Existence of positive solutions for boundary-value problems for singular higher-order functional differential equations

We study the existence of positive solutions for the boundary-value problem of the singular higher-order functional differential equation $$displaylines{ (L y^{(n-2)})(t)+h(t)f(t, y_t)=0, quad hbox{for } tin [0, 1],cr y^{(i)}(0) = 0, quad 0 leq i leq n - 3, cr alpha y^{(n-2)}(t)-eta y^{(n-1)} (t)=eta (t), quad hbox{for } t in [- au, 0],cr gamma y^{(n-2)}(t) + delta y^{(n-1)}(t) = xi (t), quad hbox{for } t in [1, 1 + a], }$$ where $ Ly := -(p y')' + q y$, $p in C([0, 1],(0, + infty))$, and $q in C([0, 1], [0, + infty))$. Our main tool is the fixed point theorem on a cone.