POSITIVE SOLUTIONS TO SINGULAR m-POINT BOUNDARY VALUE PROBLEMS OF A COUPLED SYSTEM OF DIFFERENTI
具有变号非线性项的脉冲微分方程边值问题的正解
具有变号非线性项的脉冲微分方程边值问题的正解江卫华;张强;郭巍巍【摘要】运用Avery-Peterson不动点定理,研究了具有变号非线性项的脉冲微分方程边值问题正解的存在性.%By using Avery-Peterson fixed-point theorem, the existence of positive solutions of the boundary value problem of impulsive differential equation with sign-changing nonlinearterm was studied.【期刊名称】《河北科技大学学报》【年(卷),期】2013(034)001【总页数】6页(P1-6)【关键词】脉冲微分方程;边值问题;正解;变号非线性项【作者】江卫华;张强;郭巍巍【作者单位】河北科技大学理学院,河北石家庄050018;河北科技大学理学院,河北石家庄050018;河北科技大学理学院,河北石家庄050018【正文语种】中文【中图分类】O175.8脉冲微分方程在经济、生物、生态学等领域有着广泛的应用[1-3]，考虑到其影响，很多学者常将微分方程边值问题推广到脉冲微分方程上去，通过运用锥拉伸与锥压缩不动点定理、Leray-Schauder不动点定理、不动点指数理论等方法，得到了脉冲微分方程边值问题解的存在性[4-21]。
在文献[4]中，AGARWAL等利用非线性的Leray-Schauder不动点定理和Krasnoselskii's不动点定理得到了二阶脉冲微分方程边值问题：至少存在1个解和2个解的充分条件。
在文献[5]中，AGARWAL等又利用Legget-Williams不动点定理得出了脉冲微分方程边值问题至少存在3个正解的充分条件。
在文献[6]中，李高山等运用锥拉伸与锥压缩不动点定理，得到了带有变号非线性项的二阶三点微分方程边值问题：其中α,η∈(0,1)，至少1个正解的充分条件。
带变号格林函数的四阶三点边值问题的多个正解的存在性
带变号格林函数的四阶三点边值问题的多个正解的存在性达举霞;霍梅;韩晓玲【摘要】应用Leggett-Williams不动点定理研究了四阶三点边值问题u(4)(t)=f(t,u(t)) (t∈[0,1]),u’(0)=u”(η)=u'''(0)=u(1)=0多个正解的存在性,其中f:[0,1]×[0,+∞)→[0,+∞)连续,η∈[√3/3,1]为常数.尽管Green函数是变号的,对任意的正整数m,该问题仍有正解且至少有2m-1个正解.%By applying Leggett-Williams fixed point theorem,the fourth-order three-point boundary value problem is studied:u(a)(t)=f(t,u(t)) (t∈[0,1]),u′(0)=un(η)=u"(0)=u(1)=0,where f:[0,1] ×[0,+∞)→[0,+∞) is continuous,η∈ [√3/3,1].The existence of at least 2m-1 positive solutions for arbitrary positive integer m is obtained while the problem has the sign-changing Green's function.【期刊名称】《华南师范大学学报（自然科学版）》【年(卷),期】2017(049)003【总页数】5页(P109-113)【关键词】四阶三点边值问题;变号Green函数;多个正解【作者】达举霞;霍梅;韩晓玲【作者单位】西北师范大学数学与统计学院,兰州730070;西北师范大学数学与统计学院,兰州730070;西北师范大学数学与统计学院,兰州730070【正文语种】中文【中图分类】O175.8多个正解的存在性，其中f:[0,1]×[0,+∞)→[0,+∞)连续,η为常数. 尽管Green函数是变号的,对任意的正整数m,该问题仍有正解且至少有2m-1个正解.where f:[0,1]×[0,+∞)→[0,+∞) is continuous,η. The existence of at least 2m-1 positive solutions for arbitrary positive integer m is obtained while the problem has the sign-changing Green’s function.Key words： fourth-order three-point boundary value problem; sign-changing Green’s function; multiple positive solutions弹性梁是工程建筑的基本构件,弹性力学和工程物理常用四阶常微分方程边值问题来刻画弹性梁的平衡状态,由于这类问题的普遍性和重要性,四阶两点边值问题和四阶多点边值问题深受学者关注.2008年,YAO[1]运用Krasnoselskli不动点定理获得了四阶三点边值问题x″(0)=x″(1)=0,x(η)=B,x′(η)=C (0<η<1)n个正解的存在性结果.2009年,GRAEF等[2]运用锥上的不动点定理研究了四阶三点边值问题正解的存在性,这里β为常数,gC([0,1],[0,∞)),在非线性项f满足超线性或次线性条件下获得了问题(1)、(2)至少存在一个正解.2014年,ZHOU等[3]运用不动点指数理论获得了四阶三点边值问题正解的存在性结果,这里β为常数,gC([0,1],[0,∞)).