Electron. J. Diff. Eqns., Vol. 2006(2006), No. 62, pp. 1-19.

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

Hoan Hoa Le, Thi Phuong Ngoc Le

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.

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Hoan Hoa Le
Department of Mathematics
Ho Chi Minh City University of Education
280 An Duong Vuong Str., Dist. 5, Ho Chi Minh City, Vietnam
Thi Phuong Ngoc Le
Nha Trang Educational College
01 Nguyen Chanh Str., Nha Trang City, Vietnam
email: phuongngoccdsp@dng.vnn.vn   lephuongngoc@netcenter-vn.net

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