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Positive periodic solutions of a class of functional differential systems with feedback controls
Nonlinear Analysis 57 (2004) 655 – 666
www.elsevier.com/locate/na
Positive periodic solutions of a class of functional di!erential systems with feedback controls Yongkun Li∗ , Ping Liu, Lifei Zhu Department of Mathematics, Yunnan University, Cuihu Beilu 2, Kunming, Yunnan 650091, People’s Republic of China Received 5 April 2003; accepted 10 March 2004
Abstract In this paper, by using the upper and lower solutions method, we investigate the existence and nonexistence of positive periodic solutions of in1nite delay functional di!erential system with a parameter and feedback controls x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))): ? 2004 Elsevier Ltd. All rights reserved. Keywords: Delay di!erential equation; Feedback control; Upper and lower solutions method; Positive periodic solution; State-dependent delay
1. Introduction Recently, Huo and Li [5] have studied the existence of positive periodic solutions for the following delay di!erential system with feedback control: dx = f(t; x(t − 1 (t)); : : : ; x(t − n (t)); u(t − (t))); dt du = −(t)u(t) + a(t)x(t − (t)); dt
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10361006 and the Natural Sciences Foundation of Yunnan Province under Grant 2003A0001M. ∗
Corresponding author. Tel.: +86-08715033701; fax: +86-08715147713. E-mail address: [email protected] (Y. Li).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.03.006
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where F(t; z1 ; z2 ; : : : ; zn ; zn+1 ) ∈ C(Rn+2 ; R), F(t + !; z1 ; z2 ; : : : ; zn ; zn+1 ) = F(t; z1 ; z2 ; : : : ; zn ; zn+1 ), i (t); i = 1; 2; : : : ; n; ; ∈ C(R; R); ; a ∈ C(R; (0; ∞)), all of the above functions are !-periodic functions and ! ¿ 0 is a constant; and Weng [12] has studied the existence of positive periodic solutions for the following integrodi!erential di!erential systems with feedback controls: ! n dyi (t) = yi (t) ri (t) − aii (t)yi (t) − aij (t) Kij (s)yj (t − s) ds dt 0 i=1 ! (s)u (t − s) ds ; (t) H − i i i 0 ! dui (t) = −i (t) + ai (t) Ki (s)yi (t − s) ds; i = 1; 2; : : : ; n; dt 0 where ri ; aij ; i ; i ; ai ∈ C(R; [0; ∞)) are continuous !-periodic functions, Ki ; Kij ; Hi : [0; !] → [0; ∞); i; j=1; 2; : : : ; n are piecewise continuous and normalized. Their methods are based on the continuation theorem of coincidence degree theory which was proposed by Gaines and Mawhin’s [3]. In this paper, we are concerned with the following system of in1nite delay equations with a parameter and feedback controls: x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); (1) u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))); where A(t) = diag[a1 (t); a2 (t); : : : ; an (t)], B(t) = diag[b1 (t); b2 (t); : : : ; bn (t)], aj ; bj ∈ C(R; R) are !-periodic, g(·; ·), h(·; ·), (·; ·) ∈ C(R × Rn ; R) satisfy g(t + !; y) = g(t; y), h(t + !; y) = h(t; y), (t + !; y) = (t; y) for all t ∈ R, y ∈ Rn ; ! ¿ 0 and ¿ 0 are constants, F(t; xt ; y; z) is a function de1ned on R × BC × Rn × Rn , satisfying F(t + !; xt+! ; y; z) = F(t; xt ; y; z) for all t ∈ R, x ∈ BC; y; z ∈ Rn , E ∈ C(R × BC × Rn ; Rn ) satis1es E(t + !; xt+! ; y) = E(t; xt ; y) for all t ∈ R; x ∈ BC; y ∈ Rn , where BC denotes the Banach space of bounded continuous functions : R → Rn with the norm =
n T j=1 sup#∈R |j (#)| where = (1 ; 2 ; : : : ; n ) . If x ∈ BC, then xt ∈ BC for any t ∈ R is de1ned by xt (#) = x(t + #) for # ∈ R. In the sequel, we denote F = (F1 ; F2 ; : : : ; Fn )T . System (1) was extensively investigated in literature as bio-mathematics models. It contains many bio-mathematics models of delay di!erential equations or systems with feedback control, such as the following so-called Michaelis–Menton single-species model with feedback control (see [7]) n dy(t) ai (t)x(t − i (t)) − c(t)u(t − (t)) ; dt = r(t)y(t) 1 − 1 + ci (t)x(t − i (t)) i=1 du(t) = −(t) + a(t)x(t − (t)); dt where i , i=1; 2; : : : ; n; ; (t) ∈ C(R; R); r; c; ai ; : : : ; n; ; a; ∈ C(R; (0; ∞)), all of the above functions are !-periodic functions and ! ¿ 0 is a constant.
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When (t)=a(t) ≡ 0, system (1) also contains the following cellular neural networks (CNNs) with distributed delays and periodic coeLcients [2]: ∞
n n d xi bij (t)fj %j kij (u)xj (t − u) du aij (t)fj (xj (t))+ = −bi (t)xi (t)+ dt 0 j=1
j=1
+Ii (t): For more information about the applications of system (1) to a variety of population models, we refer the reader to [1,2,4–7,9,10,12,13] and the references cited therein. Our aim in this paper is by using the upper and lower solutions method [8] to study the existence and nonexistence of periodic solutions when the parameter varies. For this purpose, we call a continuously di!erentiable and !-periodic function a periodic solution of (1) associated with ∗ if it satis1es (1) when = ∗ . We will give the suLcient conditions to show that there exists ∗ ¿ 0 such that (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not have any !-periodic positive solutions for ¿ ∗ . Our technique is based on the upper and lower solutions method. For convenience, we need to introduce a few notations. Let R = (−∞; +∞), R+ = [0; +∞), R− = (−∞; 0] and Rn+ = {(x1 ; x2 ; : : : ; xn )T ∈ Rn : xj ¿ 0; j = 1; 2; : : : ; n},
nrespectively. For each x = (x1 ; x2 ; : : : ; xn )T ∈ Rn , the norm of x is de1ned as |x| = j=1 |xj |. BC(X; Y ) denotes the set of bounded continuous functions + : X → Y . 2. Some preparation It is easy to show that each !-periodic solution of the equation u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))) is equivalent to that of the following equation t+! u(t) = H (t; s)E(s; xs ; x(h(s; x(s)))) ds := (,x)(t) t
(2)
and vice versa, where H (t; s) = diag[H1 (t; s); H2 (t; s); : : : ; Hn (t; s)] and
s exp( t bj (r) dr) ! Hj (t; s) = ; exp( 0 bj (r) dr) − 1
s ∈ [t; t + !]; j = 1; 2; : : : ; n:
In what follows, we always assume that ! ! (P1 ) 0 aj (s) ds = 0, 0 bj (s) ds = 0 for j = 1; 2; : : : ; n. ! (P2 ) Fj (t; +t ; +(g(t; +(t); (,+)( (t; +(t)))) 0 aj (s) ds ¿ 0 for all (t; +) ∈ R×BC(R; Rn+ ), j = 1; 2; : : : ; n. (P3 ) F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))) is a continuous function of t for each x ∈ BC (R; Rn+ ).
