一类分数阶微分方程边值问题正解的存在性和多重性

he aim of this paper is to consider the following fractionaldifferential equation$$left{ligned & D^{lpha}_{0+}u(t)+a(t) f(t,u(t))=0 , 0<t<1,\& u(0)=u'(0)=u''(0)=u''(1)=0,endaligned ight.$$where $3 < lpha leq 4$ is a real number, $D^{lpha}_{0+}$ isthe standard Riemann-Liouville derivative, $f:[0,1] imes[0,+infty) ightarrow [0,+infty)$ is continuous, $a(t)inC((0,1),[0,+infty))$ may be singular at $t=0,1$. Bymeans of the fixed point index theory, a number of theorems on theexistence and multiplicity of positive solutions are obtained andsome previous results are improved. Finally one example is workedout to demonstrate our main results.