The method of lower and upper solutions for third order singular four point boundary value problems

We mainly study the existence of positive solutions for the following third order singular four point boundary value problem $$\begin{cases}x^{(3)}(t)+f(t,x,x',-x'')=0,\quad 0<t<1,\\x(0)-\alpha x(\xi)=0,\quad x'(1)-\beta x'(\eta)=0,\quad x''(0)=0.\end{cases}$$ where 0≤α<1, 0≤β<1, 0<ξ<1,0<η<1. And we obtain some necessary and sufficient conditions for the existence of C ²0,1] positive solutions by means of the lower and upper solution method. Our nonlinearity f(t,x,y,z) may be singular at x,y,z,t=0 and/or t=1.