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Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control
Applied Mathematics and Computation 158 (2004) 45–68 www.elsevier.com/locate/amc
Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control Fengde Chen *, Faxin Lin, Xiaoxin Chen Department of Mathematics, Fuzhou University, Fuzhou, Fujian 350002, People’s Republic of China
Abstract With the help of a continuation theorem based on Gaines and MawhinÕs coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a neutral model with periodic delays and feedback control 8 n n P P > > dn ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ ci ðtÞn0 ðt ci ðtÞÞ > dt > > i¼1 i¼1 > < n P dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; > i¼1 > > n > P > > : du ¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ gi ðtÞnðt gi ðtÞÞ: dt i¼1
Our results extend and improve existing results, and have further applications in population dynamics. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Positive periodic solutions; Neutral delay model; Feedback control; Coincidence degree
1. Introduction In the recent years, the application of theories of functional differential equations in mathematical ecology has been developed rapidly. Various
*
Corresponding author. E-mail addresses: [email protected], [email protected] (F. Chen).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.063
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
mathematical models with delays have been proposed in the study of population dynamics, ecology and epidemic. Recently, by using the powerful technique––a continuation theorem based on Gaines and MawhinÕs coincidence degree, many good results concerned with the global existence of positive periodic solutions of the biological model are obtained (see Refs. [1–10]). However, only a few papers (except Hopf bifurcations) have been published on the existence of periodic solutions of the neutral delay population models. In particular, Gopalsamy et al. [11,13] have established the existence of a positive periodic solution for a periodic neutral delay logistic equation N ðt mxÞ þ cðtÞN 0 ðt mxÞ 0 N ðtÞ ¼ rðtÞN ðtÞ 1 ; ð1:1Þ KðtÞ where KðtÞ, rðtÞ, cðtÞ are positive continuous T -periodic functions with T > 0, and m is a positive integer. In 1993, Kuang [14] proposed the following open problem (Open Problem 9.2): How to obtain sufficient conditions for the existence of a periodic solution for equation N 0 ðtÞ ¼ N ðtÞ½aðtÞ bðtÞN ðtÞ bðtÞN ðt sðtÞÞ cðtÞN 0 ðt sðtÞÞ;
ð1:2Þ
where aðtÞ, bðtÞ, bðtÞ, cðtÞ are non-negative continuous T -periodic functions. Li Yongkun [15] tried to give an affirmative answer to the above open problem, however, there is a mistake in the proof of Theorem 2 in [15]. With the aim of giving a right answer to the above open problem, Li and Cao [16], Fang and Li [17] and Lu and Ge [18] have investigated the above question. However, it is more complex to check the sufficient conditions of [17,18]. Moreover, Li [37] studied the existence of positive periodic solution of the neutral Lotka–Volterra equation with several delays " # n n X X N 0 ðtÞ ¼ N ðtÞ aðtÞ bi ðtÞN ðt si Þ ci ðtÞN 0 ðt ci Þ ; ð1:3Þ i¼1
i¼1
where aðtÞ, bi ðtÞ, ci ðtÞ are positive continuous T -periodic functions and si ; ci ði ¼ 1; 2; . . . ; nÞ are non-negative constants. Recently, Yang and Cao [19] and Lu [38] further investigate the existence of positive periodic solution of the neutral delays models " # n n X X 0 0 N ðtÞ ¼ N ðtÞ aðtÞ bðtÞN ðtÞ bi ðtÞN ðt si ðtÞÞ ci ðtÞN ðt ci ðtÞÞ ; i¼1
i¼1
ð1:4Þ where aðtÞ, bi ðtÞ, ci ðtÞ, si ðtÞ, ci ðtÞ are non-negative continuous T -periodic functions and i ¼ 1; 2; . . . ; n, T is a positive constant. For more works on this direction, one could refer to [12,20,21,33–35].
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
47
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 for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at 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, see [36]. Recently, many scholars focus on the existence and global attractivity of the positive periodic solution of the system. Xiao et al. [25] investigate the two species non-autonomous competition system with feedback control, by using differential inequality and Lyapunov function, they obtained sufficient condition which guarantee the existence and global attractivity of the periodic solution of the system. Weng [26] further investigate the system with constant delay. For more works on this direction, one could refer to [22–29,31,32]. To the best of the authors knowledge, although there are many works concerned with the existence of positive equilibrium of the neutral delay population dynamics with feedback control (see, for example [36]), no works have been done to investigate the existence of positive periodic solution of the non-autonomous neutral delay population dynamics with feedback control. With the aim of it, in this paper, we will investigate the following neutral delay models with feedback control of the form 8 " n n X X > dn > > ¼ nðtÞ aðtÞ bðtÞnðtÞ > bi ðtÞnðt si ðtÞÞ ci ðtÞn0 ðt ci ðtÞÞ > > dt > i¼1 i¼1 > # > < n P dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; > > i¼1 > > > n X > > du > ¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ gi ðtÞnðt gi ðtÞÞ; > : dt i¼1
ð1:5Þ where aðtÞ, bðtÞ, bi ðtÞ, ci ðtÞ, dðtÞ, di ðtÞ, eðtÞ, f ðtÞ, gi ðtÞ, si ðtÞ, ci ðtÞ, gi ðtÞ are nonnegative continuous T -periodic functions and i ¼ 1; 2; . . . ; n, T is a positive constant. When f ðtÞ ¼ 0, gi ðtÞ ¼ 0, dðtÞ ¼ 0, di ðtÞ ¼ 0, i ¼ 1; 2; . . . ; n, (1.5) reduces to (1.4). Comparing the systems (1.4) and (1.5), one could see that we introduce the control variables uðtÞ so as to implement a feedback control mechanism. The main purpose of this paper is to derive a set of sufficient conditions for the global existence of positive periodic solution of (1.5). Obviously, Eq. (1.5) is a generalization of Eqs. (1.1)–(1.4), therefore, by using the theory of the continuation theorem of the coincidence degree theory which proposed in [30] by Mawhin, we shall obtain several sufficient conditions for existence of a periodic solution of Eq. (1.5) in this paper.
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
2. Notation and lemma In order to obtain the existence of positive periodic solutions of (1.5), for the readerÕs convenience, we shall summarize in the following a few concepts and results from [30] that will be basic for this paper. Let X , Z be normed vector spaces, L : DomL X ! Z be a linear mapping, N : X ! Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL ¼ CodimImL < þ1 and ImL is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projectors P : X ! X and Q : Z ! Z such that ImP ¼ KerL; ImL ¼ KerQ ¼ ImðI QÞ. It follows that LjDomL \ KerP : ðI P ÞX ! ImL is invertible. We denote the inverse of that map by KP . If X be an open bounded subset of X , the mapping N if QN ðXÞ is bounded and KP ðI QÞN : X ! X is will be called L-compact on X compact. Since ImQ is isomorphic to KerL, there exists an isomorphisms J : ImQ ! KerL. In the proof of our existence theorem below, we will use the continuation theorem of Gaines and Mawhin [30, p. 40]. Lemma 2.1 (Continuation theorem). Let L be a Fredholm mapping of index zero and let N be L-compact on X. Suppose (a) For each k 2 ð0; 1Þ, every solution x of Lx ¼ kNx is such that x 6¼ @X; (b) QNx ¼ 6 0 for each x 2 @X \ KerL and degfJQN ; X \ KerL; 0g 6¼ 0: Then the equation Lx ¼ Nx has at least one solution lying in DomL \ X. Lemma 2.2. ðnðtÞ; uðtÞÞT is an T -periodic solution of (1.5) if and only if it is also an T -periodic solution of " 8 n X > dn > > ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ > > dt > > i¼1 > # > > n n < P P 0 ci ðtÞn ðt ci ðtÞÞ dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; ð2:1Þ i¼1 i¼1 > > > > n R tþT P > > > f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ Gðt; sÞ ds uðtÞ ¼ t > > > i¼1 : :¼ ðUnÞðtÞ; where Rs expf t eðhÞ dhg Rx Gðt; sÞ ¼ : expf 0 eðhÞ dhg 1
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
49
T
Proof. First, let ðnðtÞ; uðtÞÞ be an T -periodic solution of (1.5). From the second equation of (1.5) and the variation-of-constant formulas, it follows that Z t uðtÞ ¼ uð0Þ exp eðhÞ dh 0 # Z s Z t" n X þ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ exp eðhÞ dh ds: 0
t
i¼1
Then, uðt þ T Þ ¼ uð0Þ exp Z "
Z
t
f ðsÞnðsÞ þ
0
exp þ
Z
Z
tþT
t
exp
eðhÞ dh exp
Z
0
t
þ
Z eðhÞ dh exp
"t
Z
eðhÞ dh ds #
t tþT
f ðsÞnðsÞ þ
n X
s
#
gi ðsÞnðs gi ðsÞÞ
i¼1
s
eðhÞ dh t
n X
tþT
gi ðsÞnðs gi ðsÞÞ
i¼1
Z
t
eðhÞ dh exp eðhÞ dh ds tþT Z T ¼ uðtÞ exp eðhÞ dh 0 # Z tþT " n X þ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ t
t
exp
Z
s
i¼1
eðhÞ dh exp
Z
T
eðhÞ dh ds 0
t
and hence using the periodicity of uðtÞ, we get # Z tþT " n X uðtÞ ¼ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ t
i¼1
n R o T eðhÞ dh exp eðhÞ dh t 0 n R o ds T 1 exp 0 eðhÞ dh " # n X f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ Gðt; sÞ ds:
exp
¼
Z t
tþT
Rs
i¼1
This proves one part of the lemma.
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68 T
Next, let ðnðtÞ; uðtÞÞ be an T -periodic solution of (2.1). Then, " # n X 0 u ðtÞ ¼ f ðt þ T Þnðt þ T Þ þ gi ðt þ T Þnðt þ T gi ðt þ T ÞÞ Gðt; t þ T Þ i¼1
" f ðtÞnðtÞ þ
þ
Z
tþT
n X
#
gi ðtÞnðt gi ðtÞÞ Gðt; tÞ
i¼1
"
f ðsÞnðsÞ þ
t
"
n X
# gi ðsÞnðs gi ðsÞÞ ðeðtÞÞGðt; sÞ ds
i¼1
¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ
n X
# gi ðtÞnðt gi ðtÞÞ ðGðt; t þ T Þ Gðt; tÞÞ
i¼1
¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ
n X
gi ðtÞnðt gi ðtÞÞ:
i¼1
The proof is complete.
h
By Lemma 2.2, in order to show the existence of strictly positive T -periodic solutions of (1.5), we only need to show that (2.1) possesses at least one T periodic solution with strictly positive component. Now, (2.1) can be reformulated as " n X dn ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ dt i¼1 # n n X X 0 ci ðtÞn ðt ci ðtÞÞ dðtÞðUnÞðtÞ di ðtÞðUnÞðt ri ðtÞÞ : i¼1
i¼1
ð2:2Þ Let nðtÞ ¼ expfxðtÞg, then (2.2) is the same as n X dx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ:
ð2:3Þ
i¼1
For the rest of this paper, we shall devote ourselves to study the existence of T -periodic solution of (2.3).
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
51
For convenience, we also use the notations
u ¼ min fuðtÞg; t2½0;T
U0 ¼
n X i¼1
ui0 ;
kuk0 ¼ max fuðtÞg; t2½0;T
v ¼
min vi ðtÞ;
kui k0 ¼ max fui ðtÞg;
and
t2½0;T i2f1;...;ng
t2½0;T
g ¼
1 T
Z
U¼
n X
ui ;
i¼1
T
gðtÞ dt: 0
3. Existence of positive periodic solutions The objective of this section is to derive sufficient conditions for the existence of positive periodic solutions in Eq. (1.5), by use Lemma 2.1 (i.e. continuation theorem). Theorem 3.1. If si ðtÞ ¼ ci ðtÞ, assume that
aðtÞ; bðtÞ; dðtÞ; f ðtÞ; eðtÞ 2 CðR; ð0; þ1ÞÞ; bi ðtÞ; di ðtÞ; gi ðtÞ 2 CðR; ½0; þ1ÞÞ; ci ðtÞ 2 C 1 ðR; ½0; þ1ÞÞ;
si ðtÞ 2 C 2 ðR; ½0; þ1ÞÞ;
ci ðtÞ bi ðtÞ c0i0 ðtÞ ; vi ðtÞ ¼ ; 1 si ðtÞ 1 s0i ðtÞ ( ) kC0 k0 a a þ 2T kak0 : R1 ¼ ln þ h1 ðb þ V Þ h2 vð1 ks0i k0 Þ ci0 ðtÞ ¼
Also assume that s0i ðtÞ < 1, bi ðtÞ c0i0 ðtÞ P 0 and there exist at least one i such that bi ðtÞ > c0i0 ðtÞ, also kCk0 eR1 < 1. Then Eq. (1.5) has at least one positive T -periodic solution. Proof. To finish the proof of the theorem, it is enough to show the existence of T -periodic solution of system (2.3). In order to use Lemma 2.1 to Eq. (2.3), we take X ¼ fxðtÞ 2 C 1 ðR; RÞ; xðt þ T Þ ¼ xðtÞg, and Z ¼ fzðtÞ 2 CðR; RÞ; zðt þ T Þ ¼ zðtÞg, and denote kxk0 ¼ maxt2½0;T jxðtÞj; kxk1 ¼ maxt2½0;T fkxk0 ; kx0 k0 g. Then X and Z are Banach spaces when they are endowed with the k k1 and k k0 , respectively.
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Let L : DomL X ! Z and N : X ! Z defined by Lx ¼ x_ and Nx ¼ aðtÞ bðtÞ expfxðtÞg
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
n X
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ
i¼1
for any x 2 X and two project Z 1 T xðtÞ dt; x 2 X ; Px ¼ T 0
QZ ¼
1 T
Z
T
zðtÞ dt;
z 2 Z:
0
It can be found that KerL ¼ fxjx 2 X ; x ¼ h; h 2 Rg; Z T zðtÞ dt ¼ 0 is closed in Z; ImL ¼ zjz 2 Z; 0
then dimKerL ¼ 1 ¼ codimImL and hence, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that ImP ¼ KerL;
KerQ ¼ ImL ¼ ImðI QÞ:
Furthermore, the generalized inverse (to L) KP : ImL ! KerP \ DomL exists, which is given by Z Z t Z Z t Z t 1 T 1 T zðsÞ ds zðsÞ ds dt ¼ zðsÞ ds þ szðsÞ ds: KP ðzÞ ¼ T 0 T 0 0 0 T Then QN : X ! Z and KP ðI QÞN : X ! X read Z " n X 1 T aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg QNx ¼ T 0 i¼1
n X i¼1
n X i¼1
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ dt:
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
53
ci ðtÞ Since ci0 ðtÞ ¼ 1s 0 ðtÞ, it follows that i Z T" n X 1 QNx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1
n X
ci0 ðtÞð1 s0i ðtÞÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ 1 ¼ T
Z
1 ¼ T
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
aðtÞ bðtÞ expfxðtÞg
0
1 T Z
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ
#
i¼1
"
T
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z
T
n X
"
i¼1
0 T
n X
ci0 ðtÞd expfxðt si ðtÞÞg
aðtÞ bðtÞ expfxðtÞg
0
n X
ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
and followed by some computation, one has Z " t
KP ðI QÞNx ¼
aðsÞ bðsÞ expfxðsÞg
T
Z tX n T
1 þ T
T
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
ci0 ðsÞd expfxðs ci ðsÞÞg
"
s aðsÞ bðsÞ expfxðsÞg 0
n X i¼1
n X
T
n X i¼1
bi ðsÞ expfxðs si ðsÞÞg #
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
s 0
#
i¼1
Z
Z
n X i¼1
dðsÞðU expfxgÞðsÞ
bi ðsÞ expfxðs si ðsÞÞg
i¼1
dðsÞðU expfxgÞðsÞ
n X
ci0 ðsÞd expfxðs ci ðsÞÞg
54
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
1 t þ 2 T
" Z
T
" aðsÞ bðsÞexpfxðsÞg
0
n X
bi ðsÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
Z
n X
T 0
¼
Z t"
# ci0 ðsÞdexpfxðs ci ðsÞÞg
i¼1
aðsÞ bðsÞexpfxðsÞg
T
n X
ðbi ðsÞ c0i0 ðsÞÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
n X i¼1
n X
1 ci0 ðtÞexpfxðt ci ðtÞÞg þ T
Z
T
" s aðsÞ bðsÞexpfxðsÞg
0
ðbi ðsÞ ci0 ðsÞ sc0i0 ðsÞÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
1 t þ 2 T
n X
Z
T
" aðsÞ bðsÞexpfxðsÞg
0
ðbi ðsÞ c0i0 ðsÞÞexpfxðs si ðsÞÞg dðsÞðUexpfxgÞðsÞ
i¼1
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds:
i¼1
Clearly, QN and KP ðI QÞN are continuous. It is not difficult to show that KP ðI QÞN ðXÞ is compact for any open bounded X X by using Arzela– Ascoli theorem. Moreover, QN ðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X X . Now we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
55
(Lemma 2.1). Corresponding to the operator equation Lx ¼ kNx; k 2 ð0; 1Þ, we have " n X dx ¼ k aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ :
ð3:1Þ
i¼1
If xðtÞ 2 X is a solution of Eq. (3.1) for a certain k 2 ð0; 1Þ, then Z
T
" aðtÞ bðtÞ expfxðtÞg
0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
n X
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:2Þ
i¼1
Integrating this identity we have Z " T
aðtÞ bðtÞ expfxðtÞg
0
n X ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg i¼1
dðtÞðU expfxgÞðtÞ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:3Þ
i¼1
That is Z 0
T
" bðtÞ expfxðtÞg þ
n X
ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X i¼1
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt 0
ð3:4Þ
56
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
and hence by using (3.4), one has #0 Z T " n X xðtÞ þ k ci0 ðtÞ expfxðt si ðtÞÞg dt 0 i¼1 Z T n X ¼k ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg aðtÞ bðtÞ expfxðtÞg 0 i¼1 n X di ðtÞðU expfxgÞðt ri ðtÞÞ dt dðtÞðU expfxgÞðtÞ i¼1 Z T Z T" n X ¼k aðtÞdt þ bðtÞ expfxðtÞg þ ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg 0
0
þ dðtÞðU expfxgÞðtÞ þ
i¼1 n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z
T
aðtÞdt 6 2T kak0 :
<2
ð3:5Þ
0
Since xðtÞ 2 X , there exist a n 2 ½0; T such that xðnÞ ¼ min xðtÞ:
ð3:6Þ
t2½0;T
From (3.4) and (3.6), we see that " þ b
n X
# bi c0i0 T expfxðnÞg 6 aT ;
i¼1
that is ( xðnÞ 6 ln
a
þ Pn bi c 0 b i0 i¼1
) :¼ M1 :
ð3:7Þ
On the other hand, there also exists a g 2 ½0; T such that xðgÞ ¼ max xðtÞ: t2½0;T
ð3:8Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
57
Notice that Z
ðU1ÞðtÞ ¼
"
tþT
n X
Gðt; sÞ f ðsÞ þ
t
Z
¼
tþT
f ðsÞ þ
kf k0 þ
Pn
i¼1
kgi k0
Z
i¼1
gi ðsÞ
ds
tþT
Gðt; sÞeðsÞ ds
e kf k0 þ
Pn
eðsÞ
t
¼
gi ðsÞ ds
i¼1
Gðt; sÞeðsÞ
6
#
t
Pn
i¼1
kgi k0
e
ð3:9Þ
:
From (3.4), (3.8) and (3.9) it follows that þ aT 6 b
n X
! bi c0i0
T expfxðgÞg þ kdk0
Z
þ
ðUexpfxgÞðtÞdt
0
i¼1 n X
T
kdi k0
Z
T
ðUexpfxgÞðt ri ðtÞÞdt 0
i¼1
þ 6 b
n X
! bi c0i0
T expfxðgÞg
i¼1
þ kdk0 þ
n X
! kdi k0 expfxðgÞg
n X i¼1
bi c0i0 þ
T
ðU1ÞðtÞdt 0
i¼1
þ 6 b
Z
kdk0 þ
n X i¼1
! jjdi jj0
kf k0 þ
Pn
i¼1 kgi k0
e
! T expfxðgÞg;
that is
xðgÞ P ln
8 > <
9 > =
a P :¼ M2 :
kf k0 þ n kgi k0 > P Pn > n 0 ; :b i¼1 þ b c þ kdk þ kd k i 0 i0 0 i¼1 i i¼1 e ð3:10Þ
58
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
It follows from (3.4) that Z T" n X h1 bðtÞ expfxðtÞg þ ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg 0
i¼1
þ dðtÞðU expfxgÞðtÞ þ þ h2
Z
T
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
"
bðtÞ expfxðtÞg þ 0
n X ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt: 0
i¼1
ð3:11Þ Then similar to the analysis of (3.6), (3.7) and (3.9) of [19], by using the mean value theorem, from (3.11) there exists g0 ; gi ; d0 ; di 2 ½0; T ; i ¼ 1; 2; . . . ; n such that Z T Z T n X aðtÞ dt P h1 ðbðg0 Þ þ vi ðgi ÞÞ expfxðtÞg dt 0
0
i¼1
þ h2 bðd0 sj ðd0 ÞÞð1 s0j ðd0 ÞÞ
Z
T
expfxðt sj ðtÞÞg dt 0
þ
n X
ðbi ðdi Þ
c0i0 ðdi ÞÞ
Z
!
