Positive periodic solution for a neutral Logarithmic population model with feedback control

Applied Mathematics and Computation 217 (2011) 7692–7702

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Positive periodic solution for a neutral Logarithmic population modelwith feedback control q

Rui Wang ⇑, Xiaosheng ZhangSchool of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

a r t i c l e i n f o

Keywords:Neutral Logarithmic population modelFeedback controlPositive periodic solutionGlobal asymptotically stability

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.02.072

q Supported by the Natural Science Foundation o⇑ Corresponding author.

E-mail address: [email protected] (R. Wan

a b s t r a c t

In this paper, a neutral delay Logarithmic population model with feedback control is stud-ied. By using the abstract continuous theorem of k-set contractive operator, some newresults on the existence of the positive periodic solution are obtained; after that, by con-structing a suitable Lyapunov functional, a set of easily applicable criteria is establishedfor the global asymptotically stability of the positive periodic solution.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In the past few years, many authors have studied the existence of the Logarithmic population model.Gopalsamy [3] and Kirlinger [5] had proposed the following single species Logarithmic model:

dNðtÞdt¼ NðtÞ½a� b � ln NðtÞ � c � ln Nðt � sÞ�: ð1:1Þ

System (1.1) is then generalized by Li [8] to the non-autonomous case

dNðtÞdt¼ NðtÞ½aðtÞ � bðtÞ ln NðtÞ � cðtÞ ln Nðt � sðtÞÞ�: ð1:2Þ

In [8], by using the coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of sys-tem (1.2) are established.

As was pointed out by Gopalsamy [3], in some case, the neutral delay population models are more realistic. There weremany scholars done works on the periodic solution of neutral type Logistic model or Lotka–Volterra model, but only a littlescholars considered the neutral Logarithmic model. Li [9] had studied the following single species neutral Logarithmicmodel:

dNðtÞdt¼ NðtÞ½rðtÞ � aðtÞ ln Nðt � rÞ � bðtÞðln Nðt � gÞÞ0�: ð1:3Þ

Lu and Ge [6] investigated the following system:

dNðtÞdt¼ NðtÞ rðtÞ �

Xn

j¼1

ajðtÞ ln Njðt � rðtÞÞ �Xn

j¼1

bjðtÞðln Nðt � gjðtÞÞÞ0

" #: ð1:4Þ

. All rights reserved.

f China (10871005) and the Foundation of Beijing City’s Educational Committee (KM200610028001).

g).

http://dx.doi.org/10.1016/j.amc.2011.02.072

mailto:[email protected]

http://dx.doi.org/10.1016/j.amc.2011.02.072

http://www.sciencedirect.com/science/journal/00963003

http://www.elsevier.com/locate/amc

R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702 7693

Some results on the existence of positive periodic solution of system (1.4) are obtained by employing an abstract continuoustheorem of k-set contractive operator.

The authors [13], by using an abstract continuous theorem of k-set contractive operator, investigated the existence, glob-ally attractive of the positive periodic solution of the following neutral multi-delays Logarithmic population model:

dNðtÞdt¼ NðtÞ rðtÞ � aðtÞ ln NðtÞ �

Xn

j¼1

bjðtÞ ln Nðt � sjðtÞÞ �Xn

j¼1

cjðtÞZ t

�1kjðt � sÞ ln NðsÞds�

Xn

j¼1

djðtÞðln Nðt � gjðtÞÞÞ0

" #:

ð1:5Þ
For the neutral multi-species Logarithmic population model, we refer the reader to [1].On the other hand, in some situation, people may wish to change the position of the existing periodic solution but to keep
its stability. This is of significance in the control of ecology balance. One of the methods to realize such a control is to alter thesystem structurally by introducing some feedback control variables so as to get the population stability of another periodicsolution. The realization of the feedback control mechanism might be implemented by means of some biological controlscheme or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics withfeedback control have been studied extensively. Recently, many scholars focus on the existence and global attractivity of thepositive periodic solution of the system. The authors [12] have studied the existence, uniqueness and global attractivity ofthe periodic solution of the delay multispecies Logarithmic population model with feedback control, by using the contractionmapping principle and a suitable Lyapunov functional. By means of the Cauchy matrix, the authors [14] investigate the exis-tence and exponential stability of almost periodic solutions and periodic solutions for the delay impulsive Logarithmic pop-ulation. The impulses can be considered as a kind of control.

However, to this day, there are few papers published on the existence and global asymptotically stability of positive peri-odic solution of the neutral Logarithmic population model with feedback control. This motivates us to consider the followingequation:

dNðtÞdt ¼ NðtÞ rðtÞ � aðtÞ ln NðtÞ �

Pnj¼1

bjðtÞ ln Nðt � s1jðtÞÞ �Pnj¼1

cjðtÞR t�1 kjðt � sÞ ln NðsÞds

(

�Pnj¼1

djðtÞðln Nðt � g1jðtÞÞÞ0 �Pnj¼1

fjðtÞuðt � s2jðtÞÞ)

;

duðtÞdt ¼ �a0ðtÞuðtÞ þ

Pnj¼1

gjðtÞNðt � g2jðtÞÞ;

8>>>>>>>>>><>>>>>>>>>>:

ð1:6Þ

where u(t) denotes the feedback control variable. It is assumed that:

(H1) r(t), cj(t), fj(t), a0(t), gj(t) are continuous, positive x-periodic functions on R a(t), bj(t), dj(t) are continuous differentiable,positive x-periodic functions on R.

