陳春香

(中國(guó)礦業(yè)大學(xué) 理學(xué)院,江蘇 徐州 221116)

2n階邊值問(wèn)題正解的存在性與多解性

陳春香

(中國(guó)礦業(yè)大學(xué) 理學(xué)院,江蘇 徐州 221116)

為了研究一類(lèi)非線性 2n階兩點(diǎn)邊值問(wèn)題正解的存在性,通過(guò)建立一個(gè)特殊錐,利用錐壓縮與錐拉伸不動(dòng)點(diǎn)定理,得到了該問(wèn)題一個(gè)或多個(gè)正解存在的充分條件,拓展了已有結(jié)果。

邊值問(wèn)題;不動(dòng)點(diǎn)定理;正解;存在性

Abstract:This paper is an attempt to investigate the existence and multiplicity of positive solutions for 2nth-order boundary value problemswith two points.The construction of special cone and the use of the fixed-point theorem of cone expansion or compression type lead to the sufficient conditionsof the problem,expanding the existing results.

Key words:boundary-value problem;fix-point theorem;positive solution;existence

0 引 言

在自然科學(xué)、應(yīng)用科學(xué)等領(lǐng)域,常微分方程邊值問(wèn)題正解的存在性與多解性已經(jīng)成為一個(gè)重要的研究熱點(diǎn),很多學(xué)者對(duì)其進(jìn)行了研究[1-10]。其中,文獻(xiàn)[3]利用錐壓縮與錐拉伸不動(dòng)點(diǎn)定理討論了非線性四階邊值問(wèn)題

正解與多個(gè)正解存在的充分條件。文獻(xiàn)[4]利用同一定理討論了非線性四階兩點(diǎn)積分邊值問(wèn)題

正解與多個(gè)正解存在的充分條件以及正解不存在的情況。此外,也有學(xué)者討論了高階邊值問(wèn)題[5-10]。文獻(xiàn)[5]和[8]運(yùn)用上下解方法分別給出了 2n階兩點(diǎn)邊值問(wèn)題正解存在的充要條件和常微分方程正解存在的充分條件。受文獻(xiàn)[3]和[4]的啟發(fā),筆者利用錐壓縮與錐拉伸不動(dòng)點(diǎn)定理研究 2n階邊值問(wèn)題

1 預(yù)備知識(shí)

為了書(shū)寫(xiě)方便,引入下列符號(hào):假設(shè) (H1)成立,由方程 (8)及格林函數(shù)的性質(zhì)知:當(dāng)x∈C2n-2[0,1]時(shí),Ax∈C2n-2[0,1],由式 (11)知Ax∈C2n-2[0,1],由式 (12)知 Ax∈C2n[0,1]。由式(11)及格林函數(shù)的性質(zhì)得:A:C2n-2[0,1]→C2n[0,1]為連續(xù)映射,根據(jù)嵌入 C2n[0,1]→C2n-2[0,1]的緊性知:C2n-2[0,1]→C2n-2[0,1]為全連續(xù)映射。由式 (12)知 x∈C2n-2[0,1]為邊值問(wèn)題 (1)解的充要條件為:x是 A的不動(dòng)點(diǎn)。

引理4 假設(shè) (H1)成立,則 A(P)? P且 A:P→P是全連續(xù)的。

2 主要結(jié)果

定理1 假設(shè) (H1)成立,若 η1f0<1<η2f∞,則問(wèn)題 (1)至少有一個(gè)正解。

[1] CHEN SHI HUA,N IWEI,WANG CHANGP ING.Positive solution of fourth-order ordinary differential equation with four-point boundary conditions[J].Appl.Math.Lett,2006,19(2):161-168.

[2] ZHANGQ IN,CHEN SHIHUA,LV J INHU.Upper and lower solution method for fourth-order four-point boundary value problems[J].J.Comput.Appl.Math,2006,196(2):387-393.

[3] 李永祥.四階邊值問(wèn)題正解的存在性與多解性[J].應(yīng)用數(shù)學(xué)學(xué)報(bào),2003,26(1):109-116.

[4] ZHANG XUEMEI,GEWEIGAO.Positive solutions for a class of boundary-value problems with integral boundary conditions[J].Appl.Math.Comput,2009,58(2):203-215.

[5] WEI ZHONGL I.Existence of positive solutions for 2nth-order singular sublinear boundary value problems[J].J.Math.Anal.Appl,2005,306(2):619-636.

[6] BAI ZHANB ING,GEWEIGAO.Solutionsof 2nth lidstone boundary value problems and dependence on higher order derivatives[J].J.Math.Anal.Appl,2003,279(2):442-450.

[7] CHYAN C J,HENDERSON J.Positive solutions of 2mth-order boundary value problems[J].Appl.Math.Lett,2002,15(6):767-774.

[8] YANG X IAOJ ING.Existence ofpositive solutions for2m-order nonlinear differential systems[J].Nonlinear Anal,2005,61(1/2):77-95.

[9] YAO Q INGL IU.Existence and iteration ofnsymmetric positive solutions for a singular two-point boundary value problem[J].Comput.Math.Appl,2004,47(8/9):1 195-1 200.

[10] L IANG SI HUA,ZHANG JI HU I.Positive solutions of 2nth-order ordinary differential equations with multi-point boundary conditions[J].Appl.Math.Comput,2008,197(1):262-270.

[11] 暴寧偉.奇異一階微分方程周期邊值問(wèn)題的正解[J].河北工程大學(xué)學(xué)報(bào):自然科學(xué)版,2008,25(2):98-100.

(編輯 王 冬)

Existence and multiplicity of positive solutions for 2nth-order boundary value problem s

CHEN Chunxiang(College of Sciences,China University ofMining&Technology,Xuzhou 221116,China)

O175.8

A

1671-0118(2010)05-0399-04

2010-09-07

陳春香 (1984-),女,江蘇省銅山人,碩士,研究方向:常微分方程邊值問(wèn)題,E-mail:chunxiang-chen@sina.com。