POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

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1 POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER GUANG ZHANG AND SUI SUN CHENG Received 5 November 21 This aricle invesigaes he exisence of posiive periodic soluions for a firsorder funcional differenial equaions of he form y ) = a)y)+λh) f y τ) )), 1) where a = a), h = h), and τ = τ) are coninuous T-periodic funcions. We will also assume ha T>, λ>, f = f ) aswellash = h) are posiive, a)d >. Funcional differenial equaions wih periodic delays appear in a number of ecological models. In paricular, our equaion can be inerpreed as he sandard Malhus populaion model y = a)y subjec o perurbaion wih periodical delay. One imporan quesion is wheher hese equaions can suppor posiive periodic soluions. Such quesions have been sudied exensively by a number of auhors cf. [1, 2, 3, 4, 6, 7] and he references herein). In his paper, we are concerned wih he exisence and nonexisence of periodic soluions when he parameer λ varies. For his purpose, we call a coninuously differeniable and T-periodic funcion a periodic soluion of 1) associaed wih λ if i saisfies 1) whenλ = λ. We show ha here exiss λ > suchha1) has a leas one posiive T-periodic soluion for λ,λ ] and does no have any T-periodic posiive soluions for λ>λ. Our echnique is based on he well-known upper and lower soluions mehod cf. [5]). We proceed from 1) and obain [ ) y)exp as)ds)] = λexp as)ds h) f y τ) )). 2) Copyrigh 22 Hindawi Publishing Corporaion Absrac and Applied Analysis 7:5 22) Mahemaics Subjec Classificaion: 34B15, 34K13 URL: hp://dx.doi.org/1.1155/s

2 28 Bifurcaion in funcionaldifferenial equaions Afer inegraion from o + T,weobain +T y) = λ G,s)hs) f y s τs) )) ds, 3) where s ) exp au)du G,s) = ). 4) exp au)du 1 Noe ha he denominaor in G,s) is no zero since we have assumed ha a)d >. I is no difficul o check ha any T-periodic funcion y) ha saisfies 3) is also a T-periodic soluion of 1). Noe furher ha <N min G,s) G,s) max G,s) M, s + T, s, T,s T G,s) 1 max s, T G,s) min s, T G,s) max s, T G,s) = N 5) M >. Now le X be he se of all real T-periodic coninuous funcions, endowed wih he usual linear srucure as well as he norm y = sup y). 6) T Then X is a Banach space wih cones Φ = { y) X : y) }, Ω = { y):y) σ y, R }, 7) where σ = N/M. Define a mapping F : X X by +T Fy)) = λ G,s)hs) f y s τs) )) ds. 8) Then i is easily seen ha F is compleely coninuous on bounded subses of Ω and for y Φ, so ha Fy)) λm hs) f y s τs) )) ds 9) Fy)) λn hs) f y s τs) )) ds σ Fy. 1)

3 G. Zhang and S. S. Cheng 281 Tha is, FΦ is conained in Ω. Lemma 1. The mapping F maps Φ ino Ω. Lemma 2. Suppose ha f u) lim = +. 11) u + u Le I be a compac subse of,+ ). Then here exiss a consan b I > such ha u <b I for all λ I and all possible T-periodic posiive soluions u of 1) associaed wih λ. Proof. Suppose o he conrary ha here is a sequence {u n } of T-periodic posiive soluions of 1) associaed wih {λ n } such ha λ n I for all n and u n + as n.sinceu n Ω, min u n) σ u n. 12) T By 11), we may choose R f > suchha f u) ηu for all u R f, and here exiss n such ha σ u n R f,whereη saisfies Thus, we have σηnλ n hs) ds > 1. 13) +T u n un ) = λ n G,s)hs) f )) u n s τs) ds σηnλ n hs) u n ds > u n. This is a conradicion. The proof is complee. Lemma 3. Suppose ha 14) f is nondecreasing on [,+ ) and f ) >. 15) Le 1)haveaT-periodic posiive soluion y) associaedwihλ>.then1)also has a posiive T-periodic soluion associaed wih λ,λ). Proof. In view of 3)and15), we have +T y) = λ G,s)hs) f y s τs) )) ds +T λ G,s)hs) f y s τs) )) +T 16) ds, <λ G,s)hs) f )ds.

