Positive solution to a special singular second-order boundary value problem

Let λλ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem u″(t)=λq(t)f(t,u(t),u′(t)),αu(0)−βu′(0)=d,u(1)=0, where d>0,α≥0,β≥0,α+β>0d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞)[0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v)f(t,u,v) is allowed to be singular at t=0,t=1t=0,t=1 and u=0u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.