達(dá)舉霞, 霍 梅, 韓曉玲

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 蘭州 730070)

帶變號(hào)格林函數(shù)的四階三點(diǎn)邊值問題的多個(gè)正解的存在性

達(dá)舉霞, 霍 梅, 韓曉玲*

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 蘭州 730070)

應(yīng)用Leggett-Williams不動(dòng)點(diǎn)定理研究了四階三點(diǎn)邊值問題

多個(gè)正解的存在性，其中f:[0,1]×[0,+∞)→[0,+∞)連續(xù),η為常數(shù). 盡管Green函數(shù)是變號(hào)的,對(duì)任意的正整數(shù)m,該問題仍有正解且至少有2m-1個(gè)正解.

四階三點(diǎn)邊值問題; 變號(hào)Green函數(shù); 多個(gè)正解

u(4)(t)=f(t,u(t)) (t[0,1]),

u′(0)=u″(η)=u?(0)=u(1)=0,

wheref:[0,1]×[0,+∞)→[0,+∞) is continuous,η.Theexistenceofatleast2m-1positivesolutionsforarbitrarypositiveintegermisobtainedwhiletheproblemhasthesign-changingGreen’sfunction.

Keywords：fourth-orderthree-pointboundaryvalueproblem;sign-changingGreen’sfunction;multiplepositivesolutions

彈性梁是工程建筑的基本構(gòu)件,彈性力學(xué)和工程物理常用四階常微分方程邊值問題來刻畫彈性梁的平衡狀態(tài),由于這類問題的普遍性和重要性,四階兩點(diǎn)邊值問題和四階多點(diǎn)邊值問題深受學(xué)者關(guān)注.

2008年,YAO[1]運(yùn)用Krasnoselskli不動(dòng)點(diǎn)定理獲得了四階三點(diǎn)邊值問題

u(4)(t)+F(t,x(t),x″(t))=0 (t[0,1]),

x″(0)=x″(1)=0,x(η)=B,x′(η)=C (0<η<1)

n個(gè)正解的存在性結(jié)果.

2009年,GRAEF等[2]運(yùn)用錐上的不動(dòng)點(diǎn)定理研究了四階三點(diǎn)邊值問題

u(4)(t)=g(t)f(u(t)) (t[0,1]),

(1)

u(0)=u′(0)=u″(β)=u?(1)=0

(2)

2014年,ZHOU等[3]運(yùn)用不動(dòng)點(diǎn)指數(shù)理論獲得了四階三點(diǎn)邊值問題

u(4)(t)=g(t)f(u(t)) (t[0,1]),

u(0)=u′(0)=u″(β)=u″(1)=0

以上結(jié)果都是在Green函數(shù)非負(fù)的情況下獲得的. 2012年,SUN和ZHAO[4]運(yùn)用Leggett-Williams不動(dòng)點(diǎn)定理在Green函數(shù)變號(hào)時(shí)獲得了問題

u?(t)=f(t,u(t)) (t[0,1]),

u′(0)=u″(η)=u(1)=0多個(gè)正解的存在性結(jié)果，這里fC([0,1]×[0,+∞)),η).

更多詳細(xì)結(jié)果見文獻(xiàn)[5-11]. 受前人啟發(fā)，本文在Green函數(shù)變號(hào)的情況下運(yùn)用Leggett-Williams不動(dòng)點(diǎn)定理研究問題

(3)多個(gè)正解的存在性，這里fC([0,1]×[0,+∞),[0,+∞)),η,推廣了文獻(xiàn)[4]的主要結(jié)果.

1 預(yù)備知識(shí)

設(shè)E是Banach空間,P是E上的錐. 對(duì)任意的x,yP,t[0,1],若有δ(tx+(1-t)y)≥tδ(x)+(1-t)δ(y),則映射δ:P→(-∞,+∞)是一個(gè)凹函數(shù). 設(shè)a和b是2個(gè)常數(shù)且0

Pa={xP:‖x‖≤a},

P(δ,a,b)={xP:a≤δ(x),‖x‖≤b}.

(2)‖Ax‖

‖x1‖d,δ(x3)

u″(t)≤0,t[0,η]; u″(t)≥0,t[η,1].

(5)在 u(1)≤0的情況下,由式(5)可知u′(t)≤0,t[0,1], u(1)=0 ,這意味著 u(t)≥0,t[0,1]. 因此,在 E 上定義一個(gè)錐如下:E:u(t)≥0且u單調(diào)遞減,t[0,1],u′(1)≤0}.

u(4)(t)=y(t) (t[0,1]),

(6)

u′(0)=u″(η)=u?(0)=u(1)=0.

