J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics and Saisics, KLAS, KLVE, Norheas Normal Universiy, 5268 Renmin Sree, Changchun, Jilin 1324, PR China b Academy of Mahemaics and Sysems Sciences, Chinese Academy of Sciences, Beijing 18, PR China Received 11 February 26 Available online 29 March 26 Submied by Seven G. Kranz Absrac In our paper, by employing Krasnoselskii fixed poin heorem, we invesigae he exisence of muliple posiive periodic soluions for funcional differenial equaions ẋ() = A (,x() ) x() + λf (, x ), where λ> is a parameer. Some easily verifiable sufficien crieria are esablished. 26 Elsevier Inc. All righs reserved. Keywords: Posiive periodic soluion; Funcional differenial equaions; Krasnoselskii fixed poin heorem 1. Inroducion Le R = (, + ), R + =[, + ), R = (, ] and R n + ={(x 1,...,x n ) T : x i, 1 i n}, respecively. For each x = (x 1,x 2,...,x n ) T R n, he norm of x is defined as x = ni=1 x i.lebc denoe he Banach space of bounded coninuous funcions φ : R R n wih he norm φ =sup θ R ni=1 φ i (θ), where φ = (φ 1,φ 2,...,φ n ) T. As i is well known, he exisence of posiive periodic soluions for funcional equaions was exensively sudied, see [1,2,4 1] and references herein. One of effecive approaches o fulfill * Corresponding auhor. E-mail address: zhzzj@yeah.ne (Z. Zeng). 22-247X/$ see fron maer 26 Elsevier Inc. All righs reserved. doi:1.116/j.jmaa.26.2.67
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1379 such a problem is employing fixed poin heorem, and some prior esimaions of possible periodic soluions are obained. For example, Wang [9] sudied he following funcional equaion: x () = a()g ( x() ) x() λb()f ( x ( τ() )). (1.1) In [1], Ye, Fan and Wang discussed in deail a class of more general funcional equaions ẋ() = a()x()+ f (,u() ), ẋ() = a()x() f (,u() ), (1.2) where u() = (x(g 1 ()),...,x(g n 1 )(), k( θ)x(θ)dθ) and + k(r)dr = 1. In paper [5], Jiang e al. invesigaed he exisence, mulipliciy and nonexisence of posiive periodic soluions o a sysem of infinie delay equaions ẋ() = A()x()+ λf (, x ). (1.3) In he presen paper, by uilizing he fixed poin heorem due o Krasnoselskii, we aim o sudy he exisence of muliple posiive periodic soluions for he following funcional differenial equaions ẋ() = A (,x() ) x() + λf (, x ) (1.4) in which λ> is a parameer, A(, x()) = diag[a 1 (, x()), a 2 (, x()),..., a n (, x())], a i C(R R,R) is ω-periodic, f = (f 1,f 2,...,f n ) T, f(,x ) is a funcional defined on R BC and f(,x ) is ω-periodic whenever x is ω-periodic. If x BC, hen x BC for any R, where x is defined by x (θ) = x( + θ) for θ R. In our paper, we will discuss he exisence of posiive periodic soluions in more cases han he above menioned papers and obain some easily verifiable sufficien crieria. Throughou he paper, we make he assumpions: (H 1 ) There exis coninuous ω-periodic funcions b i (), c i (), such ha b i () a i (, x()) c i (), for1 i n. (H 2 ) f i (, φ ) a i(s, x(s)) ds for all (, φ) R BC(R, R+ n ),1 i n. (H 3 ) f(,x ) is