T
expfxðt si ðtÞÞg dt : 0
i¼1
That is a P h1 ðbðg0 Þ þ
n X
vi ðgi ÞÞ expfxð1Þg þ h2 bðd0 sj ðd0 ÞÞð1 s0j ðd0 ÞÞ
i¼1
expfxð1 sj ð1ÞÞg þ
n X
! ðbi ðdi Þ
c0i0 ðdi ÞÞ expfxð1
si ð1ÞÞg
i¼1
for some 1 2 ½0; T . Therefore, we have ( ) ( ) a a P xð1Þ < ln 6 ln ; h1 ðbðg0 Þ þ ni¼1 vi ðgi ÞÞ h1 ðb þ V Þ expfxð1 si ð1ÞÞg 6
ð3:12Þ
a a a 6 6 : h2 ðbi ðdi Þ c0i0 ðdi ÞÞ h2 vi ð1 s0i Þ h2 vð1 ks0i k0 Þ ð3:13Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
59
From (3.5), (3.12) and (3.13) it follows that xðtÞ þ k
n X
ci0 ðtÞ expfxðt si ðtÞÞg
i¼1
6 xð1Þ þ k
n X
ci0 ð1Þ expfxð1 si ð1ÞÞg
i¼1
þ
Z
T
0
( < ln
" #0 n X xðtÞ þ k ci0 ðtÞ expfxðt si ðtÞÞg dt i¼1
a h1 ðb þ V Þ
) þ
kDk0 a def þ 2T kak0 ¼ R1 : h2 vð1 ks0i k0 Þ
Hence, by using (3.9), we have n X 0 bi ðtÞ expfxðt si ðtÞÞg jx ðtÞj ¼ k aðtÞ bðtÞ expfxðtÞg i¼1
n X i¼1
n X i¼1
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ
< K þ kCk0 kx0 ðt si ðtÞÞk0 eR1 ;
where K ¼ kak0 þ kbk0 eR1 þ kBk0 eR1 þ kdk0akbk0 þ kDk0akbk0 eR1 . Therefore, it follows from kCk0 eR1 < 1 that kx0 k0 <
K def ¼ M3 : 1 kCk0 eR1
ð3:14Þ
Also, from (3.7), (3.10) and (3.12) we know that there exist t0 2 ½0; T such that ( ( ) ) a jxðt0 Þj 6 max ln ð3:15Þ ; M1 ; M2 :¼ M4 : h1 ðb þ V Þ Now combining (3.15) and (3.14), we obtain Z t def kxk0 6 jxðt0 Þj þ kx0 k0 dt < M4 þ M3 T ¼ M5 : 0
60
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
When x 2 R, from the condition of the theorem it follows " # Z Z tþT " n n X X 1 T x bi þ dðtÞ f ðsÞ þ gi ðsÞ Gðt; sÞ ds dt QNx ¼ ae bþ T 0 t i¼1 i¼1 # # Z tþT " Z n n X 1 T X di ðtÞ f ðsÞ þ gi ðsÞ Gðt; sÞ ds dt þ T 0 i¼1 t i¼1 have unique solution x . Now let H ¼ maxfM3 ; M 4 ; M5 g þ M , where M is large enough such that jx j < M and take X ¼ xðtÞ 2 X ; kxk1 < H , then obviously X satisfies the condition (a) of Lemma 2.1. and when x 2 @X \ KerL ¼ @X \ R, then x ¼ H and " n X x bi QNx ¼ ae bþ i¼1
þ
Z
T
dðtÞ þ 0
n X
di ðtÞ
!Z
tþT
" f ðsÞ þ
t
i¼1
n X
#
#
gi ðsÞ ds dt 6¼ 0
i¼1
also, similar to the analysis of [19, p. 11], one has degfJQNx; X \ KerL; 0g ¼ degfx; X \ KerL; 0g 6¼ 0: Here J can be the identity mapping since ImP ¼ KerL. By now we have proved that X verifies all the requirements in Lemma 2.1. Hence (2.3) has at least one Set n ðtÞ ¼ expfx ðtÞg, then we know that n ðtÞ is solution x ðtÞ in DomL \ X. an T -periodic solution of (2.2) with strictly positive components. Then by the medium of (2.1) we know that system (1.5) has an T -periodic solution ðn ðtÞ; u ðtÞÞ. The proof of Theorem 2.1 is complete. h The next theorem deals with the case there at least one i 2 f1; 2; ::; ng such that si ðtÞ 6 gi ðtÞ. Theorem 3.2. If ci ðtÞ ¼ ci ; c00i ðtÞ ¼ 0 and ci are non-negative constants. Assume that aðtÞ; bðtÞ; dðtÞ; f ðtÞ; eðtÞ 2 CðR; ð0; þ1ÞÞ; bi ðtÞ; di ðtÞ; gi ðtÞ 2 CðR; ½0; þ1ÞÞ; ~ci0 ðtÞ ¼ e 1 ¼ ln R
ci ; 1 s0i ðtÞ ( a
~vi ðtÞ ¼
h1 ðb þ Ve Þ
)
si ðtÞ; ci ðtÞ 2 C 2 ðR; ½0; þ1ÞÞ;
bi ðtÞ ; 1 s0i ðtÞ
þþ
f0 k kC 0a þ 2T kak0 : h2 bð1 kc0i k0 Þ
e And assume that s0i ðtÞ < 1, c0i ðtÞ < 1 and kCk0 e R 1 < 1. Then Eq. (1.1) has at least one positive T -periodic solution.
