Multiple positive periodic solutions of nonlinear functional differential system with feedback control

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1 J. Mah. Anal. Appl. 288 (23) Muliple posiive periodic soluions of nonlinear funcional differenial sysem wih feedback conrol Ping Liu and Yongkun Li Deparmen of Mahemaics, Yunnan Universiy, Kunming, Yunnan 6591, PR China Received 21 April 23 Submied by S.R. Grace Absrac In his paper, we employ Avery Henderson fixed poin heorem o sudy he exisence of posiive periodic soluions o he following nonlinear nonauonomous funcional differenial sysem wih feedback conrol: { dx d = r()x()+ F(,x,u( δ())), du d = h()u()+ g()x( σ ()). We show ha he sysem above has a leas wo posiive periodic soluions under cerain growh condiion imposed on F. 23 Elsevier Inc. All righs reserved. Keywords: Posiive periodic soluion; Funcional differenial equaion; Feedback conrol; Fixed-poin heorem 1. Inroducion Recenly, periodic populaion dynamics has become a very popular subjec. Many periodic models have been exensively sudied wih various mehods by many auhors [1 4]. However, he sudies of hese models wih conrol variables are relaively few, and conrol This work was suppored by Naional Naural Sciences Foundaion of People s Republic of China under Gran 1614 and Naural Sciences Foundaion of Yunnan Province. * Corresponding auhor. address: yklie@ynu.edu.cn (Y. Li) X/$ see fron maer 23 Elsevier Inc. All righs reserved. doi:1.116/j.jmaa

2 82 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) variables are usually considered as consans or ime dependen [5,6]. In a recen paper [7], by employing he coninuaion heorem of Gaines and Mawhin s coincidence degree heory [8, p. 4], he auhors invesigaed he exisence of posiive periodic soluions of he following sysem wih feedback conrol: { dx d = F(,x( τ 1 ()),..., x( τ n ()), u( δ())), du d = ()u()+ a()x( σ ()). Moivaed by he paper above, in his paper, we are concerned wih he following nonlinear nonauonomous funcional differenial sysem wih feedback conrol: { dx d = r()x()+ F(,x,u( δ())), du d = h()u()+ g()x( σ ()), where δ(),σ() C(R,R), r(), h(), g() C(R,(, + )), all of he above funcions are -periodic funcions and > is a consan. F(,x,z) is a funcion defined on R BC R and F( +,x +,z)= F(,x,z),whereBC denoes he Banach space of bounded coninuous funcions ϕ : R R wih he norm ϕ =sup θ R ϕ(θ). Ifx BC, hen x BC for any R is defined by x (θ) = x( + θ) for θ R. Sysem (1) has been exensively invesigaed in lieraure as bio-mahemaics models. I conains many bio-mahemaics models of delay differenial equaions wih feedback conrol, such as he following muliplicaive delay logisic model wih feedback conrol (see [7]): { dx d = r()x() [ 1 n x( τ i ()) i=1 K() c()u( δ()) ], du d = ()u()+ a()x( σ ()), where τ i, i = 1, 2,...,n, δ, σ C(R,R), r, a, c,, K C(R,(, + )), all of he above funcions are -periodic funcions and > is a consan. For more informaion abou he applicaions of sysem (1) o a variey of populaion models, we refer o [7,9 11] and he references cied herein. Our purpose of his paper is by using a fixed-poin heorem, which is an appreciaive generalized form of he well-known Legge Williams fixed poin heorem [12] due o Avery and Henderson [13], o invesigae he exisence of muliple posiive -periodic soluions of sysem (1). To he bes of our knowledge, few auhors have sudied he exisence of muliple posiive periodic soluions of delay differenial equaions wih feedback conrol. 2. Some lemmas For convenience, we shall inroduce he noaions: R = (, + ), R + =[, + ), I =[,], f = max I f(), f = min I f(),wheref is a coninuous posiive periodic funcion wih period. BC(X, Y) denoes he se of bounded coninuous funcions ϕ : X Y. In addiion, we provide here some definiions cied from cone heory in Banach space.