以上结果都是在Green函数非负的情况下获得的. 2012年,SUN和ZHAO[4]运用Leggett-Williams不动点定理在Green函数变号时获得了问题更多详细结果见文献[5-11]. 受前人启发，本文在Green函数变号的情况下运用Leggett-Williams不动点定理研究问题设E是Banach空间,P是E上的锥. 对任意的x,yP,t[0,1],若有δ(tx+(1-t)y)≥tδ(x)+(1-t)δ(y),则映射δ:P→(-∞,+∞)是一个凹函数. 设a和b是2个常数且0<a<b,δ是P上的非负连续凹函数. 定义凸集如下:定理1[5] (Leggett-Williams 不动点定理)设A:c→c是全连续的且δ是P上的非负连续凹函数，使得对所有的xc，都有δ(x)≤‖x‖. 假设存在0<d<a<b≤c，使得(1){xP(δ,a,b):δ(x)>a}≠∅且δ(Ax)>a,xP(δ,a,b);(2)‖Ax‖<d,‖x‖≤d;(3) δ(Ax)>a,xP(δ,a,c)和‖Ax‖>b,则A在c上至少有3个不动点x1、x2、x3且满足定义空间E=C[0,1]的范数‖u‖|.为了得到式(3)正解的存在性,需要在Banach空间E 上定义一个合适的锥. 假设u是式(3)的解,很容易证明在u′(1)≤0的情况下,u(t)≥0,t[0,1]. 事实上,由于f是非负的,所以u(4)(t)≥0,可知 u‴(t)单调递增,t[0,1]. 又u‴(0)=0,从而 u‴(t)≥0,t[0,1]. 又u″(η)=0意味着对于任意的 yE,考虑边值问题在[0,t]上给式(6)两边积分,得到在[0,t]上给式(7)两边积分,得到继续在[0,t]上给式(8)两边积分,得到最后再在[0,t]上给式(9)两边积分,得到‴(0)+s.又由u′(0)=u″(η)=u‴(0)=u(1)=0,得到s.从而式(6)的格林函数的表达式G(t,s)如下:(1)当s≥η时,有(2)当s<η 时,有显然,当0≤s<η时,G(t,s)≥0;当η≤s≤1时,G(t,s)≤0. 因此,对于s≥η,有对于s<η,有引理1 设y且[0,1]),则u且u是式(6)的唯一解. 此外u满足这里θ(1/2,η)和θ*=(η-θ)/η.证明由式(4)可知,当t[0,η]时,u″(t)≤0,从而u(t)在[0,η]上是上凸的,有可得‖u‖.则同理可得,当t[η,1]时,结论也成立.本文假定f:[0,1]×[0,+∞)→[0,+∞)是连续的并且满足如下条件:(C1)对每一个x[0,+∞),映射t→f(t,x)是递减的;(C2)对每一个t[0,1],映射x→f(t,x)是递增的.设P={u‖u‖},易证P是E上的锥. 在P上定义算子A:显然,如果u是A在P上的不动点,则u是式(3)的非负解.为了方便,记s.定理2 设存在数d、a和c，0<d<a<a/θ*≤c,使得[0,η],u[0,d]),[1-θ,θ],u,[0,η],u[0,c]),则式(3)至少有3个正解u、v和w，满足‖u‖<‖w‖,<a.证明设uP,定义). 易证当uP时,δ是P上的非负连续凹函数且δ(u)≤‖u‖,并有A:P→ P是全连续的.首先,当t[0,η]和u[0,r]时,如果存在一个正数r,使得f(t,u)<r/H1,则A:r→Pr. 事实上,若ur,则‖Au‖<,也就是说,AuPr. 同理可得,如果式(10)和式(11)成立,则A映d到Pd,A映c到Pc. 其次,我们断言对所有的uP(δ,a,a/θ*),{uP(δ,a,a/θ*):δ(u)>a}≠∅且δ(Au)>a. 事实上,常函数(a+a/θ*)/2属于{uP(δ,a,a/θ*):δ(u)>a}.另一方面,当uP(δ,a,a/θ*)时,有‖u‖≤a/θ*,t[1-θ,θ].同样,对任意的uP和t[1-θ,θ],有×,再加上式(11)、(12),当uP(δ,a,a/θ*)时,有a.最后证明:如果有uP(δ,a,c)和‖Au‖>a/θ*,则δ(Au)>a. 为了证明此结论,假设uP(δ,a,c)和‖Au‖>a/θ*成立,若AuP,有综上,Leggett-Williams不动点定理的所有条件都被满足. 因此,A至少有3个不动点,即式(3)至少有3个正解u、v和w并满足定理3 设m是任意正整数,假设存在di(1≤i≤m)和aj(1≤j≤m-1),且0<d1<a1<a1/θ*<d2<a2<a2/θ*<…<dm-1<am-1<am-1/θ*<dm使得[1-θ,θ],u,则式(3)在dm上至少有个2m-1正解.证明使用归纳法. 首先,当m=1,由式(11)有A:d1→d1. 由Schauder不动点定理可知在d1上至少有1个正解. 其次,假设m=k成立,证明m=k+1也成立. 设存在数di(1≤i≤k+1)、aj(1≤j≤k)且0<d1<a1<a1/θ*<d2<a2<a2/θ*<…<dk<ak<ak/θ*<dk+1,使得[0,η],u[0,di],1≤i≤k+1),例1 考虑BVP其中f(t,u)=设θ=3/5,则θ*=1/10. 通过简单的计算可知H1=216/1944,H2=6/625. 选取d=1,a=2,c=1 068,则定理2的条件都被满足. 因此,由定理2可知式(15)、(16)至少有3个正解.【相关文献】[1] YAO Q L. Existence and multiplicity of positive solutions to nonlinear fourth-order three-point boundary value problem[J]. Journal of Zhejiang University,2008,35:378-380.