王平友, 賈 梅, 竇麗霞, 金京福

(上海理工大學(xué)理學(xué)院,上海 200093)

具有p- Laplace算子的積分微分方程積分邊值問題正解的存在性

王平友, 賈 梅, 竇麗霞, 金京福

(上海理工大學(xué)理學(xué)院,上海 200093)

研究了帶有p- Laplace算子的微分積分方程積分邊值問題正解的存在性,利用范數(shù)形式的錐拉伸與錐壓縮不動(dòng)點(diǎn)定理,得到了邊值問題至少存在一個(gè)正解的結(jié)論.

積分邊值問題;不動(dòng)點(diǎn)定理;正解;錐

1 問題的提出

由于邊值問題具有廣泛的應(yīng)用背景,因此已被深入的研究[1],尤其對(duì)于邊值問題

其中,α,γ,β,δ≥0,并且αγ+αδ+βγ＞0,f∈C([0,1]×[0,+∞),[0,+∞))正解的存在性,許多作者對(duì)其做了大量的工作,并且獲得了上述邊值問題有解、一定為正解等結(jié)果[1-5].對(duì)于具p- Laplace算子的邊值問題正解存在性的研究,也得到了許多有意義的結(jié)果[6-11].

但是,如果邊界條件中涉及的點(diǎn)多于兩個(gè),如四點(diǎn)邊值問題

其中,0＜ξ,η＜1,并且αγ+αδ+βγ＞0,f∈C([0,1]×[0,+∞),[0,+∞)),由于其邊界條件的復(fù)雜性,即使得到其解存在,也不能保證其解一定為正解[6].

文獻(xiàn)[6]研究了邊值問題

其中,φp(s)=|s|p-2s,p＞1,α,β＞0,0＜ξ＜η＜1,利用范數(shù)形式的錐拉伸與錐壓縮定理,得到了邊值問題至少存在一個(gè)正解、多個(gè)正解的結(jié)論.

文獻(xiàn)[7]研究了邊值問題

其中,φp(s)=|s|p-2s,p＞1,α＞0,0＜ξ＜η＜1,ξ+η=1,λ＞0,利用不動(dòng)點(diǎn)指數(shù)理論,Leray -Schauder度結(jié)合上下解方法,得到了邊值問題多個(gè)對(duì)稱正解的存在和不存在的充分條件.

本文進(jìn)一步討論具p- Laplace算子的微分積分方程邊值問題

2 預(yù)備知識(shí)

考慮邊值問題

引理4設(shè)(H1)成立,則邊值問題(3)的解u具有以下性質(zhì):

a.u是嚴(yán)格凹函數(shù);

b.u(t)≥0,t∈[0,1];

證明假設(shè)u是邊值問題(3)的解.

a.由于(φp(u′))′=-y(t)≤0,且y在(0,1)的任意子區(qū)間上不恒等于零,則φp(u′)是嚴(yán)格單調(diào)減函數(shù),從而u′(t)也是嚴(yán)格減函數(shù),故u(t)在0≤t≤1上是嚴(yán)格凹函數(shù).

b.由a可得,對(duì)任意t∈[0,1],有u(t)≥min{u(0),u(1)}.不失一般性,可設(shè)u(0)=min{u(0),u(1)},則

與文獻(xiàn)[6]類似,可證引理6成立.

引理6T:K→K為全連續(xù)算子.

3 主要結(jié)果

由引理7得算子T在K-RK r至少存在一個(gè)不動(dòng)點(diǎn)u,由于u是邊值問題(1)的解當(dāng)且僅當(dāng)u是T的不動(dòng)點(diǎn),故邊值問題(1)至少存在一個(gè)正解u,且r≤u≤R.

定理2設(shè)(H1)和(H2)成立,且存在常數(shù)0＜ε＜1,使得下面假設(shè)成立.

由引理7得算子T在K-RK r至少存在一個(gè)不動(dòng)點(diǎn)u,由于u是邊值問題(1)的解當(dāng)且僅當(dāng)u是T的不動(dòng)點(diǎn),故邊值問題(1)至少存在一個(gè)正解u,且r≤u≤R.

[1] 葛渭高.非線性常微分方程邊值問題[M].北京:科學(xué)出版社,2007.

[2] GE Wei-gao,REN Jing-li.New existence theorem positive solutions for the Sturm-Liouville boundary-value problem[J].Appl Math Comput,2004,148:631-644.

[3] WEBB J R L.Positive solutions of some three point boundary-value problem via fixed point index theory[J].Nonlinear Analysis,2001,47:4319-4332.

[4] FENGHan-ying,GE Wei-gao.Multiple position for m-point boundary-value problems with aone-dimensionalp- Laplacian[J].Nonlinear Analysis,2008,68:2269-2279.

[5] MA Ru-yun,REN Li-shun.Positive solutions for nonlinear m-point boundary value problems of Dirichlet type via fixed-point index theory[J].Applied Mathematics Letters,2003,99:863-869.

[6] LIAN Hai-rong,GE Wei-gao.Positive solutions for a four-point boundary value problem with thep- Laplacian[J].Nonlinear Analysis,2008,68:3493-3503.

[7] JI De-hong,GE Wei-gao.The existence of sysmetric positive solutions for Sturm-Liouville like four-point boundary value problem with ap- Laplacian operator[J].Applied Mathematics and Computation,2007,189:1087-1098.

[8] LI Xiang-feng.Multiple positive solutions for some four-point boundary value problems withp- Laplacian[J].Applied Mathematics and Computation,2008,202:413-426.

[9] BAI Chuan-zhi,FANG Jin-xuan.Existence of multiple positive solutions for nonlinearm-point boundary value problems[J].Applied Mathematics and Computation,2003,140:297-305.

[10] WANG You-yu,HOU Cheng-min.Existence of multiple positive solutions for one-dimensionalp- Laplacian[J].J Math Anal Appl,2006,315:144-153.

[11] SU Wei-hua,WEI Zhong-li,WANG Bao-he.The existence of positive solutions for a nonlinear four-point singular boundary value problem with ap- Laplacian operator[J].Nonlinear Analysis,2007,66:2204-2217.

[12] 郭大鈞,孫經(jīng)先,劉兆理.非線性常微分方程泛函方法[M].2版.濟(jì)南:山東科學(xué)技術(shù)出版社,2006.

Existence of positive solutions of integral boundary value problems for nonlinear integro-differential equations withp- Laplacian

WANGPing-you, JIA Mei, DOULi-xia, JINJing-fu
(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China)

The existence of positive solution of integral boundary value problems for nonlinear integro-differential equations with the one-dimensionalp- Laplacian was studied.By using the expansion and compression fixed point theorem of norm in cone,the existence of at least one positive solution for this kind of integral boundary value problems was concluded.

integral boundary value problem;fixed point theorem;positive solutions;cone

O 175.8

A

1007-6735(2011)04-0391-06

2011-01-14

上海市教育委員會(huì)科研創(chuàng)新基金重點(diǎn)資助項(xiàng)目(10ZZ93)

王平友(1981-),男,碩士研究生.研究方向:應(yīng)用微分方程.E-mail:wpingy2008@163.com賈 梅(聯(lián)系人),女,副教授.研究方向:應(yīng)用微分方程.E-mail:jiamei_usst@163.com