白 婧, 李永祥(西北師范大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 甘肅 蘭州 730070)

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含一階導(dǎo)數(shù)項(xiàng)的三階周期邊值問題解的存在唯一性

白 婧, 李永祥*
(西北師范大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 甘肅 蘭州 730070)

三階常微分方程的周期邊值問題一直是常微分方程研究的熱點(diǎn).研究非線性項(xiàng)含一階導(dǎo)數(shù)項(xiàng)的三階周期邊值問題

三階周期邊值問題; 存在性與唯一性; Leray-Schauder不動(dòng)點(diǎn)定理

本文利用Leray-Schauder不動(dòng)點(diǎn)定理,討論了三階周期邊值問題

(1)

解的存在唯一性.其中I=[0,ω],f:I×R2→R連續(xù).

三階常微分方程周期邊值問題是微分方程中人們關(guān)注的問題,在這方面已有很多研究成果,見文獻(xiàn)[1-14].A. Cabada[1]用上下解方法獲得了三階周期邊值問題

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為Carathéodory函數(shù),a、b、λi∈R.L. Kong[2]等利用Schauder不動(dòng)點(diǎn)定理考慮了三階周期邊值問題

f滿足下列條件:

(A1)f(t,u)為[0,2π]×(0,+∞)上的非負(fù)函數(shù),并且f(t,u)在[0,2π]上可積;

(A2) 對(duì)?t∈[0,2π],u>0,f(t,u)不增且滿足

u?(t)+αu″(t)+βu′(t)=

f(t,u(t)),t∈[0,2π],

(4)

其中α、β為正常數(shù),在滿足特定的條件下,利用錐上的Krasnoselskii不動(dòng)點(diǎn)定理,得出方程(4)正周期解的存在性.

在上述文獻(xiàn)中非線性均不含導(dǎo)數(shù)項(xiàng),本文在非線性含有一階導(dǎo)數(shù)項(xiàng)的情形下,利用Leray-Schauder不動(dòng)點(diǎn)定理得到方程(1)解的存在唯一性.

1 預(yù)備工作

對(duì)?h∈L2(I),a0,a1≠0,考慮三階線性周期邊值問題

在文獻(xiàn)[8]中有:

引理 1 假設(shè)如下條件成立

|k∈Z}=?,

則線性周期邊值問題(5)有唯一解u:=Th∈W3,2(I),且解算子T:L2(I)→W3,2(I)為線性全連續(xù)算子.

又由嵌入W3,2(I)W1,2(I)的緊性,則T:L2(I)→W1,2(I)為線性全連續(xù)算子.

W1,2(I)中按范數(shù)

‖

構(gòu)成Banach空間,為方便起見,用X表示該Banach空間.

引理 2 設(shè)a0,a1≠0,則線性周期邊值問題(5)的解算子T:L2(I)→X的范數(shù)滿足

‖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)

也可展為Fourier級(jí)數(shù)

(7)

所以

u?

從而

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不動(dòng)點(diǎn)定理) 設(shè)F:X→X全連續(xù),若方程簇

λF(x)=x, 0<λ<1

的解集在X中有界,則F在X中有不動(dòng)點(diǎn).

2 主要結(jié)論

定理 1 假設(shè)下列條件成立:

(H1) 存在常數(shù)α0、α1≥0及C>0,使得

(H2)M(α0+α1)<1;

則周期邊值問題(1)至少存在一個(gè)解.

證明 令

F(u)(t)=f(t,u(t),u′(t))+

a1u′(t)+a0u(t),

(11)

則F:X→L2(I)連續(xù),把有界集映為有界集.因此,復(fù)合算子A=T°F:X→X為全連續(xù)算子.根據(jù)算子T的定義,三階周期邊值問題(1)的解等價(jià)于算子A的不動(dòng)點(diǎn).對(duì)A應(yīng)用Leray-Schauder不動(dòng)點(diǎn)定理,考慮方程簇

u=λAu, 0<λ<1,

(12)

要證同倫簇方程的解集在X中有界.

設(shè)u∈X為方程簇(12)中某個(gè)方程的解,則

u=T(λF(u))

為

h=λF(u)

相應(yīng)的線性方程(5)的解.由假設(shè)條件(H1)得如下的范數(shù)估計(jì)

(13)

所以

‖u‖X=‖Th‖X≤‖T‖L(L2(I),X)‖h‖2≤

M‖h‖2

(14)

故

(15)

因此,同倫簇方程(12)的解集在X中有界.由引理3中Leray-Schauder不動(dòng)點(diǎn)定理知,算子A在X中存在不動(dòng)點(diǎn)u,u為三階周期邊值問題(1)的解.

下面討論三階周期邊值問題(1)解的存在唯一性.為此,需加強(qiáng)條件(H1)為(H1)′存在常數(shù)α0、α1≥0,使得

(16)

定理 2 假設(shè)條件(H1)′、(H2)成立,則三階周期邊值問題(1)存在唯一解.

證明 由定理1知三階周期邊值問題(1)至少存在一個(gè)解.設(shè)u1、u2為三階周期邊值問題(1)的2個(gè)解.令

u=u2-u1,

則

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

為

h=F(u2)-F(u1)

對(duì)應(yīng)的線性周期邊值問題(5)的解.因此,由引理1及假設(shè)條件(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(xiàn)(u2)-F(u1)‖2≤

M(α0+α1)‖u2-u1‖X,

(18)

從而

‖u2-u1‖X=0,

即

u2=u1.

所以周期邊值問題(1)存在唯一解.

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2010 MSC:34A12

(編輯 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

國家自然科學(xué)基金(11261053)和甘肅省自然科學(xué)基金(1208R-JZA129)

O175.8

A

1001-8395(2015)06-0834-04

10.3969/j.issn.1001-8395.2015.06.008

*通信作者簡(jiǎn)介:李永祥(1963—),男,教授,主要從事非線性泛函分析的研究,E-mail:liyx@nwnu.edu.cn

其中,f:[0,ω]×R2→R連續(xù),通過Fourier分析的方法及非線性項(xiàng)的擾動(dòng)技巧,利用Leray-Schauder不動(dòng)點(diǎn)定理得出解的存在性與唯一性.之前的許多研究主要集中在非線性項(xiàng)不含導(dǎo)數(shù)項(xiàng)的情形,對(duì)之前所得結(jié)論進(jìn)行了推廣.