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(P4 ) For any L ¿ 0 and . ¿ 0, there exists ¿ 0 such that for +; ∈ BC; + 6 L, 6 L, and + − ¡ for s ∈ [0; !] imply |F(s; +s ; +(g(s; +(s))); (,+)( (s; +(s)))) − F(s;
s;
(g(s; (s)));
(, )( (s; (s))))| ¡ .: (P5 ) For (t; s) ∈ R2 , u ∈ R, and i = 1; 2; : : : ; n, Gi (t; s)Fi (u; +u ; +(g(u; +(u))); (,+)( (u; ˜ then +(u)))) is nondecreasing in + for the following sense that if + ¿ +, ˜ ˜ ˜ Gi (t; s)Fi (u; +u ; +(g(u; +(u))); (,+) ( (u; +(u)))) ¿ Gi (t; s)Fi (u; +u ; +(g(u; +(u))); ˜ ˜ (,+)( (u; +(u)))). (P6 ) For t ∈ [0; !], i=1; 2; : : : ; n, |Fi (t; +t ; +(g(t; +(t)); (,+)( (t; +(t)))))| is nondecreas˜ then |Fi (t; +t ; +(g(t; +(t)); (,+) ing in + for the following sense that if + ¿ +, ˜ ˜ ˜ ˜ ˜ ( (t; +(t))))| ¿ |Fi (t; +t ; +(g(t; +(t))); (,+)( (t; +(t)))))|. In (P2 )–(P6 ), , is de1ned by (2). We note that the denominator in Hj (t; s) is not zero by (P1 ). It is clear that H (t; s) = H (t + !; s + !) for all (t; s) ∈ R2 and that u(t) de1ned by (2) satis1es u(t + !) = u(t) when x is !-periodic function. We denote (,x) = (,1 x; ,2 x; : : : ; ,n x)T . Therefore, the existence problem of !-periodic solution of (1) is equivalent to that of the following equation: x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); (,x)( (t; x(t)))): It follows from (3) that: t+! x(t) = − G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds; t
(3)
(4)
where G(t; s) = diag[G1 (t; s); G2 (t; s); : : : ; Gn (t; s)] and
s exp(− t aj (r) dr) ! Gj (t; s) = ; exp(− 0 aj (r) dr) − 1
s ∈ [t; t + !]; j = 1; 2; : : : ; n:
By (P1 ), it is clear that the denominator in Gj (t; s) is not zero. It is also clear that G(t; s) = G(t + !; s + !) for all (t; s) ∈ R2 , and by (P2 ), we have Gj (t; s)Fj (u; xu ; x(g(u; x(u))); (,x)( (u; x(u)))) 6 0 2
(5)
BC(R; Rn+ ).
for all (t; s) ∈ R and (u; x) ∈ R × Note further that ! exp(− 0 |aj (r)| dr) ! 0 ¡ Nj := |exp(− 0 aj (r) dr) − 1| ! exp( 0 |aj (r)| dr) ! 6 |Gj (t; s)| 6 |exp(− 0 aj (r) dr) − 1| := Mj ; s ∈ [t; t + !]
(6)
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and we denote N = min Nj and M = max Mj : 16j6n
16j6n
From the above discussion, one can see that the existence problem of positive !-periodic solution of (1) is equivalent to that of (4). Denition. Let X be a Banach space and K be a closed, nonempty subset of X . K is a cone if (i) u + 4v ∈ K for all u; v ∈ K and all ; 4 ¿ 0; (ii) u; −u ∈ K imply u = 0. Let X be the set X = {x ∈ C(R; Rn ) : x(t + !) = x(t); t ∈ R} with the linear structure as well as the norm n x = |xj |0 ; |xj |0 = sup |xj (t)|; t∈[0;!]
j=1
where x = (x1 ; x2 ; : : : ; xn )T ∈ Rn . Then X is a Banach space with cone K = {x ∈ X : x(t) ¿ 0; xj (t) ¿ |xj |0 ; t ∈ [0; !]; x = (x1 ; x2 ; : : : ; xn )T }; ! where := min{exp(−2 0 |aj (s)| ds); j = 1; 2; : : : ; n}, one may readily verify that K is a cone. Now, we de1ne an operator 6 : K → K as t+! (6x)(t) = − G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds t
for x ∈ K, t ∈ R, where G(t; s) is de1ned as that in (4), and (6x)=(61 x; 62 x; : : : ; 6n x)T . 3. Main results Lemma 1. The mapping 6 maps K into K. Proof. For each x ∈ K, it follows from (H3 ) that (6x)(t) is continuous in t and we 1nd (6x)(t + !) t+2! =− G(t + !; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds t+!
=−
t
t+!
G(t + !; v + !)F(v + !; xv+! ; x(g(v + !; x(v + !)));
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(,x)( (v + !; x(v + !)))) dv t+! =− G(t; v)F(v; xv ; x(g(v; x(v)); (,x)( (v; x(v))))) dv = (6x)(t): t
Hence, 6x ∈ X . In view of (5), for x ∈ K, we have ! |6j x|0 6 Mj |Fj (s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds 0
and
(6j x)(t) ¿ Nj
!
|Fj (s; xs ; x(g(t; x(t)); (,x)( (s; x(s)))))| ds ¿
0
(7)
Nj |Fj x|0 ¿ |Fj x|0 : Mj
Therefore, (6x) ∈ K. This completes the proof of Lemma 1. Lemma 2. 6 : K → K is completely continuous, and x = x(t) is an !-periodic solution of (3) whenever x is a 5xed point of 6. Proof. We 1rst show that 6 is continuous. By (P4 ), for any L ¿ 0 and . ¿ 0, there exists a ¿ 0 such that for +; ∈ X; + 6 L; 6 L, and + − ¡ imply sup |F(s; +s ; +(g(s; +(s))); (,+)( (s; +(s))))
06s6!
