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Positive periodic solution for a neutral Logarithmic population model with feedback control
Applied Mathematics and Computation 217 (2011) 7692–7702
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Positive periodic solution for a neutral Logarithmic population model with feedback control q Rui Wang ⇑, Xiaosheng Zhang School 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 model Feedback control Positive periodic solution Global asymptotically stability
a b s t r a c t In this paper, a neutral delay Logarithmic population model with feedback control is studied. By using the abstract continuous theorem of k-set contractive operator, some new results on the existence of the positive periodic solution are obtained; after that, by constructing a suitable Lyapunov functional, a set of easily applicable criteria is established for 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:
dNðtÞ ¼ NðtÞ½a b ln NðtÞ c ln Nðt sÞ: dt
ð1:1Þ
System (1.1) is then generalized by Li [8] to the non-autonomous case
dNðtÞ ¼ NðtÞ½aðtÞ bðtÞ ln NðtÞ cðtÞ ln Nðt sðtÞÞ: dt
ð1:2Þ
In [8], by using the coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of system (1.2) are established. As was pointed out by Gopalsamy [3], in some case, the neutral delay population models are more realistic. There were many scholars done works on the periodic solution of neutral type Logistic model or Lotka–Volterra model, but only a little scholars considered the neutral Logarithmic model. Li [9] had studied the following single species neutral Logarithmic model:
dNðtÞ ¼ NðtÞ½rðtÞ aðtÞ ln Nðt rÞ bðtÞðln Nðt gÞÞ0 : dt
ð1:3Þ
Lu and Ge [6] investigated the following system:
" # n n X X dNðtÞ ¼ NðtÞ rðtÞ aj ðtÞ ln Nj ðt rðtÞÞ bj ðtÞðln Nðt gj ðtÞÞÞ0 : dt j¼1 j¼1
q
ð1:4Þ
Supported by the Natural Science Foundation of China (10871005) and the Foundation of Beijing City’s Educational Committee (KM200610028001).
⇑ Corresponding author.
E-mail address: [email protected] (R. Wang). 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.072
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 continuous theorem of k-set contractive operator. The authors [13], by using an abstract continuous theorem of k-set contractive operator, investigated the existence, globally attractive of the positive periodic solution of the following neutral multi-delays Logarithmic population model:
" # Z t n n n X X X dNðtÞ ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ bj ðtÞ ln Nðt sj ðtÞÞ cj ðtÞ kj ðt sÞ ln NðsÞds dj ðtÞðln Nðt gj ðtÞÞÞ0 : dt 1 j¼1 j¼1 j¼1 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 the system structurally by introducing some feedback control variables so as to get the population stability of another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively. Recently, many scholars focus on the existence and global attractivity of the positive periodic solution of the system. The authors [12] have studied the existence, uniqueness and global attractivity of the periodic solution of the delay multispecies Logarithmic population model with feedback control, by using the contraction mapping principle and a suitable Lyapunov functional. By means of the Cauchy matrix, the authors [14] investigate the existence and exponential stability of almost periodic solutions and periodic solutions for the delay impulsive Logarithmic population. 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 periodic solution of the neutral Logarithmic population model with feedback control. This motivates us to consider the following equation:
8 ( n n > Rt P P > dNðtÞ > > dt ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ bj ðtÞ ln Nðt s1j ðtÞÞ cj ðtÞ 1 kj ðt sÞ ln NðsÞds > > > j¼1 j¼1 > > ) > < n n P P dj ðtÞðln Nðt g1j ðtÞÞÞ0 fj ðtÞuðt s2j ðtÞÞ ; > > j¼1 j¼1 > > > > n > P > duðtÞ > > : dt ¼ a0 ðtÞuðtÞ þ g j ðtÞNðt g2j ðtÞÞ;
ð1:6Þ
j¼1
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)) R þ1 R þ1 j = 1, . . . , n with 1 s01j ðtÞ > 0; 1 g01j ðtÞ P 0; 1 s02j ðtÞ > 0; 1 g02j ðtÞ > 0 and 0 kj ðsÞds ¼ 1; 0 skj ðsÞds < þ1. We consider (1.6) together with following initial conditions:
8 < NðtÞ ¼ uðtÞ; N0 ðtÞ ¼ u0 ðtÞ; uð0Þ > 0; u 2 C 1 ðð1; 0; ½0; 1ÞÞ; max : uðhÞ ¼ wðhÞ P 0; h 2 ½s2 ; 0; wð0Þ > 0; where s2 ¼
fs2j ðtÞg:
ð1:7Þ
t2½0;x;j¼1;...;n
The aim of this paper is to give a set of new conditions to guarantee the existence and global stability of the positive periodic 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, some lemmas are needed. Since each x-periodic solution of the equation n X duðtÞ ¼ a0 ðtÞuðtÞ þ g j ðtÞNðt g2j ðtÞÞ dt j¼1
is equivalent to that of the equation
uðtÞ ¼
Z
tþw
Gðt; sÞ
t
where, Gðt; sÞ ¼
n X j¼1
Rs a ðhÞdh R xt 0 ; s 2 ½t; t þ x; t 2 R.
exp exp
!
g j ðsÞNðs g2j ðsÞÞ ds :¼ ðUNÞðtÞ;
0
a0 ðhÞdh1
ð2:1Þ
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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 positive
x-periodic solution of the equation: ( Z t n n n X X X dNðtÞ bj ðtÞ ln Nðt s1j ðtÞÞ cj ðtÞ kj ðt sÞ ln NðsÞds dj ðtÞðln Nðt g1j ðtÞÞÞ0 ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ dt 1 j¼1 j¼1 j¼1 ! ) Z ts2j ðtÞþw n n X X fj ðtÞ G t s2j ðtÞ; s g j ðsÞNðs g2j ðsÞÞ ds : ð2:2Þ ts2j ðtÞ
j¼1
j¼1
Taking the transformation N(t) = ey(t), then (2.2) can be rewritten as
y0 ðtÞ ¼ rðtÞ aðtÞyðtÞ
n X j¼1
n X
fj ðtÞ
Z
ts2j ðtÞþw
ts2j ðtÞ
j¼1
bj ðtÞyðt s1j ðtÞÞ
n X
cj ðtÞ
Z
t
kj ðt sÞyðsÞds 1
j¼1
n X
dj ðtÞy0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
! n X yðsg2j ðsÞÞ ds: G t s2j ðtÞ; s g j ðsÞe
ð2:3Þ
j¼1
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 Let E be a Banach space. For a bounded subset A E, denote the Kuratoskii measure of non-compactness:
ðtÞ
.
aE ðAÞ ¼ inffd > 0j There is a finite number of subsets fAi g A such that A ¼ [ðAi Þ and diamðAi Þ 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 continuous and bounded map N : X ! Y is called k-set contractive if for any bounded set A X, we have
aY ðNðAÞÞ 6 kaX ðAÞ; 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 : r aX ðAÞ 6 aY ðLðAÞÞ; 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 is called a k-set contractive with k < l(L), where X X is bounded, open and symmetric about 0 2 X. 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 C x ¼ fxjx 2 CðR; Rn Þ; xðt þ xÞ ¼ xðtÞg with the norm defined by kxk = jxj0 = maxt2[0,x]{jx(t)j}, and
X ¼ C 1x ¼ fxjx 2 C 1 ðR; Rn Þ; xðt þ xÞ ¼ xðtÞg with the norm defined by jxj1 = max{jxj0, jx0 j0}. Then both Cx and C 1x are Banach spaces. We also denote:
¼ 1 h
Z
x
x
0
jhðsÞjds; ðhÞm ¼ min hðtÞ; ðhÞM ¼ max hðtÞ: t2½0;x
t2½0;x
Let L : C 1x ! C x defined by Ly = y0 (t) and N : C 1x ! C x defined by
Ny ¼ aðtÞyðtÞ
n X
bj ðtÞyðt s1j ðtÞÞ
j¼1
n X j¼1
fj ðtÞ
Z
ts2j ðtÞþw
ts2j ðtÞ
n X
cj ðtÞ
j¼1
Z
t
kj ðt sÞyðsÞds 1
! n X yðsg2j ðsÞÞ G t s2j ðtÞ; s g j ðsÞe ds:
n X
dj ðtÞy0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
j¼1
Thus (2.3) has a positive x-periodic solution if and only if Ly = Ny + r for some y 2 C 1x , 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.
ð2:4Þ
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R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Lemma 2.3. Let c0, c1 be two positive constants, and X ¼ fxjx 2 C 1x ; jxj0 < c0 ; jx0 j0 < c1 g, if k ¼ X ? Cx is a k-set contractive map.
