Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 423416, 16 pages
doi:10.1155/2012/423416
Research Article
Existence of Two Positive Periodic Solutions
for a Neutral Multi-Delay Logarithmic Population
Model with a Periodic Harvesting Rate
Hui Fang
Department of Mathematics, Kunming University of Science and Technology, Yunnan 650500, China
Correspondence should be addressed to Hui Fang, kmustfanghui@hotmail.com
Received 14 July 2012; Accepted 12 November 2012
Academic Editor: Wing-Sum Cheung
Copyright q 2012 Hui Fang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
By using an abstract existence result based on a coincidence degree theory for k-set contractive
mapping, a new result is obtained for the existence of at least two positive periodic solutions for
a neutral multidelay logarithmic population model with a periodic harvesting rate. An example is
given to illustrate the effectiveness of the result.
1. Introduction
In recent years, many papers have been published on the existence of positive periodic
solutions for neutral delay logarithmic population models by using a topological degree
theory for k-set contractive mapping see, e.g., 1–5. Recently, Xia 6 obtained some
new sufficient conditions for the existence and uniqueness of an almost periodic solution
of a multispecies logarithmic population model with feedback controls. However, few
papers deal with the existence of multiple positive periodic solutions for neutral multidelay
logarithmic population models with harvesting. The main difficulty is hard to obtain a priori
bounds on solutions for neutral multi-delay models with harvesting.
In this paper, we consider the following neutral multi-delay logarithmic population
model of single-species population growth with a periodic harvesting rate
⎡
⎤
n
m
dyt
d
yt⎣at − bj t ln y t − σj t − ci t ln yt − τi t⎦ − ht,
dt
dt
j1
i1
1.1
2
Abstract and Applied Analysis
where at, bj t, σj t j 1, 2, . . . , n, ci t, τi t i 1, 2, . . . , m, ht are nonnegative
continuous T -periodic functions, and ht denotes the harvesting rate. When ht ≡ 0, 1.1
was considered by 2–5. When n m 1, ht ≡ 0, and σ1 t, τ1 t are constants, 1.1 was
considered by 7.
The purpose of this paper is to establish the existence of at least two positive periodic
solutions for a neutral multi-delay logarithmic population model 1.1 by using a coincidence
degree theory for k-set contractions. Motivated by the work of Chen 8, some novel
techniques are employed to find a priori bounds on solutions.
2. Preliminaries
We now briefly state the part of the coincidence degree theory for k-set contractive mapping
developed by Hetzer 9, 10. For more details, we refer to 11.
Let Z be a Banach space. For a bounded subset A ⊂ Z, let ΓZ A denote the
Kuratowski measure of noncompactness defined by
ΓZ A inf δ > 0 : ∃ a finite number of subsets Ai ⊂ A, A Ai , diamAi ≤ δ . 2.1
i
Here, diam Ai denotes the maximum distance between the points in the set Ai .
Let X and Z be Banach spaces with norms ·X and ·Z , respectively and Ω a bounded
open subset of X. A continuous and bounded mapping N : Ω → Z is called k-set contractive
if for any bounded A ⊂ Ω, we have
ΓZ NA ≤ kΓX A.
2.2
Also, for a continuous and bounded map T : X → Y , we define
lT sup{r ≥ 0 : ∀ bounded subset A ⊂ X, rΓX A ≤ ΓY T A}.
2.3
Let L : dom L ⊂ X → Z be a linear mapping and N : X → Z be a continuous
mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L codim Im L < ∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero, there
then exist continuous projectors P : X → X and Q : Z → Z such that Im P Ker L, Im L Ker Q ImI − Q. If we define LP : dom L ∩ Ker P → Im L as the restriction L|dom L∩Ker P of
L to dom L ∩ Ker P , then LP is invertible. We denote the inverse of that map by KP . If Ω is an
open bounded subset of X, the mapping N will be called L-k-set contractive on Ω if QNΩ
is bounded and KP I − QN : Ω → X is k-set contractive. Since Im Q is isomorphic to Ker L,
there exists an isomorphism J : Im Q → Ker L.
Lemma 2.1 11, Proposition XI.2.. Let L be a closed Fredholm mapping of index zero and let
N : Ω → Z be k -set contractive with
0 ≤ k < lL.
