A second-order singular boundary value problem

Abstract

We study the second-order boundary value problem −″(t) = α(t)f(u(t)), o < t < 1, satisfying αu(0) − βu″(0) = 0, γu(1) + δu″(1) = 0, where a( t) = Π i=1 n a i ( t) and α, β, γ, δ ≥ 0, αγ + αδ + βγ > 0. We assume that each a i ( t) ϵ L Pi [0, 1] for p i ≥ 1 and that each a i ( t) has a singularity in (0, 1). To show the existence of countably many positive solutions, we apply Hölder's inequality and Krasnosel'ski i ∪ 's fixed-point theorem for operators on a cone.