a coninuous funcion of for each x BC(R, R+ n ). (H 4 ) For any L> and ε>, here exiss δ> such ha [φ,ψ BC, φ L, ψ L, φ ϕ <δ, s ω] imply f(s,φ s ) f(s,ψ s ) <ε. In addiion, he parameers in his paper are assumed o be no idenically equal o zero. To conclude his secion, we summarize in he following a few conceps and resuls ha will be needed in our argumens. Definiion 1.1. Le X be Banach space and E be a closed, nonempy subse of X, E is said o be a cone if (i) αu + βv E for all u, v E and all α, β > ; (ii) u, u E imply u =. Lemma 1.2 (Krasnoselskii fixed poin heorem). [3] Le X be a Banach space, and le E be a cone in X. Suppose Ω 1 and Ω 2 are open subses of X such ha Ω 1 Ω 1 Ω 2. Suppose ha T : E ( Ω 2 \ Ω 1 ) E is a compleely coninuous operaor and saisfies eiher
138 Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 (i) Tx x for any x E Ω 1 and Tx x for any x E Ω 2 ; or (ii) Tx x for any x E Ω 1 and Tx x for any x E Ω 2. Then T has a fixed poin in E ( Ω 2 \ Ω 1 ). 2. Some lemmas and In his secion, we make some preparaions for he following secions. Le B i = max { b i, c i }, 1 i n. For (, s) R 2,1 i n, we define { { } exp{ σ = min exp 2 B i (τ) dτ c } i(τ) dτ} 1 exp{ b, 1 i n i(τ) dτ} 1 G i (, s) = We also define (2.1) exp{ s a i (τ, x(τ)) dτ} exp{ a i(τ, x(τ) dτ} 1. (2.2) G(, s) = diag [ G 1 (, s), G 2 (,s),...,g n (, s) ]. I is clear ha G(, s) = G( + ω,s + ω) for all (, s) R 2 and by (H 2 ), G i (, s)f i (u, φ u ) for (, s) R 2 and (u, φ) R BC(R, R+ n ). A direc calculaion shows ha m i := exp{ B i(τ) dτ} exp{ b i(τ) dτ} 1 Gi (, s) exp{ B i(τ) dτ} exp{ c i(τ) dτ} 1 =: M i. (2.3) Le X = { x C(R,R n ): x( + ω) = x(), R } and define E = { x X: x i () σ x i, [,ω], x= (x 1,x 2,...,x n ) T }. (2.4) One may readily verify ha E is a cone. We also assume (H 5 ) inf φ =r f(s,φ s) ds > forφ E and r>. Moreover, define, for r a posiive number, Ω r by Ω r = { x E: x <r }. Noe ha Ω r ={x E: x =r}.
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1381 Le he map T λ : E E be defined by +ω (T λ x)() = λ G(, s)f (s, x s )ds for x E, R, and le (T λ x) = ( Tλ 1 x,t λ 2 x,...,tn λ x) T. To prove he exisence of posiive soluions o Eq. (1.4), we firs give he following lemmas. Lemma 2.1. Assume (H 1 ) (H 4 ) hold, hen T λ : E E is well defined and T λ is compac and coninuous. Proof. By (H 3 ),forx E,wehave(T λ x) is coninuous in and (T λ x)( + ω) = λ +2ω +ω = λ +ω +ω = λ G( + ω,s)f(s,x s )ds G( + ω,v + ω)f (v + ω,x v+ω )dv G(,v)f (v,x v )dv= (T λ x)(). Thus, (T λ x) X. In view of (2.3), for x E,wehave T i λ x ω λmi fi (s, x s ) ds, and ( T i λ x ) ω () λm i fi (s, x s ) m i ds T i M λ x σ T i λ x. i Therefore, (T λ x) E and by (H 4 ) i is easy o show ha T λ is compac. Lemma 2.2. Assume (H 1 ) (H 4 ) hold. Equaion (1.4) is equivalen o he fixed poin problem of T λ in E. Proof. If x E and T λ x = x, hen ( ẋ() = d +ω ) λ G(, s)f (s, x s )ds d = λg(, + ω)f ( + ω,x +ω ) λg(, )f (, x ) + A (,x() ) T λ x() = λ [ G(, + ω) G(, ) ] f(,x ) + A (,x() ) T λ x() = A (,x() ) x() + λf (, x ).
