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

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