− F(s;
s;
(g(s; (s))); (, )( (s; (s))))|
. : M! If x; y ∈ K with x 6 L, y 6 L, and x − y ¡ , then t+! |G(t; s)| |Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) |(6xj )(t) − (6yj )(t)|0 6 ¡
t
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; y(s))))| ds ! 6 M |Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) 0
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; y(s))))| ds for all t ∈ [0; !], where |G(t; s)| = max16j6n |Gj (t; s)|. This yields n 6x − 6y = |(6xj )(t) − (6yj )(t)|0 j=1
6
n j=1
M
0
!
|Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s))))
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; x(s))))| ds ¡ .: Thus, 6 is continuous.
Y. Li et al. / Nonlinear Analysis 57 (2004) 655 – 666
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Next, we show that 6 maps bounded sets into bounded sets. Indeed, let .=1. In view of (P4 ), for any % ¿ 0 there exists ¿ 0 such that for x; y ∈ BC, x 6 %, y 6 %, and x − y ¡ imply |F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) − F(s; ys ; y(g(s; y(s))); (,y)( (s; x(s))))| ¡ 1: Choose a positive integer N such that %=N ¡ . Let x ∈ BC and de1ne xk (t) = x(t)k=N for k = 0; 1; 2; : : : ; N . If x 6 %, then n 1 k k − 1 % 6 x 6 ¡ : xk − xk−1 = sup xj (t) − xj (t) N N N N t∈R j=1
Thus, |F(s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s)))) − F(s; (xk−1 )s ; (xk−1 )(g(s; xk−1 (s))); (,xk−1 )( (s; xk−1 (s))))| ¡ 1 for all s ∈ [0; !]. This yields |F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s))))| 6
N
|F(s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s))))
k=1
− F(s; (xk−1 )s ; (xk−1 )(g(s; xk−1 (s))); (,xk−1 )( (s; xk−1 (s))))| + |F(s; 0; 0; 0)| ¡ N + sup |F(s; 0; 0; 0)| =: M% : s∈[0;!]
(8)
It follows from (7) that for t ∈ [0; !], ! n n 6x = |6j x|0 6 Mj |Fj (s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds 6 M!M% : j=1
j=1
0
Finally, for t ∈ R we have d (6x)(t) = − [G(t; t+!)F(t+!; xt+! ; x(g(t + !; x(t+!)); (,x)( (t+!; x(t+!))))) dt − G(t; t)F(t; xt ; x(g(t; x(t))); (,x)( (t; x(t))))] + A(t)(6x)(t) = A(t)(6x)(t)−[G(t; t + !)−G(t; t)]F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))))] = A(t)(6x)(t) − F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t))))): According to (7)–(9), we 1nd ! d (6x)(t) 6 AM |F(s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds dt 0 +|F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))))| 6 [AM!M% + M% ];
(9)
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Y. Li et al. / Nonlinear Analysis 57 (2004) 655 – 666
where A = max16j6n |aj |0 . Hence, {6x : x ∈ K; x 6 %} is a family of uniformly bounded and equicontinuous functions on [0; !]. By the theorem of Ascoli–Arzela [11, p. 169], the function 6 is completely continuous. From (9), it is easy to see that x=x(t) is an !-periodic solution of (3) whenever x is a 1xed point of 6. This completes the proof of Lemma 2. Lemma 3. Suppose that lim
|uj |0→+∞
|Fj (s; us ; u(g(s; u(s)); (,u)( (s; u(s))))| = +∞ for s ∈ R and j = 1; 2; : : : ; n: |uj |0 (10)
Let I be a compact subset of (0; +∞). Then there exists a constant bI ¿ 0 such that u ¡ bI for all ∈ I and all possible !-periodic positive solution u of (3) associated with . Proof. Suppose to the contrary that there is a sequence {uk } of !-periodic positive solutions of (3) associated with {k } such that k ∈ I for all k and uk → +∞ as k → ∞. By (10), we may choose Rj ¿ 0; j = 1; 2; : : : ; n such that |Fj (s; us ; u(g(s; u(s)); (,u)( (s; u(s))))| ¿ |uj |0 for all |uj |0 ¿ Rj ; s ∈ R, and there exists k0 and some i = 1; 2; : : : ; n such that uki 0 ¿ Ri , where satis1es Nk0 ! ¿ 1: So, we have |uni 0 |0
¿ |uni 0 |
= n0
t
t+!
|Gi (t; s)Fi (s; (un0 )s ); un0 (g(s; un0 (s)); (,un0 )( (s; un0 (s))))| ds
¿ n0 N!|uki 0 |0 ¿ |uki 0 |0 : This is a contradiction. The proof is complete. Lemma 4. Suppose that (P1 )–(P3 ), (P5 ) and Fi (t; 0; 0; 0) = 0; t ∈ R; i = 1; 2; : : : ; n:
(11)
Let (3) has an !-periodic positive solution x(t) associated with P ¿ 0, then (3) P also has a positive !-periodic solution with ∈ (0; ). Proof. In view of (3) and (11), we have t+! G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds: x(t) = −P t
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For (t; s) ∈ R2 , j = 1; 2; : : : ; n, t+! Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds xj (t) = − P t
¿− Let xPj0 (t) = − P
t+!
t
t
xPjk+1 (t) = −
t+!
Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds:
Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds;
t+!
t
Gj (t; s)Fj (s; (xPk )s ); xPk (g(s; xPk (s))); (,xPk )( (s; xPk (s)))) ds;
k = 0; 1; : : : and xj0 (t) = 0; xjk+1 (t)
= −
t
t+!
Gj (t; s)Fj (s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s)))) ds;
k = 0; 1; : : : : It is clear that xPj0 (t) ¿ xPj1 (t) ¿ · · · ¿ xPjk (t) ¿ xjk (t) ¿ · · · ¿ xj1 (t) ¿ xj0 (t) = 0;
j = 1; 2; : : : ; n:
Now, if we let x˜j (t) = limk→∞ xPjk (t), then x(t) ˜ satis1es (3). And in view of (6) and (11), we have t+! x˜j (t) ¿ xj1 (t) = − Gj (t; s)Fj (s; 0; 0; 0) ds ¿ 0; j = 1; 2; : : : ; n: t
P The proof Hence, x(t) ˜ is a positive !-periodic solution of (3) when ∈ (0; ). is complete. Lemma 5. Suppose that (P1 )–(P3 ), (P5 ), (10) and (11) hold, then there exists ∗ ¿ 0 such that (3) has an !-periodic positive solution. Proof. For (t; s) ∈ R2 , i = 1; 2; : : : ; n, we let 4 = (41 ; 42 ; : : : ; 4n )T , t+! 4i (t) = |Gi (t; s)| ds; Ti = max max |Fi (t; 4t ; 4(g(t; 4(t))); (,4)( (t; 4(t))))| 16i6n t∈[0;!]
t
and ∗ = 1=Ti . We have t+! i 4 (t)= |Gi (t; s)|ds ¿ −∗ t
t
t+!