(H2) s1j(t), g1j(t) 2 C(R, [0,+1)), g2j(t), s2j(t) 2 C(R, (0,+1)) are all x-periodic functions, and kj(t) 2 C([0,+1), (0,+1))j = 1, . . . ,n with 1� s01jðtÞ > 0; 1� g01jðtÞP 0; 1� s02jðtÞ > 0; 1� g02jðtÞ > 0 and

Rþ10 kjðsÞds ¼ 1;

Rþ10 skjðsÞds < þ1.

We consider (1.6) together with following initial conditions:

NðtÞ ¼ uðtÞ; N0ðtÞ ¼ u0ðtÞ; uð0Þ > 0; u 2 C1ðð�1;0�; ½0;1ÞÞ;uðhÞ ¼ wðhÞP 0; h 2 ½�s2;0�; wð0Þ > 0; where s2 ¼ max

t2½0;x�;j¼1;...;nfs2jðtÞg:

8<: ð1:7Þ

The aim of this paper is to give a set of new conditions to guarantee the existence and global stability of the positive peri-odic solution of the system (1.6) and (1.7).

2. Main lemmas

We will investigate the existence of positive periodic solution of system (1.6) and (1.7) in this section, and to do this, somelemmas are needed.

Since each x-periodic solution of the equation

duðtÞdt¼ �a0ðtÞuðtÞ þ

Xn

j¼1

gjðtÞNðt � g2jðtÞÞ

is equivalent to that of the equation

uðtÞ ¼Z tþw

tGðt; sÞ

Xn

j¼1

gjðsÞNðs� g2jðsÞÞ !

ds :¼ ðUNÞðtÞ; ð2:1ÞR

where, Gðt; sÞ ¼exp

s

ta0ðhÞdh

expR x

0a0ðhÞdh�1

; s 2 ½t; t þx�; t 2 R.

7694 R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702

Therefore, the existence problem of positive x-periodic solution of system (1.6) and (1.7) is equivalent to that of positivex-periodic solution of the equation:

dNðtÞdt¼ NðtÞ rðtÞ � aðtÞ ln NðtÞ �

Xn

j¼1

bjðtÞ ln Nðt � s1jðtÞÞ �Xn

j¼1

cjðtÞZ t

�1kjðt � sÞ ln NðsÞds�

Xn

j¼1

djðtÞðln Nðt � g1jðtÞÞÞ0

(

�Xn

j¼1

fjðtÞZ t�s2jðtÞþw

t�s2jðtÞG t � s2jðtÞ; s� � Xn

j¼1

gjðsÞNðs� g2jðsÞÞ !

ds

): ð2:2Þ

Taking the transformation N(t) = ey(t), then (2.2) can be rewritten as

y0ðtÞ ¼ rðtÞ � aðtÞyðtÞ �Xn

j¼1

bjðtÞyðt � s1jðtÞÞ �Xn

j¼1

cjðtÞZ t

�1kjðt � sÞyðsÞds�

Xn

j¼1

djðtÞy0ðt � g1jðtÞÞð1� g01jðtÞÞ

�Xn

j¼1



j¼1

gjðsÞeyðs�g2jðsÞÞ

!ds: ð2:3Þ

It is obvious that if Eq. (2.3) has a x-periodic solution y⁄(t), then Eq. (2.2) has a positive x-periodic solution N�ðtÞ ¼ ey�ðtÞ.Let E be a Banach space. For a bounded subset A � E, denote the Kuratoskii measure of non-compactness:

aEðAÞ ¼ inffd > 0jThere is a finite number of subsets fAig � A such that A ¼ [ðAiÞ and diamðAiÞ 6 dg;

where diam(Ai) denotes the diameter of set Ai. Let X, Y be two Banach spaces and X be a bounded open subset of X. A con-tinuous and bounded map N : X! Y is called k-set contractive if for any bounded set A � X, we have

aYðNðAÞÞ 6 kaXðAÞ;

where k is a non-negative constant.Also, for a Fred-holm L : X ? Y with index zero, according to [4], we define:

lðLÞ ¼ supfr P 0 : raXðAÞ 6 aYðLðAÞÞ; for all bounded subset A � Xg:

Lemma 2.1 [11]. Let L : X ? Y be a Fred-holm operator with zero index, and r 2 Y be a fixed point. Suppose that N : X ? Y iscalled a k-set contractive with k < l(L), where X � X is bounded, open and symmetric about 0 2X. Further, we also assume that:

(1) Lx – kNx + kr, for x 2 oX, k 2 (0,1), and(2) [QN(x) + Qr,x] � [QN(�x) + Qr,x] < 0 for x 2 kerL \ oX; where [�, �] is a bilinear form on Y � X and Q is the projection of Y

onto Coker(L), where Coker(L) is the co-kernel of the operator L.
Then there is a x 2 X such that Lx � Nx = r.In order to use Lemma 2.1 to study (2.3), we set
Y ¼ Cx ¼ fxjx 2 CðR;RnÞ; xðt þxÞ ¼ xðtÞg

with the norm defined by kxk = jxj0 = maxt2[0,x]{jx(t)j}, and

X ¼ C1x ¼ fxjx 2 C1ðR;RnÞ; xðt þxÞ ¼ xðtÞg

with the norm defined by jxj1 = max{jxj0, jx0 j0}.Then both Cx and C1

x are Banach spaces. We also denote:

�h ¼ 1x

Z x

0jhðsÞjds; ðhÞm ¼ min

t2½0;x�hðtÞ; ðhÞM ¼ max

t2½0;x�hðtÞ:

Let L : C1x ! Cx defined by Ly = y0(t) and N : C1

x ! Cx defined by

Ny ¼ �aðtÞyðtÞ �Xn

j¼1

bjðtÞyðt � s1jðtÞÞ �Xn

j¼1

cjðtÞZ t

�1kjðt � sÞyðsÞds�

Xn

j¼1

djðtÞy0ðt � g1jðtÞÞð1� g01jðtÞÞ

�Xn

j¼1



j¼1

gjðsÞeyðs�g2jðsÞÞ

!ds: ð2:4Þ

Thus (2.3) has a positive x-periodic solution if and only if Ly = Ny + r for some y 2 C1x, where r = r(t).