3 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) Definiion 1. Le X be a real Banach space. A nonempy closed convex se P X is called a cone of X if i saisfies he following condiions: (i) x P, λ implies λx P ; (ii) x P, x P implies x =. If P X is a cone, we denoe he order induced by P on X by ; hais,x y if and only if y x P. Definiion 2. Amapψ : P [, + ) is called nonnegaive coninuous increasing funcional provided ψ is nonnegaive and coninuous and saisfies ψ(x) ψ(y) for all x,y P and x y. Definiion 3. Given a nonnegaive coninuous increasing funcion ϕ on a cone P of a real Banach space X, we define for each d>heses P(ϕ,d)= { x P : ϕ(x) < d }, P(ϕ,d)= { x P : ϕ(x) = d }, P(ϕ,d)= { x P : ϕ(x) d }. The following lemma is an appreciaive generalized form of Legge Williams fixed poin heorem by Avery and Henderson. Lemma 1 (Avery and Henderson [13]). Le P be a cone in a Banach space X. Leφ and γ be nonnegaive, coninuous, increasing funcionals on P, and le θ be a nonnegaive coninuous funcional on P wih θ() = such ha for some c> and M> such ha γ(x) θ(x) φ(x), x Mγ (x), for all x P(γ,c). Suppose here exiss a compleely coninuous operaor Φ : P(γ,c) P and <a<b<csuch ha θ(λx) λθ(x), for λ 1 and x P (θ, b), and (i) γ(φx)>cfor all x P(γ,c); (ii) θ(φx)<b for all x P(θ,b); (iii) φ(φx)>a and P(φ,a) for x P(φ,a). Then Φ has a leas wo fixed poins x 1 and x 2 P(γ,c)such ha a<φ(x 1 ), θ(x 1 )<b, b<θ(x 2 ), γ (x 2 )<c. The following lemma is similar o Lemma 1, so we do no prove i here.

4 822 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) Lemma 2. Le P be a cone in a Banach space X.Leφ and γ be nonnegaive, coninuous, increasing funcionals on P, and le θ be a nonnegaive coninuous funcional on P wih θ() = such ha for some c> and M> such ha γ(x) θ(x) φ(x), x Mγ (x), for all x P(γ,c). Suppose here exiss a compleely coninuous operaor Φ : P(γ,c) P and <a<b<csuch ha and θ(λx) λθ(x), for λ 1 and x P (θ, b), (i) γ(φx)<cfor all x P(γ,c); (ii) θ(φx)>b for all x P(θ,b); (iii) φ(φx)<a and P(φ,a) for x P(φ,a). Then Φ has a leas wo fixed poins x 1 and x 2 P(γ,c)such ha a<φ(x 1 ), θ(x 1 )<b, b<θ(x 2 ), γ (x 2 )<c. In order o apply Lemma 1 o esablish he exisence of muliple posiive periodic soluions of sysem (1), we mus define an operaor on a cone in a suiable Banach space. To his end, we firs ransform sysem (1) ino one equaion. By inegraing he laer equaion in sysem (1) from o +, we obain where u() = + k(,s)g(s)x ( s σ(s) ) ds := (Φx)(), (1) k(,s) = exp{ s h(v) dv} exp{ h(v) dv} 1. When x is an -periodic funcion, i is easy o see ha k( +,s +) = k(,s), u( +) = u() and n := exp{ h(v) dv} exp{ exp{ h(v) dv} k(,s) h(v) dv} 1 exp{ := m h(v) dv} 1 for (, s) R 2,wherem, n are posiive consans. Therefore, he exisence problem of -periodic soluion of sysem (1) is equivalen o ha of -periodic soluion of he following equaion: dx d = r()x()+ F (,x,(φx) ( δ() )). (2) In wha follows, we always assume ha (H ) mg > 1/, ng < 1/.

