時間尺度上三階非線性p-Laplacian邊值問題三個正解的存在性

王 穎

(安徽科技學院 信息與網絡工程學院 數學系，安徽 滁州 233100)

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時間尺度上三階非線性p-Laplacian邊值問題三個正解的存在性

王 穎

(安徽科技學院 信息與網絡工程學院 數學系，安徽 滁州 233100)

研究了時標上一個新的三階非線性p-Laplacian 三點邊值問題，利用廣義Leggett-Williams不動點定理得出該邊值問題至少有三個正解存在的結論，并給出相應的例子，其結果推廣了已有的研究.

時間尺度；p-Laplacian；邊值問題；不動點定理

1900年Stefan Hilger 發表了時間標度分析[1]，這一理論在連續與離散之間建立了一座橋梁，使人們可以同時處理連續系統和離散系統，并且有重要的應用價值.這一新興課題引起數學家們的廣泛關注，許多學者都投入到時標上微分方程的研究特別是邊值問題正解的研究中來，如文獻[2-8]及其參考文獻.隨著時標理論自身的迅速發展和不斷完善，如何利用非線性泛函分析中先進的分析工具來研究時標上含p-Laplacian的邊值問題解的存在性引起許多數學工作者的濃厚興趣，見[9-15]及其參考文獻.其中大部分研究的都是二階含p-Laplacian的邊值問題，三階及以上的方程研究的少一些，因此還有許多問題可以深入研究.

在[14]中何智敏等人研究了時標上二階p-Laplacian邊值問題

[φp(u△(t))]▽+g(t)f(u(t))=0,t∈(0,T)，

u(0)-B0(u△(0))=0,u△(0)=0或u△(0)=0,u(T)+B1(u△(T))=0，

得出至少有三個正解存在的準則.

[15]中，Hu等人研究了時標上三階三點邊值問題

[u△△(t)]▽+ω(t)f(t,u(t))=0,t∈[a,b]，

u(ρ(a))-βu△(ρ(a))=αu(η),γ(u(η))=u(b),u△△(ρ(a))=0.

該文利用錐壓縮拉伸不動點定理得到上述邊值問題正解的存在性的若干判定準則.

受上述研究的啟發，本文研究了時標上三階非線性p-Laplacian三點邊值問題

[φp(a(t)u△▽(t))]▽+ω(t),f(t,u(t))=0,t∈(0,T)∈T，

(1)

βu(0)-γu△(0)=b,u△(T)=αu(η),u△▽(0)=0,

(2)

假設以下條件在文中始終成立：

(H1) f(t,u(t)):[0,T]×R→R+是連續的；

(H2) a(t)∈C([0,T],(0,+∞))為單調遞增函數；

(H3) ω(t):[0,T]→[0,+∞)是ld-連續的，且ω(t)在[0,T]上的任意子集上不恒為零.

1 預備知識

關于時間尺度的一些基本定義和定理可見Atici,Guseinov[16]及Bohner,Peterson[17]，本節只給出主要結論要用到的常見引理和不動點定理.

引理1 如d≠0，h(t)∈Cld(T,R)則動力方程邊值問題

[φp(a(t)u△▽(t))]▽+h(t)=0,t∈(0,T),

(3)

βu(0)-γu△(0)=b,u△(T)=αu(η),u△▽(0)=0，

(4)

有唯一解

(5)

(6)

(7)

令t=T,η代入式(6)(7)有

(8)

(9)

將(8)，(9)式代入(4)式，有

將上式代入式(7)可得式(5).證畢.

證畢.

P={u∈E|u在[0,T]中是凹的，遞增且非負}.

設γ是定義在實Banach空間中的錐P的非負連續函數，對任一的c>0，集合

P(γ,c)={u∈P|γ(u)

γ(x)≤β(x)≤α(x)且‖x‖≤Mγ(x).

假設存在全連續算子T:P(γ,c)→P及正數a,b,0

(i)γ(Tx)b,x∈?P(β,b)；

(iii)P(α,a)≠?，α(Tx)

0≤α(x1)

2 主要結果

定義算子A:P→E使

(10)

易知邊值問題(1)(2)有解u(t)當且僅當u是算子方程(10)的不動點.

給定一內點l∈T,且η

則邊值問題(1)(2)至少有三個正解u1(t),u2(t),u3(t)滿足

0<φ(u1)

引理4中的條件(i)成立.

接著證明引理4中的條件(ii)成立.選取u∈?P(θ,b′)，則

由(H5),當t∈[η,T]有

從而引理4中的條件(ii)成立.

最后驗證引理4中的條件(iii)成立.當t∈[0,T]時，有

據條件(H6)可得，當t∈[0,T]時有

則引理4的條件(iii)成立.

0≤φ(u1)

即邊值問題(1)(2)至少存在三個正解.證畢.

3 應用舉例

[φp(a(t)u△▽(t)]▽+ω(t)f(t,u(t))=0,t∈(0,1)，

(11)

βu(0)-γu△(0)=b,u△(T)=αu(η),u△▽(0)=0，

(12)

據定理1，邊值問題(11)，(12)至少存在三個正解u1(t),u2(t),u3(t)滿足

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The Existence of Triple Positive Solutions for Third-Order Nonlinear p-Laplacian Boundary Value Problems on Time Scales

WANG Ying

(Department of Mathematics of College of Information and Network Engineering, University of Science and Technology, Chuzhou 233100, China)

A new third-order nonlinear p-Laplacian three-point boundary value problem on time scales is studied. By the Generalized Leggett-Williams fixed point theorem in cones, a new result of at least three positive solutions of the boundary value problem is obtained. As an application, an example is given to demonstrate the main result. The result generalized the past research.

time scales; p-Laplacian; boundary value problem; fixed point theorem

2015-09-23

安徽省高校自然科學研究重點項目(KJ2016A174)；安徽科技學院自然科學研究一般項目(ZRC2014441).

王穎(1975-)，女，江蘇徐州人，副教授，碩士，研究方向為泛函微分方程.

10.14182/J.cnki.1001-2443.2016.05.002

O175.8

A

1001-2443(2016)05-0414-06