[2] GRAEF J R,HENDERSON J,YANG B. Positive solutions to a fourth order three point boundary value problem [J]. Discrete and Continuous Dynamical,2009,285:269-275. [3] ZHOU S H,WU H P,HAN X L. Existence of positive solutions of the fourth-order three-point boundary value pro-blems [J]. Journal of Sichuan University,2014,51:11-15.[4] SUN J P,ZHAO J. Multiple positive solutions for a third-order three-point BVP with sign-changing Green’s function [J]. Journal of Mathematical Analysis and Applications,2012(118):1-7. [5] LEGGETT R W,WILLIAMS L R. Multiple positive fixed points of nonlinear operators on ordered Banach spaces [J]. Indiana University Mathematics Journal,1979,28(4):673-688. [6] SUN J P,ZHAO J. Iterative technique for a third-order three-point BVP with sign-changing Green’s function [J]. Journal of Mathematica l Analysis and Applications,2013,215:1-9. [7] SUN Y P. Positive solutions for third-order three-point nonhomogeneous boundary value problems [J]. Applied Mathematics Letters,2009,22(1):45-51.[8] 达佳丽，韩晓玲. 三阶三点边值问题3个正解的存在[J]. 华南师范大学学报(自然科学版)，2015，47(3):148-150.DA J L,HAN X L. Existence of three positive solutions for a third-order three-point boundary value problem [J]. Journal of South China Normal University(Natural Science Edition),2015,47(3):148-150.[9] YAO Q L. The existence and multiplicity of positive solutions for a third-order three-point boundary value pro-blem [J]. Journal of Mathematical Analysis and Applications,2003,288:1-14.[10]FENG X F,FENG H Y,BAI D L. Eigenvalue for a singular third-order three-point boundary value problem [J]. Applied Mathematics and Computation,2013,219(18):9783-9790.[11]DU Z J,GE W G,LIN X L. Existence of solutions for a class of third-order nonlinear boundary value problems [J]. Journal of Mathematical Analysis andApplication,2004,294(1):104-112.。
POSITIVE SOLUTIONS TO SYSTEMS OF SECOND ORDER NONLOCAL BOUNDARY VALUE PROBLEMS WITH FIRST ORDER
The ex iste nce of p ositive solu tion s to non lin ear d iffe rential eq uations for n onloc al b ou ndary valu e p roble ms h as b een stu died by man y au th ors, see [1-9] an d refere nc es th ere in . Howeve r, to th e b e st of our k nowled ge, th e stud y on p ositive solu tion s to the system for non local b ou nd ary valu e prob le ms is fe w. Rec en tly, J. He nd erson and S .K . N touyas [10] stud ie d th e existen ce of p ositive solution s to the follow in g syste m of n onlin ear n th -orde r non local b ou nd ary value prob lems: � (n ) u + λh1 ( x) f 1 ( v ) = 0 , 0 < x < 1 , � � � � v ( n ) + λh ( x ) f ( u ) = 0 , 0 < x < 1 , 2 2 � u (0) = u � (0) = · · · = u ( n − 2) = 0 , � � � v (0) = v � (0) = · · · = v ( n − 2) = 0 , u(1) = α u (η ) , v (1) = α v ( η ) ,