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
61
Proof. Take X and Z as the same of those in the proof of Theorem 3.1. Consider the Eq. (2.3), let L : DomL X ! Z and N : X ! Z, respectively, defined by Lx ¼ x_ and n X Nx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ:
i¼1
We also represent Px and Qz as above, thus, it is easy to see that Z " n X 1 T QNx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt:
i¼1
By assumptions, one has ~c0i0 ðtÞ ¼ 0, it follows that Z " n X 1 T aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg QNx ¼ T 0 i¼1 n X ~ci0 ðtÞð1 c0i ðtÞÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1 " Z n X 1 T ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1
Z n 1 T X ~ci0 ðtÞd expfxðt ci ðtÞÞg T 0 i¼1 " Z n X 1 T ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1
62
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
and followed by some computation, one has Z t" n X KP ðI QÞNx ¼ aðsÞ bðsÞ expfxðsÞg bi ðsÞ expfxðs si ðsÞÞg T
i¼1
dðsÞðU expfxgÞðsÞ
n X
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
n X
~ci0 ðsÞ expfxðs
t ci ðsÞÞgjT
i¼1
bðsÞ expfxðsÞg
n X
1 þ T
Z
"
T
s aðsÞ 0
bi ðsÞ expfxðs si ðsÞÞg
i¼1
þ
n X
~ci0 ðsÞ expfxðs ci ðsÞÞg dðsÞðU expfxgÞðsÞ
i¼1
n X i¼1
Z
T
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds þ " aðsÞ bðsÞ expfxðsÞg
0
dðsÞðU expfxgÞðsÞ
n X
1 t 2 T
bi ðsÞ expfxðs si ðsÞÞg
i¼1 n X
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
Clearly, QN and KP ðI QÞN are continuous. It is not difficult to show that KP ðI QÞN ðXÞ is compact for any open bounded X X by using Arzela– Ascoli theorem. Moreover, QN ðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X X . Now we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem (Lemma 2.1). Corresponding to the operator equation Lx ¼ kNx, k 2 ð0; 1Þ, we have " n X dx ¼ k aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ ;
i¼1
If xðtÞ 2 X is a solution of Eq. (3.16) for a certain k 2 ð0; 1Þ, then
ð3:16Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Z
T
" aðtÞ bðtÞ expfxðtÞg
0
n X i¼1
n X
n X
63
bi ðtÞ expfxðt si ðtÞÞg
i¼1
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:17Þ
i¼1
Integrating this identity we have Z T" n X aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg 0
#
i¼1
dðtÞðU expfxgÞðtÞ
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:18Þ
i¼1
That is Z
T
" bðtÞ expfxðtÞg þ
0
n X
bi ðtÞ expfxðt si ðtÞÞg þ dðtÞðU expfxgÞðtÞ
i¼1
þ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
ð3:19Þ
aðtÞ dt 0
i¼1
and hence by using (3.16), we can find #0 Z T " n X xðtÞ þ k ~ci ðtÞ expfxðt ci ðtÞÞg dt 0 i¼1 Z T n X ¼k bi ðtÞ expfxðt si ðtÞÞg aðtÞ bðtÞ expfxðtÞg 0 i¼1 n X di ðtÞðU expfxgÞðt ri ðtÞÞ dt dðtÞðU expfxgÞðtÞ i¼1 " Z T Z T n X ¼k aðtÞd t þ bðtÞ expfxðtÞg þ bi ðtÞ expfxðt si ðtÞÞg 0
0
þ dðtÞðU expfxgÞðtÞ þ
i¼1 n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z <2 0
T
aðtÞ dt 6 2T kak0 :
ð3:20Þ
64
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Since xðtÞ 2 X , there exist a ~ n 2 ½0; T such that xð~ nÞ ¼ min xðtÞ:
ð3:21Þ
t2½0;T
From (3.19) and (3.21), we see that " þ b
n X
# bi T expfxð~ nÞg 6 aT ;
i¼1
that is ( xð~ nÞ 6 ln
þ b
)
a Pn
i¼1
:¼ M10 :
bi
ð3:22Þ
On the other hand, there also exists a e g 2 ½0; T such that ð3:23Þ
xð~ gÞ ¼ max xðtÞ: t2½0;T
kf k0 þ
Also, by using ðU1ÞðtÞ 6
þ aT 6 b
n X
Pn i¼1
kgi k0
e
, from (3.19) and (3.23) it follows that
! bi T expfxð~ gÞg þ kdk0
Z
n X
kdi k0
Z
T
ðU expfxgÞðt ri ðtÞÞ dt 0
i¼1
þ 6 b
ðU expfxgÞðtÞ dt
0
i¼1
þ
T
n X
! bi T expfxð~ gÞg
i¼1
þ
kdk0 þ
n X
! kdi k0
expfxðe g Þg
Z
þ 6 b
i¼1
bi þ
ðU1ÞðtÞ dt
0
i¼1 n X
T
kdk0 þ
n X i¼1
! kdi k0
kf k0 þ
Pn
i¼1 kgi k0
e
! T expfxð~gÞg;
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
65
that is
xðe g Þ P ln
9 > =
8 > <
a Pn :¼ M20 : P P > ; :b þ n bi þ ðkdk þ n kdi k Þ kf k0 þ i¼1 kgi k0 > 0 0 i¼1 i¼1 e ð3:24Þ
It follows from (3.19) that Z " T
bðtÞ expfxðtÞg þ
h1 0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
þ h2
Z
T
" bðtÞ expfxðtÞg þ
0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt: 0
i¼1
ð3:25Þ Then similar to the analysis of (3.18), (3.19) and (3.21) of [19], by using the mean value theorem, from (3.25) there exists ~g0 , ~gi , ~d0 , e d i 2 ½0; T , i ¼ 1; 2; . . . ; n such that Z T Z T n X ~vi ð~ aðtÞ dt P h1 ðbð~ g0 Þ þ gi ÞÞ expfxðtÞg dt 0
i¼1
0
þ h2 bð~ d0 c j ð ~ d0 ÞÞð1 c0j ð~d0 ÞÞ
Z
T
expfxðt cj ðtÞÞ dt 0
þ
n X i¼1
bi ð gei Þ 1 1 s0i ðgi Þ 1 c0i ðdi Þ
Z
T
!
expfxðt ci ðtÞÞ dt :
0
That is g0 Þ þ a P h1 ðbð~
n X
~vi ð~ gi ÞÞ expfxð~1Þg þ h2 bð~d0 cj ð~d0 ÞÞð1 c0j ð~d0 ÞÞ
i¼1
expfxð~1 cj ð~1ÞÞ
n X i¼1
! bi ð gei Þ 1 expfxð~1 cj ð~1ÞÞg ; 1 s0i ð gei Þ 1 c0i ð dei Þ
66
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
for some e1 2 ½0; T . Therefore, we have ( ) ( ) a a Pn xð1Þ < ln ; 6 ln þ Ve Þ h1 ðbð~ g0 Þ þ i¼1 ~vi ð~ gi ÞÞ h1 ð b expfxðe1 ~si ð~1ÞÞg 6
a d0 cj ð~ d0 ÞÞð1 c0j ð~d0 ÞÞÞ h2 ðbð~
6
ð3:26Þ
a : h2 bð1 kc0i k0 Þ ð3:27Þ
From (3.20), (3.26) and (3.27) it follows that xðtÞ þ k
n X
~ci0 ðtÞ expfxðt ci ðtÞÞg
i¼1
6 xð~1Þ þ k
n X
~ci0 ð~1Þ expfxð~1 cei ð~1ÞÞg
i¼1
" #0 n X xðtÞ þ k ~ci0 ðtÞ expfxðt si ðtÞÞg dt þ 0 i¼1 ( ) e 0k kC a def 0a e 1: < ln þ 2T kak0 ¼ R þþ 0 e h bð1 kc k Þ h1 ðb þ V Þ 2 i 0 Z
T
The rest of the proof is similar to the proof of Theorem 3.1 and we omit it. The proof is complete. h Example. To check all the conditions for Theorem 3.2, and let h1 ¼ h2 ¼ 12, we can obtain that the following equation: " 2 X 1 0 sin2 jt n ðtÞ ¼ nðtÞ ð2 þ sin tÞe6 ecos t nðtÞ ð1 sin jtÞn t 2j j¼1 # 2 X j 0 n ðt jÞ ð3 þ cos tÞuðtÞ þ uðt sin tÞ ; 2 j¼1 u0 ðtÞ ¼ ð2 þ cos tÞuðtÞ þ ð2 þ sin tÞnðtÞ þ nðt cos tÞ þ 5nðt 3 sin tÞ: has a positive 2p-periodic solution. Acknowledgements This work is supported by the Foundation of Ability Person of Fuzhou University under the grant 0030824228, the Foundation of Developing Technology and Science of Fuzhou University under the grant 2003-XQ-21 and the Foundation of Fujian Education Bureau under the grant JB01023.
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67
References [1] Y.K. Li, Periodic solutions of a periodic delay predator–prey system, Proc. Am. Math. Soc. 127 (1999) 1331–1335. [2] H.I. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992) 689–701. [3] B.R. Tang, Y. Kuang, Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential system, Tohoku Math. 49 (1997) 217–239. [4] Y.K. Li, Periodic solutions for delay Lotka–Volterra competition systems, J. Math. Anal. Appl. 246 (2000) 230–244. [5] Y.K. Li, Y. Kuang, Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) 265–280. [6] Y.K. Li, Y. Kuang, Periodic solutions in periodic state-dependent delay equations and population models, J. Math. Anal. Appl. 255 (2001) 265–280. [7] L.L. Wang, W.T. Li, Existence and global stability of positive periodic solutions of a predator– prey system with delays, Appl. Math. Comput. 146 (2003) 167–185. [8] M. Fan, K. Wang, Positive periodic solutions of a periodic integro-differential competition system with infinite delays, Z. Angew. Math. Mech. 81 (3) (2001) 197–203. [9] M. Fan, K. Wang, Periodicity in a delayed ratio-dependent predator–prey system, J. Math. Anal. Appl. 262 (2001) 179–190. [10] M. Fan, K. Wang, Global existence of positive periodic solutions of periodic predator–prey system with infinite delay, J. Math. Anal. Appl. 262 (2001) 1–11. [11] K. Gopalsamy, B.G. Zhang, On a neutral delay logistic equation, Dyn. Stab. Syst. 2 (1998) 183–195. [12] H.I. Freedman, Y. Kuang, Stability switches in linear scalar neutral delay equations, Funk. Ekv. 34 (1991) 187–209. [13] K. Gopalsamy, X. He, L. Wen, On a periodic neutral logistic equation, Glasgow Math. J. 33 (1991) 281–286. [14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [15] Y Li, Positive periodic solution for neutral delay model, Acta Math. Sinica 39 (6) (1996) 789– 795 (in Chinese). [16] Q. Li, J. Cao, On positive periodic solutions for neutral delay model, J. Math. Expos. 20 (4) (1999) 562–565. [17] H. Fang, J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl. 259 (2001) 8–17. [18] S. Lu, W. Ge, Existence of positive periodic solutions for neutral functional differential equations with deviating arguments, Appl. Math. J. Chin. Univ. Ser. B 17 (4) (2002) 382– 390. [19] Z. Yang, J. Cao, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delays models, Appl. Math. Comput. 142 (2003) 123–142. [20] Z. Yang, J. Cao, Positive periodic solutions of neutral Lotka–Volterra system with periodic delays, Appl. Math. Comput., in press, doi:10.1016/S0096-3003(03)00170-X. [21] Y. Li, On a periodic neutral delay Lotka–Volterra system, Nonlinear Anal. 39 (2000) 767–778. [22] P. Weng, Global attractivity in a periodic competition system with feedback controls, Acta Appl. Math. 12 (1996) 11–21. [23] P. Weng, Existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls, Comput. Math. Appl. 40 (2000) 747–759. [24] P. Weng, D. Jiang, Existence and global stability of positive periodic solution of n-species periodic Lotka–Volterra competition systems with feedback controls and deviating arguments, Far East J. Math. Sci. 7 (1) (2002) 45–65.