Gi (t; s)Fi (s; 4s ; 4(g(s; 4(s))); (,4)( (s; 4(s)))) ds:
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Y. Li et al. / Nonlinear Analysis 57 (2004) 655 – 666
Let xPi0 (t) = 4i (t) = xPik+1 (t) = − ∗
t+!
t t+! t
|Gi (t; s)| ds; Gi (t; s)Fi (s; (xPk )s ); xPk (g(s; xPk (s))); (,xPk )( (s; xPk (s))) ds;
k = 0; 1; : : : ; and xi0 (t) = 0; xik+1 (t) = − ∗
t
t+!
Gi (t; s)Fi (s; (xk )s ); xk (g(s; xk (s))); (,xk )( (s; xk (s))) ds;
k = 0; 1; : : : : Clearly, we have xPi0 (t) ¿ xPi1 (t) ¿ · · · ¿ xPik (t) ¿ xik (t) ¿ · · · ¿ xi1 (t) ¿ xi0 (t) = 0; i = 1; 2; : : : ; n: ˜ satis1es (3). And according to (6) and (11), If we let x˜i (t) = limk→∞ xPik (t), then x(t) we have t+! x˜i (t) ¿ xi1 (t) = − ∗ Gi (t; s)Fi (s; 0; 0; 0) ds ¿ 0; i = 1; 2; : : : ; n: t
So, there exists ∗ ¿ 0 such that x(t) ˜ is a positive !-periodic solution of (3). This completes the proof of Lemma 5. We are now in a position to state and prove our main result of this paper. Theorem 1. Suppose that (P1 )–(P6 ), (10) and (11) hold. Then there exists ∗ ¿ 0 such that system (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not have any !-periodic positive solutions for ¿ ∗ . Proof. Suppose to the contrary that there is a sequence {uk } of !-periodic positive solutions of (3) associated with {k } such that limn→∞ k = ∞. Then either we have ukj → ∞ as j → ∞ or there is a positive constant D ¿ 0 such that uk 6 D. Assume the former case holds, by (10), we may choose Ri ¿ 0; i=1; 2; : : : ; n and 1 ¿ 0 such that |Fi (t; ut ; u(g(t; u(t)); (,u)( (t; u(t))))| ¿ 1 |ui |0 when ui ¿ Ri and t ∈ R. On the other hand, there exist {tki j } ⊂ [0; !] such that uki j (tki j ) = |uki j |0 and (uki j ) (tki j ) = 0 by the periodicity of {ukj (t)} for all large j and i = 1; 2; : : : ; n. In view of (3), we have ai (tki j )uki j (tki j ) = ai (tki j )|uki j |0 = kj Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))) j
thus, |ai (tki j )uki j |0 = kj |Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))| j
¿ kj 1 |uki j |0
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for all large j and i = 1; 2; : : : ; n. Thus, we get nj ¡ |ai (tni j )|=1 is bounded. Clearly, this is a contradiction. Assume the latter case holds. In view of Fi (t; 0; 0; 0) = 0, there exists 2 ¿ 0 such that |Fi (t; 0; 0; 0)| ¿ 2 D ¿ 0. Since uk 6 D, we have |uki |0 6 D. As above, we obtain |ai (tki j )uki |0 = k |Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))| j
¿ k |Fi (tki j ; 0; 0; 0)| ¿ n 2 D ¿ n 2 |uki |0 for tnj ∈ [0; !]. And we get n 6 |ai (tni j )|=2 is bounded, this is also a contradiction. Thus, there exists ∗ ¿ 0 such that (3) has at least one positive !-periodic solution for ∈ (0; ∗ ) and no !-periodic positive solutions for ¿ ∗ . Finally, we assert that (3) has at least one !-periodic positive solution for = ∗ . Indeed, let {k } satis1es 0 ¡ 1 ¡ · · · ¡ k ¡ ∗ and limk→∞ k = ∗ . Since uk (t) is !-periodic positive solution of (3) associated with k and Lemma 2 implies that the set {uk (t)} of solutions is uniformly bounded in K, the sequence {uk (t)} has a subsequence converging to u(t) ∈ K. We can now apply the Lebesgue convergence theorem to show that u(t) is an !-periodic positive solution of (3) associated with = ∗ . Therefore, system (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not any !-periodic positive solutions for ¿ ∗ . The proof is complete. Using the same method of this paper, one can show that Theorem 2. Suppose that (H1 )–(H6 ), (10) and (11) hold. Then there exists ∗ ¿ 0 such that the systems x(t) ˙ = −A(t)x(t) + F(t; xt ; x(g(t; x(t))); u( (t; x(t))));
u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))); x(t) ˙ = A(t)x(t) + F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t))))
and
x(t) ˙ = −A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t))))
have at least one positive !-periodic solution for ∈ (0; ∗ ] and no !-periodic positive solutions for ¿ ∗ , respectively, where A; B; ; F; E; g; h; are de5ned as those in (1). References [1] J. Cao, L. Wang, Periodic oscillation solution of bidirectional associative memory networks with delays, Phys. Rev. E 61 (2000) 1285–1828. [2] A. Chen, J. Cao, Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coeLcients, Appl. Math. Comput. 134 (2003) 125–140. [3] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Di!erential Equations, Springer, Berlin, 1977.
