东北大学学报:自然科学版  2020, Vol. 41 Issue (4): 604-608  
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张国伟, 曲雪冰. 带有变号格林函数的二阶边值问题正解[J]. 东北大学学报:自然科学版, 2020, 41(4): 604-608.
[复制中文]
ZHANG Guo-wei, QU Xue-bing. Positive Solutions of Second-Order Boundary Value Problems with Sign-Changing Green's Function[J]. Journal of Northeastern University Nature Science, 2020, 41(4): 604-608. DOI: 10.12068/j.issn.1005-3026.2020.04.026.
[复制英文]

基金项目

国家自然科学基金资助项目(61473065)

作者简介

张国伟(1965-), 男, 辽宁沈阳人, 东北大学教授。

文章历史

收稿日期:2019-09-26
带有变号格林函数的二阶边值问题正解
张国伟 , 曲雪冰     
东北大学 理学院, 辽宁 沈阳 110819
摘要:研究了一类带有变号格林函数的二阶边值问题正解的存在性, 格林函数变号由边值条件中系数的不同取值所致, 这与文献中通常由未知函数一次项系数的变化导致格林函数变号不同.没有非线性项非负的限制时, 通过对格林函数的正部和负部赋予约束条件, 证明了二阶边值问题正解的存在性.利用两个具体例子说明了理论结果的有效性, 例子中边值条件的系数包含了正的和负的两种情形.另外对两类不同的边值条件给出了说明.
关键词正解    变号格林函数    二阶边值问题    全连续算子    Leray-Schauder不动点定理    
Positive Solutions of Second-Order Boundary Value Problems with Sign-Changing Green's Function
ZHANG Guo-wei , QU Xue-bing     
School of Sciences, Northeastern University, Shenyang 110819, China
Abstract: The existence of positive solutions for a class of second-order boundary value problems with a sign-changing Green's function was studied, and the sign-changing Green's function was caused by different values of coefficients in boundary value conditions, which is different from that the change of the coefficient of the first order of the unknown function usually leads to the change of the Green's function. When there is no non-negative limitation of nonlinear term, the existence of positive solutions for second-order boundary value problems was proved by giving constraints to the positive and negative parts of Green's function. The validity of the theoretical results was illustrated by two concrete examples, in which the coefficients of boundary value condition include both positive and negative cases. In addition, two different boundary conditions were explained.
Key words: positive solution    sign-changing Green's function    second-order boundary value problem    completely continuous operator    Leray-Schauder fixed point theorem    

Torres[1]讨论了一类线性周期边值问题的格林函数, 在一定条件下证明了其格林函数是不变号的.Cabada等[2]假设格林函数是正的情形时, 证明了非线性周期边值问题正解的存在性.其他一些相关的工作可参见文献[3-8].Graef等[9]和Ma[10]分别讨论了在格林函数非负和变号情形下非线性周期边值问题正解的存在性.Zhong等[11]讨论了当时, 非线性周期边值问题式(1)的正解存在性:

(1)

实际上, 当时, 其格林函数是变号的.特别地, Gao等[12]研究了具有变号格林函数非线性周期边值问题式(2)的正解存在性:

(2)

式中, .考虑边值条件含有参数的非线性边值问题式(3)的正解存在性:

(3)

其中格林函数的变号性是由参数不同取值导致的, 结论的证明方法来自文献[12].

C[0, 2π]表示[0, 2π]上连续函数的Banach空间, 范数为.

1 格林函数及其变号性

首先假设:

H1αβ, α≠0, β≠0, α≠±1, β≠±1;

H2fC[0, ∞)是连续的,且f(0)>0;

H3gL1[0, 2π],对于任意区间不是几乎处处等于零.

f(x)再进行如下连续延拓:当x < 0时,规定f(x)=f(0),仍记为f(x).定义算子

式中,

是边值问题(3)的格林函数.利用常规方法可以证明下面的引理.

引理1  设H1,H2和H3满足.边值问题式(3)存在解yC[0, 2π],等价于y是算子S在空间C[0, 2π]的不动点.

定理1  设H1满足,则格林函数G(t, s)在[0, 2π]×[0, 2π]变号.

证明  根据参数αβ的关系,可划分为如下20个参数区域Ⅰ-XX,见图 1.

图 1 参数区域 Fig.1 Domain of parameters

仅给出对于区域0 < β < α的讨论,其余区域的讨论方法类似.