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[25] Y.N. Xiao, S.Y. Tang, J.F. Chen, Permanence and periodic solution in competition system with feedback controls, Math. Comput. Model. 27 (6) (1998) 33–37. [26] K. Gopalsamy, P.X. Weng, Feedback regulation of logistic growth, Int. J. Math. Sci. 16 (1) (1993) 177–192. [27] Y. Fan, J. Daqing, Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments, Ann. Diff. Eqs. 17 (4) (2001) 337–384. [28] X. Li, M. Fan, K. Wang, Positive periodic solution of single species model with feedback regulation and infinite delay, Appl. Math. J. Chin. Univ. Ser. A 17 (1) (2002) 13–21 (in Chinese). [29] M. Fan et al., Periodicity and stability in periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica, in press. [30] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977. [31] F. Chen, Persistence and global stability for non-autonomous cooperative system with diffusion and time delay, Acta Sci. Natural. Univ. Pekin. 39 (1) (2003) 22–28. [32] F. Chen, X. Chen, The n-competing Volterra–Lotka almost periodic systems with grazing rates, J. Biomath. 18 (3) (2003) (in Chinese). [33] F. Chen, Periodic solution of nonlinear integral-differential equations with infinite delay, Acta Math. Appl. Sinica 26 (1) (2003) 1–8 (in Chinese). [34] F. Chen, J. Shi, Periodic solution of neutral higher order periodic systems, Acta Math. Sinica 46 (6) (2003) (in Chinese). [35] F. Chen, Existence and uniqueness of almost periodic solution for a class of non-autonomous system, Math. Practice Theory 31 (5) (2001) 532–538 (in Chinese). [36] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers, Dordrecht, 1992. [37] Y.K. Li, Periodic solution of a periodic neutral delay equation, J. Math.Anal. Appl. 214 (1997) 11–21. [38] S. Lu, On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments, J. Math. Anal. Appl. 280 (2) (2003) 321–333.
Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control Fengde Chen *, Faxin Lin, Xiaoxin Chen Department of Mathematics, Fuzhou University, Fuzhou, Fujian 350002, People’s Republic of China
Abstract With the help of a continuation theorem based on Gaines and MawhinÕs coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a neutral model with periodic delays and feedback control 8 n n P P > > dn ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ ci ðtÞn0 ðt ci ðtÞÞ > dt > > i¼1 i¼1 > < n P dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; > i¼1 > > n > P > > : du ¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ gi ðtÞnðt gi ðtÞÞ: dt i¼1
Our results extend and improve existing results, and have further applications in population dynamics. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Positive periodic solutions; Neutral delay model; Feedback control; Coincidence degree
1. Introduction In the recent years, the application of theories of functional differential equations in mathematical ecology has been developed rapidly. Various
*
Corresponding author. E-mail addresses: [email protected], [email protected] (F. Chen).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.063
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
mathematical models with delays have been proposed in the study of population dynamics, ecology and epidemic. Recently, by using the powerful technique––a continuation theorem based on Gaines and MawhinÕs coincidence degree, many good results concerned with the global existence of positive periodic solutions of the biological model are obtained (see Refs. [1–10]). However, only a few papers (except Hopf bifurcations) have been published on the existence of periodic solutions of the neutral delay population models. In particular, Gopalsamy et al. [11,13] have established the existence of a positive periodic solution for a periodic neutral delay logistic equation N ðt mxÞ þ cðtÞN 0 ðt mxÞ 0 N ðtÞ ¼ rðtÞN ðtÞ 1 ; ð1:1Þ KðtÞ where KðtÞ, rðtÞ, cðtÞ are positive continuous T -periodic functions with T > 0, and m is a positive integer. In 1993, Kuang [14] proposed the following open problem (Open Problem 9.2): How to obtain sufficient conditions for the existence of a periodic solution for equation N 0 ðtÞ ¼ N ðtÞ½aðtÞ bðtÞN ðtÞ bðtÞN ðt sðtÞÞ cðtÞN 0 ðt sðtÞÞ;
ð1:2Þ
where aðtÞ, bðtÞ, bðtÞ, cðtÞ are non-negative continuous T -periodic functions. Li Yongkun [15] tried to give an affirmative answer to the above open problem, however, there is a mistake in the proof of Theorem 2 in [15]. With the aim of giving a right answer to the above open problem, Li and Cao [16], Fang and Li [17] and Lu and Ge [18] have investigated the above question. However, it is more complex to check the sufficient conditions of [17,18]. Moreover, Li [37] studied the existence of positive periodic solution of the neutral Lotka–Volterra equation with several delays " # n n X X N 0 ðtÞ ¼ N ðtÞ aðtÞ bi ðtÞN ðt si Þ ci ðtÞN 0 ðt ci Þ ; ð1:3Þ i¼1
i¼1
where aðtÞ, bi ðtÞ, ci ðtÞ are positive continuous T -periodic functions and si ; ci ði ¼ 1; 2; . . . ; nÞ are non-negative constants. Recently, Yang and Cao [19] and Lu [38] further investigate the existence of positive periodic solution of the neutral delays models " # n n X X 0 0 N ðtÞ ¼ N ðtÞ aðtÞ bðtÞN ðtÞ bi ðtÞN ðt si ðtÞÞ ci ðtÞN ðt ci ðtÞÞ ; i¼1
i¼1
ð1:4Þ where aðtÞ, bi ðtÞ, ci ðtÞ, si ðtÞ, ci ðtÞ are non-negative continuous T -periodic functions and i ¼ 1; 2; . . . ; n, T is a positive constant. For more works on this direction, one could refer to [12,20,21,33–35].
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
47
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 for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at 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, see [36]. Recently, many scholars focus on the existence and global attractivity of the positive periodic solution of the system. Xiao et al. [25] investigate the two species non-autonomous competition system with feedback control, by using differential inequality and Lyapunov function, they obtained sufficient condition which guarantee the existence and global attractivity of the periodic solution of the system. Weng [26] further investigate the system with constant delay. For more works on this direction, one could refer to [22–29,31,32]. To the best of the authors knowledge, although there are many works concerned with the existence of positive equilibrium of the neutral delay population dynamics with feedback control (see, for example [36]), no works have been done to investigate the existence of positive periodic solution of the non-autonomous neutral delay population dynamics with feedback control. With the aim of it, in this paper, we will investigate the following neutral delay models with feedback control of the form 8 " n n X X > dn > > ¼ nðtÞ aðtÞ bðtÞnðtÞ > bi ðtÞnðt si ðtÞÞ ci ðtÞn0 ðt ci ðtÞÞ > > dt > i¼1 i¼1 > # > < n P dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; > > i¼1 > > > n X > > du > ¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ gi ðtÞnðt gi ðtÞÞ; > : dt i¼1
ð1:5Þ where aðtÞ, bðtÞ, bi ðtÞ, ci ðtÞ, dðtÞ, di ðtÞ, eðtÞ, f ðtÞ, gi ðtÞ, si ðtÞ, ci ðtÞ, gi ðtÞ are nonnegative continuous T -periodic functions and i ¼ 1; 2; . . . ; n, T is a positive constant. When f ðtÞ ¼ 0, gi ðtÞ ¼ 0, dðtÞ ¼ 0, di ðtÞ ¼ 0, i ¼ 1; 2; . . . ; n, (1.5) reduces to (1.4). Comparing the systems (1.4) and (1.5), one could see that we introduce the control variables uðtÞ so as to implement a feedback control mechanism. The main purpose of this paper is to derive a set of sufficient conditions for the global existence of positive periodic solution of (1.5). Obviously, Eq. (1.5) is a generalization of Eqs. (1.1)–(1.4), therefore, by using the theory of the continuation theorem of the coincidence degree theory which proposed in [30] by Mawhin, we shall obtain several sufficient conditions for existence of a periodic solution of Eq. (1.5) in this paper.
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
2. Notation and lemma In order to obtain the existence of positive periodic solutions of (1.5), for the readerÕs convenience, we shall summarize in the following a few concepts and results from [30] that will be basic for this paper. Let X , Z be normed vector spaces, L : DomL X ! Z be a linear mapping, N : X ! Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL ¼ CodimImL < þ1 and ImL is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projectors P : X ! X and Q : Z ! Z such that ImP ¼ KerL; ImL ¼ KerQ ¼ ImðI QÞ. It follows that LjDomL \ KerP : ðI P ÞX ! ImL is invertible. We denote the inverse of that map by KP . If X be an open bounded subset of X , the mapping N if QN ðXÞ is bounded and KP ðI QÞN : X ! X is will be called L-compact on X compact. Since ImQ is isomorphic to KerL, there exists an isomorphisms J : ImQ ! KerL. In the proof of our existence theorem below, we will use the continuation theorem of Gaines and Mawhin [30, p. 40]. Lemma 2.1 (Continuation theorem). Let L be a Fredholm mapping of index zero and let N be L-compact on X. Suppose (a) For each k 2 ð0; 1Þ, every solution x of Lx ¼ kNx is such that x 6¼ @X; (b) QNx ¼ 6 0 for each x 2 @X \ KerL and degfJQN ; X \ KerL; 0g 6¼ 0: Then the equation Lx ¼ Nx has at least one solution lying in DomL \ X. Lemma 2.2. ðnðtÞ; uðtÞÞT is an T -periodic solution of (1.5) if and only if it is also an T -periodic solution of " 8 n X > dn > > ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ > > dt > > i¼1 > # > > n n < P P 0 ci ðtÞn ðt ci ðtÞÞ dðtÞuðtÞ di ðtÞuðt ri ðtÞÞ ; ð2:1Þ i¼1 i¼1 > > > > n R tþT P > > > f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ Gðt; sÞ ds uðtÞ ¼ t > > > i¼1 : :¼ ðUnÞðtÞ; where Rs expf t eðhÞ dhg Rx Gðt; sÞ ¼ : expf 0 eðhÞ dhg 1
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
49
T
Proof. First, let ðnðtÞ; uðtÞÞ be an T -periodic solution of (1.5). From the second equation of (1.5) and the variation-of-constant formulas, it follows that Z t uðtÞ ¼ uð0Þ exp eðhÞ dh 0 # Z s Z t" n X þ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ exp eðhÞ dh ds: 0
t
i¼1
Then, uðt þ T Þ ¼ uð0Þ exp Z "
Z
t
f ðsÞnðsÞ þ
0
exp þ
Z
Z
tþT
t
exp
eðhÞ dh exp
Z
0
t
þ
Z eðhÞ dh exp
"t
Z
eðhÞ dh ds #
t tþT
f ðsÞnðsÞ þ
n X
s
#
gi ðsÞnðs gi ðsÞÞ
i¼1
s
eðhÞ dh t
n X
tþT
gi ðsÞnðs gi ðsÞÞ
i¼1
Z
t
eðhÞ dh exp eðhÞ dh ds tþT Z T ¼ uðtÞ exp eðhÞ dh 0 # Z tþT " n X þ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ t
t
exp
Z
s
i¼1
eðhÞ dh exp
Z
T
eðhÞ dh ds 0
t
and hence using the periodicity of uðtÞ, we get # Z tþT " n X uðtÞ ¼ f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ t
i¼1
n R o T eðhÞ dh exp eðhÞ dh t 0 n R o ds T 1 exp 0 eðhÞ dh " # n X f ðsÞnðsÞ þ gi ðsÞnðs gi ðsÞÞ Gðt; sÞ ds:
exp
¼
Z t
tþT
Rs
i¼1
This proves one part of the lemma.