Then N : Ω → Z is a L-k-set contraction with constant k k /lL < 1.
2.4
Abstract and Applied Analysis
3
The following lemma 11, page 213 will play a key role in this paper.
Lemma 2.2. Let L be a Fredholm mapping of index zero and let N : Ω → Z be L-k-set contractive
on Ω, k < 1. Suppose
i Lx / λNx for every x ∈ dom L ∩ ∂Ω and every λ ∈ 0, 1;
ii QNx /
0 for every x ∈ ∂Ω ∩ Ker L;
0.
iii Brouwer degree degB JQN, Ω ∩ Ker L, 0 /
Then Lx Nx has at least one solution in dom L ∩ Ω.
3. Main Result
Let CT0 denote the linear space of real valued continuous T -periodic functions on R. The linear
space CT0 is a Banach space with the usual norm for x ∈ CT0 given by |x|0 maxt∈R |xt|. Let
CT1 denote the linear space of T -periodic functions with the first-order continuous derivative.
CT1 is a Banach space with norm |x|1 max{|x|0 , |x |0 }.
Let X CT1 and Z CT0 and let L : X → Z be given by Lx dx/dt. Since |Lx|0 |x |0 ≤ |x|1 , we see that L is a bounded with bound 1 linear map.
Under the transformation of yt ext , 1.1 can be rewritten as
x t at −
n
m
ht
bj tx t − σj t − ci t 1 − τi t x t − τi t − xt .
e
j1
i1
3.1
Next define a nonlinear map N : X → Z by
Nxt at −
n
m
ht
bj tx t − σj t − ci t 1 − τi t x t − τi t − xt .
e
j1
i1
3.2
Now, if Lx Nx for some x ∈ X, then the problem 3.1 has a T -periodic solution xt.
In the following, we denote
1
g
T
T
gtdt,
0
g l min gt,
t∈0,T g u max gt,
t∈0,T 3.3
where gt is a continuous nonnegative T -periodic solution.
From now on, we always assume that
H1 at, bj t ∈ CR, 0, ∞, σj t, ci t, τi t ∈ C1 R, 0, ∞, for all j ∈ {1, 2, . . . ,
n}, for all i ∈ {1, 2, . . . , m}.
H2 σj t < 1, τi t < 1, for all t ∈ R, and
n
m
bj μj t
ci γi t
Γt −
> 0,
j1 1 − σj μj t
i1 1 − τi γi t
3.4
4
Abstract and Applied Analysis
where μj t is the inverse function of t−σj t, γi t is the inverse function of t−τi t, for all j ∈
{1, 2, . . . , n}, for all i ∈ {1, 2, . . . , m}.
H3 Let
hl
au
hl
<
≤
< 1.
Γu
Γu
Γl
1 ln
3.5
H4 Let
⎛
⎞
n
m
T ⎝Γ bj ⎠ ciu < 1,
j1
i1
⎧
⎨
⎫
⎬ al − m cu 1 − τ u M0
hu
i1 i
max 1 ln n l , 0 <
< R0 ,
n u i
⎩
⎭
b
j1 bj
j1
3.6
j
where
n
bju R0 hu eR0
u , R0 q p,
1 − i1 ciu 1 − τi
hl 2T a Γ nj1 bj T q
q 2ln u , p .
Γ u
1 − T Γ nj1 bj − m
i1 ci
M0 au j1
m
3.7
We first give some technological lemmas.
Set
fx d − x − re−x ,
x ∈ −∞, ∞.
3.8
Lemma 3.1. Assume that d, r are positive constants such that
1 ln r < d.
3.9
Then there exist x∗− , x∗ such that
f x∗− fx∗ 0,
x∗− < ln r < x∗ < d,
fx > 0 for x ∈ x∗− , x∗ ;
−
3.10
fx < 0 for x ∈ −∞, x∗ ∪ x∗ , ∞.
If one assumes further that
d ≤ r < 1,
3.11
Abstract and Applied Analysis
5
then the following inequalities also hold:
2 ln r < x∗− < ln r < x∗ ≤ 0.
3.12
Proof. Clearly, f x r/ex − 1 0 if and only if x ln r. Therefore, noticing that
limx → ±∞ fx −∞, we have
sup
fx d − ln r − 1 > 0.
x∈−∞,∞
3.13
Set
gr 2 ln r 1
− r.
r
3.14
Since
g r 2 1
1 − r2
− 2 −1−
<0
r r
r2
for 0 < r < 1,
3.15
gr is monotonically decreasing on 0, 1.
Therefore, we have
gr > g1 0,
3.16
that is,
2 ln r 1
> r,
r
3.17
which implies
f2 ln r d − 2 ln r −
1
< d − r ≤ 0.
r
3.18
Again, noticing that
f0 d − r ≤ 0,
3.19
by the monotonicity of the function fx on the interval −∞, ln r and ln r, ∞, it is easy to
see that the assertion holds.
6
Abstract and Applied Analysis
Set
Fx al −
u
u
M0
i1 ci 1 − τi
n u
j1 bj
m
a
Gx n
l
j1 bj
e−x ,
e−x ,
bj
u
u
M0
au m
hl
i1 ci 1 − τi
Hx − x − n u e−x .
n l
j1 bj
j1 bj
j1
bj
h
− x − n
hu
− x − n
3.20
j1
Lemma 3.2. Assume that H1 , H2 , and H4 hold. Then the following assertions hold.
(1) There exist u− , u such that
F u− Fu 0,
u
u
M0
al − m
hu
i1 ci 1 − τi
−
,
u < ln n l < u <
n u
b
j1 j
j1 bj
Fx > 0 for x ∈ u− , u ;
3.21
Fx < 0 for x ∈ −∞, u− ∪ u , ∞.
(2) There exist x− , x such that
G x− Gx 0,
h
x− < ln n
j1
Gx > 0 for x ∈ x− , x ;
bj
a
< x < n
j1
bj
,
3.22
Gx < 0 for x ∈ −∞, x− ∪ x , ∞.
(3) There exist l− , l such that
H l− Hl 0,
u
u
M0
au m
hl
i1 ci 1 − τi
−
,
l < ln n u < l <
n l
j1 bj
j1 bj
Hx > 0 for x ∈ l− , l ;
3.23
Hx < 0 for x ∈ −∞, l− ∪ l , ∞.
(4)
l − < x − < u − < u < x < l .
3.24
Abstract and Applied Analysis
7
Proof. Noticing that
al −
u
u
M0
i1 ci 1 − τi
n u
j1 bj
m
hl
n
j1
a
< n
<
bj
j1
u
u
M0
i1 ci 1 − τi
,
n l
j1 bj
m
hu
h
bju
au ≤ n
j1
bj
≤ n
j1
bjl
3.25
,
we have
3.26
Fx < Gx < Hx.
It follows from H1 , H2 , and H4 , 3.25 that
hu
1 ln n
j1
bjl
<
al −
h
1 ln n
j1
hl
1 ln n
j1
bju
<
u
u
M0
i1 ci 1 − τi
,
n u
j1 bj
m
bj
au a
< n
j1
bj
3.27
,
u
u
M0
i1 ci 1 − τi
.
n l
j1 bj
m
Therefore, by Lemma 3.1, the assertions 1–3 hold. Furthermore, by 3.26 and the assertions 1–3, the assertion 4 also holds.
Lemma 3.3 see 12. L is a Fredholm map of index 0 and satisfies
lL ≥ 1.
3.28
Lemma 3.4 see 2. Suppose σ ∈ CT1 and σ t < 1, for all t ∈ 0, T . Then the function t − σt
has an inverse function μt satisfying μ ∈ CR, R with μa T μa T .
Lemma 3.5. Assume that (H1 )–(H4 ) hold. Let k0 m
u
i1 ci 1
− τi u , and
⎧
⎪
⎪
⎪
⎨
max xt ∈ l− − δ, R0 δ,
t∈0,T min xt ∈ l− − δ, l δ,
Ω x ∈ X t∈0,T
⎪
⎪
max |x t| < M .
⎪
⎩
0
t∈0,T ⎫
⎪
⎪
⎪
⎬
,
⎪
⎪
⎪
⎭
where 0 < δ < l− . Then N : Ω → Z is a k0 -set-contractive map.