在[0,t]上給式(6)兩邊積分,得到

(7)

在[0,t]上給式(7)兩邊積分,得到

(8)

繼續(xù)在[0,t]上給式(8)兩邊積分,得到

(9)

最后再在[0,t]上給式(9)兩邊積分,得到

從而式(6)的格林函數(shù)的表達(dá)式G(t,s)如下:

(1)當(dāng)s≥η時(shí),有

max{G(t,s):t[0,1]}=G(1,s)=0;

對(duì)于s<η,有

max{G(t,s):t.

可得

則

2 主要結(jié)果

本文假定f:[0,1]×[0,+∞)→[0,+∞)是連續(xù)的并且滿足如下條件:

(C1)對(duì)每一個(gè)x[0,+∞),映射t→f(t,x)是遞減的;

(C2)對(duì)每一個(gè)t[0,1],映射 x→f(t,x)是遞增的.

顯然,如果u是A在P上的不動(dòng)點(diǎn),則u是式(3)的非負(fù)解.

為了方便,記

定理2 設(shè)存在數(shù)d、a和c，0

(10)

(11)

(12)

(13)

綜上,Leggett-Williams不動(dòng)點(diǎn)定理的所有條件都被滿足. 因此,A至少有3個(gè)不動(dòng)點(diǎn),即式(3)至少有3個(gè)正解u、v和w并滿足

定理3 設(shè)m是任意正整數(shù),假設(shè)存在di(1≤i≤m)和aj(1≤j≤m-1),且0

(14)

(15)

(16)

例1 考慮BVP

u(4)(t)=f(t,u(t)) (t[0,1]),

(18)

(19)

其中

f(t,u)=

[1]YAOQL.Existenceandmultiplicityofpositivesolutionstononlinearfourth-orderthree-pointboundaryvalueproblem[J].JournalofZhejiangUniversity,2008,35:378-380.

[2]GRAEFJR,HENDERSONJ,YANGB.Positivesolutionstoafourthorderthreepointboundaryvalueproblem[J].DiscreteandContinuousDynamical,2009,285:269-275.

[3]ZHOUSH,WUHP,HANXL.Existenceofpositivesolutionsofthefourth-orderthree-pointboundaryvaluepro-blems[J].JournalofSichuanUniversity,2014,51:11-15.

[4]SUNJP,ZHAOJ.Multiplepositivesolutionsforathird-orderthree-pointBVPwithsign-changingGreen’sfunction[J].JournalofMathematicalAnalysisandApplications,2012(118):1-7. [5]LEGGETTRW,WILLIAMSLR.MultiplepositivefixedpointsofnonlinearoperatorsonorderedBanachspaces[J].IndianaUniversityMathematicsJournal,1979,28(4):673-688. [6]SUNJP,ZHAOJ.Iterativetechniqueforathird-orderthree-pointBVPwithsign-changingGreen’sfunction[J].JournalofMathematicalAnalysisandApplications,2013,215:1-9. [7]SUNYP.Positivesolutionsforthird-orderthree-pointnonhomogeneousboundaryvalueproblems[J].AppliedMathematicsLetters,2009,22(1):45-51.

[8] 達(dá)佳麗，韓曉玲. 三階三點(diǎn)邊值問題3個(gè)正解的存在[J]. 華南師范大學(xué)學(xué)報(bào)(自然科學(xué)版)，2015，47(3):148-150.

DAJL,HANXL.Existenceofthreepositivesolutionsforathird-orderthree-pointboundaryvalueproblem[J].JournalofSouthChinaNormalUniversity(NaturalScienceEdition),2015,47(3):148-150.

[9]YAOQL.Theexistenceandmultiplicityofpositivesolutionsforathird-orderthree-pointboundaryvaluepro-blem[J].JournalofMathematicalAnalysisandApplications,2003,288:1-14.

[10]FENGXF,FENGHY,BAIDL.Eigenvalueforasingularthird-orderthree-pointboundaryvalueproblem[J].AppliedMathematicsandComputation,2013,219(18):9783-9790.

[11]DUZJ,GEWG,LINXL.Existenceofsolutionsforaclassofthird-ordernonlinearboundaryvalueproblems[J].JournalofMathematicalAnalysisandApplication,2004,294(1):104-112.

【中文責(zé)編：莊曉瓊 英文審校：肖菁】

Existence of Multiple Positive Solutions for A Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

DAJuxia,HUOMei,HANXiaoling*

(College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

By applying Leggett-Williams fixed point theorem,the fourth-order three-point boundary value problem is studied:

2015-11-20 《華南師范大學(xué)學(xué)報(bào)(自然科學(xué)版)》網(wǎng)址：http://journal.scnu.edu.cn/n

國家自然科學(xué)基金項(xiàng)目(11561063)

u(4)(t)=f(t,u(t)) (t[0,1]),

u′(0)=u″(η)=u?(0)=u(1)=0

O175.8

A

1000-5463(2017)03-0109-05

*通訊作者:韓曉玲,教授,Email:hanxiaoling9@163.com.