1382 Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 Thus x is a posiive ω-periodic soluion of (1.4). On he oher hand, if x is a posiive ω-periodic funcion, hen λf (, x ) =ẋ() A(, x())x() and (T λ x)() = λ = +ω +ω G(, s)f (s, x s )ds G(, s) [ ẋ(s) A ( s,x(s) ) x(s) ] ds = G(,s)x(s) +ω + = x(). +ω G(, s)a ( s,x(s) ) x(s)ds Therefore, ogeher wih he proof of Lemma 2.1, we complee he proof. Lemma 2.3. Assume (H 1 ) (H 4 ) hold and here exiss η> such ha Then f(s,φ s ) ds η φ, for φ E. T λ x λmη x, for x E, where m = min 1 i n m i. Proof. If x E, hen ( T i λ x ) +ω () λm i f i (s, x s ) ω ds = λm i f i (s, x s ) ds. +ω G(, s)a ( s,x(s) ) x(s)ds Thus, we have T λ x =sup R n ( Tλ i x) () i=1 n λm i i=1 λm f i (s, x s ) ds f(s,x s ) ds λmη x. Lemma 2.4. Assume (H 1 ) (H 4 ) hold and le r>. If here exiss a sufficienly small ε> such ha f(s,φ s ) ds εr, for φ E Ω r,
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1383 hen T λ x λmε x, for x E Ω r, where M = max 1 i n M i. Proof. For any x E Ω r, we obain T λ x =sup R sup R 3. Main resuls n ( Tλ i x) () i=1 n λ i=1 n λm i i=1 +ω Gi (, s) fi (s, x s ) ds f i (s, x s ) ds λm f(s,x s ) ds λmεr = λmε x. In order o sae our main resuls, we assume he following limis exis and le sup f = lim φ sup φ E sup f = lim sup φ + f(s,φ s) ds, inf f = lim φ inf φ φ E φ E Moreover, we lis several assumpions f(s,φ s) ds, inf f = lim φ inf φ + φ E f(s,φ s) ds, φ (P 1 ) inf f =, (P 2 ) inf f =, (P 3 ) sup f =, [ ) (P 4 ) sup f =, ( ) 1 1 (P 5 ) sup f = α 1,, (P 6 ) inf f = β 1 λm λmσ,, ( ) [ ) 1 (P 7 ) inf f = α 2 λmσ, 1, (P 8 ) sup f = β 2,, λm where σ is defined in (2.1) and m, M are defined in Lemmas 2.3 and 2.4. f(s,φ s) ds. φ Theorem 3.1. If (P 1 ) and (P 3 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Proof. By (P 1 ), one can find r > such ha o f(s,φ s ) ds η φ, for φ E, < φ r, where he consan η> saisfies λmη > 1. Then by Lemma 2.3, we have T λ x λmη x > x, for x E Ω r.
1384 Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 Again, by (P 3 ), for any <ε 1/(2λM), here exiss N 1 >r such ha o Choose f(s,φ s ) ds ε φ, for φ E, φ N 1. r 1 >N 1 + 1 + 2λM If x E Ω r1, hen T λ x λm ( = λm I 1 sup φ <N 1 φ E f(s,x s ) ds f(s,x s ) ds + f(s,φ s ) ds. I 2 f(s,x s ) ) ds r 1 2 + x 2 = x, where I 1 ={x E: x <N 1 },I 2 ={x E: x N 1 }. This implies ha T λ x x for any x E Ω r1. In conclusion, under he assumpions (P 1 ) and (P 3 ), T λ saisfies all he requiremens in Lemma 1.2, hen T λ has a fixed poin in E ( Ω r1 \ Ω r ). By Lemma 2.2, we complee he proof. Theorem 3.2. If (P 2 ) and (P 4 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Proof. By (P 4 ), for any <ε 1/(λM), here exiss r 2 > such ha o f(s,φs ) ds ε φ εr2, for φ E, < φ r 2. Then by Lemma 2.4, we have T λ x λmε x x, for x E Ω r2. Nex, by (P 2 ), here exiss r 3 >r 2 > such ha o f(s,φ s ) ds η φ, for φ E, φ r 3, where η> is chosen so ha λmη > 1. I follows from Lemma 2.3 ha T λ x λmη x > x, for x E Ω r3. I follows from Lemma 1.2 ha (1.4) has a posiive ω-periodic soluion saisfying r 2 x r 3. In he following, we will inroduce wo exra assumpions o assure some basic heorems:
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1385 (P 9 ) There exiss d 1 > such ha o f(s,φ s) ds > d 1 /(λm), forσd 1 φ d 1. (P 1 ) There exiss d 2 > such ha o f(s,φ s) ds < d 2 /(λm), for φ d 2. Theorem 3.3. If (P 3 ), (P 4 ) and (P 9 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d1 < x 2. Proof. By he assumpion (P 4 ), for any <ε 1/(λM), here exiss r 4 <d 1 such ha o f(s,φ s ) ds ε φ, for φ E, < φ r 4, hen by Lemma 2.4, we obain T λ x λmε x x, for x E Ω r4. Likewise, from (P 3 ), for any <ε 1/(2λM), here exiss N 2 >d 1 such ha o Choose f(s,φs ) ds ε φ, for φ N2. r 5 >N 2 + 1 + 2λM If x E Ω r5, hen T λ x λm ( = λm max φ <N 2 φ E f(s,xs ) ds I 1 f(s,x s ) ds + f(s,φ s ) ds. I 2 f(s,x s ) ) ds r 5 2 + x 2 = x, where I 1 ={x E: x <N 2 },I 2 ={x E: x N 2 }, which shows ha T λ x x for all x E Ω r5. Se Ω d1 ={x X: x <d 1 }. Then, by (P 9 ), for any x E Ω d1,wehave T λ x λm f(s,x s ) ds > λm d 1 λm = d 1 = x, which yields T λ x > x for all x E Ω d1. By Lemma 1.2, here exis wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 1 < x 2. This complees he proof. From he argumens in he above proof, we have he following consequence immediaely.