Pn
j¼1 kdj ð1
g01j Þk, then N :
Proof. Let A X be a bounded subset and let g ¼ aC 1x ðAÞ. 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 ðUey ÞðtÞ; Fðt; x; y1 ; . . . ; yn ; z1 ; . . . ; zn ; w1 ; . . . ; wn ; v 1 ; . . . ; v n Þ ¼ aðtÞx þ
n X
bj ðtÞyj þ
j¼1
n X
cj ðtÞzj þ
j¼1
n X
dj ðtÞwj ð1 g01j ðtÞÞ þ
j¼1
n X
fj ðtÞv j :
j¼1
For convenience, in the following discussion we denote
J i ðxÞðtÞ ¼
Z
t
1
ki ðt sÞxðsÞds; for x 2 C 1x ;
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 are precompact in Cx, it follows that there is a finite family of subsets Aij of Ai such that Ai = [jAij with
jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6 e; for any x; u 2 Aij : Therefore, we have
kNx Nuk ¼ sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
x0 ðt g11 ðtÞÞ; . . . ; x0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6 sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
x0 ðt g11 ðtÞÞ; . . . ; x0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞj þ sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
0
u ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6
n X
kdj ð1 g01j Þk jx0 ðt g1j ðtÞÞ u0 ðt g1j ðtÞÞj þ e 6 kðg þ eÞ þ e:
j¼1
As e is arbitrarily small, it is easy to get that aC x ðNðAÞÞ 6 kaC 1x ðAÞ. The proof is complete. h Lemma 2.4 [7]. Suppose s 2 C 1x 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 6 x be a constant, s(t) 2 Cx, such that maxt2[0,x]js(t)j 6 a. Then for 8x 2 C 1x , we have Rx Rx jxðtÞ xðt sðtÞÞj2 dt 6 2a2 0 jx0 ðtÞj2 dt. If, in addition, sðtÞ 2 C 1x , and s0 (t) < 1, t 2 R, then for x 2 C 1x , we have the following 0 conclusions: (1) There exists a unique integer m such that l = js(t) mxj0 < x; Rx 2 Rx (2) 0 jxðtÞ xðt sðtÞÞj2 dt 6 2l 0 jx0 ðtÞj2 dt.
7696
As
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
s01j ðtÞÞ < 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
0
2
jxðtÞ xðt s1j ðtÞÞj2 dt 6 2lj
Z
x
jx0 ðtÞj2 dt;
x 2 C 1x :
0
ð2:5Þ
Lemma 2.6 [15] . Suppose x(t) is a differently continuous x-periodic function on R with x > 0. Rx maxt 6t6ðt þxÞ jxðtÞj 6 jxðt Þj þ 12 0 jx0 ðtÞjdt.
Then to any t 2 R;
3. Existence of periodic solution Since s01j ðtÞ < 1; g01j ðtÞ < 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:
C1 ðtÞ ¼ aðtÞ þ
n X j¼1
CðtÞ ¼ C1 ðtÞ þ
n X
0
n X bj ðl1j ðtÞÞ dj ðc1j ðtÞÞ ; 0 1 s1j ðl1j ðtÞÞ j¼1 1 g01j ðc1j ðtÞÞ
cj ðtÞ;
j¼1
A¼
Z
1
x
n X
x
0
I ¼ max
fj ðtÞ
t2½0;x
ts2j ðtÞþx
r ; C þ A n X
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
j¼1
Z
Z
H ¼ ln
ts2j ðtÞþx
! g j ðsÞ dsdt;
j¼1
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
n X
! g j ðsÞ ds:
j¼1
Theorem 3.1. If in addition to (H1)–(H2) assume further that: (H3):
r > 0; cj ðtÞ P 0; C1 ðtÞ > 0; t 2 ½0; x;
þ and a
n X
þ b j
j¼1
n X
cj > 0;
j¼1
(H4):
r A
CþA
6 ln
r
CþA
;
ln
r þ C r P ; CþA CþA
(H5):
2 1 2
B , 42
n X
kbj klj þ
j¼1
n X
1
1 2
k1 g01j k kdj k þ
j¼1
x2 2
Z
x
0
n X j¼1
!1 !12 312 n 2 X x 0 2 0 5 jcj ðtÞj dt þ pffiffiffi ka k þ bj < 1; 2 2 j¼1
(H6): n X
kdj kk1 g01j k < 1:
j¼1
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 kNu þ kr;
k 2 ð0; 1Þ:
ð3:1Þ
Then u(t) satisfies the following equation
" 0
u ðtÞ ¼ k rðtÞ aðtÞuðtÞ
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
n X j¼1
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
n X
cj ðtÞ
Z
j¼1
fj ðtÞ
kj ðt sÞuðsÞds
1
j¼1 n X
t
Z
ts2j ðtÞþw
ts2j ðtÞ
Gðt s2j ðtÞ; sÞ
n X j¼1
# g j ðsÞeuðsg2j ðsÞÞ Þds :
ð3:2Þ
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
7697
Integrating (3.2) on [0, x], we have
Z
rx ¼
"
x
n X
aðtÞuðtÞ þ
0 n X
þ
fj ðtÞ
cj ðtÞ
þ
n X
Gðt s2j ðtÞ; sÞ
x
n X
t
kj ðt sÞuðsÞds
0
fj ðtÞ
Z
g j ðsÞe
t
kj ðt sÞuðsÞds þ 1
! #
uðsg2j ðsÞÞ
n X
ds dt ¼
Z
n X
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
x
"
aðtÞuðtÞ þ
0
#
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
0
dj ðtÞuðt g1j ðtÞÞ dt
j¼1
ts2j ðtÞþw
ts2j ðtÞ
j¼1
Z
j¼1
1
j¼1
Z
ts2j ðtÞþw
Z
cj ðtÞ
j¼1
ts2j ðtÞ
n X
n X
j¼1
Z
j¼1
þ
bj ðtÞuðt s1j ðtÞÞ þ
! n X uðsg2j ðsÞÞ dsdt: G t s2j ðtÞ; s g j ðsÞe
ð3:3Þ
j¼1
Let t s1j(t) = s, then t = l1j(s), and
Z
Z
x
bj ðtÞuðt s1j ðtÞÞdt ¼
xs1j ðxÞ
s1j ð0Þ
0
bj ðl1j ðsÞÞ uðsÞds: 1 s01j ðl1j ðsÞÞ
From Lemma 2.4, it follows that
Z
Z
x
bj ðtÞuðt s1j ðtÞÞdt ¼
0
x
0
bj ðl1j ðsÞÞ uðsÞds: 1 s01j ðl1j ðsÞÞ
ð3:4Þ
Similarly, we have
Z
x
Z
0
dj ðtÞuðt g1j ðtÞÞdt ¼
0
x
0
0
dj ðc1j ðsÞÞ uðsÞds: 1 g01j ðc1j ðsÞÞ
ð3:5Þ
Combining (3.3)–(3.5), we have
Z
x
C1 ðtÞuðtÞdt ¼
Z
0
x
" aðtÞuðtÞ þ
0
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
n X
# 0
dj ðtÞuðt g1j ðtÞÞ dt:
ð3:6Þ
j¼1
From Lemma 2.4, we get
l1j ðxÞ ¼ l1j ð0Þ þ x; c1j ðxÞ ¼ c1j ð0Þ þ x; j ¼ 1; . . . ; n; then Z
x
0
Z 0
x
Z x Z l1j ðxÞ Z l1j ð0Þþx bj ðtÞð1 s01j ðl1j ðtÞÞÞ bj ðl1j ðtÞÞ x; b ðtÞdt ¼ bj ðtÞdt ¼ b dt ¼ dt ¼ j j 1 s01j ðl1j ðtÞÞ 1 s01j ðl1j ðtÞÞ l1j ð0Þ l1j ð0Þ 0 Z x Z c1j ðxÞ 0 0 dj ðtÞð1 g01j ðc1j ðtÞÞÞ dj ðc1j ðtÞÞ 0 dt ¼ dj ðtÞdt ¼ 0: dt ¼ 0 1 g01j ðc1j ðtÞÞ 1 g1 j ðc1j ðtÞÞ c1j ð0Þ 0
Rx
P
C1 ðtÞdt ¼ a þ nj¼1 bj x; ! Z x Z x Z x X n Cx ¼ CðtÞdt ¼ C1 ðtÞdt þ cj ðtÞ dt ¼
Then C1 x ¼
0
0
0
By (H3), we have C1(t) > 0,
rx ¼
Z
x
C1 ðtÞuðtÞdt þ
0
0
Pn
j¼1 c j ðtÞ
Z 0
Z
ts2j ðtÞþw
x
n X
n X
j¼1
bj þ
j¼1
n X
! cj x:
ð3:7Þ