1) π>t>s>0情形:因为

故恒有G(t, s)>0.事实上,,

2)π>s>t>0情形:因为

故恒有G(t, s)>0.事实上,

3) 2π>s>π>t>0情形:因为G(t, s)与情形2)相同, 所以

由于,故G(t, s)>0恒成立.

4) 2π>t>π>s>0情形:因为G(t, s)与情形1)相同, 所以

由于,故G(t, s)>0恒成立.

5) 2π>t>s>π情形:因为G(t, s)与情形1)相同, 所以

由于,则

且当t→π时,s→π;当t→2π时,s→2π.故由单调性与凹凸性可知其符号.

6) 2π>s>t>π情形:因为G(t, s)与情形2)相同, 所以

,由于,则

且当t→π时,s→π;当t→2π时,s→2π.故由单调性与凹凸性可知其符号.

综上可知,对于区域0 < β < α来说,格林函数符号情况可见图 2.

图 2 0 < β < α时格林函数的符号 Fig.2 Signs of Green's function for 0 < β < α
2 边值问题的正解

对于t∈[0, 2π], 记

再假设

H4:存在ε>0,使得对于t∈[0, 2π],

引理2  如果, H1,H3和H4满足,令

证明  如果p(t)≡0,则∀t∈[0, 2π],有[G(t, s)g(s)]+=0, a.e.s∈[0, 2π].由H4可知,∀t∈[0, 2π],[G(t, s)g(s)]-=0, a.e.s∈[0, 2π].故G(t, s)g(s)=0, a.e.s∈[0, 2π].从而由定理1可知,存在使g几乎处处等于零的区间,这与H3矛盾.证毕.

定义算子T

(Ty)(t)= , ∀yC[0, 2π].易证T:C[0, 2π]→C[0, 2π]是全连续算子.利用引理2,可得如下引理.

引理3  如果H1~H4满足,令0 < δ < 1,则存在正数λ, 使得当λ∈(0, λ)时,方程y(t)=(Ty)(t)有一个正解 ,满足当λ→0时,.并且 .

定理2  如果H1~H4满足,则存在λ0>0,使得当λ∈(0, λ0)时,式(3)有一个正解.

证明  令.由H4可知,如果q(t)≠0, 由于ε>0, 取ξ满足, 故(1+ε)ξ>1, 于是ξf(0)(1+ε)>f(0).又因为f在0处连续,存在k∈(0, 1), 使得对s∈[0, k], 有

|f(s)|≤ξf(0)(1+ε).所以

(4)

如果q(t)=0, 式(4)成立.

δ∈(ξ, 1), 由引理3可知

所以存在λ1>0, 使得当λ∈(0, λ1)时,.因为f在[-k, k]上一致连续,故对, 存在λ2>0, 使得当x, y∈[0, k]且|x-y|≤λ2δf(0)‖p0时,有.取λ0=min{λ1, λ2}, 当λ∈(0, λ0)时,

(5)

x, y∈[-k, k], 且|x-y|≤λ0δf(0)‖p0时,

(6)

λ∈(0, λ0)时,定义算子H如下:

, t∈[0, 2π], 其中 由引理3给出.容易证明H:C[0, 2π]→C[0, 2π]是全连续的.

θ∈(0, 1), yC[0, 2π],若y=θHy,下证‖y0λδf(0)‖p0.否则‖y0=λδf(0)‖p0, 则由式(5)和式(6)得

(7)
(8)

由式(4),式(7)和式(8)可知:

(9)

故与矛盾.由Leray-Schauder不动点定理知H有一个不动点 , 且.类似的推导可知满足式(9).记,于是yλ是式(3)的一个解.再由引理3得

可见yλ是式(3)的一个正解.

3 例子

例1  考虑边值问题:

可以验证定理2的条件满足, 所以存在λ0>0,当0 < λ < λ0时有正解.

例2  考虑边值问题:

可以验证定理2的条件满足,所以存在λ0>0,当0 < λ < λ0时有正解.

这里含导数部分边值的系数分别取正数和负数,并且t∈[0, 2π]上都是变号的.

4 结论

本文主要研究的是由边值条件中系数α, β不同取值导致格林函数变号的二阶非线性边值问题,其中αβ.如果H1~H4满足,则存在λ0>0,使得当λ∈(0, λ0)时,式(3)有一个正解.当α=β=0时,其格林函数不变号.而当α=β=1时,实际上是周期边值问题,其格林函数也是不变号的.

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