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68 T
Next, let ðnðtÞ; uðtÞÞ be an T -periodic solution of (2.1). Then, " # n X 0 u ðtÞ ¼ f ðt þ T Þnðt þ T Þ þ gi ðt þ T Þnðt þ T gi ðt þ T ÞÞ Gðt; t þ T Þ i¼1
" f ðtÞnðtÞ þ
þ
Z
tþT
n X
#
gi ðtÞnðt gi ðtÞÞ Gðt; tÞ
i¼1
"
f ðsÞnðsÞ þ
t
"
n X
# gi ðsÞnðs gi ðsÞÞ ðeðtÞÞGðt; sÞ ds
i¼1
¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ
n X
# gi ðtÞnðt gi ðtÞÞ ðGðt; t þ T Þ Gðt; tÞÞ
i¼1
¼ eðtÞuðtÞ þ f ðtÞnðtÞ þ
n X
gi ðtÞnðt gi ðtÞÞ:
i¼1
The proof is complete.
h
By Lemma 2.2, in order to show the existence of strictly positive T -periodic solutions of (1.5), we only need to show that (2.1) possesses at least one T periodic solution with strictly positive component. Now, (2.1) can be reformulated as " n X dn ¼ nðtÞ aðtÞ bðtÞnðtÞ bi ðtÞnðt si ðtÞÞ dt i¼1 # n n X X 0 ci ðtÞn ðt ci ðtÞÞ dðtÞðUnÞðtÞ di ðtÞðUnÞðt ri ðtÞÞ : i¼1
i¼1
ð2:2Þ Let nðtÞ ¼ expfxðtÞg, then (2.2) is the same as n X dx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ:
ð2:3Þ
i¼1
For the rest of this paper, we shall devote ourselves to study the existence of T -periodic solution of (2.3).
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
51
For convenience, we also use the notations
u ¼ min fuðtÞg; t2½0;T
U0 ¼
n X i¼1
ui0 ;
kuk0 ¼ max fuðtÞg; t2½0;T
v ¼
min vi ðtÞ;
kui k0 ¼ max fui ðtÞg;
and
t2½0;T i2f1;...;ng
t2½0;T
g ¼
1 T
Z
U¼
n X
ui ;
i¼1
T
gðtÞ dt: 0
3. Existence of positive periodic solutions The objective of this section is to derive sufficient conditions for the existence of positive periodic solutions in Eq. (1.5), by use Lemma 2.1 (i.e. continuation theorem). Theorem 3.1. If si ðtÞ ¼ ci ðtÞ, assume that
aðtÞ; bðtÞ; dðtÞ; f ðtÞ; eðtÞ 2 CðR; ð0; þ1ÞÞ; bi ðtÞ; di ðtÞ; gi ðtÞ 2 CðR; ½0; þ1ÞÞ; ci ðtÞ 2 C 1 ðR; ½0; þ1ÞÞ;
si ðtÞ 2 C 2 ðR; ½0; þ1ÞÞ;
ci ðtÞ bi ðtÞ c0i0 ðtÞ ; vi ðtÞ ¼ ; 1 si ðtÞ 1 s0i ðtÞ ( ) kC0 k0 a a þ 2T kak0 : R1 ¼ ln þ h1 ðb þ V Þ h2 vð1 ks0i k0 Þ ci0 ðtÞ ¼
Also assume that s0i ðtÞ < 1, bi ðtÞ c0i0 ðtÞ P 0 and there exist at least one i such that bi ðtÞ > c0i0 ðtÞ, also kCk0 eR1 < 1. Then Eq. (1.5) has at least one positive T -periodic solution. Proof. To finish the proof of the theorem, it is enough to show the existence of T -periodic solution of system (2.3). In order to use Lemma 2.1 to Eq. (2.3), we take X ¼ fxðtÞ 2 C 1 ðR; RÞ; xðt þ T Þ ¼ xðtÞg, and Z ¼ fzðtÞ 2 CðR; RÞ; zðt þ T Þ ¼ zðtÞg, and denote kxk0 ¼ maxt2½0;T jxðtÞj; kxk1 ¼ maxt2½0;T fkxk0 ; kx0 k0 g. Then X and Z are Banach spaces when they are endowed with the k k1 and k k0 , respectively.
52
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Let L : DomL X ! Z and N : X ! Z defined by Lx ¼ x_ and Nx ¼ aðtÞ bðtÞ expfxðtÞg
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
n X
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ
i¼1
for any x 2 X and two project Z 1 T xðtÞ dt; x 2 X ; Px ¼ T 0
QZ ¼
1 T
Z
T
zðtÞ dt;
z 2 Z:
0
It can be found that KerL ¼ fxjx 2 X ; x ¼ h; h 2 Rg; Z T zðtÞ dt ¼ 0 is closed in Z; ImL ¼ zjz 2 Z; 0
then dimKerL ¼ 1 ¼ codimImL and hence, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that ImP ¼ KerL;
KerQ ¼ ImL ¼ ImðI QÞ:
Furthermore, the generalized inverse (to L) KP : ImL ! KerP \ DomL exists, which is given by Z Z t Z Z t Z t 1 T 1 T zðsÞ ds zðsÞ ds dt ¼ zðsÞ ds þ szðsÞ ds: KP ðzÞ ¼ T 0 T 0 0 0 T Then QN : X ! Z and KP ðI QÞN : X ! X read Z " n X 1 T aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg QNx ¼ T 0 i¼1
n X i¼1
n X i¼1
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ dt:
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
53
ci ðtÞ Since ci0 ðtÞ ¼ 1s 0 ðtÞ, it follows that i Z T" n X 1 QNx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1
n X
ci0 ðtÞð1 s0i ðtÞÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ 1 ¼ T
Z
1 ¼ T
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
aðtÞ bðtÞ expfxðtÞg
0
1 T Z
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ
#
i¼1
"
T
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z
T
n X
"
i¼1
0 T
n X
ci0 ðtÞd expfxðt si ðtÞÞg
aðtÞ bðtÞ expfxðtÞg
0
n X
ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg
i¼1
dðtÞðU expfxgÞðtÞ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
and followed by some computation, one has Z " t
KP ðI QÞNx ¼
aðsÞ bðsÞ expfxðsÞg
T
Z tX n T
1 þ T
T
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
ci0 ðsÞd expfxðs ci ðsÞÞg
"
s aðsÞ bðsÞ expfxðsÞg 0
n X i¼1
n X
T
n X i¼1
bi ðsÞ expfxðs si ðsÞÞg #
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
s 0
#
i¼1
Z
Z
n X i¼1
dðsÞðU expfxgÞðsÞ
bi ðsÞ expfxðs si ðsÞÞg
i¼1
dðsÞðU expfxgÞðsÞ
n X
ci0 ðsÞd expfxðs ci ðsÞÞg
54
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
1 t þ 2 T
" Z
T
" aðsÞ bðsÞexpfxðsÞg
0
n X
bi ðsÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
Z
n X
T 0
¼
Z t"
# ci0 ðsÞdexpfxðs ci ðsÞÞg
i¼1
aðsÞ bðsÞexpfxðsÞg
T
n X
ðbi ðsÞ c0i0 ðsÞÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
n X i¼1
n X
1 ci0 ðtÞexpfxðt ci ðtÞÞg þ T
Z
T
" s aðsÞ bðsÞexpfxðsÞg
0
ðbi ðsÞ ci0 ðsÞ sc0i0 ðsÞÞexpfxðs si ðsÞÞg
i¼1
dðsÞðUexpfxgÞðsÞ
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds
i¼1
1 t þ 2 T
n X
Z
T
" aðsÞ bðsÞexpfxðsÞg
0
ðbi ðsÞ c0i0 ðsÞÞexpfxðs si ðsÞÞg dðsÞðUexpfxgÞðsÞ
i¼1
n X
# di ðsÞðUexpfxgÞðs ri ðsÞÞ ds:
i¼1
Clearly, QN and KP ðI QÞN are continuous. It is not difficult to show that KP ðI QÞN ðXÞ is compact for any open bounded X X by using Arzela– Ascoli theorem. Moreover, QN ðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X X . Now we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
55
(Lemma 2.1). Corresponding to the operator equation Lx ¼ kNx; k 2 ð0; 1Þ, we have " n X dx ¼ k aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ :
ð3:1Þ
i¼1
If xðtÞ 2 X is a solution of Eq. (3.1) for a certain k 2 ð0; 1Þ, then Z
T
" aðtÞ bðtÞ expfxðtÞg
0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