Lemma 2.2 [10]. The differential operator L is a Fred-holm operator with index zero, and satisfies l(L) P 1.


Lemma 2.3. Let c0, c1 be two positive constants, and X ¼ fxjx 2 C1x; jxj0 < c0; jx0j0 < c1g, if k ¼

Pnj¼1kdjð1� g01jÞk, then N :

X ? Cx is a k-set contractive map.

Proof. Let A � X be a bounded subset and let g ¼ aC1xðAÞ. Then, for any e > 0, there is a finite family of subsets of Ai satisfying

A = [(Ai) with diam (Ai) 6 g + e.Now we define:

ðHyÞðtÞ ¼ ðUeyÞðtÞ;

Fðt; x; y1; . . . ; yn; z1; . . . ; zn;w1; . . . ;wn;v1; . . . ;vnÞ ¼ aðtÞxþXn

j¼1

bjðtÞyj þXn

j¼1

cjðtÞzj þXn

j¼1

djðtÞwjð1� g01jðtÞÞ þXn

j¼1

fjðtÞv j:

For convenience, in the following discussion we denote

JiðxÞðtÞ ¼Z t

�1kiðt � sÞxðsÞds; for x 2 C1

x; i ¼ 1;2; . . . ;n:

Since F(t,x,y1, . . . ,yn,z1, . . . ,zn,w1, . . . ,wn,v1, . . . ,vn) is uniformly continuous on any compact subset of R � R4n+1, A and Ai areprecompact in Cx, it follows that there is a finite family of subsets Aij of Ai such that Ai = [jAij with

jFðt; xðtÞ; xðt � s11ðtÞÞ; . . . ; xðt � s1nðtÞÞ; J1ðxÞðtÞ; . . . ; JnðxÞðtÞ;u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ; ðHxÞðt � s21ðtÞÞ; . . . ;

ðHxÞðt � s2nðtÞÞÞ � Fðt;uðtÞ; uðt � s11ðtÞÞ; . . . ;uðt � s1nðtÞÞ; J1ðuÞðtÞ; . . . ; JnðuÞðtÞ;u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ;ðHuÞðt � s21ðtÞÞ; . . . ; ðHuÞðt � s2nðtÞÞÞj 6 e; for any x;u 2 Aij:

Therefore, we have

kNx� Nuk ¼ supt2½0;x�

jFðt; xðtÞ; xðt � s11ðtÞÞ; . . . ; xðt � s1nðtÞÞ; J1ðxÞðtÞ; . . . ; JnðxÞðtÞ;

x0ðt � g11ðtÞÞ; . . . ; x0ðt � g1nðtÞÞ; ðHxÞðt � s21ðtÞÞ; . . . ; ðHxÞðt � s2nðtÞÞÞ

� Fðt;uðtÞ;uðt � s11ðtÞÞ; . . . ; uðt � s1nðtÞÞ; J1ðuÞðtÞ; . . . ; JnðuÞðtÞ;

u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ; ðHuÞðt � s21ðtÞÞ; . . . ; ðHuÞðt � s2nðtÞÞÞj

6 supt2½0;x�


x0ðt � g11ðtÞÞ; . . . ; x0ðt � g1nðtÞÞ; ðHxÞðt � s21ðtÞÞ; . . . ; ðHxÞðt � s2nðtÞÞÞ

� Fðt; xðtÞ; xðt � s11ðtÞÞ; . . . ; xðt � s1nðtÞÞ; J1ðxÞðtÞ; . . . ; JnðxÞðtÞ;

u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ; ðHxÞðt � s21ðtÞÞ; . . . ; ðHxÞðt � s2nðtÞÞÞj

þ supt2½0;x�


u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ; ðHxÞðt � s21ðtÞÞ; . . . ; ðHxÞðt � s2nðtÞÞÞ

� Fðt;uðtÞ;uðt � s11ðtÞÞ; . . . ; uðt � s1nðtÞÞ; J1ðuÞðtÞ; . . . ; JnðuÞðtÞ;

u0ðt � g11ðtÞÞ; . . . ;u0ðt � g1nðtÞÞ; ðHuÞðt � s21ðtÞÞ; . . . ; ðHuÞðt � s2nðtÞÞÞj

6

Xn

j¼1

kdjð1� g01jÞk � jx0ðt � g1jðtÞÞ � u0ðt � g1jðtÞÞj þ e 6 kðgþ eÞ þ e:

As e is arbitrarily small, it is easy to get that aCx ðNðAÞÞ 6 kaC1xðAÞ. The proof is complete. h

Lemma 2.4 [7]. Suppose s 2 C1x and s0(t) < 1, t 2 [0,x]. Then the function t � s(t) has a unique inverse l(t) satisfying l 2 C(R,R)

with l(a + x) = l(a) + x, "a 2 R. And if g(�) 2 Cx then g(l(t)) 2 Cx.

