraction, we establish
criteria for the existence of positive periodic solutions for the periodic neutral
logistic equation, with distributed delays,
Z 0
Z 0
n
m
h
i
X
X
x0 (t) = x(t) a(t)−
ai (t)
x(t+θ) dµi (θ)−
bj (t)
x0 (t+θ) dνj (θ) ,
i=1
−Ti
j=1
−T̂j
where the coefficients a, ai , bj are continuous and periodic functions, with the
same period. The values Ti , T̂j are positive, and the functions µi , νj are nonR0
R0
decreasing with −T
dµi = 1 and −
dνj = 1.
T̂
i
j
1. Introduction
Consider the single species neutral logistic model, with discrete delays,
n
m
h
i
X
X
dN (t)
= N (t) a(t)−β(t)N (t)−
bi (t)Ni (t−τi (t))−
cj (t)Nj0 (t−σj (t)) , (1.1)
dt
i=1
j=1
where the functions a(t), β(t), bj (t), cj (t), τi (t), σj (t) are continuous ω-periodic, and
a(t) ≥ 0, β(t) ≥ 0, bi (t) ≥ 0, cj (t) ≥ 0 (i = 1, 2, . . . , n, j = 1, 2, . . . , m). An ecological justification of model (1.1) can be found in [3, 4, 6, 10]. Using continuation
theory for k-set-contractions, Lu [8], Lu and Ge [9] studied the existence of positive
periodic solutions of (1.1). Yang and Cao [11] used Mawhin’s continuation theorem
[2] to investigated the existence of positive periodic solutions of (1.1). The main
results obtained in [6, 9] required cj ∈ C 1 , σj ∈ C 2 and σj0 < 1 (j = 1, 2, . . . , n). To
the best of our knowledge, this is the first paper to study the existence of periodic
solutions of neutral logistic equations with distributed delays.
The main purpose of this paper is by using a fixed point theorem of strict-setcontraction [1, 5] to establish the existence of positive periodic solutions of the
2000 Mathematics Subject Classification. 34K13, 34K40.
Key words and phrases. Positive periodic solution; neutral delay logistic equation;
strict-set-contraction.
c
2006
Texas State University - San Marcos.
Submitted July 14, 2006. Published January 8, 2007.
Supported by grants 10361006 from the National Natural Sciences Foundation China,
and 2003A0001M from the the Natural Sciences Foundation of Yunnan Province.
1
2
Y. LI, G. WANG, H. WANG
EJDE-2007/13
following neutral functional differential equation, with distributed delays,
Z 0
Z 0
n
n
h
i
X
X
0
x (t) = x(t) a(t) −
ai (t)
x(t + θ) dµi (θ) −
bj (t)
x0 (t + θ) dνj (θ) ,
−Ti
i=1
−T̂j
j=1
(1.2)
where a, ai , bj ∈ C(R, R ) are ω-periodic functions, Ti , T̂j are postive constants,
R0
R0
µi , νj : [−Ti , 0] → [0, ∞) are nondecreasing functions and −Ti dµi = 1, −T̂j dνj =
1, for i = 1, 2, . . . , n, j = 1, 2, . . . , m. Note that (1.1) is a special case of (1.2). For
an ecological justification of (1.1), we refer the reader to [7].
For convenience, we introduce the following notation:
Z ωh X
n
m
i
Rω
X
λ
ai (s) −
λ = e− 0 a(s) ds , ∆ =
bj (s) ds,
+
0
Z
Π=
0
ω
n
hX
ai (s) +
i=1
f
m
X
i=1
i
bj (s) ds,
j=1
f M = max {f (t)},
j=1
m
t∈[0,ω]
= min {f (t)},
t∈[0,ω]
where f is a continuous ω-periodic function. Also we introduce the following assumptions:
Rω
(H1) λ := exp − 0 a(s) ds < 1.
Pn
Pm
(H2) λ i=1 ai (t) − j=1 bj (t) ≥ 0.
Pn
Pm
2
∆
≥ maxt∈[0,ω]
(H3) (1 + am ) λ1−λ
i=1 ai (t) +
j=1 bj (t) .
Pn
M
Pm
−1)
(H4) Π(a
i=1 ai (t) −
j=1 bj (t) .
λ(1−λ) ≤ mint∈[0,ω] λ
Pm M
(H5) λ1−λ
< 1.
