Three positive solutions for a nonlinear [formula omitted]th-order [formula omitted]-point boundary value problem

Abstract

In this paper, we consider the existence of at least three positive solutions for the nonlinear n th-order m -point boundary value problem { u ( n ) ( t ) + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , 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 . The associated Green’s function for the n th-order m -point boundary value problem is first given, and growth conditions are imposed on the nonlinearity f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem.