EXISTENCE OF TRIPLE POSITIVE PERIODIC SOLUTIONS OF A FUNCTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER

Transcription

1 EXISTENCE OF TRIPLE POSITIVE PERIODIC SOLUTIONS OF A FUNCTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER XI-LAN LIU, GUANG ZHANG, AND SUI SUN CHENG Received 15 Ocober 2002 We esablish he exisence of hree posiive periodic soluions for a class of delay funcional differenial equaions depending on a parameer by he Legge-Williams fixed poin heorem. 1. Inroducion In his paper, we will discuss he exisence of posiive periodic soluions for he firs-order funcionaldifferenial equaions y () = a()y()+λh() f ( y ( τ() )), R, (1.1) x () = a()x() λh() f ( x ( τ() )), R, (1.2) where a = a(), h = h(), and τ = τ() are coninuous T-periodic funcions, and f = f (u) is a nonnegaive coninuous funcion. We assume ha T is a fixed posiive number and ha a = a() saisfies he condiion T 0 a(u)du > 0. The number λ will be reaed as a parameer in boh equaions. Funcional differenial equaions wih periodic delays appear in a number of ecological models. In paricular, our equaions can be inerpreed as he sandard Malhusian populaion model y = a()y subjec o perurbaions wih periodic delays. One imporan quesion is wheher hese equaions can suppor posiive periodic soluions. The exisence of one or wo posiive periodic soluions for hese funcional differenial equaions has been sudied, see for examples [1, 2, 3, 5, 6, 7, 8, 9]. In his paper, we will obain some exisence crieria for hree posiive periodic soluions when he parameer λ varies. E = (E, ) in he sequel is a Banach space, and C E is a cone. By a concave nonnegaive coninuous funcional ψ on C, we mean a coninuous mapping ψ : C [0,+ ) wih ψ ( µx +(1 µ)y ) µψ(x)+(1 µ)ψ(y), x, y C, µ [0,1]. (1.3) Copyrigh 2004 Hindawi Publishing Corporaion Absrac and Applied Analysis 2004:10 (2004) Mahemaics Subjec Classificaion: 34K13 URL: hp://dx.doi.org/ /s

2 898 Triple periodic soluions of FDE Le ξ, α, β be posiive consans, we will employ he following noaions: C ξ = { y C : y <ξ }, C ξ = { y C : y ξ }, C(ψ,α,β) = { y C β : ψ(y) α }. (1.4) Our exisence crieria will be based on he Legge-Williams fixed poin heorem (see [4]). Theorem 1.1. Le E = (E, ) be a Banach space, C E aconeofe and R>0 aconsan. Suppose here exiss a concave nonnegaive coninuous funcional ψ on C wih ψ(y) y for y C R.LeA : C R C R be a compleely coninuous operaor. Assume here are numbers r, L,andK wih 0 <r<l<k R such ha (H1) he se {y C(ψ,L,K) :ψ(y) >L} is nonempy and ψ(ay) >Lfor all y C(ψ, L,K); (H2) Ay <rfor y C r ; (H3) ψ(ay) >Lfor all y C(ψ,L,R) wih Ay >K. Then A has a leas hree fixed poins y 1, y 2,andy 3 C R.Furhermore,y 1 C r, y 2 {y C(ψ,L,R):ψ(y) >L},andy 3 C R \(C(ψ,L,R) C r ). 2. Exisence of riple soluions for (1.1) A coninuously differeniable and T-periodic funcion y : R R is called a T-periodic soluion of (1.1) associaed wih ω if i saisfies (1.1)when λ = ω in (1.1). I is no difficul o check ha any T-periodic coninuous funcion y() ha saisfies y() = λ G(,s)h(s) f ( y ( s τ(s) )) ds, R, (2.1) where G(,s) = exp( s a(u)du ) exp ( T 0 a(u)du ),,s R, (2.2) 1 is also a T-periodic soluion of (1.1) associaedwih λ. Noe furher ha 0 <N min G(,s) G(,s) max G(,s) M, s + T;,s [0,T],s [0,T] G(,s) 1 max,s [0,T] G(,s) min,s [0,T] G(,s) max,s [0,T] G(,s) = N M > 0. (2.3)