2∆
j=1 bj
2. Preliminaries
To obtain the existence of periodic solutions to (1.2), we 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 the Kuratowski measure of non-compactness be
defined by
n
αE (A) = inf δ > 0 : there is a finite number of subsets Ai ⊂ A
o
such that A = ∪i Ai and diam(Ai ) ≤ δ ,
where diam(Ai ) denotes the diameter of the set Ai .
Let E, F be two Banach spaces and D ⊂ E, a continuous and bounded mapping
Φ : Ω̄ → F is called k-set contractive if for every bounded set S ⊂ D we have
αF (Φ(S)) ≤ kαE (S).
The mapping Φ is called strict-set-contractive if it is k-set-contractive for some
0 ≤ k < 1. From [1, 2], we cite the following lemma which is useful for the proof of
our main result.
EJDE-2007/13
POSITIVE PERIODIC SOLUTIONS
3
Lemma 2.1 ([1, 5]). Let K be a cone of the real Banach space X and Kr,R =
{x ∈ K : r ≤ ||x|| ≤ 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) (Φx x for all x ∈ K, kxk = r) and (Φx x for all x ∈ K, kxk = R).
(ii) (Φx x for all x ∈ K, kxk = r) and (Φx x for all x ∈ K, kxk = R).
Then Φ has at least one fixed point in Kr,R .
To apply Lemma 2.1 to (1.2), we set
Cω0 = {x ∈ C 0 (R, R) : x(t + ω) = x(t)}
with the norm |x|0 = maxt∈[0,ω] {|x(t)|}, and
Cω1 = {x ∈ C 1 (R, R) : x(t + ω) = x(t)}
with the norm |x|1 = max{|x|0 , |x0 |0 }. Then Cω0 and Cω1 are all Banach spaces.
Define the cone K in Cω1 by
K = {x ∈ Cω1 : x(t) ≥ λ|x|1 , t ∈ [0, ω]}.
Let the mapping Φ be defined by
Z t+ω
Z
n
hX
(Φx)(t) =
G(t, s)x(s)
ai (s)
t
+
i=1
m
X
Z
0
(2.1)
0
x(s + θ) dµi (θ)
−Ti
(2.2)
i
x (s + θ) dνj (θ) ds,
0
bj (s)
−T̂j
j=1
where x ∈ K, t ∈ R, and
e−
Rs
a(θ) dθ
Rω
,
− 0 a(θ) dθ
t
s ∈ [t, t + ω].
1−e
It is easy to see that G(t + ω, s + ω) = G(t, s) and
G(t, s) =
1
λ
≤ G(t, s) ≤
, s ∈ [t, t + ω].
1−λ
1−λ
Next, we give some lemmas concerning the K and Φ defined above.
Lemma 2.2. 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 Φx ∈ C 1 (R, R). In view of (2.2), for t ∈ R,
we obtain
Z t+2ω
Z 0
n
hX
(Φx)(t + ω) =
G(t + ω, s)x(s)
ai (s)
x(s + θ) dµi (θ)
t+ω
+
m
X
i=1
Z
=
t+ω
i
x0 (s + θ) dνj (θ) ds
−T̂j
j=1
Z
0
bj (s)
−Ti
Z
h
G(t + ω, u + ω)x(u + ω) a(u + ω)
0
−Ti
t
Z
0
+ b(u + ω)
−T̂j
i
x0 (u + ω + θ) dνj (θ) du
x(u + ω + θ) dµi (θ)
4
Y. LI, G. WANG, H. WANG
Z
t+ω
=
EJDE-2007/13
0
Z
h
G(t, u)x(u) a(u)
x(u + θ) dµi (θ)
−Ti
t
Z
0
i
x0 (u + θ) dνj (θ) du
+ b(u)
−T̂j
= (Φx)(t).
