Solvability of a multi-point boundary value problem of Neumann type

Let f : [ 0 , 1 ] × ℝ 2 → ℝ be a function satisfying Carathéodory's conditions and e ( t ) ∈ L 1 [ 0 , 1 ] . Let ξ i ∈ ( 0 , 1 ) , a i ∈ ℝ , i = 1 , 2 , … , m − 2 , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 be given. This paper is concerned with the problem of existence of a solution for the m -point boundary value problem x ″ ( t ) = f ( t , x ( t ) , x ′ ( t ) ) + e ( t ) , 0 < t < 1 ; x ( 0 ) = 0 , x ′ ( 1 ) = ∑ i = 1 m − 2 a i x ′ ( ξ i ) . This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of the a i ∈ ℝ , i = 1 , 2 , … , m − 2 , had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the a i ∈ ℝ , i = 1 , 2 , … , m − 2 , have the same sign. Some examples are given to illustrate this point.