Lemma 2.5 [7]. Let 0 6 a 6x be a constant, s(t) 2 Cx, such that maxt2[0,x]js(t)j 6 a. Then for 8x 2 C1x, we haveRx

0 jxðtÞ � xðt � sðtÞÞj2dt 6 2a2Rx

0 jx0ðtÞj2dt. If, in addition, sðtÞ 2 C1

x, and s0(t) < 1, t 2 R, then for x 2 C1x, we have the following

conclusions:

(1) There exists a unique integer m such that l = js(t) �mxj0 < x;(2)

Rx0 jxðtÞ � xðt � sðtÞÞj2dt 6 2l2 Rx

0 jx0ðtÞj2dt.

As s01jðtÞÞ < 1, t 2 [0,x], from Lemma 2.5, we can choose an integer mj(s1j(t)), j = 1, . . . ,n. Such that lj = js1j �mjxj0 and
Z x
0jxðtÞ � xðt � s1jðtÞÞj2dt 6 2l2

j

Z x

0jx0ðtÞj2dt; x 2 C1

x: ð2:5Þ


Lemma 2.6 [15] . Suppose x(t) is a differently continuous x-periodic function on R with x > 0. Then to any t� 2 R;maxt�6t6ðt�þxÞjxðtÞj 6 jxðt�Þj þ 1

2

Rx0 jx0ðtÞjdt.

3. Existence of periodic solution

Since s01jðtÞ < 1; g01jðtÞ < 1, t 2 [0,x] we see that t � s1j(t), t � g1j(t) all have its inverse function. In the rest of this paper,we set l1j(t), c1j(t) represent the inverse function of t � s1j(t), t � g1j(t), respectively.

For convenience, throughout this paper, we also use the notations:

C1ðtÞ ¼ aðtÞ þXn

j¼1

bjðl1jðtÞÞ1� s01jðl1jðtÞÞ

�Xn

j¼1

d0jðc1jðtÞÞ1� g01jðc1jðtÞÞ

;

CðtÞ ¼ C1ðtÞ þXn

j¼1

cjðtÞ; H ¼ ln�r

Cþ A

��;

A ¼ 1x

Z x

0

Xn

j¼1

fjðtÞZ t�s2jðtÞþx

t�s2jðtÞGðt � s2jðtÞ; sÞ

Xn

j¼1

gjðsÞ !

dsdt;

I ¼ maxt2½0;x�

Z t�s2jðtÞþx


Xn

j¼1

gjðsÞ !

ds:

Theorem 3.1. If in addition to (H1)–(H2) assume further that:(H3):

�r > 0; cjðtÞP 0; C1ðtÞ > 0; t 2 ½0;x�; and �aþXn

j¼1

�bj þXn

j¼1

�cj > 0;

(H4):

�r � ACþ A

6 lnr

Cþ A; ln

�r þ CCþ A

P�r

Cþ A;

(H5):

B , 212Xn

j¼1

kbjklj þXn

j¼1

k1� g01jk12kdjk þ

x12

2

Z x

0

Xn

j¼1

jcjðtÞj2dt

!12

24

35

12

þ x2ffiffiffi2p ka0k þ

Xn

j¼1

b0j��

!12

< 1;

(H6):

Xn

j¼1

kdjkk1� g01jk < 1:

Then (1.6) and (1.7) has at least one positive x-periodic solution.

Proof. Suppose u(t) is a x-periodic solution of the following operator equation

Lu ¼ kNuþ kr; k 2 ð0;1Þ: ð3:1Þ

Then u(t) satisfies the following equation

u0ðtÞ ¼ k rðtÞ � aðtÞuðtÞ �Xn

j¼1

bjðtÞuðt � s1jðtÞÞ �Xn

j¼1

cjðtÞZ t

�1kjðt � sÞuðsÞds

"

�Xn

j¼1

djðtÞu0ðt � g1jðtÞÞð1� g01jðtÞÞ �Xn

j¼1



Xn

j¼1

gjðsÞeuðs�g2jðsÞÞÞds

#: ð3:2Þ


Integrating (3.2) on [0,x], we have

�rx ¼Z x

0aðtÞuðtÞ þ

Xn

j¼1

bjðtÞuðt � s1jðtÞÞ þXn

j¼1

cjðtÞZ t

�1kjðt � sÞuðsÞdsþ

Xn

j¼1

djðtÞu0ðt � g1jðtÞÞð1� g01jðtÞÞ"

þXn

j¼1



Xn

j¼1

gjðsÞeuðs�g2jðsÞÞ

!ds

#dt ¼

Z x

0aðtÞuðtÞ þ

Xn

j¼1

bjðtÞuðt � s1jðtÞÞ"

þXn

j¼1

cjðtÞZ t

�1kjðt � sÞuðsÞds�

Xn

j¼1

d0jðtÞuðt � g1jðtÞÞ#

dt

þZ x

0

Xn

j¼1



j¼1


!dsdt: ð3:3Þ

Let t � s1j(t) = s, then t = l1j(s), and
Z x
0bjðtÞuðt � s1jðtÞÞdt ¼

Z x�s1jðxÞ

�s1jð0Þ

bjðl1jðsÞÞ1� s01jðl1jðsÞÞ

uðsÞds:

From Lemma 2.4, it follows that
Z x
0bjðtÞuðt � s1jðtÞÞdt ¼

Z x

0


uðsÞds: ð3:4Þ

Similarly, we have

Z x

0d0jðtÞuðt � g1jðtÞÞdt ¼

Z x

0

d0jðc1jðsÞÞ1� g01jðc1jðsÞÞ

uðsÞds: ð3:5Þ

Combining (3.3)–(3.5), we have

Z x

0C1ðtÞuðtÞdt ¼

Z x

0aðtÞuðtÞ þ

Xn

j¼1

bjðtÞuðt � s1jðtÞÞ �Xn

j¼1

d0jðtÞuðt � g1jðtÞÞ" #

dt: ð3:6Þ

From Lemma 2.4, we get

l1jðxÞ ¼ l1jð0Þ þx; c1jðxÞ ¼ c1jð0Þ þx; j ¼ 1; . . . ;n; then

Z x

0


dt ¼Z l1jðxÞ

l1jð0Þ

bjðtÞð1� s01jðl1jðtÞÞÞ1� s01jðl1jðtÞÞ

dt ¼Z l1jð0Þþx

l1jð0ÞbjðtÞdt ¼

Z x

0bjðtÞdt ¼ �bjx;

Z x

0

d0jðc1jðtÞÞ1� g01jðc1jðtÞÞ

dt ¼Z c1jðxÞ

c1jð0Þ

d0jðtÞð1� g01jðc1jðtÞÞÞ1� g1j0ðc1jðtÞÞ

dt ¼Z x

0d0jðtÞdt ¼ 0:

Then C1x ¼Rx

0 C1ðtÞdt ¼ aþPn

j¼1�bj

� �x;

Cx ¼Z x

0CðtÞdt ¼

Z x

0C1ðtÞdt þ

Z x

0

Xn

j¼1

cjðtÞ !

dt ¼ �aþXn

j¼1

bj þXn

j¼1

�cj

!x: ð3:7Þ

By (H3), we have C1(t) > 0,Pn

j¼1cjðtÞP 0, t 2 [0,x]. From (3.3)

�rx ¼Z x

0C1ðtÞuðtÞdt þ

Z x

0

Xn

j¼1

cjðtÞZ t

�1kjðt � sÞuðsÞdsdt þ

Z x

0

Xn

j¼1

fjðtÞ

�Z t�s2jðtÞþw


j¼1


!dsdt

P ðuÞmCxþ eðuÞm Ax P ðuÞmCxþ ð1þ ðuÞmÞAx:

Then �r P ðuÞmCþ ð1þ ðuÞmÞA, we obtain

ðuÞm 6�r � ACþ A

6r

Cþ A: ð3:8Þ

In the same manner, we get

�rx 6 ðuÞMCxþ eðuÞM Ax 6 eðuÞMxðCþ AÞ:


Then �r 6 eðuÞM ðCþ AÞ, we obtain

ðuÞM P ln�r

Cþ A: ð3:9Þ

By Lemma 2.6, (H4), (3.8) and (3.9), we have

uðtÞP ðuÞM �12

Z x

0ju0ðsÞjds P ln

�rCþ A

� 12

Z x

0ju0ðsÞjds;

uðtÞ 6 ðuÞm þ12

Z x

0ju0ðsÞjds 6 ln

�rCþ A

þ 12

Z x

0ju0ðsÞjds:

So

kuk 6 H þ 12

Z x

0ju0ðsÞjds: ð3:10Þ

Also, from (3.3), we have

�rx ¼Z x

0C1ðtÞuðtÞdt þ

Z x

0

Xn

j¼1

cjðtÞZ t

�1kjðt � sÞuðsÞdsdt þ

Z x

0

Xn

j¼1

fjðtÞ

�Z t�s2jðtÞþw


j¼1


!dsdt

6 ðuÞMCxþ eðuÞM Ax ¼ ððuÞM þ 1ÞCx� Cxþ eðuÞM Ax 6 eðuÞM xðCþ AÞ � Cx:

Then �r 6 eðuÞM ðCþ AÞ � C, from (H4), we obtain

ðuÞM P ln�r þ CCþ A

P�r

Cþ A: ð3:11Þ

Multiplying both sides of (3.2) by u0(t) and integrating them over [0,x], we have
Z x
0ju0ðtÞj2dt 6