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[4] K. Gopalsamy, P.X. Weng, Feedback regulation of logistic growth, J. Math. Math. Sci. 1 (1993) 177–192. [5] H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay di!erential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46. [6] D.Q. Jiang, J.J. Wei, B. Zhang, Positive periodic solutions of functional di!erential equations and population models, Elec. J. Di!erential Equations 2002 (2002) 1–13. [7] Y. Kuang, Delay Di!erential Equations with Application in Population Dynamics, Academic Press, New York, 1993. [8] Y.-H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Di!erential Equations 174 (2) (2001) 420–441. [9] Y.K. Li, Y. Kuang, Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) 260–280. [10] X.-M. Lia, H. Li, H. Zhu, Global stability of cellular neural networks with constant and variable delays, Nonlinear Anal. 53 (2003) 319–333. [11] H.L. Royden, Real Analysis, Macmillan Publishing Company, New York, 1998. [12] P. Weng, Existence and global stability of positive periodic solution of periodic integrodi!erential systems with feedback controls, Comput. Math. Appl. 40 (2000) 747–759. [13] F. Yang, D.Q. Jiang, Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments, Ann. Di!erential Equations 17 (4) (2001) 377–384.
www.elsevier.com/locate/na
Positive periodic solutions of a class of functional di!erential systems with feedback controls Yongkun Li∗ , Ping Liu, Lifei Zhu Department of Mathematics, Yunnan University, Cuihu Beilu 2, Kunming, Yunnan 650091, People’s Republic of China Received 5 April 2003; accepted 10 March 2004
Abstract In this paper, by using the upper and lower solutions method, we investigate the existence and nonexistence of positive periodic solutions of in1nite delay functional di!erential system with a parameter and feedback controls x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))): ? 2004 Elsevier Ltd. All rights reserved. Keywords: Delay di!erential equation; Feedback control; Upper and lower solutions method; Positive periodic solution; State-dependent delay
1. Introduction Recently, Huo and Li [5] have studied the existence of positive periodic solutions for the following delay di!erential system with feedback control: dx = f(t; x(t − 1 (t)); : : : ; x(t − n (t)); u(t − (t))); dt du = −(t)u(t) + a(t)x(t − (t)); dt
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10361006 and the Natural Sciences Foundation of Yunnan Province under Grant 2003A0001M. ∗
Corresponding author. Tel.: +86-08715033701; fax: +86-08715147713. E-mail address: [email protected] (Y. Li).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.03.006
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where F(t; z1 ; z2 ; : : : ; zn ; zn+1 ) ∈ C(Rn+2 ; R), F(t + !; z1 ; z2 ; : : : ; zn ; zn+1 ) = F(t; z1 ; z2 ; : : : ; zn ; zn+1 ), i (t); i = 1; 2; : : : ; n; ; ∈ C(R; R); ; a ∈ C(R; (0; ∞)), all of the above functions are !-periodic functions and ! ¿ 0 is a constant; and Weng [12] has studied the existence of positive periodic solutions for the following integrodi!erential di!erential systems with feedback controls: ! n dyi (t) = yi (t) ri (t) − aii (t)yi (t) − aij (t) Kij (s)yj (t − s) ds dt 0 i=1 ! (s)u (t − s) ds ; (t) H − i i i 0 ! dui (t) = −i (t) + ai (t) Ki (s)yi (t − s) ds; i = 1; 2; : : : ; n; dt 0 where ri ; aij ; i ; i ; ai ∈ C(R; [0; ∞)) are continuous !-periodic functions, Ki ; Kij ; Hi : [0; !] → [0; ∞); i; j=1; 2; : : : ; n are piecewise continuous and normalized. Their methods are based on the continuation theorem of coincidence degree theory which was proposed by Gaines and Mawhin’s [3]. In this paper, we are concerned with the following system of in1nite delay equations with a parameter and feedback controls: x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); (1) u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))); where A(t) = diag[a1 (t); a2 (t); : : : ; an (t)], B(t) = diag[b1 (t); b2 (t); : : : ; bn (t)], aj ; bj ∈ C(R; R) are !-periodic, g(·; ·), h(·; ·), (·; ·) ∈ C(R × Rn ; R) satisfy g(t + !; y) = g(t; y), h(t + !; y) = h(t; y), (t + !; y) = (t; y) for all t ∈ R, y ∈ Rn ; ! ¿ 0 and ¿ 0 are constants, F(t; xt ; y; z) is a function de1ned on R × BC × Rn × Rn , satisfying F(t + !; xt+! ; y; z) = F(t; xt ; y; z) for all t ∈ R, x ∈ BC; y; z ∈ Rn , E ∈ C(R × BC × Rn ; Rn ) satis1es E(t + !; xt+! ; y) = E(t; xt ; y) for all t ∈ R; x ∈ BC; y ∈ Rn , where BC denotes the Banach space of bounded continuous functions : R → Rn with the norm =
n T j=1 sup#∈R |j (#)| where = (1 ; 2 ; : : : ; n ) . If x ∈ BC, then xt ∈ BC for any t ∈ R is de1ned by xt (#) = x(t + #) for # ∈ R. In the sequel, we denote F = (F1 ; F2 ; : : : ; Fn )T . System (1) was extensively investigated in literature as bio-mathematics models. It contains many bio-mathematics models of delay di!erential equations or systems with feedback control, such as the following so-called Michaelis–Menton single-species model with feedback control (see [7]) n dy(t) ai (t)x(t − i (t)) − c(t)u(t − (t)) ; dt = r(t)y(t) 1 − 1 + ci (t)x(t − i (t)) i=1 du(t) = −(t) + a(t)x(t − (t)); dt where i , i=1; 2; : : : ; n; ; (t) ∈ C(R; R); r; c; ai ; : : : ; n; ; a; ∈ C(R; (0; ∞)), all of the above functions are !-periodic functions and ! ¿ 0 is a constant.
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When (t)=a(t) ≡ 0, system (1) also contains the following cellular neural networks (CNNs) with distributed delays and periodic coeLcients [2]: ∞
n n d xi bij (t)fj %j kij (u)xj (t − u) du aij (t)fj (xj (t))+ = −bi (t)xi (t)+ dt 0 j=1
j=1
+Ii (t): For more information about the applications of system (1) to a variety of population models, we refer the reader to [1,2,4–7,9,10,12,13] and the references cited therein. Our aim in this paper is by using the upper and lower solutions method [8] to study the existence and nonexistence of periodic solutions when the parameter varies. For this purpose, we call a continuously di!erentiable and !-periodic function a periodic solution of (1) associated with ∗ if it satis1es (1) when = ∗ . We will give the suLcient conditions to show that there exists ∗ ¿ 0 such that (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not have any !-periodic positive solutions for ¿ ∗ . Our technique is based on the upper and lower solutions method. For convenience, we need to introduce a few notations. Let R = (−∞; +∞), R+ = [0; +∞), R− = (−∞; 0] and Rn+ = {(x1 ; x2 ; : : : ; xn )T ∈ Rn : xj ¿ 0; j = 1; 2; : : : ; n},
nrespectively. For each x = (x1 ; x2 ; : : : ; xn )T ∈ Rn , the norm of x is de1ned as |x| = j=1 |xj |. BC(X; Y ) denotes the set of bounded continuous functions + : X → Y . 2. Some preparation It is easy to show that each !-periodic solution of the equation u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))) is equivalent to that of the following equation t+! u(t) = H (t; s)E(s; xs ; x(h(s; x(s)))) ds := (,x)(t) t
(2)
and vice versa, where H (t; s) = diag[H1 (t; s); H2 (t; s); : : : ; Hn (t; s)] and
s exp( t bj (r) dr) ! Hj (t; s) = ; exp( 0 bj (r) dr) − 1
s ∈ [t; t + !]; j = 1; 2; : : : ; n:
In what follows, we always assume that ! ! (P1 ) 0 aj (s) ds = 0, 0 bj (s) ds = 0 for j = 1; 2; : : : ; n. ! (P2 ) Fj (t; +t ; +(g(t; +(t); (,+)( (t; +(t)))) 0 aj (s) ds ¿ 0 for all (t; +) ∈ R×BC(R; Rn+ ), j = 1; 2; : : : ; n. (P3 ) F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))) is a continuous function of t for each x ∈ BC (R; Rn+ ).