Proof. The proof is similar to that of Lemma 3.3 in 12, so we omit it.
3.29
8
Abstract and Applied Analysis
Theorem 3.6. Assume that (H1 )–(H4 ) hold. Then 1.1 has at least two positive T -periodic solutions.
Proof. Let Lx λNx for x ∈ X, that is,
⎤
m
ht
x t λ⎣at − bj tx t − σj t − ci t 1 − τi t x t − τi t − xt ⎦,
e
j1
i1
⎡
n
λ ∈ 0, 1.
3.30
Therefore, we have
⎡
⎤
n
m
ht
x t λ⎣at − bj tx t − σj t − ci txt − τi t − xt ⎦,
e
j1
i1
λ ∈ 0, 1.
3.31
By 3.31, we have
⎤
⎡
m
n
m
ht
xt λ ci txt − τi t λ⎣at − bj tx t − σj t ci txt − τi t − xt ⎦.
e
i1
j1
i1
3.32
Integrating this identity leads to
⎡
⎤
T T
n
m
ht
⎣ bj tx t − σj t − ci txt − τi t ⎦dt atdt.
ext
0
0
j1
i1
3.33
By Lemma 3.4, we have
μj T s T μj s,
γi T s T γi s,
3.34
where t μj s is the inverse function of s t − σj t, and t γi s is the inverse function of
s t − τi t, for all j ∈ {1, 2, . . . , n} and for all i ∈ {1, 2, . . . , m}.
Then
T n
n T −σj T bj tx t − σj t dt bj μj s
0 j1
j1
n T
j1
T m
0 i1
−σj 0
0
bj μj s
xsds,
1 − σj μj s
m T −τi T ci txt − τi tdt ci γi s
i1
−τi 0
m T
i1
0
xs
ds
1 − σj μj s
xs
ds
1 − τi γi s
ci γi s
xsds.
1 − τi γi s
3.35
Abstract and Applied Analysis
9
From 3.33–3.35, we have
⎤
⎞
n
m
bj μj s
ci γi s
hs
⎣as − ⎝
−
⎠xs − xs ⎦ds 0,
e
0
j1 1 − σj μj s
i1 1 − τi γi s
T
⎡
⎛
3.36
which implies
h η
a η − Γ η x η − xη 0,
e
3.37
for some η ∈ 0, T .
Therefore, by H2 , we have
hl /Γu
au
−
x
η − xη ≥ 0.
Γl
e
3.38
By H3 and Lemma 3.1, we have
2 ln
hl
< x η ≤ 0.
u
Γ
3.39
Set
hl q : 2ln u ,
Γ 3.40
then we have
|xt| ≤ x η T
x tdt ≤ q 0
T
x tdt,
3.41
0
which implies
|x|0 ≤ q T
0
x tdt.
3.42
10
Abstract and Applied Analysis
It follows from 3.30 that
T
x tdt
0
T
n
m
ht
λ at − bj tx t − σj t − ci t 1 − τi t x t − τi t − xt dt
3.43
e 0
j1
i1
T
T
n
m
ht
≤ λ at − bj tx t − σj t − ci t 1 − τi t x t − τi tdt dt.
xt
0
0 e
j1
i1
By 3.36 and H2 , we have
T
0
ht
dt ≤
ext
T
atdt T
0
0
3.44
Γtdt|x|0 T a T Γ|x|0 .
By this and 3.43, we obtain
T
0
⎡
⎤
n
m T
x tdt ≤ T ⎣2a Γ|x| bj |x| ⎦ ci t 1 − τ t x t − τi tdt.
0
0
i
j1
3.45
0
i1
Meanwhile,
T
m T
m T
m
u
ci t 1 − τ t x t − τi tdt x sds.
ci γi s x s ds ≤
c
i1
i
0
i
0
i1
i1
3.46
0
Substituting 3.42 and 3.46 into 3.45 gives
T
0
x tdt
⎡
⎤
T
n
m
u
x sds
⎣
⎦
≤ T 2a Γ|x|0 bj |x|0 ci
j1
⎡
⎛
⎡
⎛
i1
0
⎞
⎤
T
T
n
m
≤ T ⎣2a ⎝Γ bj ⎠ q x tdt ⎦ ciu x sds
0
j1
≤ T ⎣2a ⎝Γ ⎞ ⎤
i1
⎡ ⎛
⎞
0
⎤
T
n
n
m
bj ⎠q⎦ ⎣T ⎝Γ bj ⎠ ciu ⎦ x tdt.