Z x

0rðtÞu0ðtÞdt �

Z x

0aðtÞuðtÞu0ðtÞdt �

Xn

j¼1

Z x

0bjðtÞuðt � s1jðtÞÞu0ðtÞdt

��Z x

0

Xn

j¼1

cjðtÞZ t


u0ðtÞdt

�Xn

j¼1

Z x

0djðtÞu0ðt � g1jðtÞÞð1� g01jðtÞÞu0ðtÞdt

�Z x

0

Xn

j¼1



Xn

j¼1


!ds

!u0ðtÞdt

��6

Z x


Z x


Xn

j¼1

Z x


��Z x

0

Xn

j¼1

cjðtÞZ t


u0ðtÞdt

�Xn

j¼1

Z x

0djðtÞu0ðt � g1jðtÞÞð1� g01jðtÞÞu0ðtÞdt

��þXn

j¼1

jfjj0IeHþ12

R x

0ju0 ðsÞjds

Z x

0u0ðtÞdt

¼Z x


Z x


Xn

j¼1

Z x


��Z x

0

Xn

j¼1

cjðtÞZ t


u0ðtÞdt

�Xn

j¼1

Z x

0djðtÞu0ðt � g1jðtÞÞ 1� g01jðtÞ

� �u0ðtÞdt

��


6

Z x

0jrðtÞj2dt

12Z x

0ju0ðtÞj2dt

12

þ 12ka0k

Z x

0juðtÞj2dt þ 1

2

Xn

j¼1

kb0jkZ x

0juðtÞj2dt

þXn

j¼1

kbjkZ x

0ju0ðtÞj2dt

12Z x

0juðtÞ � uðt � s1jðtÞÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞZ t

�1kjðt � sÞuðsÞdsj2dt

!12 Z x

0ju0ðtÞj2dt

12

þXn

j¼1

Z x

0ju0ðtÞj2dt

12Z x

0jdjðtÞu0ðt � g1jðtÞÞð1� g01jðtÞÞj

2dt 1

2

6

Z x

0jrðtÞj2dt

12Z x

0ju0ðtÞj2dt

12

þ 12ka0k

Z x

0juðtÞj2dt þ 1

2

Xn

j¼1

kb0jkZ x

0juðtÞj2dt

þXn

j¼1

kbjkZ x

0ju0ðtÞj2dt

12 ffiffiffi

2p

lj

Z x

0ju0ðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

kukZ x

0ju0ðtÞj2dt

12

þXn

j¼1

k1� g01jðtÞk12kdjk

Z x

0ju0ðtÞj2dt

2�12

¼ffiffiffi2p Xn

j¼1

kbjklj þXn

j¼1


" #Z x

0ju0ðtÞj2dt

þZ x

0jrðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

kuk

24

35 Z x

0ju0ðtÞj2dt

12

þ 12ka0k þ

Xn

j¼1

kb0jk !Z x

0juðtÞj2dt:

ð3:12Þ

Using the notation ku0kL2¼

Rx0 ju0ðtÞj

2dt� �1

2, by

Rx0 juðtÞj

2dt 6 kuk2x (3.10) and (3.12), we have

ku0k2L26

ffiffiffi2p Xn

j¼1

kbjklj þXn

j¼1


!ku0k2

L2

þZ x

0jrðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

H þx12

2ku0kL2

!24

35ku0kL2

þx2ka0k þ

Xn

j¼1

kb0jk !

H þx12

2ku0kL2

!2

¼ffiffiffi2p Xn

j¼1

kbjklj þXn

j¼1

k1� g01jðtÞk12kdjk þ

x12

2

Z x

0

Xn

j¼1

jcjðtÞj2dt

!12

24

35� ku0k2

L2

þZ x

0jrðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

H

24

35ku0kL2

þx2ka0k þ

Xn

j¼1

kb0jk !

H þx12

2ku0kL2

!2

: ð3:13Þ

From a26 b2 + c2 + d2) a 6 b + c + d, where a, b, c, d are all non-negative.

Then, we get

ku0kL26

ffiffiffi2p Xn

j¼1

kbjklj þXn

j¼1

k1� g01jðtÞk12kdjk þ

x12

2

Z x

0

Xn

j¼1

jcjðtÞj2dt

!12

24

35

12

� ku0kL2

þZ x

0jrðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

H

24

35

12

� ku0k12L2þ x

2ka0k þ

Xn

j¼1

kb0jk !" #1

2

H þx12

2ku0kL2

!

¼ Bku0kL2þ

Z x

0jrðtÞj2dt

12

þZ x

0

Xn

j¼1

jcjðtÞj2dt

!12

H

24

35

12

� ku0k12L2þ x1

2ffiffiffi2p ka0k þ

Xn

j¼1

kb0jk !1

2

H: ð3:14Þ


By (H5), then there exists a constant M > 0 such that ku0kL26 M, that is

Rx0 ju0ðtÞj

2dt� �1

26 M. From (3.10) and Hölder inequal-

ity, we obtain

kuk 6 H þ 12

Z x

0ju0ðtÞjdt 6 H þx1

2

2

Z x

0ju0ðtÞj2dt

12

6 H þx12

2M :¼ M1: ð3:15Þ

Again from (3.2), we get

ku0k 6 krk þ kakkuk þXn

j¼1

kbjkkuk þXn

j¼1

kcjkkuk þXn

j¼1

kdjkk1� g01jkku0k þXn

j¼1

kfjkeM1 I: ð3:16Þ

From (H6) and (3.15), we have

ku0k 6krk þ kak þ

Pnj¼1kbjk þ

Pnj¼1kcjk

� �M1 þ

Pnj¼1kfjkeM1 I

1�Pn

j¼1kdjkk1� g01jk:¼ M2: ð3:17Þ

Let X ¼ xjx 2 C1x; jxj1 > max M1;M2;

�rCþA

n on o. Then k ¼

Pnj¼1kdjð1� g01jÞk < lðLÞ. Defined a bounded bilinear form [�, �] on

Cx � C1x by ½y; x� ¼

Rx0 yðtÞxðtÞdt. Also we define Q : Y ? Coker(L) by y!

