劉三輝

(湖南汽車工程職業(yè)技術(shù)學(xué)院基礎(chǔ)課部，中國(guó) 株洲 412000)

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非線性m點(diǎn)邊值問(wèn)題的多重正解

劉三輝

(湖南汽車工程職業(yè)技術(shù)學(xué)院基礎(chǔ)課部，中國(guó) 株洲 412000)

多點(diǎn)邊值問(wèn)題的正解的存在是常微分穩(wěn)定性理論研究的一個(gè)重要問(wèn)題，引起很多學(xué)者的關(guān)注.本文運(yùn)用Krasnoselskii不動(dòng)定理論與Leggett-Williams不動(dòng)點(diǎn)理論研究二階m-點(diǎn)的邊值問(wèn)題

得到多重正解存在的一些充分條件.

正解；不動(dòng)點(diǎn)理論；m-點(diǎn)邊值問(wèn)題

本文考慮二階m-點(diǎn)邊值問(wèn)題

(1)

其中ξi∈(0,1),αi∈(0,∞)(i=1,…,m-2)是常數(shù),h,f,a和b滿足下面條件:

(H1) f∈C([0,∞),[0,∞));

(H2) h∈C([0,1],[0,∞))和存在x0∈(0,1)使得h(x0)>0;

(H3) α∈C([0,1],b∈C([0,1],(-∞,0)).

問(wèn)題1 如果f0=f∞=0或f0=f∞=∞,能否得到類似的結(jié)論或證明沒(méi)有類似的結(jié)論.

問(wèn)題2 如果f0=f∞?{0,+∞},能否得到有類似的結(jié)論或證明沒(méi)有類似的結(jié)論.

本文利用著名的Krasnoselskii不動(dòng)點(diǎn)理論[3，8-9]與Leggett-Williams不動(dòng)點(diǎn)理論[10-14]，確定了方程(1)多重正解的存在性，比文獻(xiàn)[2～3]中結(jié)論更一般化.

1 一些引理

本節(jié)首先介紹一些相關(guān)的引理.

引理1[3]假設(shè)(H3)成立.設(shè)φ1和φ2分別是下面兩問(wèn)題

(2)

(3)

引理2[3]假設(shè)(H3)成立.則問(wèn)題(2)和(3)分別有唯一的解.

引理3[3]假設(shè)(H3)和(H4)成立.y∈C[0,1].則下面問(wèn)題

(4)

(5)

(6)

且在區(qū)間[0，1]上u(t)≥0,則y≥0.

G(s,s)≥G(t,s)≥q(t)G(s,s)≥0, t∈[0,1].

取δ∈(0,1/2),則x0∈(δ,1-δ).令γ=min{q(t)|t∈[δ,1-δ]}.當(dāng)0<γ<1時(shí)，由引理1可得如下引理.

引理4[3]設(shè)(H3)和(H4)成立，設(shè)y∈C[0,1]和y≥0,則問(wèn)題(4)解u唯一且滿足

u(t)≥γ‖u‖,t∈[δ,1-δ].

2 一個(gè)或兩個(gè)正確的存在性

本節(jié)將確定問(wèn)題(1)有一個(gè)或兩個(gè)正解的存在性.

設(shè)E=C[0，1].則E是一個(gè)范數(shù)定義如下的Banach空間

現(xiàn)在(1)有一個(gè)解u=u(t)當(dāng)且僅當(dāng)u是下列算子方程的一個(gè)解

(7)

在E中p和G分別按(5)和(6)定義容易驗(yàn)證T是全連續(xù)的.在E中定義錐如下

(8)

由錐的定義及T的運(yùn)算，可得下面引理.

引理5 T(P)?P.

(i)‖Au‖≤‖u‖,u∈K∩?Ω1和‖Au‖≥‖u‖,u∈K∩?Ω2;或

(ii)‖Au‖≥‖u‖,u∈K∩?Ω1和‖Au‖≤‖u‖,u∈K∩?Ω2.

為簡(jiǎn)便，記

2.1 當(dāng)f0=f∞=∞或f0=f∞=0時(shí)，(1)結(jié)論的存在性

定理7 設(shè)下列假設(shè)滿足：

則(1)至少有兩個(gè)解u1和u2且0<‖u1‖<ρ1<‖u2‖.

證 首先，因f0=∞,取ρ0∈(0,ρ1)，使得對(duì)0

因此

‖Tu‖≥‖u‖，u∈P∩?Ωρ0.

(9)

故

(10)

最后，設(shè)Ωρ1={u∈C[0,1]:‖u‖<ρ1}.對(duì)u∈P且‖u‖=ρ1,有

這意味

‖Tu‖≤‖u‖，u∈P∩?Ωρ1.

