Existence results for semipositone boundary value problems

The authors study the existence of positive solutions to the boundary value problem (p(t)u')' + lambdaf(t,u) + e(t,u) = 0 r < t < R, au(r) - bp(r)u'(r) = 0 cu(R) + dp(R)u'(R) = 0 where f and e : [r, R] X [0, infinity) --> R are two continuous functions satisfying f greater than or equal to 0 and \e \ less than or equal to M for some M > 0. The authors show that there exists at least one positive solution in the following two cases: (i) f is superlinear at infinity and lambda > 0 is small enough; (ii) f is sublinear at infinity and lambda > 0 is large enough. Their proofs are based on fixed point theorems in cones.