含一阶导数项的三阶周期边值问题解的存在唯一性

白 婧, 李永祥(西北师范大学 数学与统计学院, 甘肃 兰州 730070)



含一阶导数项的三阶周期边值问题解的存在唯一性

白 婧, 李永祥*
(西北师范大学 数学与统计学院, 甘肃 兰州 730070)

三阶常微分方程的周期边值问题一直是常微分方程研究的热点.研究非线性项含一阶导数项的三阶周期边值问题

三阶周期边值问题; 存在性与唯一性; Leray-Schauder不动点定理

本文利用Leray-Schauder不动点定理,讨论了三阶周期边值问题

(1)

解的存在唯一性.其中I=[0,ω],f:I×R2→R连续.

三阶常微分方程周期边值问题是微分方程中人们关注的问题,在这方面已有很多研究成果,见文献[1-14].A. Cabada[1]用上下解方法获得了三阶周期边值问题

u‴(t)=f(t,u(t)),a≤t≤b,

u(i)(a)-u(i)(b)=λi,i=0,1,2

(2)

解的存在性.其中f为Carathéodory函数,a、b、λi∈R.L. Kong[2]等利用Schauder不动点定理考虑了三阶周期边值问题

f满足下列条件:

(A1)f(t,u)为[0,2π]×(0,+∞)上的非负函数,并且f(t,u)在[0,2π]上可积;

(A2) 对∀t∈[0,2π],u>0,f(t,u)不增且满足

u‴(t)+αu″(t)+βu′(t)=

f(t,u(t)),t∈[0,2π],

(4)

其中α、β为正常数,在满足特定的条件下,利用锥上的Krasnoselskii不动点定理,得出方程(4)正周期解的存在性.

在上述文献中非线性均不含导数项,本文在非线性含有一阶导数项的情形下,利用Leray-Schauder不动点定理得到方程(1)解的存在唯一性.

1 预备工作

对∀h∈L2(I),a0,a1≠0,考虑三阶线性周期边值问题

在文献[8]中有:

引理 1 假设如下条件成立

|k∈Z}=Ø,

则线性周期边值问题(5)有唯一解u:=Th∈W3,2(I),且解算子T:L2(I)→W3,2(I)为线性全连续算子.

又由嵌入W3,2(I)W1,2(I)的紧性,则T:L2(I)→W1,2(I)为线性全连续算子.

W1,2(I)中按范数

‖

构成Banach空间,为方便起见,用X表示该Banach空间.

引理 2 设a0,a1≠0,则线性周期边值问题(5)的解算子T:L2(I)→X的范数满足

‖T‖L(L2(I),X)≤M,

其中

(6)

其中

且有Parseval等式

成立.因此

u=Th∈W3,2(I)

也可展为Fourier级数

(7)

所以

u‴

从而

u‴(t)+a1u′(t)+a0u(t)=

(8)

由Fourier展式的唯一性

(9)

所以

(10)

因此,‖T‖L(L2(I),X)≤M.

引理 3[9](Leray-Schauder不动点定理) 设F:X→X全连续,若方程簇

λF(x)=x, 0<λ<1

的解集在X中有界,则F在X中有不动点.

2 主要结论

定理 1 假设下列条件成立:

(H1) 存在常数α0、α1≥0及C>0,使得

(H2)M(α0+α1)<1;

则周期边值问题(1)至少存在一个解.

证明 令

F(u)(t)=f(t,u(t),u′(t))+

a1u′(t)+a0u(t),

(11)

则F:X→L2(I)连续,把有界集映为有界集.因此,复合算子A=T∘F:X→X为全连续算子.根据算子T的定义,三阶周期边值问题(1)的解等价于算子A的不动点.对A应用Leray-Schauder不动点定理,考虑方程簇

u=λAu, 0<λ<1,

(12)

要证同伦簇方程的解集在X中有界.

设u∈X为方程簇(12)中某个方程的解,则

u=T(λF(u))

为

h=λF(u)

相应的线性方程(5)的解.由假设条件(H1)得如下的范数估计

(13)

所以

‖u‖X=‖Th‖X≤‖T‖L(L2(I),X)‖h‖2≤

M‖h‖2

(14)

故

(15)

因此,同伦簇方程(12)的解集在X中有界.由引理3中Leray-Schauder不动点定理知,算子A在X中存在不动点u,u为三阶周期边值问题(1)的解.

下面讨论三阶周期边值问题(1)解的存在唯一性.为此,需加强条件(H1)为(H1)′存在常数α0、α1≥0,使得

(16)

定理 2 假设条件(H1)′、(H2)成立,则三阶周期边值问题(1)存在唯一解.