(11)

定理8 設(shè)下列條件滿足：

則(1)至少有兩個(gè)正解u1和u2且0<‖u1‖<ρ2<‖u2‖.

證 因f0=0,取0<ρ<ρ2使得對(duì)00滿足εA1≤1.因此對(duì)u∈P和‖u‖=ρ,得到

設(shè)Ωρ={u∈C[0,1]:‖u‖<ρ},因此

‖Tu‖≤‖u‖,u∈P∩?Ωρ.

(12)

因?yàn)閒∞=0,故存在r0>ρ2，使得對(duì)u≥r0,有f(u)≤ε1u，其中ε1>0滿足ε1A1≤1.以下考慮兩種情況.

情況(i)：若f無(wú)界，則存在ρ*>r0,對(duì)0

情況(ii).如果f是有界，對(duì)所有u∈[0,∞),即f(u)≤N.取ρ*≥max{2ρ2,NA1},對(duì)u∈P且‖u‖=ρ*,有

NA1≤ρ*=‖u‖.

所以對(duì)u∈P∩?Ωρ2,‖Tu‖≥‖u‖.

2.2 當(dāng)f0,f∞?{0,∞}時(shí)，(1)結(jié)論的存在性

現(xiàn)在討論在f0,f∞?{0,∞}下條件(1)正解的存在性.

定理9 設(shè)兩個(gè)正常數(shù)ρ1≠ρ2.若

則(1)至少有一正解u且‖u‖介于ρ1和ρ2之間.

證 不失一般性，設(shè)不妨設(shè)ρ1<ρ2.設(shè)Ωρ1={u∈C[0,1]:‖u‖<ρ1},當(dāng)u∈P和‖u‖=ρ1,有

則‖Tu‖≤‖u‖, u∈P∩?Ωρ1.

又因?yàn)椤琓u‖≥‖u‖, u∈P∩?Ωρ2.由定理6，結(jié)論成立.

易證下列推論成立.

推論10 設(shè)下列條件成立.

則(1)至少有一正解.

推論11 設(shè)下列條件成立.

則(1)至少有一正解.

推論12 設(shè)條件(C1)，(C4)成立且(C5)成立.則(1)至少有兩正解u1和u2滿足0<‖u1‖<ρ1<‖u2‖.

推論13 設(shè)前面的條件(C2),(C3)成立且(C6)成立.則(1)至少有兩正解u1和u2滿足0<‖u1‖<ρ2<‖u2‖.

3 3個(gè)正解的存在性

設(shè)E是實(shí)Banach空間且有錐P.映射β:P→[0,+∞)是P上非負(fù)連續(xù)凹函數(shù)，β連續(xù)且對(duì)所有x,y∈P和t∈[0,1].有β(tx+(1-t)y)≥tβ(x)+(1-t)β(y).設(shè)a,b是兩個(gè)數(shù)且0

Pa={x∈P:‖x‖

(i) {x∈P(β,a,b):β(x)>a}≠?和β(Tx)>aforx∈P(β,a,b),

(ii) ‖Tx‖

(iii)β(Tx)>aforx∈P(β,a,c)with‖Tx‖>b.

‖x1‖

下面確定(1)存在3個(gè)正解的條件.

定理15 設(shè)條件(H1)～(H4)成立且存在數(shù)a和d，0

其中

若下面條件成立

則邊值問(wèn)題(1)至少有3個(gè)正解.

證 設(shè)P定義如(8)和T定義如(7).當(dāng)u∈P,設(shè)

則容易驗(yàn)證β是P上非連續(xù)凹函數(shù)且β(x)≤‖x‖當(dāng)x∈P和T:P→P全連續(xù).

由(21)得

最后證明，若u∈P(β,a,c)和‖Tu‖>a/γ,則β(Tu)>a.由G(t,s)≥q(t)G(s,s)和φ1(t)≥q(t)得

另一方面

綜上所述，取b=a/γ，定理4.1的條件都滿足.因此，T至少有3個(gè)不動(dòng)點(diǎn)，即(1)至少有3個(gè)正解u1,u2和u3且‖u1‖d且α(u3)

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(編輯 沈小玲)

Multiple Positive Solutions to a Nonlinearm-Point Boundary Value Problem

LIUSan-hui*

(Department of Basic Courses, Hunan Automative Engineering Vocational College, Zhuzhou 412000, China)

It is an important problem to ordinary dierential stability theory about the existence criteria for the positive solutions of multi-point boundary value problem, which is studied by many researchers. The second-orderm-point boundary value problem

is studied by using Krasnoselskii fixed point theorem and Leggett-Williams fixed-point theorem and the existence criteria for multiple positive solutions is obtained.

positive solution; fixed point theorem;m-point boundary value problem

2013-03-26

O175.8

A

1000-2537(2014)03-0080-07

*通訊作者,E-mail：465023361@qq.com