Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control

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...

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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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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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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



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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