一类分数阶微分方程边值问题解的存在性
一类分数阶微分方程边值问题解的存在性王文倩;马凡婷;孙芮【摘要】文章首先介绍了分数阶微分方及其边值问题研究现状,对分数阶常微分方程边值问题及其研究方法有一个基本的了解,并提出文章所研究的主要内容.其次,利用压缩映像原理研究了一类阶数为2<q≤3的分数阶微分方程,得到了一个解的存在定理,并且给出解的存在性和唯一性的判断依据.最后,举出例子作为文章结论的应用.【期刊名称】《阜阳师范学院学报（自然科学版）》【年(卷),期】2018(035)001【总页数】4页(P8-11)【关键词】分数阶微分方程;压缩映射原理;不动点定理;Riemann-Liouville导数【作者】王文倩;马凡婷;孙芮【作者单位】兰州交通大学数理学院,甘肃兰州730070;兰州交通大学数理学院,甘肃兰州730070;兰州交通大学数理学院,甘肃兰州730070【正文语种】中文分数阶微积分是处理任意阶微积分研究及应用的数学分析领域，其微分与积分不再局限于整数，可以是任意的实数或者复数。
1832年Liouville提出了分数阶微分定义，并用该定义成功解决了势理论问题。
1847年Riemann对分数阶微积分定义做了进一步修正和补充。
研究和应用比较多的两种定义是Caputo定义和Riemann-Liouville定义。
目前对分数阶微分方程研究主要集中在初值问题、边值问题、周期解、分数阶微分系统的稳定性、能控性以及微分包含问题等方面[1-9]。
2010年Zhou等考虑如下边界值问题[10]其中，是 Caputo 微分，是实数。
2012年Bashir Ahmad等讨论了分布边界值条件的分数阶微分方程的边值问题[11]其中，是Caputo微分，f是给定的连续函数，且α1,β1,γ1,α2,β2,γ2是实数，α1≠0。
受到上述文献的启发，主要研究下述分数阶微分方程边值问题解的存在性。
其中是Riemann-Liouville型分数导数，且a,b是实数，a≠0f:[0,1]×R→R是给定的连续函数，C[0,1]={x:[0,1]→R,x连续}构成了一个Banach空间，其范数‖x‖:=sup{|x|:t∈[0,1]}。
POSITIVE SOLUTION TO QUASI-LINEAR m-POINT BOUNDARY VALUE PROBLEM WITH SIGN CHANGING NONLINEARITY
A bs tra ct
T his pa per consider a class o f quasi-linea r m ult i-p oint b oun dary va lue p ro ble m w hose no nlinea r t erm is a llow ed to chang e sig n and d epends on t he first o rder derivat ive. By a new fixed p oint theo rem, suffi cient co ndit ions fo r t he exist ence of p osit ive solutio ns qua si-linea r m ult i-po int b oun dary value problem ; p osit ive solutio n; co ne; fixed po int 20 00 M athem atic s Sub ject C lassific ation 3 4 B1 0; 3 4B 15
积极的态度英语
积极的态度英语whatever we have undergone in our life, we shouldn't complain about it. we may get a lot or lose so much in our life journey, but keeping a positive attitude should always be together with us. nothing can defeat us if we are confident and diligent. just as saying goes, god is equal to everyone. as he closes the door, he will also open a window for us.无论我们在生活中经历过什么，我们都不应该抱怨。
在人生旅途中我们可以得到很多也可以失去很多，但我们应该永远保持积极的态度。
如果我们有信心和勤奋，没什么能打败我们。
正如俗话所说的，上帝对每个人都是平等的。
他关上一道门，就会为我们开启另一扇窗。
being enthusiastic保持热情even though we're poor in knowledge, our thinking is infinite. maybe we will make some great achievements because of our enthusias whether we're experienced or not, it doesn't matter. enthusiasm will take us further and deeper.尽管我们知识匮乏，但我们的思维是无限的。
因为我们的热忱也许我们会有很好的成就。
不管我们是否经历过，都没有关系。
热情会带我们走得更远更深入。
POSITIVE SOLUTIONS TO PERIODIC BOUNDARY VALUE PROBLEM OF SINGULAR SEMI-POSITONE NONLINEAR THIRD-
