The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem$$x\prime\prime=f(x)-\mu {{q(t)x\prime(0)} \over {\int_{0}^{1}q(s) h(x(s)){\rm d}s}} h(x), \quad x(0)= x\prime(1)=0$$where q ∈ C 0 ([0,1]) and f , h ∈ C 0 ((0,∞)), lim   x →0   +f ( x )=−∞, lim   x →0   +h ( x )=∞. We present conditions guaranteeing the existence of a solution x ∈ C 1 ([0,1]) ∩ C 2 ((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.