王 剛, 朱思念

(中國礦業(yè)大學 理學院,江蘇 徐州 221008)

偶數(shù)階微分方程邊值問題的上下解方法

王 剛, 朱思念

(中國礦業(yè)大學 理學院,江蘇 徐州 221008)

針對實際應用中高階微分方程的求解問題,討論了一類偶數(shù)階微分方程兩點邊值問題解的存在性,利用上下解方法,通過將 2n階微分方程轉(zhuǎn)化為二階積分微分方程,得到其解的存在性定理,同時,在形式上推廣了已知的四階兩點邊值問題的結(jié)果。

微分方程;上下解方法;邊值問題;存在性

0 引 言

非線性高階微分方程邊值問題在應用數(shù)學和物理領域中具有深刻背景,其解的存在性有重要的理論和現(xiàn)實意義。關于偶數(shù)階微分方程邊值問題,目前多集中于討論四階微分方程邊值問題解的存在性[1-7],其主要研究方法有上下解方法、單調(diào)迭代技術和不動點理論等。其中,白占兵[1]研究了四階微分方程兩點邊值問題,通過降階將其化為二階微分 -積分方程,再利用上下解方法證明其解的存在性;張琴等[2]利用上下解方法和單調(diào)迭代技術考慮了一類四階四點微分方程邊值問題解的存在性;張興秋[3]利用上下解方法和極值原理研究了一類具有積分邊界條件的奇異四階微分方程正解的存在性和唯一性。而對于任意偶數(shù)階微分方程邊值問題,研究其解的存在性問題較少。如鄭惠軍等[8]利用錐壓縮與錐拉伸不動點定理,證明了一類 2m階邊值問題的正解存在性;梁思華等[9]利用上下解方法考慮了一類 2n階多點邊值問題解的存在性。受文獻[1]啟發(fā),筆者研究 2n階微分方程兩點邊值問題

通過降階和上下解方法,將其轉(zhuǎn)化為積分微分方程,討論其解的存在性。其中f:[0,1]×R2n→R是連續(xù)的。易知,當n=2時,問題(1)就轉(zhuǎn)化為四階兩點邊值問題,即文獻[1]所討論的結(jié)果,因此,文獻[1]是文中的一個特例。

1 預備知識

2 主要結(jié)果

[1] BA I ZHANB ING.The upper and lower solution method for some fourth-order boundary value problems[J].Nonlinear Analysis, 2007,67(6):1 704-1 709.

[2] ZHANGQ IN,CHEN SHIHUA,LU J INHU.Upper and lower solution method for fourth-order four-point boundary value problems [J].Journal of Computational and Applied Mathematics,2006, 196(2):387-393.

[3] 張興秋.奇異四階積分邊值問題正解的存在唯一性[J].應用數(shù)學學報,2010,33(1):38-50.

[4] DEL P INOM A,MANASEV I CH R F.Existence for a fourth-order boundary value problem under a two parameter non resonance condition[J].Proc AmerMath Soc,1991,112(1):81-86.

[5] CABADA A.The method of lower and upper solutions for second, third,fourth and higher order boundary value problems[J].J Math AnalAppl,1994,185(2):302-320.

[6] WEI ZHONGL I,PANG CHANGCI.The method of lower and upper solutions for fourth order singularm-point boundary value problems[J].J Math AnalAppl,2006,322(2):675-692.

[7] FENG HANY ING,JIDEHONG,GEWEIGAO.Existence and uniqueness of solutions for a fourth-order boundary value problem [J].NonlinearAnalysis,2009,70(10):3 561-3 566.

[8] 鄭惠軍,孔令彬.2m階非線性邊值問題的 Green函數(shù)與正解[J].大慶石油學院學報,2010,34(01):106-111.

[9] L IANG SIHUA,ZHAN JI HU I.Themethod of lower and upper solutions for 2nth-order multi-point boundary value problems[J]. NonlinearAnalysis,2009,71(10):4 581-4 587.

Method of lower and upper solutions for even-order boundary value problem s

WANG Gang,ZHU Sinian
(College of Sciences,China University ofMining&Technology,Xuzhou 221008,China)

Aimed at the solution to higher differential equation in practical application,thispaper discusses the existence of solutionsof two-point boundary value problem of a class of even order differential equation and describes the use of the method of upper and lower solutions to convert the 2n-order differential equation into second order integral-differential equation to obtain the existence theorems of solutions,alongwith the extension of the known results of fourth-order two-point boundary value problem in the fo rm.

differential equation;method of lower and upper solutions;boundary value problems;existence

O175.8

A

1671-0118(2011)02-0157-04

2011-02-21

王 剛(1985-),男,安徽省安慶人,碩士,研究方向:微分方程,E-mail:wangg0824@163.com。

(編輯王 冬)