That is, (Φx)(t + ω) = (Φx)(t), t ∈ R. So Φx ∈ Cω1 . In view of (H2), for x ∈ K,
t ∈ [0, ω], we have
Z 0
Z 0
n
m
X
X
ai (t)
x(t + θ) dµi (θ) +
bj (t)
x0 (t + θ) dνj (θ)
−Ti
i=1
≥
n
X
Z
−Ti
i=1
≥
x(t + θ) dµi (θ) −
ai (t)
n
X
Z
ai (t)
i=1
n
X
m
X
λ|x|1 dµi (θ) −
−Ti
m
X
|x0 (t + θ)| dνj (θ)
−T̂j
Z
(2.3)
0
|x|1 dνj (θ)
bj (t)
−T̂j
j=1
m
X
0
Z
bj (t)
j=1
0
ai (t) −
≥λ
−T̂j
j=1
0
bj (t)]|x|1 dνj (θ) ≥ 0.
j=1
i=1
Therefore, for x ∈ K, t ∈ [0, ω], we find
Z 0
Z ω
n
hX
1
ai (s)
x(s + θ) dµi (θ)
|Φx|0 ≤
x(s)
1−λ 0
−Ti
i=1
Z 0
m
i
X
+
bj (s)
x0 (s + θ) dνj (θ) ds
−T̂j
j=1
and
(Φx)(t) ≥
λ
1−λ
+
m
X
t
Z
+
m
X
0
bj (s)
0
x(s + θ) dµi (θ)
−Ti
i=1
i
x0 (s + θ) dνj (θ) ds
−T̂j
j=1
λ
=
1−λ
Z
n
hX
ai (s)
x(s)
t+ω
Z
ω
Z
Z
n
hX
x(s)
ai (s)
0
i=1
Z
0
bj (s)
0
x(s + θ) dµi (θ)
−Ti
i
x0 (s + θ) dνj (θ) ds
−T̂j
j=1
≥ λ|Φx|0 .
Now, we show that (Φx)0 (t) ≥ λ|(Φx)0 |0 , t ∈ [0, ω]. From (2.2), we have
Z 0
h
0
x(t + ω + θ) dµi (θ)
(Φx) (t) = G(t, t + ω)x(t + ω) a(t + ω)
−Ti
Z
0
+ b(t + ω)
−T̂j
i
x0 (t + ω + θ) dνj (θ)
(2.4)
EJDE-2007/13
POSITIVE PERIODIC SOLUTIONS
Z
n
hX
− G(t, t)x(t)
ai (t)
m
X
+
Z
0
x(t + θ) dµi (θ)
−Ti
i=1
0
i
x0 (t + θ) dνj (θ) + a(t)(Φx)(t)
bj (t)
Z
n
hX
= a(t)(Φx)(t) − x(t)
ai (t)
Z
0
0
x(t + θ) dµi (θ)
−Ti
i=1
+
(2.5)
−T̂j
j=1
m
X
5
i
x0 (t + θ) dνj (θ) .
bj (t)
−T̂j
j=1
It follows from (2.3) and (2.5) that if (Φx)0 (t) ≥ 0, then
(Φx)0 (t) ≤ a(t)(Φx)(t) ≤ aM (Φx)(t) ≤ (Φx)(t).
(2.6)
On the other hand, from (2.4), (2.5) and (H3), if (Φx)0 (t) < 0, then
− (Φx)0 (t)
Z
n
hX
ai (t)
= x(t)
0
x(t + θ) dµi (θ) +
−Ti
i=1
m
X
Z
0
bj (t)
j=1
i
x0 (t + θ) dνj (θ)
−T̂j
− a(t)(Φx)(t)
m
n
i
hX
X
bj (t) − am (Φx)(t)
≤ |x|21
ai (t) +
j=1
i=1
Z ωh X
m
n
i
X
λ
2
bj (s) ds − am (Φx)(t)
|x|1
ai (s) +
λ
≤ (1 + a )
1−λ
0
j=1
i=1
Z ω
n
m
h
i
X
X
λ
= (1 + am )
λ|x|1 λ|x|1
ai (s) − |x|1
bj (s) ds − am (Φx)(t)
0 1−λ
i=1
j=1
Z 0
Z t+ω
n
hX
ai (s)
x(s + θ) dµi (θ)
≤ (1 + am )
G(t, s)x(s)
2
m
t
−
m
X
i=1
Z
0
bj (s)
−T2
j=1
m
t+ω
Z
≤ (1 + a )
Z
n
hX
G(t, s)x(s)
ai (s)
t
+
m
X
−T1
i
|x0 (θ + s)| dνj (θ) ds − am (Φx)(t)
i=1
Z
0
bj (s)
0
x(s + θ) dµi (θ)
−T1
i
x0 (θ + s) dνj (θ) ds − am (Φx)(t)
−T2
j=1
m
= (1 + a )(Φi x)(t) − am (Φx)(t)
= (Φx)(t).