3 Xi-lan Liu e al. 899 For he sake of convenience, we se A 0 = max [0,T] B 0 = min [0,T] G(,s)h(s)ds, G(,s)h(s)ds. (2.4) We use Theorem 1.1 o esablish he exisence of hree posiive periodic soluions o (2.1). To his end, one or several of he following condiions will be needed: (S1) f :[0,+ ) [0,+ ) is a coninuous and nondecreasing funcion, (S2) h() > 0for R, (S3) lim x 0 f (x)/x = l 1, (S4) lim x + f (x)/x = l 2. Le E be he se of all real T-periodic coninuous funcions endowed wih he usual operaions and he norm y =max [0,T] y().thene is a Banach space wih cone C = { y E : y() 0, (,+ ) }. (2.5) Theorem 2.1. Suppose (S1) (S4) hold such ha l 1 = l 2 = 0. Suppose furher ha here is a number L>0 such ha f (L) > 0.LeR, K, L,andr be four numbers such ha R K> LM N f (r) < f (R) r R L>r>0, (2.6) < B 0 f (L) A 0 L. (2.7) Then for each λ (L/(B 0 f (L)),R/(A 0 f (R))], here exis hree nonnegaive periodic soluions y 1, y 2,andy 3 of (1.1) associaed wih λ such ha y 1 () <r<y 2 () <L<y 3 () R for R. Proof. Firsofall,inviewof(S2),A 0,B 0 > 0. Noe furher ha if f (L) > 0, hen by (S1), f (R) > 0foranyR greaer han L. In view of (S4), we may choose R K>Lsuch ha he second inequaliy in (2.7) holds, and in view of (S3), we may choose r (0,L) suchha he firs inequaliy in (2.7) holds.weseλ 1 = L/(B 0 f (L)) and λ 2 = R/(A 0 f (R)). Then λ 1,λ 2 > 0. Furhermore, λ 1 <λ 2 in view of (2.7). We now define for each λ (λ 1,λ 2 ]a coninuous mapping A : C C by and a funcional ψ : C [0, )by (Ay)() = λ G(,s)h(s) f ( y ( s τ(s) )) ds, R (2.8) ψ(y) = min y(). (2.9) [0,T]

4 900 Triple periodic soluions of FDE In view of (S1), (S2), and (2.7), we have (Ay)() = λ G(,s)h(s) f ( y ( s τ(s) )) ds λf ( y ) G(,s)h(s)ds λf(r) G(,s)h(s)ds λa 0 f (R) λ 2 A 0 f (R) = R, (2.10) for [0,T]andally C R. Therefore, A(C R ) C R. We asser ha A is compleely coninuous on C R. Indeed, in view of he heorem of Arzela-Ascoli, i suffices o show ha A(C R ) is equiconinuous. To see his, noe ha for 1 < 2, 2+T 2 G ( 2,s ) h(s) f ( y ( s τ(s) )) 1+T ds 2+T = 1+T 1+T G ( 2,s ) h(s) f ( y ( s τ(s) )) ds { G ( 2,s ) G ( 1,s )} h(s) f ( y ( s τ(s) )) ds 1 G ( 1,s ) h(s) f ( y ( s τ(s) )) ds. 1 G ( 1,s ) h(s) f ( y ( s τ(s) )) ds (2.11) Furhermore, 2 G ( 1,s ) h(s) f ( y ( s τ(s) )) ds 1 { f ( y ) 1+T } G(,s)h(s)ds T 1+T 1+T 2 1 A 0 f ( y ) 2 1, G(,s)h(s) f ( y ( s τ(s) )) ds { f ( y ) 1+2T G(,s)h(s)ds} T A 0 f ( y ) 2 1, { ( G 2,s ) G ( 1,s )} h(s) f ( y ( s τ(s) )) ds f ( y ) 1+T max h(x) G ( 2,s ) G ( 1,s ) ds x [0,T] 2 f ( y ) 2T max h(x) G ( 2,s ) G ( 1,s ) ds. x [0,T] 0 (2.12)

