Positive solutions of singular third-order three-point boundary value problem

Abstract

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular third-order three-point boundary value problem u ‴ ( t ) − λ a ( t ) F ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ′ ( η ) = u ″ ( 1 ) = 0 , where λ is a positive parameter and η ∈ [ 1 / 2 , 1 ) is a constant. By using a fixed point theorem of cone expansion-compression type due to Krasnosel'skii, we establish various results on the existence of single and multiple positive solutions to the boundary value problem. Under various assumptions on functions F and a, we give explicitly the intervals for parameter λ in which the existence of positive solutions is guaranteed. Especially, we allow the function a ( t ) of nonlinear term to have suitable singularities.