证明 由定理1知三阶周期边值问题(1)至少存在一个解.设u1、u2为三阶周期边值问题(1)的2个解.令

u=u2-u1,

则

u=T(F(u2)-F(u1))

为

h=F(u2)-F(u1)

对应的线性周期边值问题(5)的解.因此,由引理1及假设条件(H1)′可得

(α0+α1)‖u‖X.

(17)

所以

‖u2-u1‖X=‖T(F(u2)-F(u1))‖X≤

‖T‖L(L2(I),X)‖F(u2)-F(u1)‖2≤

M(α0+α1)‖u2-u1‖X,

(18)

从而

‖u2-u1‖X=0,

即

u2=u1.

所以周期边值问题(1)存在唯一解.

[1] Cabada A. The method of lower and upper solutions for second, third, fouth and higher order boundary value problems[J]. J Math Anal Appl,1994,185:302-320.

[2] Kong L, Wang S, Wang J. Positive solution of a singular nonlinear third-order periodic boundary value problem [J]. J Comput Appl Math,2001,132:247-253.

[3] Sun J, Liu Y. Multiple positive solutions of sigular third-order periodic boundary value problem [J]. Acta Math Sci,2005,25:81-88.

[4] Feng Y. On the exixtence and multiplicity of positive periodic solutions of a nonlinear third-order equation[J]. Appl Math lett,2009,22:1220-1224.

[5] Chu J, Zhou Z. Positive solutions for singular non-linear third order periodic boundary value problems[J]. Nonlinear Anal,2006,64:1528-1542.

[6] Li Y. Positive periodic solutions for fully third-order ordinary differential equations [J]. Comput Appl Math,2010,59:3464-3471.

[7] 姚庆六. 变系数三阶周期边值问题的正解[J]. 吉首大学学报:自然科学版,2010,31(6):9-13.

[8] Li Y. Existence and uniqueness for higher order periodic boundary value problem under spetral separarion conditions[J]. J Math Anal Appl,2006,322:530-539.

[9] Ren J L, Cheng Z B, Chen Y L. Existence results of periodic solutions for third order nonlinear sigular differential equation [J]. Math Nach,2013,286:1022-1042.

[10] Cheng Z B. Exixtence of positive periodic solutions for third-order differential equation with strong singularity[J]. Adv Difference Equ,2014,162:1-12.

[11] Cheng Z B, Ren J L. Existence of positive periodic solution for variable-coefficient third order differential equation with singularity[J]. Math Meth Appl Sci,2014,37:2281-2289.

[12] 李小龙. Banach空间非线性三阶周期边值问题的正解[J]. 应用数学,2013,26(3):652-657.

[13] Wang Y, Lian H, Ge W. Periodic solutions for a second order nonlinear funtional differential equation[J]. Appl Math Lett,2007,20:110-115.

[14] Li Y H, Zhang X Y. Multiple positive solutions of boundary value problems for system of nonlinear third-order differential equations[J]. J Math Res Appl,2013,3:321-329.

[15] Chen L, Hu L G, Ma X D. Positive solutions to singular eigenvale problems for third-order differential equations[J]. Acta Anal Funct Appl,2012,14:377-387.

[16] Deimling K. Nonlinear Functional Analysis[M]. Berlin:Springer-Verlag,1985.

2010 MSC:34A12

(编辑 余 毅)

Existence and Uniqueness Result for Third-order Periodic Boundary Value Problem with the First Derivatives

BAI Jing, LI Yongxiang
(CollegeofMathematicsandStatistics,NorthwestNormalUniversity,Lanzhou730070,Gansu)

Third-order periodic boundary value problem has always been the hot of the ordinary differential equation. In this paper, we discuss the solvability of the third-order periodic boundary value problem with nonlinear termu′,u‴(t)=f(t,u(t),u′(t)),0≤t≤ω,u(i)(0)=u(i)(ω),i=0,1,2, wheref:[0,ω]× R2→ R is a continuous function. Using the method of Fourier analysis and the perturbation technique of nonlinear term, the existence and uniqueness of solution are obtained by Leray-Schauder fixed point theorem. Since many former studies have focused on nonlinear term without derivative, our results generalize the previous conclusions.

third order periodic boundary value problem; existence and uniqueness; Leray-Schauder fixed point theorem

2015-03-30

国家自然科学基金(11261053)和甘肃省自然科学基金(1208R-JZA129)

O175.8

A

1001-8395(2015)06-0834-04

10.3969/j.issn.1001-8395.2015.06.008

*通信作者简介:李永祥(1963—),男,教授,主要从事非线性泛函分析的研究,E-mail:liyx@nwnu.edu.cn

其中,f:[0,ω]×R2→R连续,通过Fourier分析的方法及非线性项的扰动技巧,利用Leray-Schauder不动点定理得出解的存在性与唯一性.之前的许多研究主要集中在非线性项不含导数项的情形,对之前所得结论进行了推广.