Positive solutions to a type of nonlinear three-point boundary value problem with sign changing nonlinearities

Abstract

In this paper, by using Krasnoselskii’s Fixed Point Theorem in a cone, we study the existence of positive solutions for the second-order three-point boundary value problem u ″ ( t ) + a ( t ) u ′ ( t ) + λ f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ′ ( 0 ) = 0 , u ( 1 ) = α u ( η ) , where 0 < α , η < 1 and f is allowed to change sign. We also give some examples to illustrate our results.