1386 Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 Corollary 3.4. If (P 1 ), (P 2 ) and (P 1 ) hold, hen (1.4) has a leas wo ω-periodic soluions x 1 and x 2 saisfying < x 1 <d2 < x 2. To obain beer resuls in his secion, we give a more general crierion in he following, which plays an imporan role in he consequence. Theorem 3.5. Suppose ha (P 9 ) and (P 1 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion x wih x lying beween d 2 and d 1, where d 1 and d 2 are defined in (P 9 ) and (P 1 ), respecively. Proof. Wihou loss of generaliy, we may assume ha d 2 <d 1. If x E Ω d2, hen by (P 1 ), one can ge T λ x λm f(s,xs ) d 2 ds < λm λm = d 2 = x. In paricular, T λ x < x for all x E Ω d2. On he oher hand, by (P 9 ), one has T λ x λm f(s,x s ) ds > λm d 1 λm = d 1 = x, which produces T λ x > x for all x E Ω d1. Therefore, by Lemma 1.2, we can obain he conclusion and his complees he proof. Theorem 3.6. If (P 5 ) and (P 6 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Proof. By assumpion (P 5 ), for any ε = λm 1 α 1 >, here exiss a sufficienly small d 2 > such ha ha is sup x E f(s,x s) ds <α 1 + ε = 1 x λm, for x d 2, f(s,xs ) 1 ds < λm x d 2 λm, for x d 2, so, (P 1 ) is saisfied. By assumpion (P 6 ),forε = β 1 1 inf x E f(s,x s) ds >β 1 ε = 1 x λmσ, for x σd 1, λmσ >, here exiss a sufficienly large d 1 > such ha
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1387 ha is, f(s,xs ) 1 1 ds > x λmσ λmσ σd 1 = d 1 λm. Therefore, (P 9 ) holds. By Theorem 3.5 we complee he proof. Theorem 3.7. If (P 7 ) and (P 8 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Proof. By assumpion (P 7 ), for any ε = α 2 λmσ 1 >, here exiss a sufficienly small d 1 > such ha inf x E Therefore, we have sup x E f(s,x s) ds >α 2 ε = 1 x λmσ, for < x d 1. f(s,x s ) ds > 1 λmσ σd 1 = d 1 λm, for σd 1 x d 1. Tha is, (P 9 ) holds. By consumpion (P 8 ),forε = λm 1 β 2 >, here exiss a sufficienly large d such ha f(s,x s) ds <β 2 + ε = 1 x λm, for x >d. In he following, we consider wo cases o prove (P 1 ) o be saisfied: sup x E f(s,x s) ds ω bounded and unbounded. The bounded case is clear. If sup x E f(s,x s) ds is unbounded, hen here exiss y R n +, y =d 2 >d, such ha f(s,x s ) ds f(s,y s ) ds, for < x y =d 2. Since y =d 2 >d, hen we have f(s,xs ) ω ds f(s,ys ) 1 ds < λm y = d 2 λm, for < x d 2, which implies he condiion (P 1 ) holds. Therefore, by Theorem 3.5 we complee he proof. Theorem 3.8. Suppose ha (P 6 ), (P 7 ) and (P 1 ) hold, hen (1.4) has a leas wo posiive ω- periodic soluions x 1 and x 2 saisfying < x 1 <d 2 < x 2, where d 2 is defined in (P 1 ). Proof. From (P 6 ) and he proof of Theorem 3.6, we know ha here exiss a sufficienly large d 1 >d 2, such ha f(s,x s ) ds > d 1 λm, for σd 1 x d 1.