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(P4 ) For any L ¿ 0 and . ¿ 0, there exists ¿ 0 such that for +; ∈ BC; + 6 L, 6 L, and + − ¡ for s ∈ [0; !] imply |F(s; +s ; +(g(s; +(s))); (,+)( (s; +(s)))) − F(s;
s;
(g(s; (s)));
(, )( (s; (s))))| ¡ .: (P5 ) For (t; s) ∈ R2 , u ∈ R, and i = 1; 2; : : : ; n, Gi (t; s)Fi (u; +u ; +(g(u; +(u))); (,+)( (u; ˜ then +(u)))) is nondecreasing in + for the following sense that if + ¿ +, ˜ ˜ ˜ Gi (t; s)Fi (u; +u ; +(g(u; +(u))); (,+) ( (u; +(u)))) ¿ Gi (t; s)Fi (u; +u ; +(g(u; +(u))); ˜ ˜ (,+)( (u; +(u)))). (P6 ) For t ∈ [0; !], i=1; 2; : : : ; n, |Fi (t; +t ; +(g(t; +(t)); (,+)( (t; +(t)))))| is nondecreas˜ then |Fi (t; +t ; +(g(t; +(t)); (,+) ing in + for the following sense that if + ¿ +, ˜ ˜ ˜ ˜ ˜ ( (t; +(t))))| ¿ |Fi (t; +t ; +(g(t; +(t))); (,+)( (t; +(t)))))|. In (P2 )–(P6 ), , is de1ned by (2). We note that the denominator in Hj (t; s) is not zero by (P1 ). It is clear that H (t; s) = H (t + !; s + !) for all (t; s) ∈ R2 and that u(t) de1ned by (2) satis1es u(t + !) = u(t) when x is !-periodic function. We denote (,x) = (,1 x; ,2 x; : : : ; ,n x)T . Therefore, the existence problem of !-periodic solution of (1) is equivalent to that of the following equation: x(t) ˙ = A(t)x(t) − F(t; xt ; x(g(t; x(t))); (,x)( (t; x(t)))): It follows from (3) that: t+! x(t) = − G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds; t
(3)
(4)
where G(t; s) = diag[G1 (t; s); G2 (t; s); : : : ; Gn (t; s)] and
s exp(− t aj (r) dr) ! Gj (t; s) = ; exp(− 0 aj (r) dr) − 1
s ∈ [t; t + !]; j = 1; 2; : : : ; n:
By (P1 ), it is clear that the denominator in Gj (t; s) is not zero. It is also clear that G(t; s) = G(t + !; s + !) for all (t; s) ∈ R2 , and by (P2 ), we have Gj (t; s)Fj (u; xu ; x(g(u; x(u))); (,x)( (u; x(u)))) 6 0 2
(5)
BC(R; Rn+ ).
for all (t; s) ∈ R and (u; x) ∈ R × Note further that ! exp(− 0 |aj (r)| dr) ! 0 ¡ Nj := |exp(− 0 aj (r) dr) − 1| ! exp( 0 |aj (r)| dr) ! 6 |Gj (t; s)| 6 |exp(− 0 aj (r) dr) − 1| := Mj ; s ∈ [t; t + !]
(6)
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and we denote N = min Nj and M = max Mj : 16j6n
16j6n
From the above discussion, one can see that the existence problem of positive !-periodic solution of (1) is equivalent to that of (4). Denition. Let X be a Banach space and K be a closed, nonempty subset of X . K is a cone if (i) u + 4v ∈ K for all u; v ∈ K and all ; 4 ¿ 0; (ii) u; −u ∈ K imply u = 0. Let X be the set X = {x ∈ C(R; Rn ) : x(t + !) = x(t); t ∈ R} with the linear structure as well as the norm n x = |xj |0 ; |xj |0 = sup |xj (t)|; t∈[0;!]
j=1
where x = (x1 ; x2 ; : : : ; xn )T ∈ Rn . Then X is a Banach space with cone K = {x ∈ X : x(t) ¿ 0; xj (t) ¿ |xj |0 ; t ∈ [0; !]; x = (x1 ; x2 ; : : : ; xn )T }; ! where := min{exp(−2 0 |aj (s)| ds); j = 1; 2; : : : ; n}, one may readily verify that K is a cone. Now, we de1ne an operator 6 : K → K as t+! (6x)(t) = − G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds t
for x ∈ K, t ∈ R, where G(t; s) is de1ned as that in (4), and (6x)=(61 x; 62 x; : : : ; 6n x)T . 3. Main results Lemma 1. The mapping 6 maps K into K. Proof. For each x ∈ K, it follows from (H3 ) that (6x)(t) is continuous in t and we 1nd (6x)(t + !) t+2! =− G(t + !; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds t+!
=−
t
t+!
G(t + !; v + !)F(v + !; xv+! ; x(g(v + !; x(v + !)));
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(,x)( (v + !; x(v + !)))) dv t+! =− G(t; v)F(v; xv ; x(g(v; x(v)); (,x)( (v; x(v))))) dv = (6x)(t): t
Hence, 6x ∈ X . In view of (5), for x ∈ K, we have ! |6j x|0 6 Mj |Fj (s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds 0
and
(6j x)(t) ¿ Nj
!
|Fj (s; xs ; x(g(t; x(t)); (,x)( (s; x(s)))))| ds ¿
0
(7)
Nj |Fj x|0 ¿ |Fj x|0 : Mj
Therefore, (6x) ∈ K. This completes the proof of Lemma 1. Lemma 2. 6 : K → K is completely continuous, and x = x(t) is an !-periodic solution of (3) whenever x is a 5xed point of 6. Proof. We 1rst show that 6 is continuous. By (P4 ), for any L ¿ 0 and . ¿ 0, there exists a ¿ 0 such that for +; ∈ X; + 6 L; 6 L, and + − ¡ imply sup |F(s; +s ; +(g(s; +(s))); (,+)( (s; +(s))))
06s6!