( D ept . of M ath ., Qufu N o rma l U nive rsity , Qufu 2 73 1 65 , Sh a nd on g, E -ma il: zhkm9 0 @12 6 .co m (K. Z hang ) )
1
Introduction
In th is pap e r, we stu dy the existen ce of non lin ear th ird -ord er d iffe rential e quation s ( u �i� � + ρ 3 u i = f i (t, u 1 , · · · , un ) + e i ( t) , 0 ≤ t ≤ 2 π, (1 . 1) u i (0) = u i (2 π ) , u �i (0) = u �i (2 π ) , u �i� (0) = u�i � (2 π ) , i = 1 , 2 , · · · , n, √ wh ere ρ ∈ (0 , 1 / 3), f i : [0 , 2 π ] × R n \{ 0 } → [0 , + ∞ ), e i : [0 , 2 π ] → R , i = 1 , 2 , · · · , n, are con tin uou s; fi ( t, u) may b e sin gular at u = 0. In re ce nt years, the p e rio dic b ou nd ary prob lems have b ee n wide ly stu died b y many auth ors, on e can re fer to [1-6] an d the re feren ce s the rein. In [6], usin g n online ar alte rnative of Le ray -Sc hau de r ty p e an d Krasn oselskii fi xed p oint the orem, the existen ce of p ositive solu tion s to th e following n online ar th ird-ord er p eriod ic b oun dary prob le m ( u � � � + ρ 3 u = f ( t, u ) , 0 ≤ t ≤ 2 π, u ( i ) (0) = u( i ) (2 π ) , √ i = 0, 1, 2,
一类线性方程组奇异边值问题的谱配置方法
一类线性方程组奇异边值问题的谱配置方法蔡伟云;王天军;殷艳红【摘要】对常微分方程组奇异边值问题进行了正则化处理，利用 Legendre-Gauss-Lobatto 节点为配置点，用Legendre谱配置法求其数值解，逼近方程组的正确解。
数值例子说明求解该类问题的具体方法和步骤。
数值实验结果证明了所提算法格式的有效性和高精度。
【期刊名称】《河南科技大学学报（自然科学版）》【年(卷),期】2014(000)005【总页数】4页(P87-89,94)【关键词】常微分方程组;奇异边值问题;Legendre配点法;Legendre-Gauss-Lobatto节点【作者】蔡伟云;王天军;殷艳红【作者单位】河南科技大学数学与统计学院，河南洛阳 471023;河南科技大学数学与统计学院，河南洛阳 471023;河南科技大学数学与统计学院，河南洛阳471023【正文语种】中文【中图分类】O241.81常微分方程（组）奇异边值问题是在多个科学领域经常出现的一类情况［1－3］。
最近，一些作者针对不同类型方程（组）的奇异边值问题提出了不同的数值方法［4－10］。
文献［5］给出了单个方程具有正则型奇异点边值问题的谱配置方法。
而文献［4］考虑如下方程组奇异边值问题的分段m次插值多项式逼近，其中，z为n维列向量，A（t）为已知n×n矩阵函数，f（t）为已知n维列向量，且可有正则型奇点。
然而，目前针对方程组奇异问题的Legendre谱配置法的相关文献鲜见。
另一方面，高阶方程可通过降阶方法化为一阶方程组求解。
所以，研究方程组的谱配置法是非常有意义的。
本文考虑上述模型问题的谱配置法，以期获得高精度的数值解，也为高阶方程的求解提供高精度数值方法。
1.1 一阶微分矩阵记LN（x）（－1≤x≤1）为N次Legendre多项式。
x0＝－1；xN＝1；xm（1≤m≤N－1）是N（x）＝0的根［11］。
以xi为节点的Lagrange插值基函数为：满足：构造Lagrange插值多项式，对pN（x）关于x求一阶导数，并令x＝xk，k＝0，1，…，N，得引理令D＝（dkj）（N＋1）×（N＋1），称D为一阶微分矩阵。
POSITIVE SOLUTION TO SINGULAR SEMIPOSITONEn,p BOUNDARY VALUE PROBLEM
∗ T his pap er wa s su ppo rt ed by t he im po rt a nt science a nd techno lo g y pro ject of Sh ando ng Pro vince (2 00 5 GG2 10 0 60 01 ) and the d oc to ra l fo unda t io n o f S hando ng Jia nzhu U niversity ( 4 24 11 1 ). † M a nuscrip t received No vem ber 1 3, 20 0 6
An n. of D iff. Eqs . 24:2(2008) , 233-238
POSITIVE SOLUTION TO SINGULAR SEMIPOSITONE ( n, p) BOUNDARY VALUE PROBLEM ∗ †
Zhang M ingchuan 1,2 , Yin Yanmin1
233
23N. OF DIFF. EQ S.