It follows from the above inequality and (2.6) that |(Φx)0 |0 ≤ |Φx|0 . So |Φx|1 =
|Φx|0 . By (2.2) we have (Φx)(t) ≥ λ|Φx|1 . Hence, Φx ∈ K. The proof of (i) is
complete.
6
Y. LI, G. WANG, H. WANG
EJDE-2007/13
(ii) In view of the proof of (i), we only need to prove that (Φx)0 (t) ≥ 0 implies
(Φx)0 (t) ≤ (Φx)(t).
From (2.3), (2.5), (H2) and (H4), we obtain
0
(Φi x) (t) ≤ a(t)(Φx)(t) − λ|x|1
n
hX
x(t + θ) dµi (θ)
ai (t)
−Ti
i=1
−
m
X
Z
0
bj (t)
0
Z
i
|x0 (t + θ)| dνj (θ)
−T̂j
j=1
n
m
h X
i
X
≤ a(t)(Φx)(t) − λ|x|21 λ
ai (t) −
bj (t)
i=1
Z
≤ aM (Φx)(t) − λ|x|21
≤ aM (Φx)(t) − (aM
≤ aM (Φx)(t) − (aM
j=1
m
i
X
a −1
ai (s) +
bj (s) ds
λ(1 − λ) 0
i=1
j=1
Z t+ω
n
m
h
i
X
X
1
− 1)
|x|1
ai (s)|x|1 +
bj (s)|x|1 ds
1−λ
t
i=1
j=1
Z t+ω
Z 0
n
hX
− 1)
ai (t)
G(t, s)x(s)
x(t + θ) dµi (θ)
M
ω
n
hX
t
+
m
X
Z
0
bj (t)
i=1
−T̂j
j=1
≤ aM (Φx)(t) − (aM − 1)
Z
t+ω
Z
n
hX
ai (t)
G(t, s)x(s)
t
+
m
X
−Ti
i
|x0 (t + θ)| dνj (θ) ds
Z
0
bj (t)
j=1
i=1
0
x(t + θ) dµi (θ)
−Ti
i
x0 (t + θ) dνj (θ) ds
−T̂j
M
= a (Φx)(t) − (aM − 1)(Φx)(t)
= (Φx)(t).
The proof of (ii) is complete.
Pm
Lemma 2.3. Assume that (H1)–(H3) hold and R j=1 bM
j < 1.
T
(i) If aM ≤ 1, then Φ : K Ω̄R → K is strict-set-contractive,
T
(ii) If (H4) holds and aM > 1, then Φ : K Ω̄R → K is strict-set-contractive,
where ΩR = {x ∈ Cω1 : |x|1 < R}.
Proof. We only need to prove (i), since the proof of (ii) is similar. It is easy
to see that Φ is continuous and bounded. Now we prove that αCω1 (Φ(S)) ≤
Pm
1 (S) for any bounded set S ⊂ Ω̄R . Let η = αC 1 (S). Then, for
R j=1 bM
j αCω
ω
Pm
η, there is a finite family of subsets {Si }
any positive number ε < R j=1 bM
j
S
satisfying S = i Si with diam(Si ) ≤ η + ε. Therefore,
|x − y|1 ≤ η + ε
for all x, y ∈ Si .
(2.7)
EJDE-2007/13
POSITIVE PERIODIC SOLUTIONS
7
As S and Si are precompactSin Cω0 , it follows that there is a finite family of subsets
{Sij } of Si such that Si = j Sij and
|x − y|0 ≤ ε
for all x, y ∈ Sij .
In addition, for any x ∈ S and t ∈ [0, ω], we have
Z t+ω
Z
n
hX
|(Φx)(t)| =
G(t, s)x(s)
ai (t)
t
+
Z
x(t + θ) dµi (θ)
i
x0 (t + θ) dνj (θ) ds
bj (t)
−T̂j
j=1
≤
0
0
−Ti
i=1
m
X
(2.8)
R2
1−λ
Z
t+ω
n
hX
t
ai (s) +
i=1
m
X
i
bj (s) ds := H
j=1
and
Z
n
hX
ai (s)
|(Φx)0 (t)| = a(t)(Φx)(t) − x(t)
+
Z
0
bj (s)
−T̂j
j=1
≤ aM H + R2
m
X
x(s + θ) dµi (θ)
−Ti
i=1
m
X
0
i
x0 (s + θ) dνj (θ) (aM + bM ).
j=1
Applying the Arzela-Ascoli Theorem, we know that Φ(S) is precompact
in Cω0 .
