Banach空間中含間斷項(xiàng)Sturm-Liouville問(wèn)題的解

陸海霞

(宿遷學(xué)院教師教育系,江蘇宿遷 223800)

Banach空間中含間斷項(xiàng)Sturm-Liouville問(wèn)題的解

陸海霞

(宿遷學(xué)院教師教育系,江蘇宿遷 223800)

利用擬上下解方法和混合單調(diào)迭代法,研究了Banach空間中含間斷項(xiàng)的非線(xiàn)性Sturm-liouville問(wèn)題解的存在唯一性,并給出逼近解迭代序列的誤差估計(jì).

擬上下解;混合單調(diào)迭代方法;Sturm-Liouville問(wèn)題

1 引言

E為Banach空間,考察E中非線(xiàn)性Sturm-Liouville問(wèn)題

解的存在性及迭代求法,其中Lu=(p(t)u′)′+q(t)u,θ是E中的零元,

這里fi(t,u):I×E→E(不假定連續(xù))(i=1,2).

Banach空間E中邊值問(wèn)題解的存在性引起了許多讀者的研究興趣[16].文獻(xiàn)[1-3]采用了拓?fù)涠确椒?文獻(xiàn)[5-6]應(yīng)用了錐理論和相關(guān)的不動(dòng)點(diǎn)理論,并對(duì)非線(xiàn)性項(xiàng)f增加了緊型條件,文獻(xiàn)[1-6]中都要求非線(xiàn)性項(xiàng)f連續(xù).本文利用擬上下解方法與混合單調(diào)迭代法,去掉f的連續(xù)性條件,在更廣泛的條件下,得到Banach空間中Sturm-Liouville問(wèn)題(1)解的存在唯一性,同時(shí)給出逼近解迭代序列及其誤差估計(jì).

2 預(yù)備知識(shí)

在C2[I,R]中只有零解.

設(shè)k(t,s)是相應(yīng)于方程(3)的Green函數(shù),即

引理2.1[8]假設(shè)(H1)滿(mǎn)足,那么由(4)定義的Green函數(shù)k(t,s)具有以下性質(zhì):

(i)k(t,s)在[0,1]×[0,1]上連續(xù);

(ii)u(t)∈C2[0,1]單調(diào)增加,u(t)＞0,t∈[0,1];

(iii)v(t)∈C2[0,1]單調(diào)減少,v(t)＞0,t∈[0,1];

(iv)ω為正常數(shù).

引理2.2假設(shè)u∈C2[I,E]滿(mǎn)足:

則u(t)≥θ,?t∈I.

證明任取泛函φ∈P?(P的共軛錐),令y(t)=φ(x(t)),則y∈C2[I,R]滿(mǎn)足:

由文獻(xiàn)[9]知y(t)≥0,?t∈I.再由φ的任意性可知u(t)≥θ,?t∈I.

定義2.1若函數(shù)對(duì)v0,w0∈C1[I,E]滿(mǎn)足條件:

則稱(chēng)v0,w0為Sturm-Liouville問(wèn)題(1)的擬下、擬上解對(duì).

若上定義中四個(gè)不等號(hào)均取等號(hào),則稱(chēng)v0,w0為Sturm-Liouville問(wèn)題(1)的擬解對(duì).

定義2.2設(shè)D?E.稱(chēng)算子A:D×D→E是混合單調(diào)的,若對(duì)任給ui,vi∈D(i=1,2), u1≤u2,v2≤v1,都有A(u1,v1)≤A(u2,v2).

3 主要結(jié)果

為了方便起見(jiàn),列出以下假設(shè):

(H2)由fi(t,u)確定的抽象算子Fiu=fi(t,u)(i=1,2)把u∈C[I,E]映為強(qiáng)可測(cè)函數(shù).

(H3)存在v0,w0是問(wèn)題(1)的擬下,擬上解對(duì),并且v0(t)≤w0(t),?t∈I.

(H4)f具有分解式(2),且Fiv0=fi(t,v0),Fiw0=fi(t,w0)∈L1[I,E].

(H5)存在常數(shù)M1,M2＞0,對(duì)一切

4 例子

例4.1考察二階非線(xiàn)性?xún)牲c(diǎn)邊值問(wèn)題:

-u′′=f(t,u),t∈[0,1], (16) u(0)=u(1)=θ,

其中非線(xiàn)性項(xiàng)f滿(mǎn)足定理1條件.

由于p(t)≡1,a=c=1,b=d=0,經(jīng)過(guò)計(jì)算可得相應(yīng)齊次邊值問(wèn)題的Green函數(shù)為:

當(dāng)非線(xiàn)性項(xiàng)f連續(xù)時(shí),問(wèn)題(16)已被許多學(xué)者采用拓?fù)涠燃跋嚓P(guān)的不動(dòng)點(diǎn)方法作過(guò)研究[13].該問(wèn)題通?；癁榈葍r(jià)的Banach空間中的積分方程

來(lái)處理,其中k(t,s)為相應(yīng)的Green函數(shù).由于在一般的Banach空間中,上述積分方程中相應(yīng)的積分算子A不再具有緊性,為對(duì)算子A應(yīng)用凝聚映象的拓?fù)涠燃跋嚓P(guān)的不動(dòng)點(diǎn)方法,需對(duì)非線(xiàn)性項(xiàng)增加“緊型條件”(如文獻(xiàn)[6]).當(dāng)非線(xiàn)性項(xiàng)f不連續(xù),上述文獻(xiàn)中的結(jié)論不再適用,而由本文結(jié)論可知,將(17)式定義的Green函數(shù)直接代入定理1結(jié)論即可得到問(wèn)題(16)的迭代解.

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Solutions of Sturm-Liouville problems with discontinuous terms in Banach spaces

Lu Haixia
(Department of Teachers Education,Suqian College,Suqian223800,China)

By using the method of quasi-upper and lower solutions and the mixed monotone iterative technique, the existence and unique solution of nonlinear Sturm-Liouville problems with discontinuous terms in Banach spaces are obtained.The error estimate of the iterative sequences of approximation solutions is given.

quasi-upper and lower solutions,mixed monotone iterative technique,Sturm-Liouville problems 2010 MSC:47H10,34B15

O175

A

1008-5513(2013)02-0125-07

10.3969/j.issn.1008-5513.2013.02.003

2011-08-05.

國(guó)家自然科學(xué)基金(10971179);宿遷學(xué)院科研基金(2011KY10).

陸海霞(1976-),碩士,講師,研究方向：非線(xiàn)性泛函分析.