The existence of countably many positive solutions for some nonlinear th order -point boundary value problems

In this paper, we consider the existence of countably many positive solutions for n th-order m -point boundary value problems consisting of the equation u ( n ) ( t ) + a ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , with one of the following boundary value conditions: u ( 0 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = 0 , and u ( 0 ) = 0 , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , where n ≥ 2 , k i > 0 ( i = 1 , 2 , … , m − 2 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 , a ( t ) ∈ L p [ 0 , 1 ] for some p ≥ 1 and has countably many singularities in [ 0 , 1 2 ) . The associated Green’s function for the n th order m -point boundary value problem is first given, and we show that there exist countably many positive solutions using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.