− F(s;
s;
(g(s; (s))); (, )( (s; (s))))|
. : M! If x; y ∈ K with x 6 L, y 6 L, and x − y ¡ , then t+! |G(t; s)| |Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) |(6xj )(t) − (6yj )(t)|0 6 ¡
t
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; y(s))))| ds ! 6 M |Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) 0
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; y(s))))| ds for all t ∈ [0; !], where |G(t; s)| = max16j6n |Gj (t; s)|. This yields n 6x − 6y = |(6xj )(t) − (6yj )(t)|0 j=1
6
n j=1
M
0
!
|Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s))))
− Fj (s; ys ; y(g(s; y(s))); (,y)( (s; x(s))))| ds ¡ .: Thus, 6 is continuous.
Y. Li et al. / Nonlinear Analysis 57 (2004) 655 – 666
661
Next, we show that 6 maps bounded sets into bounded sets. Indeed, let .=1. In view of (P4 ), for any % ¿ 0 there exists ¿ 0 such that for x; y ∈ BC, x 6 %, y 6 %, and x − y ¡ imply |F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) − F(s; ys ; y(g(s; y(s))); (,y)( (s; x(s))))| ¡ 1: Choose a positive integer N such that %=N ¡ . Let x ∈ BC and de1ne xk (t) = x(t)k=N for k = 0; 1; 2; : : : ; N . If x 6 %, then n 1 k k − 1 % 6 x 6 ¡ : xk − xk−1 = sup xj (t) − xj (t) N N N N t∈R j=1
Thus, |F(s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s)))) − F(s; (xk−1 )s ; (xk−1 )(g(s; xk−1 (s))); (,xk−1 )( (s; xk−1 (s))))| ¡ 1 for all s ∈ [0; !]. This yields |F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s))))| 6
N
|F(s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s))))
k=1
− F(s; (xk−1 )s ; (xk−1 )(g(s; xk−1 (s))); (,xk−1 )( (s; xk−1 (s))))| + |F(s; 0; 0; 0)| ¡ N + sup |F(s; 0; 0; 0)| =: M% : s∈[0;!]
(8)
It follows from (7) that for t ∈ [0; !], ! n n 6x = |6j x|0 6 Mj |Fj (s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds 6 M!M% : j=1
j=1
0
Finally, for t ∈ R we have d (6x)(t) = − [G(t; t+!)F(t+!; xt+! ; x(g(t + !; x(t+!)); (,x)( (t+!; x(t+!))))) dt − G(t; t)F(t; xt ; x(g(t; x(t))); (,x)( (t; x(t))))] + A(t)(6x)(t) = A(t)(6x)(t)−[G(t; t + !)−G(t; t)]F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))))] = A(t)(6x)(t) − F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t))))): According to (7)–(9), we 1nd ! d (6x)(t) 6 AM |F(s; xs ; x(g(s; x(s)); (,x)( (s; x(s)))))| ds dt 0 +|F(t; xt ; x(g(t; x(t)); (,x)( (t; x(t)))))| 6 [AM!M% + M% ];
(9)
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where A = max16j6n |aj |0 . Hence, {6x : x ∈ K; x 6 %} is a family of uniformly bounded and equicontinuous functions on [0; !]. By the theorem of Ascoli–Arzela [11, p. 169], the function 6 is completely continuous. From (9), it is easy to see that x=x(t) is an !-periodic solution of (3) whenever x is a 1xed point of 6. This completes the proof of Lemma 2. Lemma 3. Suppose that lim
|uj |0→+∞
|Fj (s; us ; u(g(s; u(s)); (,u)( (s; u(s))))| = +∞ for s ∈ R and j = 1; 2; : : : ; n: |uj |0 (10)
Let I be a compact subset of (0; +∞). Then there exists a constant bI ¿ 0 such that u ¡ bI for all ∈ I and all possible !-periodic positive solution u of (3) associated with . Proof. Suppose to the contrary that there is a sequence {uk } of !-periodic positive solutions of (3) associated with {k } such that k ∈ I for all k and uk → +∞ as k → ∞. By (10), we may choose Rj ¿ 0; j = 1; 2; : : : ; n such that |Fj (s; us ; u(g(s; u(s)); (,u)( (s; u(s))))| ¿ |uj |0 for all |uj |0 ¿ Rj ; s ∈ R, and there exists k0 and some i = 1; 2; : : : ; n such that uki 0 ¿ Ri , where satis1es Nk0 ! ¿ 1: So, we have |uni 0 |0
¿ |uni 0 |
= n0
t
t+!
|Gi (t; s)Fi (s; (un0 )s ); un0 (g(s; un0 (s)); (,un0 )( (s; un0 (s))))| ds
¿ n0 N!|uki 0 |0 ¿ |uki 0 |0 : This is a contradiction. The proof is complete. Lemma 4. Suppose that (P1 )–(P3 ), (P5 ) and Fi (t; 0; 0; 0) = 0; t ∈ R; i = 1; 2; : : : ; n:
(11)
Let (3) has an !-periodic positive solution x(t) associated with P ¿ 0, then (3) P also has a positive !-periodic solution with ∈ (0; ). Proof. In view of (3) and (11), we have t+! G(t; s)F(s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds: x(t) = −P t
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For (t; s) ∈ R2 , j = 1; 2; : : : ; n, t+! Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds xj (t) = − P t
¿− Let xPj0 (t) = − P
t+!
t
t
xPjk+1 (t) = −
t+!
Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds:
Gj (t; s)Fj (s; xs ; x(g(s; x(s))); (,x)( (s; x(s)))) ds;
t+!
t
Gj (t; s)Fj (s; (xPk )s ); xPk (g(s; xPk (s))); (,xPk )( (s; xPk (s)))) ds;
k = 0; 1; : : : and xj0 (t) = 0; xjk+1 (t)
= −
t
t+!