5 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) (H 1 ) F(,φ, (Φφ)( δ())) is a coninuous funcion of for each φ BC(R, R + ),and F(,φ, (Φφ)( δ())) for(, φ) R BC(R, R + ),whereφ is defined as (2). (H 2 ) For any C>andε>, here exiss µ> such ha for γ,ξ BC, γ C, ξ C, γ ξ <µand for s imply F ( s,γs,(φγ) ( s δ(s) )) F ( s,ξ s,(φξ) ( s δ(s) )) <ε, where Φ is defined as (2). Since Eq. (3) can be ransformed ino ( x()e r(s)ds) = e r(s)ds F (,x,(φx) ( δ() )), one may see ha x()e r(s)ds is nondecreasing on R when x BC(R, R + ). Now, inegraing Eq. (3) from o +, we have where x() = + G(, s)f ( s,x s,(φx) ( s δ(s) )) ds, G(, s) = exp{ s r(v)dv} exp{ (3) r(v)dv} 1. I is clear ha G( +,s + ) = G(, s) for all (, s) R 2 and p := exp{ r(v)dv} exp{ exp{ G(, s) r(v)dv} r(v)dv} 1 exp{ := q r(v)dv} 1 for all s [, + ],wherep,q are posiive consans. For (, s) R 2,wedefine := min{exp( 2 r(s)ds)}. In order o use Lemma 1, we le X be he se X = { x C(R,R): x( + ) = x(), R } wih he norm x = sup I x() ;henx BC is a Banach space. Also we define P as P = { x X: x() x, I, and x()e r(s)ds is nondecreasing on I }. One may readily verify ha P is a cone in X. Define an operaor T : P P by (T x)() = + G(, s)f ( s,x s,(φx) ( s δ(s) )) ds (4) for x P, R, whereg(, s) is defined by (4) and Φ is defined by (2). Lemma 3. T : P P is well defined. Proof. For each x P,by(H 1 ),wehaveha(t x)() is coninuous in and

6 824 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) (T x)( + ) = +2 + = + + = G( +,s)f ( s,x s,(φx) ( s δ(s) )) ds G( +,v + )F ( v +,x v+,(φx) ( v + δ(v + ) )) dv G(, v)f ( v,x v,(φx) ( v δ(v) )) dv = (T x)(). Hence, (T x) X. In addiion, for x P,wehave and Tx q (T x)() p F ( s,x s,(φx) ( s δ(s) )) ds (5) F ( s,x s,(φx) ( s δ(s) )) ds p q Tx Tx. Furhermore, we find ha ( (T x)()e r(s)ds) = e r(s)ds F (,x,(φx) ( δ() )) for x P, which implies ha (T x)()e r(s)ds is nondecreasing on I. Therefore, (T x) P. This complees he proof of Lemma 3. Lemma 4. T : P P is compleely coninuous. Proof. Firsly, we show ha T is coninuous. By (H 2 ),foranyc>andε>, here exiss a µ> such ha for γ,ξ BC, γ C, ξ C, and γ ξ <µimply ( sup F s,γs,(φγ) ( s δ(s) )) F ( s,ξ s,(φξ) ( s δ(s) )) ε < s q, where Φ is defined by (2). If x,y P wih x C, y C, and x y <µ,hen ( (T x)() (T y)() q F s,xs,(φx) ( s δ(s) )) F ( s,y s,(φy) ( s δ(s) )) ds < ε for I, which yields Tx Ty = sup I (T x)() (T y)() <ε. Thus, F is coninuous. Secondly, we show ha F maps bounded ses ino bounded ses. Indeed, le ε = 1. By (H 2 ),foranyc> here exiss µ> such ha for x,y BC, x C, y C, and x y <µimply F ( s,x s,(φx) ( s δ(s) )) F ( s,y s,(φy) ( s δ(s) )) < 1, s I.

