Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ left { begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,quad0 where (0leqalpha_{i}leqsum_{i=1}^{m_{1}}alpha_{i}<1) , i=1,2,…,m ₁, (0 , (0leqbeta_{j}leqsum_{i=1}^{m_{2}}beta_{i}<1) , j=1,2,…,m ₂, (0 . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.