一个新的分数阶微分方程边值问题正解的存在性结果

A NEW EXISTENCE RESULT OF POSITIVE SOLUTIONS FOR A ClASS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS

研究了下面分数阶微分方程边值问题正解的存在性和唯一性D_(0+)~αu(t)=f(t,u(t)),0<t<1,u(0)=u(1)=u′(0)=u′(1)=0,其中3<α≤4是实数,D_(0+)~α是标准的Riemann-Liouville微分,f:[0,1]×[0,∞)→[0,∞)连续。首先应用压缩映像原理得到解的唯一性,其次应用不动点指数得到正解的存在性,证明中借助了特征值理论。

In this paper, we consider the existence and uniqueness of positive solutions for a nonlinear fractional differential equation boundary-value problem D0^αu（t）=f（t,u（t））,0〈t〈1, u（0）=u（1）=u＇（0）=u＇（1）=0where 3 〈 α ≤ 4, and D0＋α＋ is the stanctara ruem^uu [0, ∞） → [0,∞） is continuous. Firstly, the uniqueness of positive solution is obtained by use of contraction map principle. Then, some existence results of positive solutions are obtained. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.