DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

BOUNDARY VALUE PROBLEMS ON THE HALF-LINE
WITHOUT DICHOTOMIES
JASON R. MORRIS AND PATRICK J. RABIER
Given a piecewise continuous function A : R + → (C N )andaprojectionP 1 onto a subspace
X 1 of C N , we investigate the injectivity, surjectivity and, more generally, the Fredholmness
of the differential operator with boundary condition ( ˙u + Au,P 1 u(0)) acting on
the “natural” space WA
1,2 ={u : ˙u ∈ L 2 , Au ∈ L 2 }. It is not assumed that A is bounded or
that ˙u + Au = 0 has any dichotomy, except to discuss the impact of the results on this special
case. All the functional properties of interest, including the Fredholm index, can be
related to a selfadjoint solution H of the Riccati differential inequality HA+ A ∗ H − Ḣ ≥
ν(A ∗ A + H 2 ).
Copyright © 2006 J. R. Morris and P. J. Rabier. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The problem of interest is the existence and possible uniqueness of solutions in Sobolevlike
spaces of the linear boundary value problem on R + = [0,∞),
˙u + Au = f ,
P 1 u(0) = ξ,
(1.1)
where A : R + → (C N )islocallyboundedandP 1 and P 2 are the projections associated
with a given splitting C N = X 1 ⊕ X 2 .
When P 1 = I, that is, X 1 = C N and X 2 ={0}, the familiar Cauchy problem is recovered,
but, even in this case, the existence question in spaces constraining the possible
behavior of the solutions at infinity (such as W 1,2 (R + ,C N )) does not follow from local
existence and uniqueness and is far from being fully resolved. Furthermore, many concrete
problems arise in the form (1.1) withP 1 ≠ I. For instance, second-order equations
¨v + B ˙v + Cv = g with Dirichlet condition v(0) = ξ or Neumann condition ˙v(0) = ξ correspond
to first-order systems (1.1) withN = 2M, C N = C M × C M , u = (v, ˙v), and P 1 the
projection onto the first factor (Dirichlet) or the second one (Neumann). We will denote
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 825–833