带p-Laplacian算子的分数阶微分方程多点边值问题的解的存在性

吕秋燕,刘文斌,唐　敏*,申腾飞,程玲玲

(1.苏州市吴中区东山中学，中国 苏州　215107；2.中国矿业大学理学院，中国 徐州　221116)



带p-Laplacian算子的分数阶微分方程多点边值问题的解的存在性

吕秋燕1,刘文斌2,唐敏2*,申腾飞2,程玲玲2

(1.苏州市吴中区东山中学，中国 苏州215107；2.中国矿业大学理学院，中国 徐州221116)

摘要利用不动点定理,研究带有p-Laplacian算子的分数阶微分方程多点边值问题解的存在性,得到边值问题至少存在一个解的充分条件.

关键词分数阶微分方程;p-Laplacian算子;存在性;不动点定理

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

近年来,分数阶微分方程被广泛应用于物理学、生物学、控制论等诸多领域[1-3],因此,分数阶微分方程受到许多学者的广泛关注，并取得了很多有意义的结果.文献[4]研究了分数阶微分方程三点边值问题

解的存在性,其中1<α≤2,0≤β≤1.

为了研究流体力学中相关问题,文献[5]介绍了一类带有p-Laplacian算子的微分方程,其一维形式如下

(φp(x′(t)))′=f(t,x(t),x′(t)),

(1)

文献[13]研究了下面分数阶微分方程反周期边值问题

受以上文献的启示,我们研究如下一类带有p-Laplacian算子分数阶微分方程多点边值问题解的存在性，

(2)

1基本定义和预备知识

显然,E是Banach空间.

定义1.1[14]函数u:(0,∞)→R的α>0阶Riemann-Liouville型积分是指

其中右边在(0,∞)上逐点定义.

定义1.2[14]函数u:(0,∞)→R的α>0阶Caputo型微分是指

其中n为大于或等于α的最小整数,右边是在(0,∞)上逐点定义的.

引理1.1[14]设函数u∈C(0,1)有α>0阶的Caputo型微分,则

其中n是大于或等于α的最小整数.

引理1.3设g(t)∈C[0,1]，

(3)

2主要结论

引理2.1算子F:E→E是全连续的.

定理2.1对任意的常数r>0,Ω={u|‖u‖

(H)|f(t,u,v,w)|≤σφp(mr),

因此得到

由引理1.2知F满足Rothe条件,原方程至少存在一个解.

3例子

例3.1考虑如下带有p-Laplacian算子的分数阶微分方程多点边值问题

(4)

不妨设r=6,则有界集Ω={u‖|u‖E<6,u∈E}.

显然,问题(4)满足定理2.1的假设条件.因此,至少存在一个解.

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(编辑HWJ)

中图分类号O175.8

文献标识码A

文章编号1000-2537(2016)01-0080-05

*通讯作者,E-mail：wblium@163.com

基金项目：国家自然科学基金资助项目(11271364)

收稿日期：2013-12-02

DOI:10.7612/j.issn.1000-2537.2016.01.014