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

In this paper, we deal with the boundary value problem x = f t,x,xt ,x ,x t , x0,x 0 ∈ (φ +c1,ψ + c2): c1, c2 ∈ R , α(x|J ) = 0, β x (1) − δx |J = 0, where f ∈ Car(J × R × Cr × R × Cr ), φ, ψ ∈ Cr , J := [0, 1], δ = 1, α ∈ AJ , β ∈ A[0,1), AJ 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.  2003 Elsevier Inc. All rights reserved