j¼1
P 0, t 2 [0, x]. From (3.3)
cj ðtÞ
Z
t
kj ðt sÞuðsÞdsdt þ
n X
! g j ðsÞe
Z 0
1
j¼1
G t s2j ðtÞ; s
ts2j ðtÞ
þ a
uðsg2j ðsÞÞ
x
n X
fj ðtÞ
j¼1
dsdt
j¼1
P ðuÞm Cx þ eðuÞm Ax P ðuÞm Cx þ ð1 þ ðuÞm ÞAx: Then r P ðuÞm C þ ð1 þ ðuÞm ÞA, we obtain
ðuÞm 6
r A r 6 : CþA CþA
In the same manner, we get
rx 6 ðuÞM Cx þ eðuÞM Ax 6 eðuÞM xðC þ AÞ:
ð3:8Þ
7698
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Then r 6 eðuÞM ðC þ AÞ, we obtain
ðuÞM P ln
r
CþA
ð3:9Þ
:
By Lemma 2.6, (H4), (3.8) and (3.9), we have
uðtÞ 6 ðuÞm þ
Z
1 2
uðtÞ P ðuÞM
1 2
x
ju0 ðsÞjds P ln
0
Z
x
ju0 ðsÞjds 6 ln
0
r 1 CþA 2
Z
ju0 ðsÞjds; 0
Z
r 1 þ CþA 2
x
x
ju0 ðsÞjds:
0
So
kuk 6 H þ
1 2
Z
x
ju0 ðsÞjds:
Also, from (3.3), we have
rx ¼
Z
Z
x
C1 ðtÞuðtÞdt þ
0
ð3:10Þ
0
x
0
Z
ts2j ðtÞþw
n X
cj ðtÞ
Z
kj ðt sÞuðsÞdsdt þ
Z
G t s2j ðtÞ; s
n X
ts2j ðtÞ
x
0
1
j¼1
t
! uðsg2j ðsÞÞ
g j ðsÞe
n X
fj ðtÞ
j¼1
dsdt
j¼1
6 ðuÞM Cx þ eðuÞM Ax ¼ ððuÞM þ 1ÞCx Cx þ eðuÞM Ax 6 eðuÞM xðC þ AÞ Cx: Then r 6 eðuÞM ðC þ AÞ C, from (H4), we obtain
ðuÞM P ln
r þ C r P : CþA CþA
ð3:11Þ
Multiplying both sides of (3.2) by u0 (t) and integrating them over [0, x], we have
Z 0
x
Z Z x x n Z x X ju ðtÞj dt 6 rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt 0
2
0
n X
Z
x
0
x
0
j¼1
1
j¼1
Z
n X
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞu0 ðtÞdt fj ðtÞ
Z
ts2j ðtÞþw
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
j¼1
n X j¼1
! ! g j ðsÞeuðsg2j ðsÞÞ ds u0 ðtÞdt
Z Z x x n Z x X 6 rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt
0
j¼1
n Z X
x
j¼1
þ
n X
1
dj ðtÞu ðt g1j ðtÞÞð1 g
0
jfj j0 Ie
Hþ12
Rx 0
0 0 1j ðtÞÞu ðtÞdt
0
ju0 ðsÞjds
Z
x
u0 ðtÞdt
0
j¼1
Z Z x x n Z x X rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt ¼ 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt
0
j¼1
n Z X
x
j¼1
0
1
dj ðtÞu0 ðt g1j ðtÞÞ 1 g01j ðtÞ u0 ðtÞdt
7699
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Z
x
6
jrðtÞj2 dt
12 Z
0
þ
kbj k
Z
Z
n X
x
0
ju0 ðtÞj2 dt Z
x
x
6
jcj ðtÞ
2
kj ðt sÞuðsÞdsj dt
ju0 ðtÞj2 dt
jrðtÞj2 dt Z
n X
12 Z
x
12
2
0
jrðtÞj2 dt
x
12
Z
Using the notation ku kL2 ¼
0
12
ju0 ðtÞj2 dt
þ
x
Z
x
0
2
0
juðtÞj2 dt þ
ju ðtÞj dt
12
Z
þ
x
0
ju0 ðtÞj2 dt
212
n X
!12 jcj ðtÞj2 dt
12
Z x n 1X 0 kbj k juðtÞj2 dt 2 j¼1 0 n X
!12 2
jcj ðtÞj dt
Z kuk
x
ju0 ðtÞj2 dt
3 Z kuk5
ju0 ðtÞj2 dt
x
ju0 ðtÞj2 dt
0
j¼1
x
12
0
j¼1
" #Z n n X pffiffiffi X 1 ¼ 2 kbj klj þ k1 g01j ðtÞk2 kdj k
12 þ
0
j¼1
!Z n x X 1 0 ka0 k þ kbj k juðtÞj2 dt: 2 0 j¼1 ð3:12Þ
12
2
0
12
j¼1
0
x
x
0
0
R
x
0
Z
j¼1
0
!12 Z
1 þ ka0 k 2
12 pffiffiffi Z ju ðtÞj dt 2l j
x
1
x
0
Z x n 1X 0 kbj k juðtÞj2 dt 2 j¼1 0
jdj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞj2 dt
ju0 ðtÞj2 dt
k1 g01j ðtÞk2 kdj k
2 Z þ4
ku0 k2L2
x
0
j¼1
juðtÞj2 dt þ
0
12 Z 0
kbj k
x
juðtÞ uðt s1j ðtÞÞj2 dt
0
n X
Z
1
0
þ
x
1 þ ka0 k 2
t
0
j¼1
þ
12
0
j¼1
n Z X
Z
x
12 Z
0
j¼1
þ
ju0 ðtÞj2 dt
0
n X
þ
x
ju ðtÞj dt , by
Rx 0
2
2
juðtÞj dt 6 kuk x (3.10) and (3.12), we have
! n n X pffiffiffi X 1 0 2 2 kbj klj þ k1 g1j ðtÞk kdj k ku0 k2L2
6
2 þ4
þ
j¼1
Z
j¼1
x
2
jrðtÞj dt
12 þ
Z
0
x 2
x
0
ka0 k þ
n X
0
!12 2
jcj ðtÞj dt
1
Hþ
j¼1
!
j¼1
x2 2
x2 2
!3 ku kL2 5ku0 kL2 0
!2
1
Hþ
kbj k
n X
ku0 kL2
2 !12 3 Z xX 1 n n n X pffiffiffi X 1 x2 2 0 4 2 ¼ 2 kbj klj þ k1 g1j ðtÞk kdj k þ jcj ðtÞj dt 5 ku0 k2L2 2 0 j¼1 j¼1 j¼1 2 !2 !12 3 ! Z x
12 Z xX 1 n n X x x2 0 0 2 2 0 0 4 5 þ jrðtÞj dt þ jcj ðtÞj dt H ku kL2 þ kbj k Hþ ka k þ ku kL2 : 2 2 0 0 j¼1 j¼1
ð3:13Þ
From a2 6 b2 + c2 + d2 ) a 6 b + c + d, where a, b, c, d are all non-negative. Then, we get
2
!12 312 Z xX 1 n n n X pffiffiffi X 2 1 x 2 0 ku kL2 6 4 2 kbj klj þ k1 g1j ðtÞk2 kdj k þ jcj ðtÞj dt 5 ku0 kL2 2 0 j¼1 j¼1 j¼1 0
2 þ4
Z
x
jrðtÞj2 dt
12 þ
Z
0
2 Z 0 ¼ Bku kL2 þ 4
x
0
x
2
jrðtÞj dt 0
n X j¼1
12 þ
! !12 312 " !#12 1 n X 1 x x2 0 0 2 0 2 0 5 jcj ðtÞj dt H ku kL2 þ kbj k Hþ ka k þ ku kL2 2 2 j¼1
Z
x 0
n X j¼1
!12 312 !12 1 n X 1 2 x 0 0 0 2 jcj ðtÞj dt H5 ku kL2 þ pffiffiffi ka k þ kbj k H: 2 j¼1 2
ð3:14Þ
7700
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
By (H5), then there exists a constant M > 0 such that ku0 kL2 6 M, that is ity, we obtain
1 2
kuk 6 H þ
Z
1
x
ju0 ðtÞjdt 6 H þ
x2
Z
2
0
x
ju0 ðtÞj2 dt
12
R x 0
ju0 ðtÞj2 dt
12
6 M. From (3.10) and Hölder inequal-
1
6Hþ
x2
0
2
M :¼ M1 :
ð3:15Þ
Again from (3.2), we get
ku0 k 6 krk þ kakkuk þ
n X
kbj kkuk þ
j¼1
n X
kcj kkuk þ
n X
j¼1
kdj kk1 g01j kku0 k þ
j¼1
n X
kfj keM1 I:
ð3:16Þ
j¼1
From (H6) and (3.15), we have
P P P krk þ kak þ nj¼1 kbj k þ nj¼1 kcj k M 1 þ nj¼1 kfj keM1 I 0 P :¼ M 2 : ku k 6 1 nj¼1 kdj kk1 g01j k
ð3:17Þ
n n oo P r . Then k ¼ nj¼1 kdj ð1 g01j Þk < lðLÞ. Defined a bounded bilinear form [, ] on Let X ¼ xjx 2 C 1x ; jxj1 > max M 1 ; M 2 ; CþA R Rx x C x C 1x by ½y; x ¼ 0 yðtÞxðtÞdt. Also we define Q : Y ? Coker(L) by y ! 0 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 Q ðrÞ; u ½QNðuÞ þ Q ðrÞ; u "Z ! Z x n n x X X 2 rðtÞdt r0 aðtÞ þ bj ðtÞ þ cj ðtÞ dt ¼ r0 0
e
r0
0
Z
"Z
n x X
0
fj ðtÞ
Z
j¼1
ts2j ðtÞþw
x
rðtÞdt ðr 0 Þ
0
er0
"
G t s2j ðtÞ; s
Z
0
n X
fj ðtÞ
Z
x
aðtÞ þ
¼ r 20 x2 r r 0 a þ " < r 20 x2 r r 0 a þ
n X
bj þ
j¼1
g j ðsÞ dsdt
bj þ
j¼1
n X
bj ðtÞ þ
#
! cj ðtÞ dt
j¼1
! # n X G t s2j ðtÞ; s g j ðsÞ dsdt
n X
!