S
Then, there is a finite family of subsets {Sijk } of Sij such that Sij = k Sijk and
|Φx − Φy|0 ≤ ε
for allx, y ∈ Sijk .
(2.9)
From (2.3), (2.5) and (2.7)-(2.9) and (H2), for any x, y ∈ Sijk , we obtain
|(Φx)0 − (Φy)0 |0
n
= max a(t)(Φx)(t) − a(t)(Φy)(t)
t∈[0,ω]
Z
n
hX
− x(t)
ai (t)
i=1
n
hX
+ y(t)
i=1
0
x(t + θ) dµi (θ) +
−Ti
Z
0
ai (t)
y(t + θ) dµi (θ) +
−Ti
m
X
j=1
m
X
Z
0
i
x0 (t + θ) dνj (θ)
bj (t)
−T̂j
Z
0
bj (t)
−T̂j
j=1
io
y 0 (t + θ) dνj (θ) ≤ max {|a(t)[(Φx)(t) − (Φy)(t)]|}
t∈[0,ω]
Z
n
n
hX
+ max x(t)
ai (t)
t∈[0,ω]
i=1
Z
n
hX
− y(t)
ai (t)
i=1
x(t + θ) dµi (θ) +
−Ti
0
−Ti
≤ aM |(Φx) − (Φy)|0
0
y(t + θ) dµi (θ) +
m
X
Z
j=1
m
X
j=1
Z
0
bj (t)
0
bj (t)
−T̂j
i
x0 (t + θ) dνj (θ)
−T̂j
io
y 0 (t + θ) dνj (θ) 8
Y. LI, G. WANG, H. WANG
Z
n
h X
n
ai (t)
+ max x(t)
t∈[0,ω]
+
−
m
X
0
−T̂j
0
Z
ai (t)
y(t + θ) dµi (θ) +
−Ti
i=1
t∈[0,ω]
m
X
Z
0
−T̂j
j=1
t∈[0,ω]
+
m
X
Z
0
y(t + θ) dµi (θ)
Z
0
|x(s + θ) − y(s + ω)| dµi (s)
ai (t)
Ti
i=1
|x0 (s + θ) − y 0 (s + ω)| dνj (s)
T̂j
j=1
+ ε max
n
nX
t∈[0,ω]
≤ aM ε + Rε
Z
0
ai (t)
y(t + θ) dµi (θ) +
−Ti
i=1
n
X
aM
i + R(η + ε)
i=1
= Rη
−T̂j
io
y 0 (t + θ) dνj (θ) 0
bj (t)
m
X
bj (t)
o
i
y 0 (t + θ) dνj (θ) [x(t) − y(t)]
n
nX
≤ aM ε + R max
0
Z
−Ti
i=1
bj (t)
m
X
j=1
Z
n
hX
n
+ max y(t)
ai (t)
+
x(t + θ) dµi (θ)
x0 (t + θ) dνj (θ)
bj (t)
j=1
n
X
0
−Ti
i=1
Z
EJDE-2007/13
m
X
bM
j + Rε
j=1
m
X
Z
0
o
y 0 (t + θ) dνj (θ)
bj (t)
j=1
n
X
−T̂j
aM
i + Rε
i=1
m
X
bM
j
j=1
bM
j + Ĥε,
j=1
where Ĥ = aM + 2R
we have
Pn
i=1
aM
i + 2R
Pn
M
j=1 bj .
From the above equation and (2.9),
m
X
|Φx − Φy|1 ≤ R
bM
η + Ĥε
j
for all x, y ∈ Sijk .
j=1
Since ε is arbitrary small, it follows that
m
X
αCω1 (Φ(S)) ≤ R
bM
αCω1 (S).
j
j=1
Therefore, Φ is strict-set-contractive. The proof of Lemma 2.3 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 ωperiodic solution.
EJDE-2007/13
POSITIVE PERIODIC SOLUTIONS
9
Proof. We only need to prove (i), since the proof of (ii) is similar. Let R = λ1−λ
2∆
λ(1−λ)
and 0 < r < Π . Then we have 0 < r < R. From Lemma 2.2 and 2.3, we know
that Φ is strict-set-contractive on Kr,R . In view of (2.5), we see that if there exists
x∗ ∈ K such that Φx∗ = x∗ , then x∗ is one positive ω-periodic solution of (1.2).