Vol.24
(H2 ) Let lim
u→ ∞
= ∞ b e u niform on a com pact sub in terval [α , β ] of (0,1).
Ma also estab lish ed th e following e xisten ce resu lt f or (1.2): Theorem A Ass ume that (H1 ) an d (H2 ) hold. T hen problem (1 . 2) has at least on e posi tive solu tion given t hat λ > 0 is small en ough. In th is p ap er, we gen eralize su ch p rob lem to a sin gular case, u nd er assump tion that q in (1.1) is allowe d to h ave a fin ite nu mb er of singu larities. Essen tially, ou r p ap er gen eralize s the relative th eorem of [12] an d [9]. The follow in g we ll-k nown fi xed p oint the orem is use d to prove our m ain re sult. Lemm a 1. 1[ 13] Let E be a Ban ach space, P ⊂ E be a con e. Assu me th at Ω 1 an d Ω 2 are boun ded open subsets of E wi th 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 . Fu rther, let T : P ∩ (Ω 2 \ Ω 1 ) → P be a co mpletely con tin uous operator su ch that either (i) � T u � ≤ � u � , u ∈ P ∩ ∂ Ω 1 , and � T u � ≥ � u � , u ∈ P ∩ ∂ Ω 2 ; or (ii) � T u� ≥ � u� , u ∈ P ∩ ∂ Ω 1 , and � T u � ≤ � u � , u ∈ P ∩ ∂ Ω 2 . Then T h as a fi xed po int in P ∩ (Ω 2 \ Ω 1 ) .
EXISTENCE AND MULTIPLE EXISTENCE OF POSITIVE SOLUTIONS TO SECOND-ORDER m-POINT BOUNDARY VALUE PR
αi x � (ξ i ) ,
x (1) =
i=1
m −2 X
β i x (ξ i ) ,
(1.4)
∗ T he wo rk w as sp onso red by the N a tu ra l Science Fo unda t io n of A nhui Ed ucat iona l Dep artm ent (KJ2 00 9 B1 00 ; KJ20 10 B1 6 3) . † M anuscript received July 29 , 20 08 ; Revised Ma rch 21 , 2 0 09
234
No.2
L. Yang, etc., BVP O N TIME SCA LE S
235
wh ere ξi ∈ [0 , ρ (1)]T , 0 < ξ1 < ξ2 < · · · < ξm − 2 < ρ (1) an d the followin g con ditions holds: (C 1 ) f ∈ C ([0 , + ∞ ) , [0 , +∞ )) an d th e f un ction a is le ft de nse con tin uou s on [0,1] and is R1 non ne gative with 0 a( s) � s existing as a strictly p ositive re al numb e r; m −2 m −2 P P (C 2 ) αi ≥ 0, β i ≥ 0 , i = 1 , 2 , · · · , m − 2 satisfy 0 < αi < 1 an d 0 < βi < 1.