Rx0 yðtÞdt. It is obvious that {u : u 2 Ker L \ oX} =

{u : u � r0 or �r0}, without loss of generality, suppose that u � r0, then

½QNðuÞ þ QðrÞ;u� � ½QNð�uÞ þ QðrÞ; u�

¼ r20

Z x

0rðtÞdt � r0

Z x

0aðtÞ þ

Xn

j¼1

bjðtÞ þXn

j¼1

cjðtÞ !

dt

"

�er0

Z x

0

Xn

j¼1



j¼1

gjðsÞ !

dsdt

#

�Z x

0rðtÞdt � ð�r0Þ

Z x

0aðtÞ þ

Xn

j¼1

bjðtÞ þXn

j¼1

cjðtÞ !

dt

"

�e�r0

Z x

0

Xn

j¼1



j¼1

gjðsÞ !

dsdt

#ð3:18Þ

¼ r20x

2 �r � r0 aþXn

j¼1

bj þXn

j¼1

cj

!� er0 A

" #� �r � ð�r0Þ aþ

Xn

j¼1

bj þXn

j¼1

cj

!� e�r0 A

" #

< r20x

2 �r � r0 aþXn

j¼1

bj þXn

j¼1

cj

!� r0A

" #� �r þ r0 aþ

Xn

j¼1

bj þXn

j¼1

cj

!þ r0A

" #

< r20x

2 �r � r0 Cþ A� ��

� �r þ r0 Cþ A� ��

:

Since r0 >�r

CþA

�� , then r0 >�r

CþA

�� P �rCþA

, and �r0 < � rCþA

�� 6 �rCþA

.Thus

�r � r0ðCþ AÞ < 0;

r þ r0ðCþ AÞ > 0:

By (3.18), we get [QN(u) + Q(r),u] � [QN(�u) + Q(r),u] < 0.Then all of the conditions required in Lemma 2.1 are hold. It follows from Lemma 2.1 that Eq. (2.3) has at least one x-

periodic solution. Therefore, system (1.6) and (1.7) has at least one positive x-periodic solution. The proof is complete. h

4. Global asymptotic stability

In this section, we devote ourselves to the study of the global asymptotic stability of periodic solution of system (1.6) and(1.7). Our method involves the construction of a suitable Lyapunov functional, which is based on the Lyapunov functionalintroduced by [12].

Definition 4.1. Let (N⁄(t),u⁄(t))T be a periodic solution of (1.6) and (1.7). We say (N⁄(t),u⁄(t))T is globally asymptotically stableif any other solution (N(t),u(t))T of (1.6) and (1.7) has the property:

limt!þ1

jN�ðtÞ � NðtÞj ¼ 0; limt!þ1

ju�ðtÞ � uðtÞj ¼ 0: ð4:1Þ

Now we state our main results of this section below.


Theorem 4.1. Assume that the conditions in Theorem 3.1 hold. Moreover, if there is a positive constant k such that:

inft2½0;þ1Þ

k aðtÞ �Pnj¼1

bjðl1jðtÞÞ1�s0

1jðl1jðtÞÞ

�Pnj¼1

Rþ10 kjðsÞcjðt þ sÞds

" #( )> 0;

inft2½0;þ1Þ

k a0ðtÞ �Pnj¼1

fjðl2jðtÞÞ1�s0

2jðl2jðtÞÞ

" #( )> 0;

Rþ10

gjðc2jðsÞÞ1�g0

2jðc2jðsÞÞ

ds < þ1; j ¼ 1; . . . ;n;Rþ10

djðc1jðsÞÞ1�g0

1jðc1jðsÞÞ

ds < þ1; j ¼ 1; . . . ;n:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð4:2Þ

where l1j(t), l2j(t), c1j(t), c2j(t) are the inverse function of t � s1j(t), t � s2j(t), t � g1j(t), t � g2j(t), respectively. Then system(1.6) and (1.7) has a unique periodic solution which is globally asymptotically stable.

Proof. By Theorem 3.1, there exists a periodic solution of (1.6) and (1.7), say (N⁄(t),u⁄(t))T. To complete the proof, we onlyneed to show that (N⁄(t),u⁄(t))T is globally asymptotically stable. Let (N(t),u(t))T be any solution of (1.6) and (1.7). Consider aLyapunov functional V(t) = V(t, (N⁄(t),u⁄(t))T, (N(t),u(t))T) defined by

VðtÞ ¼ V1ðtÞ þ V2ðtÞ þ V3ðtÞ þ V4ðtÞ þ V5ðtÞ þ V6ðtÞ; for t P 0;

where

V1ðtÞ ¼ kðj ln N�ðtÞ � ln NðtÞj þ ju�ðtÞ � uðtÞjÞ;

V2ðtÞ ¼ kXn

j¼1

Z t

t�s1jðtÞ


j ln N�ðsÞ � ln NðsÞjds;

V3ðtÞ ¼ kXn

j¼1

Z þ1

0kjðsÞ

Z t

t�scjðhþ sÞj ln N�ðhÞ � ln NðhÞjdhds;

V4ðtÞ ¼ kXn

j¼1

Z t

t�s2jðtÞ

fjðl2jðsÞÞ1� s02jðl2jðsÞÞ

ju�ðsÞ � uðsÞjds;

V5ðtÞ ¼ kXn

j¼1

Z þ1

t�g2jðtÞ

gjðc2jðsÞÞ1� g02jðc2jðsÞÞ

jN�ðsÞ � NðsÞjds;

V6ðtÞ ¼ kXn

j¼1

Z þ1

t�g1jðtÞ

djðc1jðsÞÞ1� g01jðc1jðsÞÞ

jðln N�ðsÞÞ0 � ðln NðsÞÞ0jds:

From the definition of V(t), it is easy to see that
Vð0Þ < þ1 ð4:3Þ
and

VðtÞP kðj ln N�ðtÞ � ln NðtÞj þ ju�ðtÞ � uðtÞjÞ; t P 0: ð4:4Þ

Calculating the upper right derivative D+V(t) of V(t) along the solution of (1.6) and (1.7), by computation, one could obtain