1388 Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 From (P 7 ) and he proof of Theorem 3.7, we can find a sufficienly small d 1 (,d 2) such ha f(s,xs ) d ds > 1 λm, for σd 1 x d 1. Therefore, from he proof of Theorem 3.5, here exis wo posiive soluions x 1 and x 2 saisfying d1 < x1 <d 2 < x 2 <d 1. From he argumens in he above proof, we have he following consequence, oo. Corollary 3.9. Suppose ha (P 5 ), (P 8 ) and (P 9 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 2 < x 2, where d 1 is defined in (P 9 ). Theorem 3.1. If (P 1 ) and (P 8 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Proof. From he assumpion (P 1 ) and he proof of Theorem 3.1, we know ha T λ x x for all x E Ω r. From (P 8 ) and Theorem 3.7, as x r 1, we know ha f(s,x s) ds < r 1 λm and T λ x λm f(s,xs ) r 1 ds < λm λm = r 1 = x, which implies T λ x < x for all x E Ω r1. This complees he proof. Similar o Theorem 3.1, one immediaely has he following consequences. Theorem 3.11. If (P 2 ) and (P 5 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Theorem 3.12. If (P 3 ) and (P 7 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Theorem 3.13. If (P 4 ) and (P 6 ) hold, hen (1.4) has a leas one posiive ω-periodic soluion. Theorem 3.14. If (P 1 ), (P 6 ) and (P 1 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d2 < x 2, where d 2 is defined in (P 1 ). Proof. Le Ω r ={x X: x <r }, where r <d 2. By assumpion (P 1 ) and he proof of Theorem 3.1, we know T λ x x for all x E Ω r. Le Ω d1 ={x X: x <d 1 }. By he assumpion (P 6 ) and he proof of Theorem 3.6, we can see ha f(s,x s) ds > d 1 λm for σd 1 x d 1. Incorporaing (P 1 ) and he proof of Theorem 3.5, we know ha here exis wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 2 < x 2.
Z. Zeng e al. / J. Mah. Anal. Appl. 325 (27) 1378 1389 1389 Theorem 3.15. If (P 2 ), (P 7 ) and (P 1 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 2 < x 2, where d 2 is defined in (P 1 ). Theorem 3.16. If (P 3 ), (P 5 ) and (P 9 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 1 < x 2, where d 1 is defined in (P 9 ). Theorem 3.17. If (P 4 ), (P 8 ) and (P 9 ) hold, hen (1.4) has a leas wo posiive ω-periodic soluions x 1 and x 2 saisfying < x 1 <d 1 < x 2, where d 1 is defined in (P 9 ). References [1] S.-N. Chow, Exisence of periodic soluions of auonomous funcional differenial equaions, J. Differenial Equaions 15 (1974) 35 378. [2] K. Deimling, Nonlinear Funcional Analysis, Springer-Verlag, New York, 1985. [3] M.A. Krasnoselskii, Posiive Soluion of Operaor Equaion, Noordhoff, Gröningen, 1964. [4] D.Q. Jiang, J.J. Wei, B. Zhang, Posiive periodic soluions of funcional differenial equaions and populaion models, Elecron. J. Differenial Equaions 71 (22) 1 13. [5] D.Q. Jiang, D. O Regan, R.P. Agarwal, X.J. Xu, On he number of posiive periodic soluions of funcional differenial equaions and populaion models, Mah. Models Mehods Appl. Sci. 15 (4) (25) 555 573. [6] G. Makay, Periodic soluions of dissipaive funcional differenial equaions, J. Tohoku Mah. 46 (1994) 417 426. [7] M.J. Ma, J.S. Yu, Exisence of muliple posiive periodic soluions for nonlinear funcional difference equaions, J. Mah. Anal. Appl. 35 (25) 483 49. [8] S.G. Peng, S.M. Zhu, Posiive periodic soluions for funcional differenial equaions wih infinie delay, Chinese Ann. Mah. Ser. A 25 (3) (24) 285 292 (in Chinese). [9] H.Y. Wang, Posiive periodic soluions of funcional differenial equaions, J. Differenial Equaions 22 (24) 354 366. [1] D. Ye, M. Fan, H.Y. Wang, Periodic soluions for scalar funcional differenial equaions, Nonlinear Anal. 62 (25) 1157 1181.