一類非線性三階三點邊值問題的可解性*

許也平

(杭州廣播電視大學(xué),浙江 杭州 310012)

一類非線性三階三點邊值問題的可解性*

許也平

(杭州廣播電視大學(xué),浙江 杭州 310012)

討論了一類非線性項含一階和二階導(dǎo)數(shù)的三階三點邊值問題的可解性,在非線性項f滿足線性增長的限制條件下,通過構(gòu)造適當(dāng)?shù)腂anach空間,并利用Leray-Schauder非線性抉擇,證明了一個存在定理.

三階三點邊值問題；解；存在性；Leray-Schauder非線性抉擇

三階邊值問題在應(yīng)用數(shù)學(xué)和物理中有著非常重要的意義，對此已有許多研究成果[1-5].本文研究三階三點邊值問題

筆者討論上述非線性項含一階和二階導(dǎo)數(shù)的三階三點邊值問題(1)解的存在性,得到如下結(jié)論：

定理1假設(shè)f:[0,1]×R×R×R→R連續(xù)并存在非負(fù)函數(shù)a,b,c,d∈L1[0,1],使得

|f(t,u,v,w)|≤a(t)|u|+b(t)|v|+c(t)|w|+d(t),0≤t≤1.

如果

則問題(1)至少有1個解u*∈C2[0,1].

E={u∈C2[0,1]:αu(0)=βu′(0)=0,u′(1)=αu′(η)}

是一個關(guān)于范數(shù)|‖u‖|=max{‖u‖,‖u′‖,‖u"‖}的Banach空間.

引理1[1]設(shè)G(t,s)是齊次三階三點邊值問題

的Green函數(shù)，那么

引理2對G(t,s),有

另外,由式(2)知

下面證明

接下來證明

于是引理2得證.

定理1的證明 對于u∈E，定義T算子為

容易證明T:E→E全連續(xù)且算子T在E中的不動點均為問題(1)的解.因此,證明的目標(biāo)就是尋找T在E中的不動點.顯然,零函數(shù)為問題(1)的解當(dāng)且僅當(dāng)

f(t,0,0,0)≡0,0≤t≤1.

假設(shè)存在u∈?Vρ,λgt;1,使得Tu=λu.因為|‖u‖|=ρ，那么‖u‖≤ρ,‖u′‖≤ρ,‖u"‖≤ρ.這表明

|u(s)|≤ρ,|u′(s)|≤ρ,|u"(s)|≤ρ,0≤s≤1,

故

這與λgt;1,ρgt;0矛盾.根據(jù)引理3,算子T至少有1個不動點u*∈Vρ.也就是說,問題(1)至少有1個解u*∈E.定理1證畢.

[1]Zhao Weili.Singular perturbations for third order nonlinear boundary value problems[J].Nonl Anal,1994,23(10):1225-1242.

[2]姚慶六.線性增長限制下一類三階邊值問題的可解性[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2005,21(2):164-167.

[3]王素云.三階非線性常微分方程正解的存在性[J].甘肅科學(xué)學(xué)報,2003,15(3):1-5.

[4]Guo Lijun,Sun Jianping,Zhao Yahong.Existence of positive solution for nonlinear third-order three-point boundary value problem[J].Nonlinear Analysis:Theory Methods amp; Applications,2008,68(10):3151-3158.

[5]Yao Qingliu.The existence and multiplicity of positive solutions for a third-order three-point boundary value problem[J].Acta Math Appl Sinica,2003,19(1):117-122.

(責(zé)任編輯 陶立方)

Solvabilityofathird-orderthree-pointboundaryvalueproblem

XU Yeping

(HangzhouRadioandTVUniversity,HangzhouZhejiang310012,China)

The solvability was considered for a class of third-order three-point boundary value problem with first and second derivatives. With the nonlinear termfsatisfied a restriction of linear growth, by constructing a suitable Banach space and applying the Leray-Schauder nonlinear alternative, an existence theorem was established.

third-order three-point boundary value problem; solutions; existence; Leray-Schauder nonlinear alternative

1001-5051(2010)01-0022-05

2009-09-06

浙江省教育廳科研項目(Y200804663)

許也平(1962-),男,浙江杭州人，副教授.研究方向:微分方程及其應(yīng)用.

O175.8

A