Global behavior of positive solutions of nonlinear three-point boundary value problems

Abstract

We investigate the structure of the positive solution set for nonlinear three-point boundary value problems of the form u ″ + h ( t ) f ( u ) = 0 , u ( 0 ) = 0 , u ( 1 ) = λ u ( η ) , where η ∈ ( 0 , 1 ) is given, λ ∈ [ 0 , 1 η ) is a parameter, f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) satisfies f ( s ) > 0 for s > 0 , and h ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) is not identically zero on any subinterval of [ 0 , 1 ] . Our main results demonstrate the existence of continua of positive solutions of the above problem.