莫宜春,孫晉易,王珍燕

(西北師范大學數(shù)學與信息科學學院,甘肅 蘭州 730070)

一類泛函微分方程半正問題的正周期解

莫宜春,孫晉易,王珍燕

(西北師范大學數(shù)學與信息科學學院,甘肅 蘭州 730070)

運用Krasnosel′skii不動點理論研究了一類含參泛函微分方程半正問題正周期解的存在性,獲得了當參數(shù)充分小時正周期解的存在性結(jié)果以及半正問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;不動點定理;正周期解;存在性

1 引言及主要結(jié)果

帶有周期時滯的泛函微分方程在生物學、經(jīng)濟學、生態(tài)學和人口動力系統(tǒng)等實際問題中有著廣泛的應用,例如動物血紅細胞存在模型,人口動力系統(tǒng)模型等.因此,對帶有周期時滯的泛函微分方程周期解存在性的研究就更具有現(xiàn)實意義.近年來,許多學者對泛函微分方程周期解的存在性進行了深入而細致的研究,并取得了相當豐富的研究成果[18].

正周期解的存在性.顯然,文獻[7-8]中要求非線性項f是非負的.當然,很自然的問題是:非線性項f允許取負值時,方程(1.1)是否仍然存在正周期解?據(jù)筆者所知,此問題還沒有被討論過.鑒于此,本文試圖回答此問題.本文的研究將會進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 預備知識

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題[M].北京:科學出版社,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.

國家自然科學基金(10671158);甘肅省自然科學基金(3ZS051-A25-016);NWNU-KJCXGC-03-17;春輝計劃(Z2004-1-62033);高等學校博士學科點專項基金(20060736001);教育部留學回國人員啟動資金(2006[311]).

莫宜春(1987-),碩士,研究方向：常微分方程邊值問題.

2010 MSC:15A42