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A bs tra ct
A pplying t op olo gical m et ho d, this pa per investig at es the exist ence o f po sit ive so lut io ns t o singu la r m -p oint b ound ary va lue prob lem s o f a co upled syste m o f diffe rent ia l equa tio ns. Key wo rds syst em o f differential equat io ns; po sit ive so lut ion; to po lo g ica l m etho d 20 00 M athem atic s Sub jec t C lassific ation 3 4 B1 5; 3 4B 25 ; 45 G1 5
L i Hongyu1 , Sun J ingxian 2 , Cui Yujun 1
( 1 . C o ll eg e o f I nform atio n S cience a nd E ngineering, Sh a nd on g U niversity of Sc ie nce a nd Tech no logy , Qingd a o 26 6 51 0 , Sh a ndo ng ; 2 . D ept . of M ath ., X uzho u N o rma l Un iv ersity , X uzho u 2 2 11 1 6, Jia ngsu )
1
Introduction and Preliminaries
Th is pap er e xp lore s the ex iste nc e of p ositive solution s to the followin g cou ple d sin gular system of se con d ord er ord inary differe ntial equ ation s � − L 1 u ( x ) = a( x ) f (v , u ) , 0 < x < 1 , � � � � � − L 2 v (x ) = b (x ) g ( u ) , 0 < x < 1 , � � � m −2 P (1 . 1) u (0) = 0 , u(1) = ai u( ξ i ) , � � � i = 1 � � m −2 P � � � v (0) = 0 , v (1) = bi u( ξ i ) ,
i=1
wh ere L i u = ( p i u ) u + qi u, f ∈ C [R × R , R + ], g ∈ C [ R + , R+ ]; 0 < ξ1 < ξ2 < · · · < ξm − 2 < 1; ai ∈ [0 , + ∞ ) , bi ∈ [0 , + ∞ ) , i = 1 , 2 , · · · , m − 2; a, b : (0 , 1) → [0 , + ∞ ) are con tinu ou s, an d prob ably sin gular at t = 0 or t = 1. Th e m -p oint bou n dary value proble ms of ordin ary differe ntial e quation s arose from differe nt are as of ap plied m ath em atic s an d p hysics. For e xamp le, many p roblem s c onc ernin g the th eory of elastic stab ility can b e han d le d by m -p oint p roble ms (se e [13]). In rec ent years, m -p oin t b ou nd ary value p roblem s to secon d orde r d iffe rential e qu ation s h ave b e en stu died ex te nsively (for ex ample [2-5] an d re fere nc es th erein ). Re cen tly, in [6,7,9-11], the authors c onside red th e two-p oint bou n dary value p roble ms for a sy ste m of sec ond ord er ord in ary differe ntial e quation s. [12] stud ied th e b oun dary value
∗ †
�
�
+
+
S uppo rt ed by t he N at iona l Na t ura l Scie nce Fo unda tion of C hin a ( 10 6 71 1 67 ). M a nuscrip t received No vemf D iff. Eqs . 24:2(2008) , 163-170
POSITIVE SOLUTIONS TO SINGULAR m-POINT BOUNDARY VALUE PROBLEMS OF A COUPLED SYSTEM OF DIFFERENTIAL EQUATIONS ∗ †
164
A NN. OF DIFF. EQ S.
Vol.24
prob le ms on a syste m of fourth -orde r ord inary differe ntial equ ations. Howeve r, to th e be st of our kn ow le dge , f ewer p ape rs stu die d the m- p oin t b ou nd ary valu e prob lems for system s of differe ntial equ ations. To b ridge this gap , we will give an ex iste nc e re sult for p ositive solu tions to m- point bou nd ary value p roblem (1.1) usin g top ological meth od. ( u, v ) ∈ C 2 [(0 , 1) , R + ] × C 2 [(0 , 1), R+ ] is said to b e a p ositive solu tion to (1.1) if and only if ( u, v ) satisfie s the bou nd ary value prob le m (1.1) and u( x ) > 0 , v ( x ) > 0 , for any x ∈ (0 , 1) . In th is pap er, we sup p ose th at (H1 ) p i ∈ C 1 [0 , 1], p i ( x ) > 0; qi ∈ C [0 , 1] , q i ( x ) ≤ 0 , i = 1 , 2 . Lemm a 1.1 [2] Su ppose t hat (H 1 ) i s satis fied. Let Φ 1 i ( x ) an d Φ 2 i ( x) be t he solu tion s to ( ( L i ϕ )( x ) = 0 , 0 < x < 1 , (1 .2 i ) ϕ (0) = 0 , ϕ(1) = 1 and ( ( L i ϕ )( x ) = 0 , ϕ (0) = 1 , 0 < x < 1, ϕ(1) = 0 , (1 .3 i )