7 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) Choose a posiive ineger N such ha C/N < µ. Lex BC and define x k () = x()k/n for k =, 1, 2,...,N.If x C, hen x k x k 1 =sup x() k N x()k 1 N = x 1 N C N <µ. R Thus, F ( s,x k s,(φxk ) ( s δ(s) )) F ( s,xs k 1,(Φx k 1 ) ( s δ(s) )) < 1 for all s I. This yields F ( s,x s,(φx) ( s δ(s) )) N ( F s,x k s,(φx k ) ( s δ(s) )) F ( s,xs k 1,(Φx k 1 ) ( s δ(s) )) k=1 + F(s,, ) <N+ sup F(s,, ) := MC. (6) s I I follows from (6) ha for I, Tx q F ( s,x s,(φx) ( s δ(s) )) ds < qm C. (7) Finally, for R we have d d (T x)() = G(, + )F( +,x +,(Φx) ( + δ( + ) )) G(,)F (,x,(φx) ( δ() )) r()(t x)() = r()(t x)() + [ G(, + ) G(,) ] F (,x,(φx) ( δ() )) = r()(t x)() + F (,x,(φx) ( δ() )). (8) According o (6) (9), we obain d (T x)() d r Tx + ( F,x,(Φx) ( δ() )) r qm C + M C. Hence, {(T x): x P, x C} is a family of uniformly bounded and equiconinuous funcions on I. By he Ascoli Arzela heorem [14, p. 169], he funcion T is compleely coninuous. The proof is complee. Lemma 5. x is a posiive -periodic soluion of (3) if and only if x is a fixed poin of he operaor T on P,whereT is defined by (5). Proof. The only if par has been shown above. For he if par, we assume ha x P is a posiive -periodic soluion of equaion x = Tx.For I,wehave + x() = (T x)() = G(, s)f ( s,x s,(φx) ( s δ(s) )) ds,

8 826 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) where G(, s) is defined by (4) and Φ is defined by (2). Then, x () = r()x()+ F (,x,(φx) ( δ() )). Thus, x saisfies (3). The proof is complee. Evidenly, x is a posiive -periodic soluion of sysem (1) if and only if x is a fixed poin of he operaor T on P. From now on, we fix <l and define he nonnegaive, increasing, coninuous funcionals γ, θ and φ by r(s)ds x(), γ(x)= min e r(s)ds x() = e l θ(x)= max e r(s)ds x() = e r(s)ds x(), φ(x)= min e r(s)ds l x() = e r(s)ds x(l) l for every x P. Obviously, γ(x)= θ(x) φ(x). In addiion, for each x P, γ(x)= e r(s)ds x() e r(s)ds x. Hence, x e r(s)ds 1 γ(x) for all x P. (9) We also find ha θ(λx)= λθ(x) for λ [, 1] and x P. 3. Main resuls Before presening our firs resul, we denoe λ, ξ and λ l by λ = e ξ = e λ l = e r(s)ds r(s)ds l r(s)ds G(, s) ds, ( ) G(, s) + G(,s) ds, l G(l, s) ds. We are now in a posiion o sae and prove our firs resul. Theorem 1. Assume (H ) holds and le a>, b>, c> saisfy <a< λ l b< ng ξ mg e r(s)ds λ l 2 c, ξ and suppose ha F saisfies he following condiions:

9 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) (A) F(,φ, (Φφ)( δ())) > c/λ for ce r(s)ds φ (θ) c e r(s)ds, ng ce r(s)ds (Φφ) ( δ() ) mg c e r(s)ds, [,], θ R; (B) F(,φ, (Φφ)( δ())) < b/ξ for be r(s)ds φ (θ) b e r(s)ds, ng be r(s)ds (Φφ) ( δ() ) mg b e r(s)ds, [,], θ R; (C) F(,φ, (Φφ)( δ())) > a/λ l for ae r(s)ds φ (θ) a e l r(s)ds, ng ae r(s)ds (Φφ) ( δ() ) mg a e l r(s)ds, [l,], θ R. Then Eq. (3) admis a leas wo posiive -periodic soluions x 1,x 2 in P(γ,c)such ha x 1 (l) > ae l r(s)ds, x 1 () < be r(s)ds, x 2 () > be r(s)ds, x 2 () < ce r(s)ds. Proof. As a resul of Lemmas 3 and 4, we conclude T : P(γ,c) P and ha T is compleely coninuous. We proceed o verify he condiions of Lemma 1 are me. Firsly, we prove ha he condiion (i) of Lemma 1 is saisfied. For each x P(γ,c), γ(x)= e r(s)ds x() = c. Then for [,], here exiss a poin [,] such ha x (θ) = x( + θ)= x( ) x x() ce r(s)ds ce r(s)ds. Recalling (1) ha x e r(s)ds 1 γ(x)= e r(s)ds c, we have ce r(s)ds x (θ) e r(s)ds c for and θ R. For [,],