5 Xi-lan Liu e al. 901 In view of he uniform coninuiy of G in {(,s) 0, s 2T}, foranyε>0, here is δ which saisfies { 0 <δ<min T, and for 0 < 2 1 <δ,wehave ε 3λ 2 A 0 f (R), ε 3λ 2 f (R) [ max 0, s 2T G(,s) ][ max x [0,T] h(x) ] }, (2.13) G ( 1,s ) G ( 2,s ) ε <, s [0,2T]. (2.14) 6λ 2 Tf(R)max 0 T h() Thus (Ay) ( ) ( ) 1 (Ay) 2 2+T = λ G(,s)h(s) f ( y ( s τ(s) )) 1+T ds G(,s)h(s) f ( y ( s τ(s) )) ds T λ 2 G ( 2,s ) h(s) f ( y ( s τ(s) )) ds 1+T 1+T { ( + λ 2 G 2,s ) G ( 1,s )} h(s) f ( y ( s τ(s) )) ds λ 2 G ( 1,s ) h(s) f ( y ( s τ(s) )) ds 1 2λ 2 A 0 f (R) λ 2 f (R)2T max h() ε [0,T] 6λ 2 Tf(R)max 0 T h() <ε (2.15) for any y() C R. This means ha A(C R )isequiconinuous. We now asser ha Theorem 1.1(H2) holds. Indeed, (Ay)() = λ G(,s)h(s) f ( y ( s τ(s) )) ds λf ( y ) G(,s)h(s)ds λf(r) G(,s)h(s)ds λ 2 A 0 f (r) <r (2.16) for all y C r, where he las inequaliy follows from (2.7). In addiion, we can show ha he condiion (H1) of Theorem 1.1 holds. Obviously, ψ(y) is a concave coninuous funcion on C wih ψ(y) y for y C R. We noice ha if u() = (1/2)(L + K)for (,+ ), hen u {y C(ψ,L,K):ψ(y) >L} which impliesha {y C(ψ,L,K): ψ(y) >L} is nonempy. For y C(ψ,L,K), we haveψ(y) = min [0,T] y() L and y K. In view of (S1) (S4), we have ψ(ay) = λ min G(,s)h(s) f ( y ( s τ(s) )) ds λb 0 f (L) >λ 1 B 0 f (L) = L (2.17) [0,T] for all y C(ψ,L,K).

6 902 Triple periodic soluions of FDE Finally, we prove condiion (H3) in Theorem 1.1. Le y C(ψ,L,R) wih Ay > K. We noice ha (2.8) implies Thus T Ay λm h(s) f ( y ( s τ(s) )) ds. (2.18) 0 ψ(ay) = λ min λn [0,T] T 0 G(,s)h(s) f ( y ( s τ(s) )) ds h(s) f ( y ( s τ(s) )) ds N M Ay > N M K>L. (2.19) An applicaion of Theorem1.1 saedabove now yields our proof. We remark ha he assumpions of Theorem 2.1 are no vacuous as can be seen by leing T = 3, a() 1, h() 1, and xln(1 + x), 0 x<1, f (x) = e x e +ln2, 1 x<6, (2.20) x 6+e 6 e +ln2, x 6. Then by aking r = 3/2, L = 5, K = 101, and R = , we easily check ha A 0 = , B 0 = and all he condiions of Theorem 2.1 hold. Theorem 2.2. Suppose (S1) (S4) hold such ha 0 <l 1 <l 2. Suppose here is a number L>0 such ha f (L) > 0 and 0 <l 1 <l 2 <B 0 f (L)/(A 0 L).LeR, K, L,andr be four numbers such ha R K> LM L>r>0, N (2.21) f (R) R <l 2 + ε, (2.22) f (r) <l 1 + ε, r (2.23) where ε is a posiive number such ha l 2 + ε< B 0 f (L) A 0 L. (2.24) Then for each λ (L/(B 0 f (L)),1/(A 0 l 2 )),(1.1) has a leas hree nonnegaive periodic soluions y 1, y 2,andy 3 associaed wih λ such ha y 1 () <r<y 2 () <L<y 3 () R for R. Proof. Firs of all, A 0,B 0 > 0 by (S2). Noe furher ha if f (L) > 0, hen by (S1), f (R) > 0 for any R greaer han L.Leλ 1 = L/(B 0 f (L)) and λ 2 = 1/(A 0 l 2 ). Then λ 1,λ 2 > 0. Furhermore, 0 <λ 1 <λ 2 in view of he condiion 0 <l 1 <l 2 <B 0 f (L)/(A 0 L). For posiive ε ha