n X
ci ðtÞx0 ðt si ðtÞÞ expfxðt si ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:2Þ
i¼1
Integrating this identity we have Z " T
aðtÞ bðtÞ expfxðtÞg
0
n X ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg i¼1
dðtÞðU expfxgÞðtÞ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:3Þ
i¼1
That is Z 0
T
" bðtÞ expfxðtÞg þ
n X
ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X i¼1
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt 0
ð3:4Þ
56
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
and hence by using (3.4), one has #0 Z T " n X xðtÞ þ k ci0 ðtÞ expfxðt si ðtÞÞg dt 0 i¼1 Z T n X ¼k ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg aðtÞ bðtÞ expfxðtÞg 0 i¼1 n X di ðtÞðU expfxgÞðt ri ðtÞÞ dt dðtÞðU expfxgÞðtÞ i¼1 Z T Z T" n X ¼k aðtÞdt þ bðtÞ expfxðtÞg þ ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg 0
0
þ dðtÞðU expfxgÞðtÞ þ
i¼1 n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z
T
aðtÞdt 6 2T kak0 :
<2
ð3:5Þ
0
Since xðtÞ 2 X , there exist a n 2 ½0; T such that xðnÞ ¼ min xðtÞ:
ð3:6Þ
t2½0;T
From (3.4) and (3.6), we see that " þ b
n X
# bi c0i0 T expfxðnÞg 6 aT ;
i¼1
that is ( xðnÞ 6 ln
a
þ Pn bi c 0 b i0 i¼1
) :¼ M1 :
ð3:7Þ
On the other hand, there also exists a g 2 ½0; T such that xðgÞ ¼ max xðtÞ: t2½0;T
ð3:8Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
57
Notice that Z
ðU1ÞðtÞ ¼
"
tþT
n X
Gðt; sÞ f ðsÞ þ
t
Z
¼
tþT
f ðsÞ þ
kf k0 þ
Pn
i¼1
kgi k0
Z
i¼1
gi ðsÞ
ds
tþT
Gðt; sÞeðsÞ ds
e kf k0 þ
Pn
eðsÞ
t
¼
gi ðsÞ ds
i¼1
Gðt; sÞeðsÞ
6
#
t
Pn
i¼1
kgi k0
e
ð3:9Þ
:
From (3.4), (3.8) and (3.9) it follows that þ aT 6 b
n X
! bi c0i0
T expfxðgÞg þ kdk0
Z
þ
ðUexpfxgÞðtÞdt
0
i¼1 n X
T
kdi k0
Z
T
ðUexpfxgÞðt ri ðtÞÞdt 0
i¼1
þ 6 b
n X
! bi c0i0
T expfxðgÞg
i¼1
þ kdk0 þ
n X
! kdi k0 expfxðgÞg
n X i¼1
bi c0i0 þ
T
ðU1ÞðtÞdt 0
i¼1
þ 6 b
Z
kdk0 þ
n X i¼1
! jjdi jj0
kf k0 þ
Pn
i¼1 kgi k0
e
! T expfxðgÞg;
that is
xðgÞ P ln
8 > <
9 > =
a P :¼ M2 :
kf k0 þ n kgi k0 > P Pn > n 0 ; :b i¼1 þ b c þ kdk þ kd k i 0 i0 0 i¼1 i i¼1 e ð3:10Þ
58
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
It follows from (3.4) that Z T" n X h1 bðtÞ expfxðtÞg þ ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg 0
i¼1
þ dðtÞðU expfxgÞðtÞ þ þ h2
Z
T
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
"
bðtÞ expfxðtÞg þ 0
n X ðbi ðtÞ c0i0 ðtÞÞ expfxðt si ðtÞÞg i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt: 0
i¼1
ð3:11Þ Then similar to the analysis of (3.6), (3.7) and (3.9) of [19], by using the mean value theorem, from (3.11) there exists g0 ; gi ; d0 ; di 2 ½0; T ; i ¼ 1; 2; . . . ; n such that Z T Z T n X aðtÞ dt P h1 ðbðg0 Þ þ vi ðgi ÞÞ expfxðtÞg dt 0
0
i¼1
þ h2 bðd0 sj ðd0 ÞÞð1 s0j ðd0 ÞÞ
Z
T
expfxðt sj ðtÞÞg dt 0
þ
n X
ðbi ðdi Þ
c0i0 ðdi ÞÞ
Z
!
T
expfxðt si ðtÞÞg dt : 0
i¼1
That is a P h1 ðbðg0 Þ þ
n X
vi ðgi ÞÞ expfxð1Þg þ h2 bðd0 sj ðd0 ÞÞð1 s0j ðd0 ÞÞ
i¼1
expfxð1 sj ð1ÞÞg þ
n X
! ðbi ðdi Þ
c0i0 ðdi ÞÞ expfxð1
si ð1ÞÞg
i¼1
for some 1 2 ½0; T . Therefore, we have ( ) ( ) a a P xð1Þ < ln 6 ln ; h1 ðbðg0 Þ þ ni¼1 vi ðgi ÞÞ h1 ðb þ V Þ expfxð1 si ð1ÞÞg 6
ð3:12Þ
a a a 6 6 : h2 ðbi ðdi Þ c0i0 ðdi ÞÞ h2 vi ð1 s0i Þ h2 vð1 ks0i k0 Þ ð3:13Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
59
From (3.5), (3.12) and (3.13) it follows that xðtÞ þ k
n X
ci0 ðtÞ expfxðt si ðtÞÞg
i¼1
6 xð1Þ þ k
n X
ci0 ð1Þ expfxð1 si ð1ÞÞg
i¼1
þ
Z
T
0
( < ln
" #0 n X xðtÞ þ k ci0 ðtÞ expfxðt si ðtÞÞg dt i¼1
a h1 ðb þ V Þ
) þ
kDk0 a def þ 2T kak0 ¼ R1 : h2 vð1 ks0i k0 Þ
Hence, by using (3.9), we have n X 0 bi ðtÞ expfxðt si ðtÞÞg jx ðtÞj ¼ k aðtÞ bðtÞ expfxðtÞg i¼1
n X i¼1
n X i¼1
ci ðtÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ
< K þ kCk0 kx0 ðt si ðtÞÞk0 eR1 ;
where K ¼ kak0 þ kbk0 eR1 þ kBk0 eR1 þ kdk0akbk0 þ kDk0akbk0 eR1 . Therefore, it follows from kCk0 eR1 < 1 that kx0 k0 <
K def ¼ M3 : 1 kCk0 eR1
ð3:14Þ
Also, from (3.7), (3.10) and (3.12) we know that there exist t0 2 ½0; T such that ( ( ) ) a jxðt0 Þj 6 max ln ð3:15Þ ; M1 ; M2 :¼ M4 : h1 ðb þ V Þ Now combining (3.15) and (3.14), we obtain Z t def kxk0 6 jxðt0 Þj þ kx0 k0 dt < M4 þ M3 T ¼ M5 : 0
60
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
When x 2 R, from the condition of the theorem it follows " # Z Z tþT " n n X X 1 T x bi þ dðtÞ f ðsÞ þ gi ðsÞ Gðt; sÞ ds dt QNx ¼ ae bþ T 0 t i¼1 i¼1 # # Z tþT " Z n n X 1 T X di ðtÞ f ðsÞ þ gi ðsÞ Gðt; sÞ ds dt þ T 0 i¼1 t i¼1 have unique solution x . Now let H ¼ maxfM3 ; M 4 ; M5 g þ M , where M is large enough such that jx j < M and take X ¼ xðtÞ 2 X ; kxk1 < H , then obviously X satisfies the condition (a) of Lemma 2.1. and when x 2 @X \ KerL ¼ @X \ R, then x ¼ H and " n X x bi QNx ¼ ae bþ i¼1
þ
Z
T
dðtÞ þ 0
n X
di ðtÞ
!Z
tþT
" f ðsÞ þ
t
i¼1
n X
#
#
gi ðsÞ ds dt 6¼ 0
i¼1
also, similar to the analysis of [19, p. 11], one has degfJQNx; X \ KerL; 0g ¼ degfx; X \ KerL; 0g 6¼ 0: Here J can be the identity mapping since ImP ¼ KerL. By now we have proved that X verifies all the requirements in Lemma 2.1. Hence (2.3) has at least one Set n ðtÞ ¼ expfx ðtÞg, then we know that n ðtÞ is solution x ðtÞ in DomL \ X. an T -periodic solution of (2.2) with strictly positive components. Then by the medium of (2.1) we know that system (1.5) has an T -periodic solution ðn ðtÞ; u ðtÞÞ. The proof of Theorem 2.1 is complete. h The next theorem deals with the case there at least one i 2 f1; 2; ::; ng such that si ðtÞ 6 gi ðtÞ. Theorem 3.2. If ci ðtÞ ¼ ci ; c00i ðtÞ ¼ 0 and ci are non-negative constants. Assume that aðtÞ; bðtÞ; dðtÞ; f ðtÞ; eðtÞ 2 CðR; ð0; þ1ÞÞ; bi ðtÞ; di ðtÞ; gi ðtÞ 2 CðR; ½0; þ1ÞÞ; ~ci0 ðtÞ ¼ e 1 ¼ ln R
ci ; 1 s0i ðtÞ ( a
~vi ðtÞ ¼
h1 ðb þ Ve Þ
)
si ðtÞ; ci ðtÞ 2 C 2 ðR; ½0; þ1ÞÞ;
bi ðtÞ ; 1 s0i ðtÞ
þþ
f0 k kC 0a þ 2T kak0 : h2 bð1 kc0i k0 Þ
e And assume that s0i ðtÞ < 1, c0i ðtÞ < 1 and kCk0 e R 1 < 1. Then Eq. (1.1) has at least one positive T -periodic solution.