10 828 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) and Thus, (Φx) ( δ() ) = m (Φx) ( δ() ) n δ()+ δ() δ()+ δ() δ()+ δ() k ( δ(),s ) g(s)x ( s σ(s) ) ds g(s) x ds mg c e r(s)ds g(s)x ( s σ(s) ) ds ng ce r(s)ds. ng ce r(s)ds (Φx) ( δ() ) mg c e r(s)ds, [,]. As a consequence of (A), Therefore, F (,φ,(φφ) ( δ() )) >c/λ, [,]. γ(tx)= e r(s)ds (T x)() + = e r(s)ds G(, s)f ( s,x s,(φx) ( s δ(s) )) ds e r(s)ds > c e λ r(s)ds G(, s)f ( s,x s,(φx) ( s δ(s) )) ds G(, s) ds = c. Secondly, we show ha he condiion (ii) of Lemma 1 is saisfied. We choose x P(θ,b),henθ(x) = e r(s)ds x() = b. This implies ha for I, here exiss a poin 1 I such ha x (θ) = x( + θ)= x( 1 ) x x() = be r(s)ds and x (θ) = x( 1 ) x b e r(s)ds. So be r(s)ds x (θ) e r(s)ds b for and θ R.

11 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) Similarly, for I,wehave ng be r(s)ds (Φx) ( δ() ) mg b e r(s)ds. By (B), we have F(,φ, (Φφ)( δ())) < b/ξ, [,].So, + θ(tx)= e r(s)ds (T x)() = e r(s)ds G(, s)f ( s,x s,(φx) ( s δ(s) )) ds [( = e r(s)ds + + ) G(, s)f ( s,x s,(φx) ( s δ(s) )) ] ds [ e r(s)ds G(, s)f ( s,x s,(φx) ( s δ(s) )) ds + G(,s)F ( s,x s,(φx) ( s δ(s) )) ds [ e r(s)ds G(, s)f ( s,x s,(φx) ( s δ(s) )) ds + G(,s)F ( s,x s,(φx) ( s δ(s) )) ds [ <e r(s)ds G(, s) ds + G(,s)ds ] ] ] b ξ = b. Finally, we verify ha he condiion (iii) of Theorem 1 is also saisfied. I is obvious ha P(φ,a).Nowwelex P(φ,a),henφ(x) = e l r(s)ds x(l) = a. By a similar mehod used o verify he condiion (i) of Theorem 1, we obain he fac ha ae r(s)ds x (θ) e r(s)ds a for l, θ R and ng ae r(s)ds (Φx) ( δ() ) mg a e l r(s)ds, [l,]. Under he assumpion of (C), F(,φ, (Φφ)( δ())) > a/λ l for [l,] holds. Therefore, l φ(tx)= e r(s)ds l (T x)(l) = e r(s)ds l+ l G(l, s)f ( s,x s,(φx) ( s δ(s) )) ds

12 83 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) l e > a λ l e r(s)ds l l r(s)ds G(l, s)f ( s,x s,(φx) ( s δ(s) )) ds l G(l, s) ds = a. By Lemmas 1 and 5, we see ha Eq. (3) has a leas wo posiive -periodic soluions x 1 and x 2 in P(γ,c)such ha x 1 (l) > ae l r(s)ds, x 1 () < be r(s)ds, x 2 () > be r(s)ds, x 2 () < ce r(s)ds. The proof is complee. From Eq. (2) and Theorem 1 i follows ha sysem (1) has a leas wo posiive - periodic soluions (x 1,u 1 ), (x 2,u 2 ),wherex 1,x 2 are he same as hose in Theorem 1 and u 1 = Φx 1, u 2 = Φx 2, Φ is defined by (2). Before saing our second resul, we make some preparaions. Fix <l and define he nonnegaive increasing coninuous funcionals γ, θ and φ on P by γ(x)= max e r(s)ds x() = e r(s)ds x(), θ(x)= min e r(s)ds x() = e r(s)ds x(), l φ(x)= max e r(s)ds l x() = e r(s)ds x(l) l for every x P and we can see ha γ(x)= θ(x) φ(x). Now we choose noaions λ, ξ and λ by λ = e ξ = e l λ = e r(s)ds r(s)ds r(s)ds ( G(, s) + G(,s) ) ds, G(, s) ds, ( G(l, s) + G(l,s) ) ds. Theorem 2. Assume (H ) holds and le a>, b>, c> saisfy <a< ng mg 2 be l r(s)ds < ng mg 2 c ξ λ e l r(s)ds, and suppose ha F saisfies he following condiions:

