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.