莫宜春,孫晉易,王珍燕

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅 蘭州 730070)

一類(lèi)泛函微分方程半正問(wèn)題的正周期解

莫宜春,孫晉易,王珍燕

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅 蘭州 730070)

運(yùn)用Krasnosel′skii不動(dòng)點(diǎn)理論研究了一類(lèi)含參泛函微分方程半正問(wèn)題正周期解的存在性,獲得了當(dāng)參數(shù)充分小時(shí)正周期解的存在性結(jié)果以及半正問(wèn)題正周期解存在的充分條件.豐富了一階泛函微分方程解的存在性理論.

泛函微分方程;不動(dòng)點(diǎn)定理;正周期解;存在性

1 引言及主要結(jié)果

帶有周期時(shí)滯的泛函微分方程在生物學(xué)、經(jīng)濟(jì)學(xué)、生態(tài)學(xué)和人口動(dòng)力系統(tǒng)等實(shí)際問(wèn)題中有著廣泛的應(yīng)用,例如動(dòng)物血紅細(xì)胞存在模型,人口動(dòng)力系統(tǒng)模型等.因此,對(duì)帶有周期時(shí)滯的泛函微分方程周期解存在性的研究就更具有現(xiàn)實(shí)意義.近年來(lái),許多學(xué)者對(duì)泛函微分方程周期解的存在性進(jìn)行了深入而細(xì)致的研究,并取得了相當(dāng)豐富的研究成果[18].

正周期解的存在性.顯然,文獻(xiàn)[7-8]中要求非線性項(xiàng)f是非負(fù)的.當(dāng)然,很自然的問(wèn)題是:非線性項(xiàng)f允許取負(fù)值時(shí),方程(1.1)是否仍然存在正周期解?據(jù)筆者所知,此問(wèn)題還沒(méi)有被討論過(guò).鑒于此,本文試圖回答此問(wèn)題.本文的研究將會(huì)進(jìn)一步豐富一階泛函微分方程(1.1)解的存在性理論.

本文總假定:

2 預(yù)備知識(shí)

3 主要結(jié)果的證明

[1]Wang H.Positive periodic solutions for functional di ff erential equations[J].J.Di ff erential Equations,2004, 202:354-366.

[2]Jiang D,Wei J,Zhang B.Positive periodic solutions of functional di ff erential equations and population models[J].Electron J.Di ff erential Equations,2002,71:1-13.

[3]Anuradha V,Hai D D,Shivaji R.Existence results for superlinear semipositone BVP[J].Proc.Amer.Math., 1996,124(3):757-763.

[4]Ma R.Positive solutions for semipositone(k,n-k)conjugate boundary value problems[J].J.Math.Anal. Appl.,2000,252:220-229.

[5]Zhang G,Cheng S.Positive periodic solutions of nonautonomous functional di ff erential equations depending on a parameter[J].Abstract Appl.Anal.,2002,7:279-286.

[6] Padhi S,Shilpee Srivastava.Multiple periodic solutions for nonlinear fi rst order functional di ff erential equations with applications to population dynamics[J].Appl.Math.Comput.,2008,203(1):1-6.

[7]Cheng S,Zhang G.Existence of positive periodic solutions for non-autonomous functional di ff erential equations[J].Electron.J.Di ff erential Equations,2001,59:1-8.

[8]Wan A,Jiang D,Xu X.A new existence theory for positive periodic solutions to functional di ff erential equations[J].Comput.Math.Appl.,2004,4:1257-1262.

[9]馬如云.非線性常微分方程非局部問(wèn)題[M].北京:科學(xué)出版社,2004.

A positive periodic solutions for semipositone problems of functional di ff erential equations

Mo Yichun,Sun Jinyi,Wang Zhenyan
(College of Mathematics and Information Science,Northwest Normal University,Lanzhou 730070,China)

By using Krasnosel'skii fi xed-point theorem in cones,this paper studies the existence of positive periodic solutions for semipositone problems of functional di ff erential equations.We obtain the existence of positive periodic solutions when the parameter is small enough,and the sufficient conditions for existence of positive periodic solutions for semipositone problems,enriching the theory for existence of solutions of functional di ff erential equations.

functional di ff erential equations, fi xed-point theorem,positive periodic solutions,existence

O178

A

1008-5513(2012)01-0137-06

2011-03-18.

國(guó)家自然科學(xué)基金(10671158);甘肅省自然科學(xué)基金(3ZS051-A25-016);NWNU-KJCXGC-03-17;春輝計(jì)劃(Z2004-1-62033);高等學(xué)校博士學(xué)科點(diǎn)專(zhuān)項(xiàng)基金(20060736001);教育部留學(xué)回國(guó)人員啟動(dòng)資金(2006[311]).

莫宜春(1987-),碩士,研究方向：常微分方程邊值問(wèn)題.

2010 MSC:15A42