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

Le, Hoan Hoa ; Le, Thi Phuong Ngoc

Electronic Journal of Differential Equations, Tome 2006 (2006).

Résumé

Summary: 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.

Détail

Classification : 34M50, 34K40
Keywords: three-point boundary-value problem, topological degree, Leray-Schauder nonlinear alternative, contraction mapping principle