一类Caputo分数阶微分方程边值问题多解的存在性  

Existence of Multiple Solutions for a Caputo Fractional Difference Equation Boundary Value Problem

作　　者：郭彩霞[1] 任玉岗[1] 郭建敏[1] 

机构地区：[1]山西大同大学数学与计算机科学学院,山西大同037009

出　　处：《广西科学》2016年第4期374-377,共4页Guangxi Sciences

基　　金：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);河南省高等学校重点科研计划项目(15A110047)资助

摘　　要：研究一类Caputo分数阶微分方程边值问题：{D0^α＋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.We investigate the existence and multiplicity of positive solutions for nonlinear Caputo fractional differential equation boundary value problem {D0^α＋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 o f proof,we first transform it into integral equation, then differ-ential 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, ex-istence 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.

关 键 词：分数阶微分方程 边值问题 LEGGETT-WILLIAMS不动点定理 

分 类 号：O175.8[理学—数学]