7 Xi-lan Liu e al. 903 saisfies (2.24) andanyλ (λ 1,λ 2 ), in view of (S4) (and he fac ha λ 1/(A 0 (l 2 + ε))), here is R K>Lsuch ha (2.22) holds, and in view of (S3), here is r (0,L)suchha (2.23)holds. We now define for each λ (λ 1,λ 2 ) a coninuous mapping A : C C by (2.8) anda funcional ψ : C [0,+ ) by(2.9). As in he proof of Theorem 2.1, i is easy o see ha A is compleely coninuous on C R and maps C R ino C R.Forally C R,wehave (Ay)() = λ G(,s)h(s) f ( y ( s τ(s) )) ds λf ( y ) G(,s)h(s)ds λf(r) G(,s)h(s)ds λa 0 f (R) <λa 0 ( l2 + ε ) R R. (2.25) Furhermore, condiion (H2) of Theorem 1.1 holds. Indeed, for y C r,wehave (Ay)() = λ G(,s)h(s) f ( y ( s τ(s) )) ds λf ( y ) G(,s)h(s)ds λf(r) G(,s)h(s)ds λa 0 f (r) λa 0 ( l1 + ε ) r<r. (2.26) Similarly, we can prove ha he condiions (H1) and (H3) of Theorem 1.1 hold. An applicaionof Theorem 1.1 now yields our proof. Theorem 2.3. Suppose(S1)and(S2)holdand f (0) > 0. Suppose here exis four numbers L, R, K,andr such ha (2.6)and(2.7) hold. Then for each λ (L/(B 0 f (L)),R/(A 0 f (R))], (1.1) has a leas hree posiive periodic soluions y 1, y 2,andy 3 associaed wih λ such ha 0 <y 1 () <r<y 2 () <L<y 3 () R for R. The proof is similar o Theorem 2.1 and is hence omied. 3. Exisence of riple soluions for (1.2) Equaion (1.2) can be regarded as a dual of (1.1). Therefore, dual exisence heorems can be found. Their proofs are obained by argumens parallel o hose for our previous heorems. Therefore, only a shor summary will be given. Firs, (1.2)isransformedino x() = λ H(,s)h(s) f ( x ( s τ(s) )) ds, (3.1)

8 904 Triple periodic soluions of FDE where H(,s) = exp( s a(u)du ) 1 exp ( T 0 a(u)du ) = exp ( s a(u)du ) exp ( T 0 a(u)du ), 1 s + T, (3.2) which saisfies Le M max H(,s) H(,s) min,s [0,T],s [0,T] H(,s) N, s + T. (3.3) A = max [0,T] B = min [0,T] H(,s)h(s)ds, H(,s)h(s)ds. (3.4) Theorem 3.1. Suppose (S1) (S4) hold such ha l 1 = l 2 = 0. Suppose furher ha here is a number L>0 such ha f (L) > 0. Le R, K, L,andr be four numbers such ha R K> LM L>r>0, f (r) r N < f (R) R < B f (L) A L. (3.5) Then for each λ (L/(B f (L)),R/(A f (R))], here exis hree nonnegaive periodic soluions x 1, x 2,andx 3 of (1.2) associaed wih λ such ha x 1 () <r<x 2 () <L<x 3 () R for R. Theorem 3.2. Suppose (S1) (S4) hold such ha 0 <l 1 <l 2. Suppose here is a number L>0 such ha f (L) > 0 and 0 <l 1 <l 2 <B f (L)/(A L).LeR, K, L,andr be four numbers such ha where ε is a posiive number such ha R K> LM L>r>0, N f (R) R <l 2 + ε, f (r) <l 1 + ε, r (3.6) l 2 + ε< B f (L) A L. (3.7) Then for each λ (L/(B f (L)),1/(A l 2 )),(1.2) has a leas hree nonnegaive periodic soluions x 1, x 2,andx 3 associaed wih λ such ha x 1 () <r<x 2 () <L<x 3 () R for R. Theorem 3.3. Suppose(S1)and(S2)holdand f (0) > 0. Suppose here exis four numbers L, R, K, andr such ha (3.5) hold. Then for each λ (L/(B f (L)),R/(A f (R))], (1.2) has a leas hree posiive periodic soluions x 1, x 2,andx 3 associaed wih λ such ha 0 <x 1 () < r<x 2 () <L<x 3 () R for R.