j1
j1
i1
0
3.47
Abstract and Applied Analysis
11
Since
⎞
n
m
T ⎝Γ bj ⎠ ciu < 1,
⎛
j1
3.48
i1
we have
T
x tdt <
0
2T a Γ nj1 bj T q
: p.
u
1 − T Γ nj1 bj − m
i1 ci
3.49
Then,
|x|0 ≤ q T
x tdt ≤ q p : R0 .
3.50
0
Again from 3.30, we get
n
m
x < au bu |x| cu 1 − τ u x hu eR0 .
0
i
j
i
0
0
j1
Since
m
u
i1 ci 1
3.51
i1
− τi u < 1, we have
au nj1 bju R0 hu eR0
x <
: M0 .
0
u
u
1− m
i1 ci 1 − τi
3.52
Choose tM , tm ∈ 0, T , such that
xtM max xt,
t∈0,T xtm min xt.
3.53
x tm 0.
3.54
t∈0,T Then, it is clear that
x tM 0,
From this and 3.30, we obtain that
atM n
m
htM bj tM x tM − σj tM ci tM 1 − τi tM x tM − τi tM xt ,
e M
j1
i1
3.55
atm n
m
htm bj tm x tm − σj tm ci tm 1 − τi tm x tm − τi tm xt .
e m
j1
i1
3.56
12
Abstract and Applied Analysis
It follows from 3.55 that
atM −
m
n
htM ci tM 1 − τi tM M0 − bj tM xtM − xt ≤ 0,
e M
i1
j1
3.57
which implies
al −
u
u
M0
i1 ci 1 − τi
n u
j1 bj
m
hu
− xtM − n
l
j1 bj
e−xtM ≤ 0.
3.58
By the assertion 1 of Lemma 3.2, we have
xtM ≤ u−
or xtM ≥ u .
3.59
It follows from 3.56 that
atm m
n
htm ci tm 1 − τi tm M0 − bj tm xtm − xt ≥ 0,
e m
i1
j1
3.60
which implies
au u
u
M0
i1 ci 1 − τi
n l
j1 bj
m
hl
− xtm − n
u
j1 bj
e−xtm ≥ 0.
3.61
By the assertion 3 of Lemma 3.2, we have
l− ≤ xtm ≤ l .
3.62
Hence, it follows from 3.50, 3.59, and 3.62 that
!
xtM ∈ l− , u− ∪ u , R0 ,
!
xtm ∈ l− , l .
3.63
Clearly, l± , u± are independent of λ. Now, let us consider QNx with x ∈ R. Note that
QNx a −
n
h
bj x − x .
e
j1
3.64
It follows from the assertion 2 of Lemma 3.2 that QNx 0 has two distinct solutions:
u
" 1 x− ,
u
" 2 x .
3.65
Abstract and Applied Analysis
13
By the assertion 4 of Lemma 3.2, one can take v− , v > 0 such that
u− < v− < v < u .
3.66
Let
⎧
⎪
⎪
⎪
⎨
max xt ∈ l− − δ, v− ,
t∈0,T min xt ∈ l− − δ, l δ,
Ω1 x ∈ X t∈0,T
⎪
⎪
max |x t| < M .
⎪
⎩
0
t∈0,T ⎧
⎪
⎪
⎪
⎨
max xt ∈ v , R0 δ,
t∈0,T min xt ∈ l− − δ, l δ,
Ω2 x ∈ X t∈0,T
⎪
⎪
max |x t| < M .
⎪
⎩
0
t∈0,T ⎫
⎪
⎪
⎪
⎬
,
⎪
⎪
⎪
⎭
3.67
⎫
⎪
⎪
⎪
⎬
.
⎪
⎪
⎪
⎭
Then Ω1 , Ω2 are bounded open subsets of X. Clearly, Ωi ⊂ Ω i 1, 2, where Ω as defined in
Lemma 3.5. It follows from Lemma 3.5 that N : Ωi → Z is a k0 -set-contractive map i 1, 2.