n X
#
! cj
ð3:18Þ
j¼1
#
"
er0 A r ðr 0 Þ a þ
cj
j¼1
n X
n X j¼1
ts2j ðtÞþw
ts2j ðtÞ
j¼1
!
j¼1
0
x
n X
ts2j ðtÞ
j¼1
Z
j¼1
"
r 0 A r þ r0 a þ
j¼1
< r 20 x2 r r 0 C þ A r þ r 0 C þ A :
n X
bj þ
j¼1 n X j¼1
bj þ
n X
! cj
# er0 A
j¼1 n X
#
!
cj
þ r0 A
j¼1
r r r r r Since r 0 > CþA , and r0 < CþA . , then r 0 > CþA P CþA 6 CþA Thus
r r0 ðC þ AÞ < 0; r þ r0 ðC þ 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 xperiodic 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 functional introduced 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 stable if any other solution (N(t), u(t))T of (1.6) and (1.7) has the property:
lim jN ðtÞ NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
Now we state our main results of this section below.
ð4:1Þ
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
7701
Theorem 4.1. Assume that the conditions in Theorem 3.1 hold. Moreover, if there is a positive constant k such that:
8 ( " #) n n R > P P bj ðl1j ðtÞÞ > þ1 > inf k aðtÞ kj ðsÞcj ðt þ sÞds > 0; > > 0 1s01j ðl1j ðtÞÞ > t2½0;þ1Þ > j¼1 j¼1 > > ( " # ) > > n > P < fj ðl2j ðtÞÞ > 0; k a0 ðtÞ inf 1s02j ðl2j ðtÞÞ t2½0;þ1Þ j¼1 > > > R þ1 gj ðc2j ðsÞÞ > > ds < þ1; j ¼ 1; . . . ; n; > > 0 1g02j ðc2j ðsÞÞ > > > R > d ð c ðsÞÞ þ1 j 1j > : ds < þ1; j ¼ 1; . . . ; n: 0 1g0 ðc ðsÞÞ 1j
ð4:2Þ
1j
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 only need 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 a Lyapunov functional V(t) = V(t, (N⁄(t), u⁄(t))T, (N(t), u(t))T) defined by
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ þ V 6 ðtÞ; for t P 0; where
V 1 ðtÞ ¼ kðj ln N ðtÞ ln NðtÞj þ ju ðtÞ uðtÞjÞ; n Z t X bj ðl1j ðsÞÞ j ln N ðsÞ ln NðsÞjds; V 2 ðtÞ ¼ k s01j ðl1j ðsÞÞ 1 ts1j ðtÞ j¼1 Z t n Z þ1 X kj ðsÞ cj ðh þ sÞj ln N ðhÞ ln NðhÞjdhds; V 3 ðtÞ ¼ k j¼1
V 4 ðtÞ ¼ k
j¼1
V 5 ðtÞ ¼ k
þ1
tg2j ðtÞ
n Z X j¼1
ts t
ts2j ðtÞ
n Z X j¼1
V 6 ðtÞ ¼ k
0
n Z X
þ1
tg1j ðtÞ
fj ðl2j ðsÞÞ ju ðsÞ uðsÞjds; 1 s02j ðl2j ðsÞÞ g j ðc2j ðsÞÞ jN ðsÞ NðsÞjds; 1 g02j ðc2j ðsÞÞ dj ðc1j ðsÞÞ jðln N ðsÞÞ0 ðln NðsÞÞ0 jds: 1 g01j ðc1j ðsÞÞ
From the definition of V(t), it is easy to see that
Vð0Þ < þ1
ð4:3Þ
and
VðtÞ P 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
Dþ VðtÞ 6 kaðtÞj ln N ðtÞ ln NðtÞj Z n n X X þk bj ðtÞj ln N ðt s1j ðtÞÞ ln Nðt s1j ðtÞÞj þ k cj ðtÞ j¼1
þk
n X
0
dj ðtÞjðln N ðt g1j ðtÞÞÞ ðln Nðt g1j ðtÞÞÞ j þ k
j¼1
kj ðt sÞj ln N ðsÞ ln NðsÞjds
1
j¼1 0
t
n X
fj ðtÞju ðt s2j ðtÞÞ uðt s2j ðtÞÞj
j¼1
ka0 ðtÞju ðtÞ uðtÞj þ k
n X
g j ðtÞjN ðt g2j ðtÞÞ Nðt g2j ðtÞÞj
j¼1 n X bj ðl1j ðtÞÞ bj ðtÞj ln N ðt s1j ðtÞÞ ln Nðt s1j ðtÞÞj þk j ln N ðtÞ ln NðtÞj k 0 s ð l ðtÞÞ 1 1j 1j j¼1 j¼1 n Z þ1 n Z þ1 X X þk kj ðsÞcj ðt þ sÞj ln N ðtÞ ln NðtÞjds k kj ðsÞcj ðtÞj ln N ðt sÞ ln Nðt sÞjds n X
j¼1
0
j¼1
n X
0
n X fj ðl2j ðtÞÞ þk fj ðtÞju ðt s2j ðtÞÞ uðt s2j ðtÞÞj ju ðtÞ uðtÞj k 0 s ð l ðtÞÞ 1 2j 2j j¼1 j¼1 n n X X k g j ðtÞjN ðt g2j ðtÞÞ Nðt g2j ðtÞÞj k dj ðtÞjðln N ðt g1j ðtÞÞÞ0 ðln Nðt g1j ðtÞÞÞ0 j j¼1
¼ S1 j ln N ðtÞ ln NðtÞj S2 ju ðtÞ uðtÞj;
j¼1
7702
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
where,
"
n X
n X bj ðl1j ðtÞÞ 0 1 s1j ðl1j ðtÞÞ j¼1 j¼1 " # n X fj ðl2j ðtÞÞ : S2 ¼ k a0 ðtÞ 1 s02j ðl2j ðtÞÞ j¼1
S1 ¼ k aðtÞ
Z
#
þ1
kj ðsÞcj ðt þ sÞds ;
0
From (4.2), it follows that there exists a constant K > 0 such that
S1 > K;
S2 > K:
Hence, it follows that
Dþ VðtÞ < K ðj ln N ðtÞ ln NðtÞj þ ju ðtÞ uðtÞjÞ:
ð4:5Þ
Then, by using (4.3) and (4.5) and the analysis of that in [2, p. 816], one could obtain:
lim j ln N ðtÞ ln NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
From this, one could easily obtain:
lim jN ðtÞ NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
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 and deviating 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, Kluwer Academic 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.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Positive periodic solution for a neutral Logarithmic population model with feedback control q Rui Wang ⇑, Xiaosheng Zhang School 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 model Feedback control Positive periodic solution Global asymptotically stability
a b s t r a c t In this paper, a neutral delay Logarithmic population model with feedback control is studied. By using the abstract continuous theorem of k-set contractive operator, some new results on the existence of the positive periodic solution are obtained; after that, by constructing a suitable Lyapunov functional, a set of easily applicable criteria is established for 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:
dNðtÞ ¼ NðtÞ½a b ln NðtÞ c ln Nðt sÞ: dt
ð1:1Þ
System (1.1) is then generalized by Li [8] to the non-autonomous case
dNðtÞ ¼ NðtÞ½aðtÞ bðtÞ ln NðtÞ cðtÞ ln Nðt sðtÞÞ: dt
ð1:2Þ
In [8], by using the coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of system (1.2) are established. As was pointed out by Gopalsamy [3], in some case, the neutral delay population models are more realistic. There were many scholars done works on the periodic solution of neutral type Logistic model or Lotka–Volterra model, but only a little scholars considered the neutral Logarithmic model. Li [9] had studied the following single species neutral Logarithmic model:
dNðtÞ ¼ NðtÞ½rðtÞ aðtÞ ln Nðt rÞ bðtÞðln Nðt gÞÞ0 : dt
ð1:3Þ
Lu and Ge [6] investigated the following system:
" # n n X X dNðtÞ ¼ NðtÞ rðtÞ aj ðtÞ ln Nj ðt rðtÞÞ bj ðtÞðln Nðt gj ðtÞÞÞ0 : dt j¼1 j¼1
q
ð1:4Þ
Supported by the Natural Science Foundation of China (10871005) and the Foundation of Beijing City’s Educational Committee (KM200610028001).
