Existence of solutions of boundary value problems for second order functional differential equations

Abstract In this paper, we deal with the boundary value problem x″=f t,x,x t ,x′,x′ t , x 0 ,x′ 0 ∈ (φ+c 1 ,ψ+c 2 ): c 1 ,c 2 ∈R , α(x| J )=0, β x′(1)−δx′| J =0, where f ∈Car( J × R × C r × R × C r ), φ , ψ ∈ C r , J :=[0,1], δ ≠1, α∈ A J , β∈ A [0,1) , A J and A [0,1) are two special sets of functionals, and x | J is the restriction of x to J . We find sufficient conditions for the existence of solutions of the above problem. The proof is based on the Leray–Schauder degree theory.