中文摘要：

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.

英文摘要：

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.