9 Acknowledgmen Xi-lan Liu e al. 905 This projec is suppored by he Naural Science Foundaion of Shanxi Province and Yanbei Normal Insiue. References [1] S. S. Cheng and G. Zhang, Exisence of posiive periodic soluions for non-auonomous funcional differenial equaions, Elecron. J. Differenial Equaions 2001 (2001), no. 59, 1 8. [2] S.N.Chow,Remarks on one-dimensional delay-differenial equaions, J. Mah. Anal. Appl. 41 (1973), [3] H.M.Gibbs,F.A.Hopf,D.L.Kaplan,andR.L.Shoemaker,Observaion of chaos in opical bisabiliy, Phys. Rev. Le. 46 (1981), [4] D. J. Guo, Nonlinear Funcional Analysis, Shandong Science and Technology Press, Shandong, [5] K. P. Hadeler and J. Tomiuk, Periodic soluions of difference-differenial equaions,arch.raional Mech. Anal. 65 (1977), no. 1, [6] D.Q.JiangandJ.J.Wei,Exisence of posiive periodic soluions for nonauonomous delay differenial equaions, Chinese Ann. Mah. Ser. A 20 (1999), no. 6, (Chinese). [7] J. Malle-Pare and R. D. Nussbaum, Global coninuaion and asympoic behaviour for periodic soluions of a differenial-delay equaion, Ann. Ma. Pura Appl. (4) 145 (1986), [8], Adifferenial-delay equaion arising in opics and physiology, SIAM J. Mah. Anal. 20 (1989), no. 2, [9] G. Zhang and S. S. Cheng, Posiive periodic soluions of nonauonomous funcional differenial equaions depending on a parameer, Absr. Appl. Anal. 7 (2002), no. 5, Xi-lan Liu: Deparmen of Mahemaics, Yanbei Normal Insiue, Daong, Shanxi , China address: liuchl03@s.lzu.edu.cn Guang Zhang: Qingdao Polyechnic Universiy, Qingdao, Shandong , China address: qd gzhang@126.com Sui Sun Cheng: Deparmen of Mahemaics, Naional Tsing Hua Universiy, Hsinchu 30043, Taiwan address: sscheng@mah.nhu.edu.w

10 Mahemaical Problems in Engineering Special Issue on Modeling Experimenal Nonlinear Dynamics and Chaoic Scenarios Call for Papers Thinking abou nonlineariy in engineering areas, up o he 70s, was focused on inenionally buil nonlinear pars in order o improve he operaional characerisics of a device or sysem. Keying, sauraion, hysereic phenomena, and dead zones were added o exising devices increasing heir behavior diversiy and precision. In his conex, an inrinsic nonlineariy was reaed jus as a linear approximaion, around equilibrium poins. Inspired on he rediscovering of he richness of nonlinear and chaoic phenomena, engineers sared using analyical ools from Qualiaive Theory of Differenial Equaions, allowing more precise analysis and synhesis, in order o produce new vial producs and services. Bifurcaion heory, dynamical sysems and chaos sared o be par of he mandaory se of ools for design engineers. This proposed special ediion of he Mahemaical Problems in Engineering aims o provide a picure of he imporance of he bifurcaion heory, relaing i wih nonlinear and chaoic dynamics for naural and engineered sysems. Ideas of how his dynamics can be capured hrough precisely ailored real and numerical experimens and undersanding by he combinaion of specific ools ha associae dynamical sysem heory and geomeric ools in a very clever, sophisicaed, and a he same ime simple and unique analyical environmen are he subjec of his issue, allowing new mehods o design high-precision devices and equipmen. Auhors should follow he Mahemaical Problems in Engineering manuscrip forma described a hp:// Prospecive auhors should submi an elecronic copy of heir complee manuscrip hrough he journal Manuscrip Tracking Sysem a hp:// ms.hindawi.com/ according o he following imeable: Gues Ediors José Robero Casilho Piqueira, Telecommunicaion and Conrol Engineering Deparmen, Polyechnic School, The Universiy of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elber E. Neher Macau, Laboraório Associado de Maemáica Aplicada e Compuação (LAC), Insiuo Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, São Paulo, Brazil ; elber@lac.inpe.br Celso Grebogi, Deparmen of Physics, King s College, Universiy of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscrip Due February 1, 2009 Firs Round of Reviews May 1, 2009 Publicaion Dae Augus 1, 2009 Hindawi Publishing Corporaion hp://