周媛媛

(徐州師范大學(xué) 科文學(xué)院,江蘇 徐州 221116)

二階微分方程周期邊值問(wèn)題解的存在性

周媛媛

(徐州師范大學(xué) 科文學(xué)院,江蘇 徐州 221116)

為了進(jìn)一步研究常微分方程周期邊值問(wèn)題解的存在性,利用上下解方法和拓?fù)涠壤碚?構(gòu)造兩個(gè)新的比較定理,獲得了二階常微分方程周期邊值問(wèn)題解的兩個(gè)存在性定理,此時(shí)僅要求f滿足比單邊 Lipschitz條件更弱的條件,且不要求上下解滿足常見(jiàn)的邊界條件。對(duì)于上下解反向給定時(shí),亦建立了相應(yīng)的解的存在性定理。文中給出的數(shù)值表達(dá)式在形式上更簡(jiǎn)潔,更易驗(yàn)證,且條件更寬,改進(jìn)了已有結(jié)果。

周期邊值問(wèn)題;上下解;反向上下解

微分方程周期解問(wèn)題一直受到人們的廣泛關(guān)注[1-5]。文中主要討論二階微分方程周期邊值問(wèn)題

解的存在性,其中f∈C[I×R,R]。對(duì)這個(gè)問(wèn)題的研究方法較多,其中利用上下解是常見(jiàn)的重要方法之一[6-8],但以往的上下解定義的形式多為:

α(t)≤β(t),t∈I,而對(duì)于下解α與上解β不滿足邊界條件

以及上下解為反向形式:β(t)≤α(t),t∈I的情形,關(guān)于周期邊值問(wèn)題(1)、(2)解的存在性討論,相關(guān)工作不多見(jiàn),主要因?yàn)榇藭r(shí)的比較定理的建立較為困難。一些學(xué)者[7-8]利用單調(diào)迭代法證明了在上下解不滿足條件(4)的情況下問(wèn)題(1)、(2)的解的存在性。周友明[6]在單邊 Lipschitz條件和存在滿足式(3)的反向上下解等條件時(shí)利用迭代方法證明了問(wèn)題(1)、(2)解的存在性。筆者在已有研究的基礎(chǔ)上,針對(duì)該問(wèn)題進(jìn)行了深入討論。

1 預(yù)備知識(shí)

2 主要結(jié)果及證明

3 應(yīng)用舉例

就正向上、下解和反向上、下解分別舉例。

例1 考慮周期邊值問(wèn)題

[1] 李 波,劉文斌.三階非線性常微分方程的周期邊值問(wèn)題[J].數(shù)學(xué)研究,2005,38(2):163-168.

[2] 羅治國(guó),王衛(wèi)兵.二階微分方程反周期邊值問(wèn)題解的存在性[J].應(yīng)用數(shù)學(xué)學(xué)報(bào),2006,29(6):1 111-1 117.

[3] YAO Q INGL IU.Positive solutions of nonlinear second-order periodic boundary value problems[J].Applied Mathamatics Letters, 2007,20(5):583-590.

[4] 郭大鈞,孫經(jīng)先,劉兆理.非線性常微分方程泛函方法[M].濟(jì)南:山東科學(xué)技術(shù)出版社,1995.

[5] 陳 潔.一階微分方程周期邊值問(wèn)題的解的存在性[J].數(shù)學(xué)物理學(xué)報(bào):A輯,2003,23(2):129-134.

[6] 周友明.Banach空間中二階微分方程的周期邊值問(wèn)題[J].應(yīng)用數(shù)學(xué)學(xué)報(bào),2006,29:436-444.

[7] LAKSHM IKANTHAM V,LEELA S.Remarks on first and second order periodic boundary value problems[J].Nonlinear Analysis, 1984,8(3):281-287.

[8] CABADA A,N IETO J J.A generalization of the monotone iterative technique for nonlinear second-order periodic boundary value problems[J].J Math AnalApplc,1990,151(1):181-189.

[9] 暴寧偉.一類(lèi)高階微分方程邊值問(wèn)題正解的存在性[J].河北工程大學(xué)學(xué)報(bào):自然科學(xué)版,2007,24(2):108-110.

On existence of solutions for PBVP of second order differential equations

ZHOU Yuanyuan
(Kewen Institute,Xuzhou NormalUniversity,Xuzhou 221116,China)

This paper presents an attempt to investigate the existence of solutions for periodic boundary value problems(PBVP)of ordinary differential equations,and introduces two existence theorems of solutions of the PBVP for second order differential equations are obtained by using the method of upper and lower solutions and the topological degree theory and establishing two new comparison theorem,wherefsatisfies some weaker conditions than the one-side Lipschitz condition and the lower and upper solutions need not satisfy the common boundary relations.Furthe rmore,the s imilar existence theorems are also obtained in the case when upper and lower solutions are in the reversed order.The paper features the mathematical expressions,s impler in fo rm and easier to verify.The results improve the relative conclusions in the early t imes.

periodic boundary value problems;upper and lower solutions;upper and lower solutions in reversed order

O175.8

A

1671-0118(2010)01-0159-05

2009-11-07

周媛媛(1981-),女,江蘇省宿遷人,講師,碩士,研究方向:微分方程理論與應(yīng)用,E-mail:z4081@sina.com。

(編輯王 冬)