DþVðtÞ 6 �kaðtÞj ln N�ðtÞ � ln NðtÞj

þ kXn

j¼1

bjðtÞj ln N�ðt � s1jðtÞÞ � ln Nðt � s1jðtÞÞj þ kXn

j¼1

cjðtÞZ t

�1kjðt � sÞj ln N�ðsÞ � ln NðsÞjds

þ kXn

j¼1

djðtÞjðln N�ðt � g1jðtÞÞÞ0 � ðln Nðt � g1jðtÞÞÞ

0j þ kXn

j¼1

fjðtÞju�ðt � s2jðtÞÞ � uðt � s2jðtÞÞj

� ka0ðtÞju�ðtÞ � uðtÞj þ kXn

j¼1

gjðtÞjN�ðt � g2jðtÞÞ � Nðt � g2jðtÞÞj

þ kXn

j¼1


j ln N�ðtÞ � ln NðtÞj � kXn

j¼1

bjðtÞj ln N�ðt � s1jðtÞÞ � ln Nðt � s1jðtÞÞj

þ kXn

j¼1

Z þ1

0kjðsÞcjðt þ sÞj ln N�ðtÞ � ln NðtÞjds� k

Xn

j¼1

Z þ1

0kjðsÞcjðtÞj ln N�ðt � sÞ � ln Nðt � sÞjds

þ kXn

j¼1

fjðl2jðtÞÞ1� s02jðl2jðtÞÞ

ju�ðtÞ � uðtÞj � kXn

j¼1

fjðtÞju�ðt � s2jðtÞÞ � uðt � s2jðtÞÞj

� kXn

j¼1

gjðtÞjN�ðt � g2jðtÞÞ � Nðt � g2jðtÞÞj � kXn

j¼1

djðtÞjðln N�ðt � g1jðtÞÞÞ0 � ðln Nðt � g1jðtÞÞÞ

0j

¼ �S1j ln N�ðtÞ � ln NðtÞj � S2ju�ðtÞ � uðtÞj;


where, " #
S1 ¼ k aðtÞ �
Xn

j¼1


�Xn

j¼1

Z þ1

0kjðsÞcjðt þ sÞds ;

S2 ¼ k a0ðtÞ �Xn

j¼1

fjðl2jðtÞÞ1� s02jðl2jðtÞÞ

" #:

From (4.2), it follows that there exists a constant K > 0 such that
S1 > K; S2 > K:
Hence, it follows that

DþVðtÞ < �K j ln N�ðtÞ � ln NðtÞj þ ju�ðtÞ � uðtÞjð Þ: ð4:5Þ

Then, by using (4.3) and (4.5) and the analysis of that in [2, p. 816], one could obtain:

limt!þ1

j ln N�ðtÞ � ln NðtÞj ¼ 0; limt!þ1

ju�ðtÞ � uðtÞj ¼ 0:

From this, one could easily obtain:

limt!þ1

jN�ðtÞ � NðtÞj ¼ 0; limt!þ1

ju�ðtÞ � uðtÞj ¼ 0:

which means (N⁄(t),u⁄(t))T is globally asymptotically stable. This completes the proof. h

Remark. From Theorems 3.1 and 4.1, we can get the system (1.6) and (1.7) has only one positive periodic solution.

References

[1] F.D. Chen, Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model, Appl. Math. Comput. 176 (2006)431–441.

[2] M. Fan, P.J.Y. Wong, Ravi P. Agarwal, Periodicity and stability in Periodic n-species Lotka–Volterra competition system with feedback controls anddeviating arguments, Acta. Math. Sin. 19 (4) (2003) 801–822.

[3] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, vol. 74, KluwerAcademic Publishers Group, Dordrecht, 1992.

[4] R.E. Gaines, J.L. Mawhin, Lecture Notes in Mathematics, Springer, Berlin, 1977. vol. 586.[5] G. Kirlinger, Permanence in Lotka–Volterra equations linked prey-predator systems, Math. Biosci. 82 (1986) 165–169.[6] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral Logarithmic population model with multiple delays, J. Comput. Appl. Math. 166 (2)

(2004) 371–383.[7] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral population model with multiple delays, Appl. Math. Comput 153 (2004) 885–902.[8] Y.K. Li, Attractivity of a positive periodic solution for all other positive solution in a delay population model, Appl. Math. -JCU. 12 (3) (1997) 279–282. in

Chinese.[9] Y.K. Li, On a periodic neutral delay Logarithmic population model, J. Syst. Sci. Math. Sci. 19 (1) (1999) 34–38. in Chinese.

[10] Z.D. Liu, Y.P. Mao, Existence theorem for periodic solutions of higher order nonlinear differential equations, J. Math. Anal. Appl. 216 (1997) 481–490.[11] W.V. Petryshyn, Z.S. Yu, Existence theorems for higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6 (9) (1982) 943–969.[12] C.Z. Wang, J.L. Shi, Periodic solution for a delay multispecies Logarithmic population model with feedback control, Appl. Math. Comput 193 (2007)

257–265.[13] Q. Wang, Y. Wang, B.X. Dai, Existence and uniqueness of positive periodic solutions for a neutral Logarithmic population model, Appl. Math. Comput.

213 (2009) 137–147.[14] Q. Wang, H.Y. Zhang, Y. Wang, Existence and stability of positive almost periodic solutions and periodic solutions for a logarithmic population model,

Nonlinear Anal. 72 (2010) 4384–4389.[15] Y.G. Zhou, X.H. Tang, On existence of periodic solutions of Rayleigh equation of retarded type, J. Comput. Appl. Math. 203 (2007) 1–5.

Documents

Positive periodic solution for a neutral Logarithmic population model with feedback control