Therefore, it follows from Lemmas 2.1 and 3.3 that N : Ωi → Z is L-k-set contractive on
Ωi i 1, 2 with k k0 /lL ≤ k0 < 1.
By the assertion 4 of Lemma 3.2, 3.52, 3.63, 3.65, and 3.66, it is easy to verify
that Ωi satisfies the assumptions i and ii in Lemma 2.2 i 1, 2. By the assertion 2 of
Lemma 3.2, a direct computation gives
⎛
⎞
h
degB {JQN, Ω1 ∩ Ker L, 0} sgn⎝− bj x− ⎠ 1,
e
j1
n
⎞
n
h
degB {JQN, Ω2 ∩ Ker L, 0} sgn⎝− bj x ⎠ −1.
e
j1
⎛
3.68
Here, J is taken as the identity mapping since Im Q Ker L. So far we have proved that Ωi
satisfies all the assumptions in Lemma 2.2 i 1, 2. Hence, 3.1 has at least two T -periodic
solutions xi∗ t and xi∗ ∈ dom L ∩ Ωi i 1, 2. Since Ω1 ∩ Ω2 ∅, xi∗ i 1, 2 are different. Let
∗
yi∗ t exi t i 1, 2. Then yi∗ t i 1, 2 are two different positive T -periodic solutions of
1.1. The proof is complete.
Example 3.7. Consider the following equation:
#
$
dyt
d
yt at − bt ln yt − σt − ct ln yt − τt − ht,
dt
dt
3.69
14
Abstract and Applied Analysis
where
%
at &
1
1
sin2 t ,
4.2e 4.2e
ct 0.9 0.1 cos t,
bt 2.35 0.15 sin t,
σt ≡ 2π,
τt ≡ 4π,
3
sin t,
h
2e 2e
3.70
and the constant > 0. Clearly, H1 is satisfied.
Let μt be the inverse function of t − σt, and γt be the inverse function of t − τt.
Then we have
μt t 2π,
γt t 4π,
c γt
b μt
−
Γt bt 2π − c t 4π 2.35 0.25 sin t.
1 − σ μt
1 − τ γt
3.71
Hence, H2 is satisfied.
It is easy to see that
T 2π,
au ,
2.1e
Γ 2.6,
u
al Γ 2.1,
l
,
4.2e
a
,
2.8e
Γ 2.35,
bu 2.5,
c ,
u
2
,
h e
u
bl 2.2,
h ,
e
l
b 2.35,
3
.
h
2e
3.72
Therefore, we have
1
hl
,
u
Γ
2.6e
au
1
,
l
4.41e
Γ
hl
1 ln u − ln 2.6.
Γ
3.73
So,
1 ln
Hence, H3 is satisfied.
hl
au
hl
<
<
< 1.
Γu
Γu
Γl
3.74
Abstract and Applied Analysis
15
Also, it is easy to see that
al − cu 1 − τ u M0 /4.2e − M0
1
M0
−
< 1,
bu
2.5
10.5e 2.5
hl 2T a Γ b T q
π/0.7e 18.8ln 2.6 1π
p
,
q 2ln u 2ln 2.6 1 > 1,
Γ 1 − 9.4π − 1 − T Γ b − cu
R0 q p,
M0 au bu R0 hu eR0
/2.1e 2.5R0 2eR0 −1
.
u
1−
1 − cu 1 − τ 3.75
Therefore, we obtain that
R0 > q >
al − cu 1 − τ u M0
.
bu
3.76
Noticing that lim → 0 R0 q, we have lim → 0 M0 0. Therefore, for some sufficiently small
> 0, the following inequalities hold:
'
(
hu
al − cu 1 − τ u M0
max 1 ln l , 0 max{− ln 1.1, 0} 0 <
,
bu
b
T Γ b cu 9.4π < 1.
3.77
By 3.76-3.77, H4 is also satisfied. Therefore, all necessary conditions of Theorem 3.6 are
satisfied. By Theorem 3.6, 3.69 has at least two positive 2π-periodic solutions.
Acknowledgment
This paper is supported by the National Natural Science Foundation of China Grant nos.
10971085 and 11061016.
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