⇑ Corresponding author.
E-mail address: [email protected] (R. Wang). 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.072
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 continuous theorem of k-set contractive operator. The authors [13], by using an abstract continuous theorem of k-set contractive operator, investigated the existence, globally attractive of the positive periodic solution of the following neutral multi-delays Logarithmic population model:
" # Z t n n n X X X dNðtÞ ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ bj ðtÞ ln Nðt sj ðtÞÞ cj ðtÞ kj ðt sÞ ln NðsÞds dj ðtÞðln Nðt gj ðtÞÞÞ0 : dt 1 j¼1 j¼1 j¼1 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 the system structurally by introducing some feedback control variables so as to get the population stability of another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively. Recently, many scholars focus on the existence and global attractivity of the positive periodic solution of the system. The authors [12] have studied the existence, uniqueness and global attractivity of the periodic solution of the delay multispecies Logarithmic population model with feedback control, by using the contraction mapping principle and a suitable Lyapunov functional. By means of the Cauchy matrix, the authors [14] investigate the existence and exponential stability of almost periodic solutions and periodic solutions for the delay impulsive Logarithmic population. 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 periodic solution of the neutral Logarithmic population model with feedback control. This motivates us to consider the following equation:
8 ( n n > Rt P P > dNðtÞ > > dt ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ bj ðtÞ ln Nðt s1j ðtÞÞ cj ðtÞ 1 kj ðt sÞ ln NðsÞds > > > j¼1 j¼1 > > ) > < n n P P dj ðtÞðln Nðt g1j ðtÞÞÞ0 fj ðtÞuðt s2j ðtÞÞ ; > > j¼1 j¼1 > > > > n > P > duðtÞ > > : dt ¼ a0 ðtÞuðtÞ þ g j ðtÞNðt g2j ðtÞÞ;
ð1:6Þ
j¼1
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)) R þ1 R þ1 j = 1, . . . , n with 1 s01j ðtÞ > 0; 1 g01j ðtÞ P 0; 1 s02j ðtÞ > 0; 1 g02j ðtÞ > 0 and 0 kj ðsÞds ¼ 1; 0 skj ðsÞds < þ1. We consider (1.6) together with following initial conditions:
8 < NðtÞ ¼ uðtÞ; N0 ðtÞ ¼ u0 ðtÞ; uð0Þ > 0; u 2 C 1 ðð1; 0; ½0; 1ÞÞ; max : uðhÞ ¼ wðhÞ P 0; h 2 ½s2 ; 0; wð0Þ > 0; where s2 ¼
fs2j ðtÞg:
ð1:7Þ
t2½0;x;j¼1;...;n
The aim of this paper is to give a set of new conditions to guarantee the existence and global stability of the positive periodic 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, some lemmas are needed. Since each x-periodic solution of the equation n X duðtÞ ¼ a0 ðtÞuðtÞ þ g j ðtÞNðt g2j ðtÞÞ dt j¼1
is equivalent to that of the equation
uðtÞ ¼
Z
tþw
Gðt; sÞ
t
where, Gðt; sÞ ¼
n X j¼1
Rs a ðhÞdh R xt 0 ; s 2 ½t; t þ x; t 2 R.
exp exp
!
g j ðsÞNðs g2j ðsÞÞ ds :¼ ðUNÞðtÞ;
0
a0 ðhÞdh1
ð2:1Þ
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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 positive
x-periodic solution of the equation: ( Z t n n n X X X dNðtÞ bj ðtÞ ln Nðt s1j ðtÞÞ cj ðtÞ kj ðt sÞ ln NðsÞds dj ðtÞðln Nðt g1j ðtÞÞÞ0 ¼ NðtÞ rðtÞ aðtÞ ln NðtÞ dt 1 j¼1 j¼1 j¼1 ! ) Z ts2j ðtÞþw n n X X fj ðtÞ G t s2j ðtÞ; s g j ðsÞNðs g2j ðsÞÞ ds : ð2:2Þ ts2j ðtÞ
j¼1
j¼1
Taking the transformation N(t) = ey(t), then (2.2) can be rewritten as
y0 ðtÞ ¼ rðtÞ aðtÞyðtÞ
n X j¼1
n X
fj ðtÞ
Z
ts2j ðtÞþw
ts2j ðtÞ
j¼1
bj ðtÞyðt s1j ðtÞÞ
n X
cj ðtÞ
Z
t
kj ðt sÞyðsÞds 1
j¼1
n X
dj ðtÞy0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
! n X yðsg2j ðsÞÞ ds: G t s2j ðtÞ; s g j ðsÞe
ð2:3Þ
j¼1
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 Let E be a Banach space. For a bounded subset A E, denote the Kuratoskii measure of non-compactness:
ðtÞ
.
aE ðAÞ ¼ inffd > 0j There is a finite number of subsets fAi g A such that A ¼ [ðAi Þ and diamðAi Þ 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 continuous and bounded map N : X ! Y is called k-set contractive if for any bounded set A X, we have
aY ðNðAÞÞ 6 kaX ðAÞ; 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 : r aX ðAÞ 6 aY ðLðAÞÞ; 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 is called a k-set contractive with k < l(L), where X X is bounded, open and symmetric about 0 2 X. 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 C x ¼ fxjx 2 CðR; Rn Þ; xðt þ xÞ ¼ xðtÞg with the norm defined by kxk = jxj0 = maxt2[0,x]{jx(t)j}, and
X ¼ C 1x ¼ fxjx 2 C 1 ðR; Rn Þ; xðt þ xÞ ¼ xðtÞg with the norm defined by jxj1 = max{jxj0, jx0 j0}. Then both Cx and C 1x are Banach spaces. We also denote:
¼ 1 h
Z
x
x
0
jhðsÞjds; ðhÞm ¼ min hðtÞ; ðhÞM ¼ max hðtÞ: t2½0;x
t2½0;x
Let L : C 1x ! C x defined by Ly = y0 (t) and N : C 1x ! C x defined by
Ny ¼ aðtÞyðtÞ
n X
bj ðtÞyðt s1j ðtÞÞ
j¼1
n X j¼1
fj ðtÞ
Z
ts2j ðtÞþw
ts2j ðtÞ
n X
cj ðtÞ
j¼1
Z
t
kj ðt sÞyðsÞds 1
! n X yðsg2j ðsÞÞ G t s2j ðtÞ; s g j ðsÞe ds:
n X
dj ðtÞy0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
j¼1
Thus (2.3) has a positive x-periodic solution if and only if Ly = Ny + r for some y 2 C 1x , 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.
ð2:4Þ
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R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Lemma 2.3. Let c0, c1 be two positive constants, and X ¼ fxjx 2 C 1x ; jxj0 < c0 ; jx0 j0 < c1 g, if k ¼ X ? Cx is a k-set contractive map.