Gj (t; s)Fj (s; (xk )s ; xk (g(s; xk (s))); (,xk )( (s; xk (s)))) ds;
k = 0; 1; : : : : It is clear that xPj0 (t) ¿ xPj1 (t) ¿ · · · ¿ xPjk (t) ¿ xjk (t) ¿ · · · ¿ xj1 (t) ¿ xj0 (t) = 0;
j = 1; 2; : : : ; n:
Now, if we let x˜j (t) = limk→∞ xPjk (t), then x(t) ˜ satis1es (3). And in view of (6) and (11), we have t+! x˜j (t) ¿ xj1 (t) = − Gj (t; s)Fj (s; 0; 0; 0) ds ¿ 0; j = 1; 2; : : : ; n: t
P The proof Hence, x(t) ˜ is a positive !-periodic solution of (3) when ∈ (0; ). is complete. Lemma 5. Suppose that (P1 )–(P3 ), (P5 ), (10) and (11) hold, then there exists ∗ ¿ 0 such that (3) has an !-periodic positive solution. Proof. For (t; s) ∈ R2 , i = 1; 2; : : : ; n, we let 4 = (41 ; 42 ; : : : ; 4n )T , t+! 4i (t) = |Gi (t; s)| ds; Ti = max max |Fi (t; 4t ; 4(g(t; 4(t))); (,4)( (t; 4(t))))| 16i6n t∈[0;!]
t
and ∗ = 1=Ti . We have t+! i 4 (t)= |Gi (t; s)|ds ¿ −∗ t
t
t+!
Gi (t; s)Fi (s; 4s ; 4(g(s; 4(s))); (,4)( (s; 4(s)))) ds:
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Let xPi0 (t) = 4i (t) = xPik+1 (t) = − ∗
t+!
t t+! t
|Gi (t; s)| ds; Gi (t; s)Fi (s; (xPk )s ); xPk (g(s; xPk (s))); (,xPk )( (s; xPk (s))) ds;
k = 0; 1; : : : ; and xi0 (t) = 0; xik+1 (t) = − ∗
t
t+!
Gi (t; s)Fi (s; (xk )s ); xk (g(s; xk (s))); (,xk )( (s; xk (s))) ds;
k = 0; 1; : : : : Clearly, we have xPi0 (t) ¿ xPi1 (t) ¿ · · · ¿ xPik (t) ¿ xik (t) ¿ · · · ¿ xi1 (t) ¿ xi0 (t) = 0; i = 1; 2; : : : ; n: ˜ satis1es (3). And according to (6) and (11), If we let x˜i (t) = limk→∞ xPik (t), then x(t) we have t+! x˜i (t) ¿ xi1 (t) = − ∗ Gi (t; s)Fi (s; 0; 0; 0) ds ¿ 0; i = 1; 2; : : : ; n: t
So, there exists ∗ ¿ 0 such that x(t) ˜ is a positive !-periodic solution of (3). This completes the proof of Lemma 5. We are now in a position to state and prove our main result of this paper. Theorem 1. Suppose that (P1 )–(P6 ), (10) and (11) hold. Then there exists ∗ ¿ 0 such that system (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not have any !-periodic positive solutions for ¿ ∗ . Proof. Suppose to the contrary that there is a sequence {uk } of !-periodic positive solutions of (3) associated with {k } such that limn→∞ k = ∞. Then either we have ukj → ∞ as j → ∞ or there is a positive constant D ¿ 0 such that uk 6 D. Assume the former case holds, by (10), we may choose Ri ¿ 0; i=1; 2; : : : ; n and 1 ¿ 0 such that |Fi (t; ut ; u(g(t; u(t)); (,u)( (t; u(t))))| ¿ 1 |ui |0 when ui ¿ Ri and t ∈ R. On the other hand, there exist {tki j } ⊂ [0; !] such that uki j (tki j ) = |uki j |0 and (uki j ) (tki j ) = 0 by the periodicity of {ukj (t)} for all large j and i = 1; 2; : : : ; n. In view of (3), we have ai (tki j )uki j (tki j ) = ai (tki j )|uki j |0 = kj Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))) j
thus, |ai (tki j )uki j |0 = kj |Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))| j
¿ kj 1 |uki j |0
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for all large j and i = 1; 2; : : : ; n. Thus, we get nj ¡ |ai (tni j )|=1 is bounded. Clearly, this is a contradiction. Assume the latter case holds. In view of Fi (t; 0; 0; 0) = 0, there exists 2 ¿ 0 such that |Fi (t; 0; 0; 0)| ¿ 2 D ¿ 0. Since uk 6 D, we have |uki |0 6 D. As above, we obtain |ai (tki j )uki |0 = k |Fi (tki j ; (ukj )tki ; ukj (g(tki j ; ukj (tki j ))); (,ukj )( (tki j ; ukj (tki j )))| j
¿ k |Fi (tki j ; 0; 0; 0)| ¿ n 2 D ¿ n 2 |uki |0 for tnj ∈ [0; !]. And we get n 6 |ai (tni j )|=2 is bounded, this is also a contradiction. Thus, there exists ∗ ¿ 0 such that (3) has at least one positive !-periodic solution for ∈ (0; ∗ ) and no !-periodic positive solutions for ¿ ∗ . Finally, we assert that (3) has at least one !-periodic positive solution for = ∗ . Indeed, let {k } satis1es 0 ¡ 1 ¡ · · · ¡ k ¡ ∗ and limk→∞ k = ∗ . Since uk (t) is !-periodic positive solution of (3) associated with k and Lemma 2 implies that the set {uk (t)} of solutions is uniformly bounded in K, the sequence {uk (t)} has a subsequence converging to u(t) ∈ K. We can now apply the Lebesgue convergence theorem to show that u(t) is an !-periodic positive solution of (3) associated with = ∗ . Therefore, system (1) has at least one positive !-periodic solution for ∈ (0; ∗ ] and does not any !-periodic positive solutions for ¿ ∗ . The proof is complete. Using the same method of this paper, one can show that Theorem 2. Suppose that (H1 )–(H6 ), (10) and (11) hold. Then there exists ∗ ¿ 0 such that the systems x(t) ˙ = −A(t)x(t) + F(t; xt ; x(g(t; x(t))); u( (t; x(t))));
u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t)))); x(t) ˙ = A(t)x(t) + F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t))))
and
x(t) ˙ = −A(t)x(t) − F(t; xt ; x(g(t; x(t))); u( (t; x(t)))); u(t) ˙ = −B(t)u(t) + E(t; xt ; x(h(t; x(t))))
have at least one positive !-periodic solution for ∈ (0; ∗ ] and no !-periodic positive solutions for ¿ ∗ , respectively, where A; B; ; F; E; g; h; are de5ned as those in (1). References [1] J. Cao, L. Wang, Periodic oscillation solution of bidirectional associative memory networks with delays, Phys. Rev. E 61 (2000) 1285–1828. [2] A. Chen, J. Cao, Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coeLcients, Appl. Math. Comput. 134 (2003) 125–140. [3] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Di!erential Equations, Springer, Berlin, 1977.
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