帶p-Laplacian算子的分?jǐn)?shù)階微分方程多點(diǎn)邊值問(wèn)題的解的存在性

呂秋燕,劉文斌,唐　敏*,申騰飛,程玲玲

(1.蘇州市吳中區(qū)東山中學(xué)，中國(guó) 蘇州　215107；2.中國(guó)礦業(yè)大學(xué)理學(xué)院，中國(guó) 徐州　221116)

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帶p-Laplacian算子的分?jǐn)?shù)階微分方程多點(diǎn)邊值問(wèn)題的解的存在性

呂秋燕1,劉文斌2,唐敏2*,申騰飛2,程玲玲2

(1.蘇州市吳中區(qū)東山中學(xué)，中國(guó) 蘇州215107；2.中國(guó)礦業(yè)大學(xué)理學(xué)院，中國(guó) 徐州221116)

摘要利用不動(dòng)點(diǎn)定理,研究帶有p-Laplacian算子的分?jǐn)?shù)階微分方程多點(diǎn)邊值問(wèn)題解的存在性,得到邊值問(wèn)題至少存在一個(gè)解的充分條件.

關(guān)鍵詞分?jǐn)?shù)階微分方程;p-Laplacian算子;存在性;不動(dòng)點(diǎn)定理

Exitence of Solutions for Fractions Multi-point Boundary Value Problem withp-Laplacian Operator

LVQiu-yan1,LIUWen-bin2*,TANGMin2,SHENTeng-fei2,CHENGLing-ling2

(1.Dongshan High School, Suzhou 215107, China;2.College of Science, China University of Mining and Technology, Xuzhou 221116, China)

AbstractThis paper presents a study on the existence of solutions for the fractional multi-point boundary value problem withp-Laplacian operator. Making use of the fixed-point theorem, we obtained sufficient conditions to guarantee the existence of at least one solution for the boundary value problem.

Key wordsfractional differential equation;p-Laplacian operator; existence; fixed point theorem

近年來(lái),分?jǐn)?shù)階微分方程被廣泛應(yīng)用于物理學(xué)、生物學(xué)、控制論等諸多領(lǐng)域[1-3],因此,分?jǐn)?shù)階微分方程受到許多學(xué)者的廣泛關(guān)注，并取得了很多有意義的結(jié)果.文獻(xiàn)[4]研究了分?jǐn)?shù)階微分方程三點(diǎn)邊值問(wèn)題

解的存在性,其中1<α≤2,0≤β≤1.

為了研究流體力學(xué)中相關(guān)問(wèn)題,文獻(xiàn)[5]介紹了一類(lèi)帶有p-Laplacian算子的微分方程,其一維形式如下

(φp(x′(t)))′=f(t,x(t),x′(t)),

(1)

文獻(xiàn)[13]研究了下面分?jǐn)?shù)階微分方程反周期邊值問(wèn)題

受以上文獻(xiàn)的啟示,我們研究如下一類(lèi)帶有p-Laplacian算子分?jǐn)?shù)階微分方程多點(diǎn)邊值問(wèn)題解的存在性，

(2)

1基本定義和預(yù)備知識(shí)

顯然,E是Banach空間.

定義1.1[14]函數(shù)u:(0,∞)→R的α>0階Riemann-Liouville型積分是指

其中右邊在(0,∞)上逐點(diǎn)定義.

定義1.2[14]函數(shù)u:(0,∞)→R的α>0階Caputo型微分是指

其中n為大于或等于α的最小整數(shù),右邊是在(0,∞)上逐點(diǎn)定義的.

引理1.1[14]設(shè)函數(shù)u∈C(0,1)有α>0階的Caputo型微分,則

其中n是大于或等于α的最小整數(shù).

引理1.3設(shè)g(t)∈C[0,1]，

(3)

2主要結(jié)論

引理2.1算子F:E→E是全連續(xù)的.

定理2.1對(duì)任意的常數(shù)r>0,Ω={u|‖u‖

(H)|f(t,u,v,w)|≤σφp(mr),

因此得到

由引理1.2知F滿(mǎn)足Rothe條件,原方程至少存在一個(gè)解.

3例子

例3.1考慮如下帶有p-Laplacian算子的分?jǐn)?shù)階微分方程多點(diǎn)邊值問(wèn)題

(4)

不妨設(shè)r=6,則有界集Ω={u‖|u‖E<6,u∈E}.

顯然,問(wèn)題(4)滿(mǎn)足定理2.1的假設(shè)條件.因此,至少存在一個(gè)解.

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(編輯HWJ)

中圖分類(lèi)號(hào)O175.8

文獻(xiàn)標(biāo)識(shí)碼A

文章編號(hào)1000-2537(2016)01-0080-05

*通訊作者,E-mail：wblium@163.com

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目(11271364)

收稿日期：2013-12-02

DOI:10.7612/j.issn.1000-2537.2016.01.014