4 282 Bifurcaion in funcionaldifferenial equaions Le y ) = y), y ) =, and Clearly, we have +T y k+1 ) = λ G,s)hs) f )) y k s τs) ds, k =,1,2,..., 17) +T y ) = λ G,s)hs) f ) y ) s τs) ds, k =,1,2,... 18) k+1 k y ) y 1 ) y k ) y k ) y 1 ) y ). 19) If we now le y) = lim k y k ), hen y) saisfies 3). Clearly, we have +T y) y ) = λ 1 G,s)hs) f )ds >. 2) This complees our proof. Lemma 4. Suppose ha 11)and15) hold. Then here exiss λ > such ha 1) hasat-periodic posiive soluion. Proof. Le +T β) = G,s)hs)ds, M f = max T f β τ) )), λ = 1 M f. 21) We have +T β) = +T G,s)hs)ds λ G,s)hs) f β s τs) )) ds, +T <λ G,s)hs) f )ds. 22) Le y ) = β), y ) =, and +T y k+1 ) = λ G,s)hs) f )) y k s τs) ds, k =,1,2,..., 23) +T y ) = λ G,s)hs) f ) y ) s τs) ds, k =,1,2,... 24) k+1 k

5 G. Zhang and S. S. Cheng 283 Clearly, we have y ) y 1 ) y k ) y k ) y 1 ) y ). 25) If we now le y) = lim k y k ), hen y) saisfies 3). Clearly, we have The proof is complee. +T y) y ) = λ G,s)hs) f )ds >. 26) 1 Theorem 5. Suppose ha 11) and15) hold. Then here exiss λ > such ha 1) has a leas one posiive T-periodic soluion for λ,λ ] and does no have any T-periodic posiive soluions for λ>λ. Proof. Suppose o he conrary ha here is a sequence {u n } of T-periodic posiive soluions of 1) associaed wih {λ n } such ha lim n λ n =. Then eiher we have u nj + as j or here is M >suchha u n M. Assume he former case holds. Noe ha u n Ω and hus min T u n) σ u n. 27) By 11), we may choose R f > andη 1 > suchha f u) η 1 u when σu R f. On he oher hand, here exis { n } [,T]suchhau nj nj ) = u nj and u n j nj ) = by he periodiciy of {u nj )}.Inviewof1), we have a ) ) ) nj u nj = a nj u nj = λnj h ) nj f unj nj τ ))) nj λ nj η 1 σh ) 28) nj u nj for all large j. Tha is, we have λ nj a nj )/η 1 σh nj )). Noe ha a)/h) is bounded. Thus, we obain a conradicion. Nex, suppose ha he laer case holds. In view of 15), here exiss η 2 > such ha f ) η 2 M. Then as above, we will obain a n ) u n = a n ) u n ) = λn h n ) f un n τ n ))) λ n η 2 h n ) M λ n η 2 h n ) u n 29) for all n. A conradicion will again be reached.