Pn
j¼1 kdj ð1
g01j Þk, then N :
Proof. Let A X be a bounded subset and let g ¼ aC 1x ðAÞ. 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 ðUey ÞðtÞ; Fðt; x; y1 ; . . . ; yn ; z1 ; . . . ; zn ; w1 ; . . . ; wn ; v 1 ; . . . ; v n Þ ¼ aðtÞx þ
n X
bj ðtÞyj þ
j¼1
n X
cj ðtÞzj þ
j¼1
n X
dj ðtÞwj ð1 g01j ðtÞÞ þ
j¼1
n X
fj ðtÞv j :
j¼1
For convenience, in the following discussion we denote
J i ðxÞðtÞ ¼
Z
t
1
ki ðt sÞxðsÞds; for x 2 C 1x ;
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 are precompact in Cx, it follows that there is a finite family of subsets Aij of Ai such that Ai = [jAij with
jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6 e; for any x; u 2 Aij : Therefore, we have
kNx Nuk ¼ sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
x0 ðt g11 ðtÞÞ; . . . ; x0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6 sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
x0 ðt g11 ðtÞÞ; . . . ; x0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞj þ sup jFðt; xðtÞ; xðt s11 ðtÞÞ; . . . ; xðt s1n ðtÞÞ; J 1 ðxÞðtÞ; . . . ; J n ðxÞðtÞ; t2½0;x
0
u ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHxÞðt s21 ðtÞÞ; . . . ; ðHxÞðt s2n ðtÞÞÞ Fðt; uðtÞ; uðt s11 ðtÞÞ; . . . ; uðt s1n ðtÞÞ; J 1 ðuÞðtÞ; . . . ; J n ðuÞðtÞ; u0 ðt g11 ðtÞÞ; . . . ; u0 ðt g1n ðtÞÞ; ðHuÞðt s21 ðtÞÞ; . . . ; ðHuÞðt s2n ðtÞÞÞj 6
n X
kdj ð1 g01j Þk jx0 ðt g1j ðtÞÞ u0 ðt g1j ðtÞÞj þ e 6 kðg þ eÞ þ e:
j¼1
As e is arbitrarily small, it is easy to get that aC x ðNðAÞÞ 6 kaC 1x ðAÞ. The proof is complete. h Lemma 2.4 [7]. Suppose s 2 C 1x 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 6 x be a constant, s(t) 2 Cx, such that maxt2[0,x]js(t)j 6 a. Then for 8x 2 C 1x , we have Rx Rx jxðtÞ xðt sðtÞÞj2 dt 6 2a2 0 jx0 ðtÞj2 dt. If, in addition, sðtÞ 2 C 1x , and s0 (t) < 1, t 2 R, then for x 2 C 1x , we have the following 0 conclusions: (1) There exists a unique integer m such that l = js(t) mxj0 < x; Rx 2 Rx (2) 0 jxðtÞ xðt sðtÞÞj2 dt 6 2l 0 jx0 ðtÞj2 dt.
7696
As
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
s01j ðtÞÞ < 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
0
2
jxðtÞ xðt s1j ðtÞÞj2 dt 6 2lj
Z
x
jx0 ðtÞj2 dt;
x 2 C 1x :
0
ð2:5Þ
Lemma 2.6 [15] . Suppose x(t) is a differently continuous x-periodic function on R with x > 0. Rx maxt 6t6ðt þxÞ jxðtÞj 6 jxðt Þj þ 12 0 jx0 ðtÞjdt.
Then to any t 2 R;
3. Existence of periodic solution Since s01j ðtÞ < 1; g01j ðtÞ < 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:
C1 ðtÞ ¼ aðtÞ þ
n X j¼1
CðtÞ ¼ C1 ðtÞ þ
n X
0
n X bj ðl1j ðtÞÞ dj ðc1j ðtÞÞ ; 0 1 s1j ðl1j ðtÞÞ j¼1 1 g01j ðc1j ðtÞÞ
cj ðtÞ;
j¼1
A¼
Z
1
x
n X
x
0
I ¼ max
fj ðtÞ
t2½0;x
ts2j ðtÞþx
r ; C þ A n X
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
j¼1
Z
Z
H ¼ ln
ts2j ðtÞþx
! g j ðsÞ dsdt;
j¼1
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
n X
! g j ðsÞ ds:
j¼1
Theorem 3.1. If in addition to (H1)–(H2) assume further that: (H3):
r > 0; cj ðtÞ P 0; C1 ðtÞ > 0; t 2 ½0; x;
þ and a
n X
þ b j
j¼1
n X
cj > 0;
j¼1
(H4):
r A
CþA
6 ln
r
CþA
;
ln
r þ C r P ; CþA CþA
(H5):
2 1 2
B , 42
n X
kbj klj þ
j¼1
n X
1
1 2
k1 g01j k kdj k þ
j¼1
x2 2
Z
x
0
n X j¼1
!1 !12 312 n 2 X x 0 2 0 5 jcj ðtÞj dt þ pffiffiffi ka k þ bj < 1; 2 2 j¼1
(H6): n X
kdj kk1 g01j k < 1:
j¼1
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 kNu þ kr;
k 2 ð0; 1Þ:
ð3:1Þ
Then u(t) satisfies the following equation
" 0
u ðtÞ ¼ k rðtÞ aðtÞuðtÞ
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
n X j¼1
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
n X
cj ðtÞ
Z
j¼1
fj ðtÞ
kj ðt sÞuðsÞds
1
j¼1 n X
t
Z
ts2j ðtÞþw
ts2j ðtÞ
Gðt s2j ðtÞ; sÞ
n X j¼1
# g j ðsÞeuðsg2j ðsÞÞ Þds :
ð3:2Þ
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
7697
Integrating (3.2) on [0, x], we have
Z
rx ¼
"
x
n X
aðtÞuðtÞ þ
0 n X
þ
fj ðtÞ
cj ðtÞ
þ
n X
Gðt s2j ðtÞ; sÞ
x
n X
t
kj ðt sÞuðsÞds
0
fj ðtÞ
Z
g j ðsÞe
t
kj ðt sÞuðsÞds þ 1
! #
uðsg2j ðsÞÞ
n X
ds dt ¼
Z
n X
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞ
j¼1
x
"
aðtÞuðtÞ þ
0
#
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
0
dj ðtÞuðt g1j ðtÞÞ dt
j¼1
ts2j ðtÞþw
ts2j ðtÞ
j¼1
Z
j¼1
1
j¼1
Z
ts2j ðtÞþw
Z
cj ðtÞ
j¼1
ts2j ðtÞ
n X
n X
j¼1
Z
j¼1
þ
bj ðtÞuðt s1j ðtÞÞ þ
! n X uðsg2j ðsÞÞ dsdt: G t s2j ðtÞ; s g j ðsÞe
ð3:3Þ
j¼1
Let t s1j(t) = s, then t = l1j(s), and
Z
Z
x
bj ðtÞuðt s1j ðtÞÞdt ¼
xs1j ðxÞ
s1j ð0Þ
0
bj ðl1j ðsÞÞ uðsÞds: 1 s01j ðl1j ðsÞÞ
From Lemma 2.4, it follows that
Z
Z
x
bj ðtÞuðt s1j ðtÞÞdt ¼
0
x
0
bj ðl1j ðsÞÞ uðsÞds: 1 s01j ðl1j ðsÞÞ
ð3:4Þ
Similarly, we have
Z
x
Z
0
dj ðtÞuðt g1j ðtÞÞdt ¼
0
x
0
0
dj ðc1j ðsÞÞ uðsÞds: 1 g01j ðc1j ðsÞÞ
ð3:5Þ
Combining (3.3)–(3.5), we have
Z
x
C1 ðtÞuðtÞdt ¼
Z
0
x
" aðtÞuðtÞ þ
0
n X
bj ðtÞuðt s1j ðtÞÞ
j¼1
n X
# 0
dj ðtÞuðt g1j ðtÞÞ dt:
ð3:6Þ
j¼1
From Lemma 2.4, we get
l1j ðxÞ ¼ l1j ð0Þ þ x; c1j ðxÞ ¼ c1j ð0Þ þ x; j ¼ 1; . . . ; n; then Z
x
0
Z 0
x
Z x Z l1j ðxÞ Z l1j ð0Þþx bj ðtÞð1 s01j ðl1j ðtÞÞÞ bj ðl1j ðtÞÞ x; b ðtÞdt ¼ bj ðtÞdt ¼ b dt ¼ dt ¼ j j 1 s01j ðl1j ðtÞÞ 1 s01j ðl1j ðtÞÞ l1j ð0Þ l1j ð0Þ 0 Z x Z c1j ðxÞ 0 0 dj ðtÞð1 g01j ðc1j ðtÞÞÞ dj ðc1j ðtÞÞ 0 dt ¼ dj ðtÞdt ¼ 0: dt ¼ 0 1 g01j ðc1j ðtÞÞ 1 g1 j ðc1j ðtÞÞ c1j ð0Þ 0
Rx
P
C1 ðtÞdt ¼ a þ nj¼1 bj x; ! Z x Z x Z x X n Cx ¼ CðtÞdt ¼ C1 ðtÞdt þ cj ðtÞ dt ¼
Then C1 x ¼
