We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations u ′′(x) +∑ i pi(x)u ′ (hi(x)) +∑ i qi(x)u(gi(x)) = 0 without the delay conditions hi(x), gi(x) ≤ x, i = 1, 2, . . ., and u ′′(x) + ∫ ∞ 0 u ′ (s)dsr1(x, s) + ∫ ∞ 0 u(s)dsr0(x, s) = 0.
We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.