6 284 Bifurcaion in funcionaldifferenial equaions Thus, here exiss λ > suchha1) has a leas one posiive T-periodic soluion for λ,λ ) and no T-periodic posiive soluions for λ>λ. Finally, we asser ha 1) has a leas one T-periodic posiive soluion for λ = λ. Indeed, le {λ n } saisfy <λ 1 < <λ k <λ and lim k λ k = λ.since u n ) ist-periodic posiive soluion of 1) associaed wih λ n and Lemma 2 implies ha he se {u n )} of soluions is uniformly bounded in Ω, he sequence {u n )} has a subsequence converging o u) Ω. Wecannowapply he Lebesgue convergence heorem o show ha u) isat-periodic posiive soluion of 1) associaed wih λ = λ.theproofiscomplee. Example 6. Consider he equaion x )+a)x) = λh) { x γ τ) ) +1 }, γ>1, 3) where a, h,andτ saisfy he same assumpions saed for 1). In view of Theorem 5, here exiss a λ > suchha3) has a leas one T-periodic posiive soluion for λ,λ ] and no T-periodic posiive soluion for λ>λ. Example 7. Consider he equaion y ) = ay)+λb y 2 )+ε ), 31) where a,b,ε >. Noe ha he funcion f x) = x 2 + ε) saisfies 11)and15)in Theorem 5. Therefore Theorem 5 may be applied. However, we may give a direc proof ha, for λ>a/2b ε), his equaion canno have any posiive 2π-periodic soluions associaed wih λ. Indeed, assume o he conrary ha y) issucha soluion. Then y ξ) = forsomeξ [,2π]. Hence ayξ)+λby 2 ξ)+λbε =. 32) However, since he discriminan of he quadraic equaion saisfies λbx 2 ax + λbε = 33) a 2 4λ 2 b 2 ε<, 34) a conradicion is obained. We remark ha when ε =, our equaion reduces o he well-known logisic equaion. Similarly, we can consider he equaion x ) = a)x) λh) f x τ) )), 35)

7 G. Zhang and S. S. Cheng 285 where a = a), h = h), and f = f ) saisfy he same assumpions saed for 1). By 35), we have +T x) = H,s)hs) f x s τs) )) ds, 36) where H,s) = s ) exp au)du 1 exp ) = au)du exp +T s exp ) au)du ) 37) au)du 1 which saisfies for some M and N>, and σ = N/M 1. M H,s) N, s + T, 38) Theorem 8. Suppose ha 11) and15) hold. Then here exiss λ > such ha 35) has a leas one posiive T-periodic soluion for λ,λ ] and no T-periodic posiive soluion for λ>λ. Acknowledgmen This work was suppored by he Naural Science Foundaion of Shanxi Province and Yanbei Normal College. Par of his work was done during he firs auhor s visi o he Insiue of Applied Mahemaics, Academy of Mahemaics and Sysem Sciences, Chinese Academy of Sciences. The firs auhor wishes o express his hanks o Professor Daomin Cao for his kind inviaion and nice hospialiy. We also hank he referee for his helpful criicisms. References [1] S. S. Cheng and G. Zhang, Exisence of posiive periodic soluions for non-auonomous funcional differenial equaions, Elecron.J.Differenial Equaions 21 21), no. 59, 1 8. [2] M. Fan and K. Wang, Opimal harvesing policy for single populaion wih periodic coefficiens, Mah. Biosci ), no. 2, [3], Uniform ulimae boundedness and periodic soluions of funcionaldifferenial equaions wih infinie delay, J. SysemsSci. Mah. Sci ), no. 3, Chinese). [4] D.Q.JiangandJ.J.Wei,Exisence of posiive periodic soluions for nonauonomous delay differenial equaions, Chinese Ann. Mah. Ser. A ), no. 6, Chinese). [5] Y.-H. Lee, Mulipliciy of posiive radial soluions for muliparameer semilinear ellipic sysems on an annulus, J. Differenial Equaions ), no. 2,

8 286 Bifurcaion in funcionaldifferenial equaions [6] Y. K. Li, Exisence and global araciviy of posiive periodic soluions for a class of delay differenial equaions, Science in China Series A ), no. 2, Chinese). [7] S. N. Zhang, Periodiciy in funcional-differenial equaions, Ann. Differenial Equaions ), no. 2, Guang Zhang: Deparmen of Mahemaics, Yanbei Normal College and Daong College, Daong, Shanxi 37,China address: dgzhang@yahoo.com.cn Sui Sun Cheng: Deparmen of Mahemaics, Tsing Hua Universiy, Hsinchu, Taiwan 343, Taiwan address: sscheng@mah.nhu.edu.w