Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 62, pp. 1-19.

Title: Boundary and initial value problems for  second-order 
       neutral functional differential equations

Authors: Hoan Hoa Le (Ho Chi Minh City Univ., Vietnam)
         Thi Phuong Ngoc Le (Nha Trang Educational College, Vietnam)

Abstract: 
 In this paper, we consider the three-point boundary-value problem
 for the second order neutral functional differential equation
 $$
 u''+ f(t,u_t, u'(t))= 0, \quad  0 \leq t\leq 1,
 $$
 with the three-point boundary condition
 $u_0= \phi$, $u(1) = u(\eta)$. Under suitable
 assumptions on the function $f$ we prove the existence,
 uniqueness and continuous dependence of solutions.
 As an application of the methods used, we study the existence
 of solutions for the same equation with a ``mixed" boundary
 condition
 $u_0 = \phi, u(1) = \alpha [u'(\eta) - u'(0)]$,
 or with an initial condition $ u_0 = \phi, u'(0) =0$.
 For the initial-value problem, the uniqueness and
 continuous dependence of solutions are also considered.
 Furthermore, the paper shows that the solution
 set of the initial-value problem is nonempty, compact and
 connected. Our approach is based on the fixed point theory.
 
Submitted February 20, 2006. Published May 11, 2006.
Math Subject Classifications: 34M50, 34K40.
Key Words: Three-point boundary-value problem; topological degree;
 Leray-Schauder nonlinear alternative; Contraction mapping principle.