Existence of triple positive solutions for a third-order three-point boundary value problem

In this paper we investigate the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem u ‴ ( t ) = a ( t ) f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , 0 < t < 1 , u ( 0 ) = δ u ( η ) , u ′ ( η ) = 0 , u ″ ( 1 ) = 0 , where δ ∈ ( 0 , 1 ) , η ∈ [ 1 / 2 , 1 ) are constants. f : [ 0 , 1 ] × [ 0 , ∞ ) × R 2 → [ 0 , ∞ ) , q : ( 0 , 1 ) → [ 0 , ∞ ) are continuous. First, Green’s function for the associated linear boundary value problem is constructed, and then, by using a fixed-point theorem due to Avery and Peterson, we establish results on the existence of triple positive solutions to the boundary value problem.