0
0
0
By (H3), we have C1(t) > 0,
rx ¼
Z
x
C1 ðtÞuðtÞdt þ
0
0
Pn
j¼1 c j ðtÞ
Z 0
Z
ts2j ðtÞþw
x
n X
n X
j¼1
bj þ
j¼1
n X
! cj x:
ð3:7Þ
j¼1
P 0, t 2 [0, x]. From (3.3)
cj ðtÞ
Z
t
kj ðt sÞuðsÞdsdt þ
n X
! g j ðsÞe
Z 0
1
j¼1
G t s2j ðtÞ; s
ts2j ðtÞ
þ a
uðsg2j ðsÞÞ
x
n X
fj ðtÞ
j¼1
dsdt
j¼1
P ðuÞm Cx þ eðuÞm Ax P ðuÞm Cx þ ð1 þ ðuÞm ÞAx: Then r P ðuÞm C þ ð1 þ ðuÞm ÞA, we obtain
ðuÞm 6
r A r 6 : CþA CþA
In the same manner, we get
rx 6 ðuÞM Cx þ eðuÞM Ax 6 eðuÞM xðC þ AÞ:
ð3:8Þ
7698
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Then r 6 eðuÞM ðC þ AÞ, we obtain
ðuÞM P ln
r
CþA
ð3:9Þ
:
By Lemma 2.6, (H4), (3.8) and (3.9), we have
uðtÞ 6 ðuÞm þ
Z
1 2
uðtÞ P ðuÞM
1 2
x
ju0 ðsÞjds P ln
0
Z
x
ju0 ðsÞjds 6 ln
0
r 1 CþA 2
Z
ju0 ðsÞjds; 0
Z
r 1 þ CþA 2
x
x
ju0 ðsÞjds:
0
So
kuk 6 H þ
1 2
Z
x
ju0 ðsÞjds:
Also, from (3.3), we have
rx ¼
Z
Z
x
C1 ðtÞuðtÞdt þ
0
ð3:10Þ
0
x
0
Z
ts2j ðtÞþw
n X
cj ðtÞ
Z
kj ðt sÞuðsÞdsdt þ
Z
G t s2j ðtÞ; s
n X
ts2j ðtÞ
x
0
1
j¼1
t
! uðsg2j ðsÞÞ
g j ðsÞe
n X
fj ðtÞ
j¼1
dsdt
j¼1
6 ðuÞM Cx þ eðuÞM Ax ¼ ððuÞM þ 1ÞCx Cx þ eðuÞM Ax 6 eðuÞM xðC þ AÞ Cx: Then r 6 eðuÞM ðC þ AÞ C, from (H4), we obtain
ðuÞM P ln
r þ C r P : CþA CþA
ð3:11Þ
Multiplying both sides of (3.2) by u0 (t) and integrating them over [0, x], we have
Z 0
x
Z Z x x n Z x X ju ðtÞj dt 6 rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt 0
2
0
n X
Z
x
0
x
0
j¼1
1
j¼1
Z
n X
dj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞu0 ðtÞdt fj ðtÞ
Z
ts2j ðtÞþw
Gðt s2j ðtÞ; sÞ
ts2j ðtÞ
j¼1
n X j¼1
! ! g j ðsÞeuðsg2j ðsÞÞ ds u0 ðtÞdt
Z Z x x n Z x X 6 rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt
0
j¼1
n Z X
x
j¼1
þ
n X
1
dj ðtÞu ðt g1j ðtÞÞð1 g
0
jfj j0 Ie
Hþ12
Rx 0
0 0 1j ðtÞÞu ðtÞdt
0
ju0 ðsÞjds
Z
x
u0 ðtÞdt
0
j¼1
Z Z x x n Z x X rðtÞu0 ðtÞdt aðtÞuðtÞu0 ðtÞdt bj ðtÞuðt s1j ðtÞÞu0 ðtÞdt ¼ 0 0 0 j¼1 Z t
Z xX n cj ðtÞ kj ðt sÞuðsÞds u0 ðtÞdt
0
j¼1
n Z X
x
j¼1
0
1
dj ðtÞu0 ðt g1j ðtÞÞ 1 g01j ðtÞ u0 ðtÞdt
7699
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
Z
x
6
jrðtÞj2 dt
12 Z
0
þ
kbj k
Z
Z
n X
x
0
ju0 ðtÞj2 dt Z
x
x
6
jcj ðtÞ
2
kj ðt sÞuðsÞdsj dt
ju0 ðtÞj2 dt
jrðtÞj2 dt Z
n X
12 Z
x
12
2
0
jrðtÞj2 dt
x
12
Z
Using the notation ku kL2 ¼
0
12
ju0 ðtÞj2 dt
þ
x
Z
x
0
2
0
juðtÞj2 dt þ
ju ðtÞj dt
12
Z
þ
x
0
ju0 ðtÞj2 dt
212
n X
!12 jcj ðtÞj2 dt
12
Z x n 1X 0 kbj k juðtÞj2 dt 2 j¼1 0 n X
!12 2
jcj ðtÞj dt
Z kuk
x
ju0 ðtÞj2 dt
3 Z kuk5
ju0 ðtÞj2 dt
x
ju0 ðtÞj2 dt
0
j¼1
x
12
0
j¼1
" #Z n n X pffiffiffi X 1 ¼ 2 kbj klj þ k1 g01j ðtÞk2 kdj k
12 þ
0
j¼1
!Z n x X 1 0 ka0 k þ kbj k juðtÞj2 dt: 2 0 j¼1 ð3:12Þ
12
2
0
12
j¼1
0
x
x
0
0
R
x
0
Z
j¼1
0
!12 Z
1 þ ka0 k 2
12 pffiffiffi Z ju ðtÞj dt 2l j
x
1
x
0
Z x n 1X 0 kbj k juðtÞj2 dt 2 j¼1 0
jdj ðtÞu0 ðt g1j ðtÞÞð1 g01j ðtÞÞj2 dt
ju0 ðtÞj2 dt
k1 g01j ðtÞk2 kdj k
2 Z þ4
ku0 k2L2
x
0
j¼1
juðtÞj2 dt þ
0
12 Z 0
kbj k
x
juðtÞ uðt s1j ðtÞÞj2 dt
0
n X
Z
1
0
þ
x
1 þ ka0 k 2
t
0
j¼1
þ
12
0
j¼1
n Z X
Z
x
12 Z
0
j¼1
þ
ju0 ðtÞj2 dt
0
n X
þ
x
ju ðtÞj dt , by
Rx 0
2
2
juðtÞj dt 6 kuk x (3.10) and (3.12), we have
! n n X pffiffiffi X 1 0 2 2 kbj klj þ k1 g1j ðtÞk kdj k ku0 k2L2
6
2 þ4
þ
j¼1
Z
j¼1
x
2
jrðtÞj dt
12 þ
Z
0
x 2
x
0
ka0 k þ
n X
0
!12 2
jcj ðtÞj dt
1
Hþ
j¼1
!
j¼1
x2 2
x2 2
!3 ku kL2 5ku0 kL2 0
!2
1
Hþ
kbj k
n X
ku0 kL2
2 !12 3 Z xX 1 n n n X pffiffiffi X 1 x2 2 0 4 2 ¼ 2 kbj klj þ k1 g1j ðtÞk kdj k þ jcj ðtÞj dt 5 ku0 k2L2 2 0 j¼1 j¼1 j¼1 2 !2 !12 3 ! Z x
12 Z xX 1 n n X x x2 0 0 2 2 0 0 4 5 þ jrðtÞj dt þ jcj ðtÞj dt H ku kL2 þ kbj k Hþ ka k þ ku kL2 : 2 2 0 0 j¼1 j¼1
ð3:13Þ
From a2 6 b2 + c2 + d2 ) a 6 b + c + d, where a, b, c, d are all non-negative. Then, we get
2
!12 312 Z xX 1 n n n X pffiffiffi X 2 1 x 2 0 ku kL2 6 4 2 kbj klj þ k1 g1j ðtÞk2 kdj k þ jcj ðtÞj dt 5 ku0 kL2 2 0 j¼1 j¼1 j¼1 0
2 þ4
Z
x
jrðtÞj2 dt
12 þ
Z
0
2 Z 0 ¼ Bku kL2 þ 4
x
0
x
2
jrðtÞj dt 0
n X j¼1
12 þ
! !12 312 " !#12 1 n X 1 x x2 0 0 2 0 2 0 5 jcj ðtÞj dt H ku kL2 þ kbj k Hþ ka k þ ku kL2 2 2 j¼1
Z
x 0
n X j¼1
!12 312 !12 1 n X 1 2 x 0 0 0 2 jcj ðtÞj dt H5 ku kL2 þ pffiffiffi ka k þ kbj k H: 2 j¼1 2
ð3:14Þ
7700
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
By (H5), then there exists a constant M > 0 such that ku0 kL2 6 M, that is ity, we obtain
1 2
kuk 6 H þ
Z
1
x
ju0 ðtÞjdt 6 H þ
x2
Z
2
0
x
ju0 ðtÞj2 dt
12
R x 0
ju0 ðtÞj2 dt
12
6 M. From (3.10) and Hölder inequal-
1
6Hþ
x2
0
2
M :¼ M1 :
ð3:15Þ
Again from (3.2), we get
ku0 k 6 krk þ kakkuk þ
n X
kbj kkuk þ
j¼1
n X
kcj kkuk þ
n X
j¼1
kdj kk1 g01j kku0 k þ
j¼1
n X
kfj keM1 I:
ð3:16Þ
j¼1
From (H6) and (3.15), we have
P P P krk þ kak þ nj¼1 kbj k þ nj¼1 kcj k M 1 þ nj¼1 kfj keM1 I 0 P :¼ M 2 : ku k 6 1 nj¼1 kdj kk1 g01j k
ð3:17Þ
n n oo P r . Then k ¼ nj¼1 kdj ð1 g01j Þk < lðLÞ. Defined a bounded bilinear form [, ] on Let X ¼ xjx 2 C 1x ; jxj1 > max M 1 ; M 2 ; CþA R Rx x C x C 1x by ½y; x ¼ 0 yðtÞxðtÞdt. Also we define Q : Y ? Coker(L) by y ! 0 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 Q ðrÞ; u ½QNðuÞ þ Q ðrÞ; u "Z ! Z x n n x X X 2 rðtÞdt r0 aðtÞ þ bj ðtÞ þ cj ðtÞ dt ¼ r0 0
e
r0
0
Z
"Z
n x X
0
fj ðtÞ
Z
j¼1
ts2j ðtÞþw
x
rðtÞdt ðr 0 Þ
0
er0
"
G t s2j ðtÞ; s
Z
0
n X
fj ðtÞ
Z
x
aðtÞ þ
¼ r 20 x2 r r 0 a þ " < r 20 x2 r r 0 a þ
n X
bj þ
j¼1
g j ðsÞ dsdt
bj þ
j¼1
n X
bj ðtÞ þ
#
! cj ðtÞ dt
j¼1
! # n X G t s2j ðtÞ; s g j ðsÞ dsdt
n X
!
n X
#
! cj
ð3:18Þ
j¼1
#
"
er0 A r ðr 0 Þ a þ
cj
j¼1
n X
n X j¼1
ts2j ðtÞþw
ts2j ðtÞ
j¼1
!
j¼1
0
x
n X
ts2j ðtÞ
j¼1
Z
j¼1
"
r 0 A r þ r0 a þ
j¼1
< r 20 x2 r r 0 C þ A r þ r 0 C þ A :
n X
bj þ
j¼1 n X j¼1
bj þ
n X
! cj
# er0 A
j¼1 n X
#
!
cj
þ r0 A
j¼1
r r r r r Since r 0 > CþA , and r0 < CþA . , then r 0 > CþA P CþA 6 CþA Thus
r r0 ðC þ AÞ < 0; r þ r0 ðC þ 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 xperiodic 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 functional introduced 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 stable if any other solution (N(t), u(t))T of (1.6) and (1.7) has the property:
lim jN ðtÞ NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
Now we state our main results of this section below.
ð4:1Þ
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
7701
Theorem 4.1. Assume that the conditions in Theorem 3.1 hold. Moreover, if there is a positive constant k such that:
8 ( " #) n n R > P P bj ðl1j ðtÞÞ > þ1 > inf k aðtÞ kj ðsÞcj ðt þ sÞds > 0; > > 0 1s01j ðl1j ðtÞÞ > t2½0;þ1Þ > j¼1 j¼1 > > ( " # ) > > n > P < fj ðl2j ðtÞÞ > 0; k a0 ðtÞ inf 1s02j ðl2j ðtÞÞ t2½0;þ1Þ j¼1 > > > R þ1 gj ðc2j ðsÞÞ > > ds < þ1; j ¼ 1; . . . ; n; > > 0 1g02j ðc2j ðsÞÞ > > > R > d ð c ðsÞÞ þ1 j 1j > : ds < þ1; j ¼ 1; . . . ; n: 0 1g0 ðc ðsÞÞ 1j
ð4:2Þ
1j
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 only need 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 a Lyapunov functional V(t) = V(t, (N⁄(t), u⁄(t))T, (N(t), u(t))T) defined by
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ þ V 6 ðtÞ; for t P 0; where
V 1 ðtÞ ¼ kðj ln N ðtÞ ln NðtÞj þ ju ðtÞ uðtÞjÞ; n Z t X bj ðl1j ðsÞÞ j ln N ðsÞ ln NðsÞjds; V 2 ðtÞ ¼ k s01j ðl1j ðsÞÞ 1 ts1j ðtÞ j¼1 Z t n Z þ1 X kj ðsÞ cj ðh þ sÞj ln N ðhÞ ln NðhÞjdhds; V 3 ðtÞ ¼ k j¼1
V 4 ðtÞ ¼ k
j¼1
V 5 ðtÞ ¼ k
þ1
tg2j ðtÞ
n Z X j¼1
ts t
ts2j ðtÞ
n Z X j¼1
V 6 ðtÞ ¼ k
0
n Z X
þ1
tg1j ðtÞ
fj ðl2j ðsÞÞ ju ðsÞ uðsÞjds; 1 s02j ðl2j ðsÞÞ g j ðc2j ðsÞÞ jN ðsÞ NðsÞjds; 1 g02j ðc2j ðsÞÞ dj ðc1j ðsÞÞ jðln N ðsÞÞ0 ðln NðsÞÞ0 jds: 1 g01j ðc1j ðsÞÞ
From the definition of V(t), it is easy to see that
Vð0Þ < þ1
ð4:3Þ
and
VðtÞ P 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
Dþ VðtÞ 6 kaðtÞj ln N ðtÞ ln NðtÞj Z n n X X þk bj ðtÞj ln N ðt s1j ðtÞÞ ln Nðt s1j ðtÞÞj þ k cj ðtÞ j¼1
þk
n X
0
dj ðtÞjðln N ðt g1j ðtÞÞÞ ðln Nðt g1j ðtÞÞÞ j þ k
j¼1
kj ðt sÞj ln N ðsÞ ln NðsÞjds
1
j¼1 0
t
n X
fj ðtÞju ðt s2j ðtÞÞ uðt s2j ðtÞÞj
j¼1
ka0 ðtÞju ðtÞ uðtÞj þ k
n X
g j ðtÞjN ðt g2j ðtÞÞ Nðt g2j ðtÞÞj
j¼1 n X bj ðl1j ðtÞÞ bj ðtÞj ln N ðt s1j ðtÞÞ ln Nðt s1j ðtÞÞj þk j ln N ðtÞ ln NðtÞj k 0 s ð l ðtÞÞ 1 1j 1j j¼1 j¼1 n Z þ1 n Z þ1 X X þk kj ðsÞcj ðt þ sÞj ln N ðtÞ ln NðtÞjds k kj ðsÞcj ðtÞj ln N ðt sÞ ln Nðt sÞjds n X
j¼1
0
j¼1
n X
0
n X fj ðl2j ðtÞÞ þk fj ðtÞju ðt s2j ðtÞÞ uðt s2j ðtÞÞj ju ðtÞ uðtÞj k 0 s ð l ðtÞÞ 1 2j 2j j¼1 j¼1 n n X X k g j ðtÞjN ðt g2j ðtÞÞ Nðt g2j ðtÞÞj k dj ðtÞjðln N ðt g1j ðtÞÞÞ0 ðln Nðt g1j ðtÞÞÞ0 j j¼1
¼ S1 j ln N ðtÞ ln NðtÞj S2 ju ðtÞ uðtÞj;
j¼1
7702
R. Wang, X. Zhang / Applied Mathematics and Computation 217 (2011) 7692–7702
where,
"
n X
n X bj ðl1j ðtÞÞ 0 1 s1j ðl1j ðtÞÞ j¼1 j¼1 " # n X fj ðl2j ðtÞÞ : S2 ¼ k a0 ðtÞ 1 s02j ðl2j ðtÞÞ j¼1
S1 ¼ k aðtÞ
Z
#
þ1
kj ðsÞcj ðt þ sÞds ;
0
From (4.2), it follows that there exists a constant K > 0 such that
S1 > K;
S2 > K:
Hence, it follows that
Dþ VðtÞ < K ðj ln N ðtÞ ln NðtÞj þ ju ðtÞ uðtÞjÞ:
ð4:5Þ
Then, by using (4.3) and (4.5) and the analysis of that in [2, p. 816], one could obtain:
lim j ln N ðtÞ ln NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
From this, one could easily obtain:
lim jN ðtÞ NðtÞj ¼ 0;
t!þ1
lim ju ðtÞ uðtÞj ¼ 0:
t!þ1
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 and deviating 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, Kluwer Academic 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.