Now, we shall prove that condition (ii) of Lemma 2.1 hold.
First, we prove that Φx x, for all x ∈ K, |x|1 = r. Otherwise, there exists
x ∈ K, |x|1 = r such that Φx ≥ x. So |x| > 0 and Φx − x ∈ K, which implies that
(Φx)(t) − x(t) ≥ λ|Φx − x|1 ≥ 0
for all t ∈ [0, ω].
Moreover, for t ∈ [0, ω], we have
Z t+ω
Z
n
hX
G(t, s)x(s)
(Φx)(t) =
ai (s)
t
i=1
m
X
+
0
Z
0
x(s + θ) dµi (θ)
−Ti
i
x0 (s + θ) dνj (θ) ds
bj (s)
−T̂j
j=1
1
≤
r|x|0
1−λ
Z
ω
(3.1)
(3.2)
n
hX
0
ai (s) +
i=1
m
X
i
bj (s) ds
j=1
1
=
Π|x|0 < λ|x|0 .
1−λ
In view of (3.1) and (3.2), we have
|x|0 ≤ |Φx|0 < λ|x|0 < |x|0 ,
which is a contradiction. Finally, we prove that Φx x for all x ∈ K, |x|1 = R
also holds. For this case, we only need to prove that
x ∈ K, |x|1 = R.
Φx ≮ x
Suppose, for the sake of contradiction, that there exists x ∈ K and |x|1 = R such
that Φx < x. Thus x − Φx ∈ K \ {0}. Furthermore, for any t ∈ [0, ω], we have
x(t) − (Φx)(t) ≥ λ|x − Φx|1 > 0.
In addition, for any t ∈ [0, ω], we find
Z t+ω
Z
n
hX
(Φx)(t) =
G(t, s)x(s)
ai (s)
t
+
i=1
m
X
Z
λ
|x|2
1−λ 0
0
x(s + θ) dµi (θ)
−Ti
i
x0 (s + θ) dνj (θ) ds
−T̂j
j=1
≥
0
bj (s)
Z
ω
(3.3)
(3.4)
n
m
h X
i
X
λ
ai (s) −
bj (s) ds
0
i=1
j=1
2
=
λ
∆R2 = R.
1−λ
From (3.3) and (3.4), we obtain |x| > |Φx|0 ≥ R, which is a contradiction. Therefore, conditions (i) and (ii) hold. By Lemma 2.1, we see that Φ has at least one
nonzero fixed point in K. Therefore, (1.2) has at least one positive ω-periodic
solution. The proof of Theorem 3.1 is complete.
10
Y. LI, G. WANG, H. WANG
EJDE-2007/13
We remark that from the proof of our results, if some (or all) T̂j (j = 1, 2, . . . , n)
are replaced by ∞ the conclusion of Theorem 3.1 remains valid.
As an example of (1.1), consider the equation
Z 0
Z
i
h 1 + cos t
1 − sin t 0 0
0
x(t+θ)dθ−
x (t+θ) dθ . (3.5)
x (t) = x(t)
−(5−2 sin t)
4π
20
−1
−1
Obviously,
1 + cos t
,
4π
Furthermore, we have
a(t) =
a1 (t) = 5 − 2 sin t,
b1 (t) =
1 − sin t
.
20
1
λ = e− 2 , max {λa1 (t) − b1 (t)} > 1.7 > 0,
0≤t≤2π
Z 2π
1
1
∆=
[λa1 (s) − b1 (s)] ds = 10πe− 2 − π > 18,
10
0
71
max {a1 (t) + b1 (t)} =
,
0≤t≤2π
10
λ2
71
(1 + am )
∆ > 16 >
> max {a1 (t) + b1 (t)},
1−λ
10 0≤t≤2π
1−λ M
b < 5.95 × 10−3 < 1.
λ2 ∆ 1
Hence, (H1)–(H3), (H5) hold and aM ≤ 1. According to Theorem 3.1, system (3.5)
has at least one positive 2π-periodic solution.
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Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
E-mail address, Yongkun Li: yklie@ynu.edu.cn
E-mail address, Guoqiao Wang: wgq81@126.com
E-mail address, Huimei Wang: wanghmei@163.com