In this paper, using a simple and classical application of the Leray-Schauder degree theory, we study the existence of solutions of the following boundary value problem for functional differential equations x ″ ( t ) + f ( t , x t , x ′ ( t ) ) = 0 ,    t ∈ [ 0 , T ] x 0 + α x ′ ( 0 ) = h x ( T ) + β x ′ ( T ) = η where f ∈ C ( [ 0 , T ] × C r × ℝ n , ℝ n ) , h ∈ C r , η ∈ ℝ n and α , β , are real constants.