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
61
Proof. Take X and Z as the same of those in the proof of Theorem 3.1. Consider the Eq. (2.3), let L : DomL X ! Z and N : X ! Z, respectively, defined by Lx ¼ x_ and n X Nx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ:
i¼1
We also represent Px and Qz as above, thus, it is easy to see that Z " n X 1 T QNx ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ
i¼1
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt:
i¼1
By assumptions, one has ~c0i0 ðtÞ ¼ 0, it follows that Z " n X 1 T aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg QNx ¼ T 0 i¼1 n X ~ci0 ðtÞð1 c0i ðtÞÞx0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1 " Z n X 1 T ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1
Z n 1 T X ~ci0 ðtÞd expfxðt ci ðtÞÞg T 0 i¼1 " Z n X 1 T ¼ aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg T 0 i¼1 # n X dðtÞðU expfxgÞðtÞ di ðtÞðU expfxgÞðt ri ðtÞÞ dt i¼1
62
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
and followed by some computation, one has Z t" n X KP ðI QÞNx ¼ aðsÞ bðsÞ expfxðsÞg bi ðsÞ expfxðs si ðsÞÞg T
i¼1
dðsÞðU expfxgÞðsÞ
n X
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
n X
~ci0 ðsÞ expfxðs
t ci ðsÞÞgjT
i¼1
bðsÞ expfxðsÞg
n X
1 þ T
Z
"
T
s aðsÞ 0
bi ðsÞ expfxðs si ðsÞÞg
i¼1
þ
n X
~ci0 ðsÞ expfxðs ci ðsÞÞg dðsÞðU expfxgÞðsÞ
i¼1
n X i¼1
Z
T
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds þ " aðsÞ bðsÞ expfxðsÞg
0
dðsÞðU expfxgÞðsÞ
n X
1 t 2 T
bi ðsÞ expfxðs si ðsÞÞg
i¼1 n X
#
di ðsÞðU expfxgÞðs ri ðsÞÞ ds
i¼1
Clearly, QN and KP ðI QÞN are continuous. It is not difficult to show that KP ðI QÞN ðXÞ is compact for any open bounded X X by using Arzela– Ascoli theorem. Moreover, QN ðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X X . Now we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem (Lemma 2.1). Corresponding to the operator equation Lx ¼ kNx, k 2 ð0; 1Þ, we have " n X dx ¼ k aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg dt i¼1
n X i¼1
n X
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ ;
i¼1
If xðtÞ 2 X is a solution of Eq. (3.16) for a certain k 2 ð0; 1Þ, then
ð3:16Þ
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Z
T
" aðtÞ bðtÞ expfxðtÞg
0
n X i¼1
n X
n X
63
bi ðtÞ expfxðt si ðtÞÞg
i¼1
ci x0 ðt ci ðtÞÞ expfxðt ci ðtÞÞg dðtÞðU expfxgÞðtÞ # di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:17Þ
i¼1
Integrating this identity we have Z T" n X aðtÞ bðtÞ expfxðtÞg bi ðtÞ expfxðt si ðtÞÞg 0
#
i¼1
dðtÞðU expfxgÞðtÞ
n X
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼ 0:
ð3:18Þ
i¼1
That is Z
T
" bðtÞ expfxðtÞg þ
0
n X
bi ðtÞ expfxðt si ðtÞÞg þ dðtÞðU expfxgÞðtÞ
i¼1
þ
n X
#
di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
ð3:19Þ
aðtÞ dt 0
i¼1
and hence by using (3.16), we can find #0 Z T " n X xðtÞ þ k ~ci ðtÞ expfxðt ci ðtÞÞg dt 0 i¼1 Z T n X ¼k bi ðtÞ expfxðt si ðtÞÞg aðtÞ bðtÞ expfxðtÞg 0 i¼1 n X di ðtÞðU expfxgÞðt ri ðtÞÞ dt dðtÞðU expfxgÞðtÞ i¼1 " Z T Z T n X ¼k aðtÞd t þ bðtÞ expfxðtÞg þ bi ðtÞ expfxðt si ðtÞÞg 0
0
þ dðtÞðU expfxgÞðtÞ þ
i¼1 n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
Z <2 0
T
aðtÞ dt 6 2T kak0 :
ð3:20Þ
64
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
Since xðtÞ 2 X , there exist a ~ n 2 ½0; T such that xð~ nÞ ¼ min xðtÞ:
ð3:21Þ
t2½0;T
From (3.19) and (3.21), we see that " þ b
n X
# bi T expfxð~ nÞg 6 aT ;
i¼1
that is ( xð~ nÞ 6 ln
þ b
)
a Pn
i¼1
:¼ M10 :
bi
ð3:22Þ
On the other hand, there also exists a e g 2 ½0; T such that ð3:23Þ
xð~ gÞ ¼ max xðtÞ: t2½0;T
kf k0 þ
Also, by using ðU1ÞðtÞ 6
þ aT 6 b
n X
Pn i¼1
kgi k0
e
, from (3.19) and (3.23) it follows that
! bi T expfxð~ gÞg þ kdk0
Z
n X
kdi k0
Z
T
ðU expfxgÞðt ri ðtÞÞ dt 0
i¼1
þ 6 b
ðU expfxgÞðtÞ dt
0
i¼1
þ
T
n X
! bi T expfxð~ gÞg
i¼1
þ
kdk0 þ
n X
! kdi k0
expfxðe g Þg
Z
þ 6 b
i¼1
bi þ
ðU1ÞðtÞ dt
0
i¼1 n X
T
kdk0 þ
n X i¼1
! kdi k0
kf k0 þ
Pn
i¼1 kgi k0
e
! T expfxð~gÞg;
F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
65
that is
xðe g Þ P ln
9 > =
8 > <
a Pn :¼ M20 : P P > ; :b þ n bi þ ðkdk þ n kdi k Þ kf k0 þ i¼1 kgi k0 > 0 0 i¼1 i¼1 e ð3:24Þ
It follows from (3.19) that Z " T
bðtÞ expfxðtÞg þ
h1 0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt
i¼1
þ h2
Z
T
" bðtÞ expfxðtÞg þ
0
n X
bi ðtÞ expfxðt si ðtÞÞg
i¼1
þ dðtÞðU expfxgÞðtÞ þ
n X
# di ðtÞðU expfxgÞðt ri ðtÞÞ dt ¼
Z
T
aðtÞ dt: 0
i¼1
ð3:25Þ Then similar to the analysis of (3.18), (3.19) and (3.21) of [19], by using the mean value theorem, from (3.25) there exists ~g0 , ~gi , ~d0 , e d i 2 ½0; T , i ¼ 1; 2; . . . ; n such that Z T Z T n X ~vi ð~ aðtÞ dt P h1 ðbð~ g0 Þ þ gi ÞÞ expfxðtÞg dt 0
i¼1
0
þ h2 bð~ d0 c j ð ~ d0 ÞÞð1 c0j ð~d0 ÞÞ
Z
T
expfxðt cj ðtÞÞ dt 0
þ
n X i¼1
bi ð gei Þ 1 1 s0i ðgi Þ 1 c0i ðdi Þ
Z
T
!
expfxðt ci ðtÞÞ dt :
0
That is g0 Þ þ a P h1 ðbð~
n X
~vi ð~ gi ÞÞ expfxð~1Þg þ h2 bð~d0 cj ð~d0 ÞÞð1 c0j ð~d0 ÞÞ
i¼1
expfxð~1 cj ð~1ÞÞ
n X i¼1
! bi ð gei Þ 1 expfxð~1 cj ð~1ÞÞg ; 1 s0i ð gei Þ 1 c0i ð dei Þ
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F. Chen et al. / Appl. Math. Comput. 158 (2004) 45–68
for some e1 2 ½0; T . Therefore, we have ( ) ( ) a a Pn xð1Þ < ln ; 6 ln þ Ve Þ h1 ðbð~ g0 Þ þ i¼1 ~vi ð~ gi ÞÞ h1 ð b expfxðe1 ~si ð~1ÞÞg 6
a d0 cj ð~ d0 ÞÞð1 c0j ð~d0 ÞÞÞ h2 ðbð~
6
ð3:26Þ
a : h2 bð1 kc0i k0 Þ ð3:27Þ
From (3.20), (3.26) and (3.27) it follows that xðtÞ þ k
n X
~ci0 ðtÞ expfxðt ci ðtÞÞg
i¼1
6 xð~1Þ þ k
n X
~ci0 ð~1Þ expfxð~1 cei ð~1ÞÞg
i¼1
" #0 n X xðtÞ þ k ~ci0 ðtÞ expfxðt si ðtÞÞg dt þ 0 i¼1 ( ) e 0k kC a def 0a e 1: < ln þ 2T kak0 ¼ R þþ 0 e h bð1 kc k Þ h1 ðb þ V Þ 2 i 0 Z
T
The rest of the proof is similar to the proof of Theorem 3.1 and we omit it. The proof is complete. h Example. To check all the conditions for Theorem 3.2, and let h1 ¼ h2 ¼ 12, we can obtain that the following equation: " 2 X 1 0 sin2 jt n ðtÞ ¼ nðtÞ ð2 þ sin tÞe6 ecos t nðtÞ ð1 sin jtÞn t 2j j¼1 # 2 X j 0 n ðt jÞ ð3 þ cos tÞuðtÞ þ uðt sin tÞ ; 2 j¼1 u0 ðtÞ ¼ ð2 þ cos tÞuðtÞ þ ð2 þ sin tÞnðtÞ þ nðt cos tÞ þ 5nðt 3 sin tÞ: has a positive 2p-periodic solution. Acknowledgements This work is supported by the Foundation of Ability Person of Fuzhou University under the grant 0030824228, the Foundation of Developing Technology and Science of Fuzhou University under the grant 2003-XQ-21 and the Foundation of Fujian Education Bureau under the grant JB01023.
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