13 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) (A ) F(,φ, (Φφ)( δ())) < c/λ for ce r(s)ds φ (θ) c e r(s)ds, ng ce r(s)ds (Φφ) ( δ() ) mg c e r(s)ds, [,], θ R; (B ) F(,φ, (Φφ)( δ())) > b/ξ for be r(s)ds φ (θ) b e r(s)ds, ng be r(s)ds (Φφ) ( δ() ) mg b e r(s)ds, [,], θ R; (C ) F(,φ, (Φφ)( δ())) < a/λ for ae l r(s)ds φ (θ) a e l r(s)ds, ng ae l r(s)ds (Φφ) ( δ() ) mg a e l r(s)ds, [,], θ R. Then Eq. (3) admis a leas wo posiive -periodic soluions x 1,x 2 in P(γ,c)such ha x 1 (l) > ae l r(s)ds, x 1 () < be r(s)ds, x 2 () > be r(s)ds, x 2 () < ce r(s)ds. In view of Lemma 2, he proof is similar o ha of Theorem 1 and will be omied. Therefore, sysem (1) has a leas wo posiive -periodic soluions (x 1,u 1 ) and (x 2,u 2 ),wherex 1,x 2 are he same as hose in Theorem 2 and u 1 = Φx 1, u 2 = Φx 2,where Φ is defined by (2). References [1] S.H. Saker, S. Agarwal, Oscillaion and global araciviy in a nonlinear delay periodic model of respiraory dynamics, Compu. Mah. Appl. 44 (22) [2] S.H. Saker, S. Agarwal, Oscillaion and global araciviy in a periodic Nicholson s blowflies model, Mah. Compu. Modelling 35 (22) [3] D.Q. Jiang, J.J. Wei, Posiive periodic soluions of funcional differenial equaions and populaion models, Elecron. J. Differenial Equaions 22 (22) [4] J. Yan, Q. Feng, Global araciviy and oscillaion in a nonlinear delay equaion, Nonlinear Anal. 43 (21) [5] S. Lefschez, Sabiliy of Nonlinear Conrol Sysem, Academic Press, New York, [6] Y. Kuang, Delay Differenial Equaions wih Applicaion in Populaion Dynamics, Academic Press, New York, 1993.

14 832 P. Liu, Y. Li / J. Mah. Anal. Appl. 288 (23) [7] H.F. Huo, W.T. Li, Posiive soluions of a class of delay differenial sysem wih feedback conrol, Compu. Mah. Appl., in press. [8] R.E. Gains, J.L. Mawhin, Coincidence Degree and Nonlinear Differenial Equaions, Springer, Berlin, [9] P. Weng, Exisence and global sabiliy of posiive periodic soluion of periodic inegrodifferenial sysems wih feedback conrols, Compu. Mah. Appl. 4 (2) [1] F. Yang, D.Q. Jiang, Exisence and global araciviy of posiive periodic soluion of a logisic growh sysem wih feedback conrol and deviaing argumens, Ann. Differenial Equaions 17 (21) [11] K. Gopalsamy, P.X. Weng, Feedback regulaion of logisic growh, J. Mah. Mah. Sci. 1 (1993) [12] R. Legge, L. Williams, Muliple posiive fixed poins of nonlinear operaors on ordered Banach spaces, Indiana Univ. Mah. J. 28 (1979) [13] R.I. Avery, J. Henderson, Two posiive fixed poins of nonlinear operaors on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8 (21) [14] H.L. Royden, Real Analysis, Macmillan, New York, 1988.