一类Caputo分数阶微分方程边值问题多解的存在性

研究一类Caputo分数阶微分方程边值问题:{D_0~α+u(t)+f(t,u(t))=0,t∈(0,1),u′(0)=u(1)=0,多解的存在性,其中1α≤2,f:0,+∞)×R→0,+∞)是连续的,D_(0+)~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.

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

We investigate the existence and multiplicity of positive solutions for nonlinear Caputo fractional differential equation boundary value problem Dα0+u(t)+f(t,u(t))=0,t∈(0,1),u′(0)=u(1)=0{ , Where 1 <α≤2,f:0,+∞)× →0,+∞)is continuous,and Dα0+is the standard Caputo differentiation.In the process of 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.