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  • 郭彩霞,任玉岗,郭建敏.一类Caputo分数阶微分方程边值问题多解的存在性[J].广西科学,2016,23(4):374-377.    [点击复制]
  • GUO Caixia,REN Yugang,GUO Jianmin.Existence of Multiple Solutions for a Caputo Fractional Difference Equation Boundary Value Problem[J].Guangxi Sciences,2016,23(4):374-377.   [点击复制]
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一类Caputo分数阶微分方程边值问题多解的存在性
郭彩霞, 任玉岗, 郭建敏
0
(山西大同大学数学与计算机科学学院, 山西大同 037009)
摘要:
研究一类Caputo分数阶微分方程边值问题:
D0α+u(t)+f(t,u(t))=0,t∈(0,1)

u'(0)=u(1)=0
多解的存在性,其中1 < α ≤ 2,f:[0,+∞)×R→[0,+∞)是连续的,D0+α是标准的Caputo微分.先将微分方程边值问题转化为积分方程,再转化为积分算子不动点问题,最后利用Leggett-Williams不动点定理得出Caputo分数阶微分方程边值问题至少有3个正解存在,其中格林函数的性质和非线性项的条件至关重要.
关键词:  分数阶微分方程  边值问题  Leggett-Williams不动点定理
DOI:10.13656/j.cnki.gxkx.20160913.002
投稿时间:2016-05-15
基金项目:国家自然科学基金项目(No.11271235),大同大学青年科研基金项目(2014Q10)和河南省高等学校重点科研计划项目(15A110047)资助。
Existence of Multiple Solutions for a Caputo Fractional Difference Equation Boundary Value Problem
GUO Caixia, REN Yugang, GUO Jianmin
(School of Mathematics and Computer Science, Datong University, Datong, Shanxi, 037009, China)
Abstract:
We investigate the existence and multiplicity of positive solutions for nonlinear Caputo fractional differential equation boundary value problem
D0α+u(t)+f(t,u(t))=0,t∈(0,1)

u'(0)=u(1)=0,
Where 1 < α ≤ 2,f:[0,+∞)×R→[0,+∞) is continuous,and D0+α is the standard Caputo differentiation.In the process of proof,we first transform it into integral equation,then differential equation boundary value problem is further converted to discuss the problem of integral operator fixed point. Finally,by means of Leggett-Williams fixed point theorems on cone,existence results of at least three positive solutions are obtained.The properties of the Green function and the conditions of the nonlinear term is very important.
Key words:  fractional difference equation  boundary value problem  Leggett-Williams fixed point theorems

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