Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 44, pp. 1-16.
Title: Fredholm linear operators associated with ordinary differential
equations on noncompact intervals
Authors: Mariella Cecchi (Univ. di Firenze, Italy)
Massimo Furi (Univ. di Firenze, Italy)
Mauro Marini (Univ. di Firenze, Italy)
Maria Patrizia Pera (Univ. di Firenze, Italy)
Abstract:
In the noncompact interval $J=[a,\infty )$ we consider a linear
problem of the form $Lx=y,\; x \in S$, where $L$ is a first order
differential operator, $y$ a locally summable function in $J$, and
$S$ a subspace of the Fr\'{e}chet space of the locally absolutely
continuous functions in $J$. In the general case, the restriction of
$L$ to $S$ is not a Fredholm operator. However, we show that, under
suitable assumptions, $S$ and $L(S)$ can be regarded as subspaces of
two quite natural spaces in such a way that $L$ becomes a Fredholm
operator between them. Then, the solvability of the problem will be
reduced to the task of finding linear functionals defined in a
convenient subspace of $L_{loc}^{1}(J,{\mathbb R}^{n})$ whose
``kernel intersection'' coincides with $L(S)$. We will prove that,
for a large class of ``boundary sets'' $S$, such functionals can
be obtained by reducing the analysis to the case when the function
$y$ has compact support. Moreover, by adding a suitable stronger
topological assumption on $S$, the functionals can be represented
in an integral form. Some examples illustrating our results are
given as well.
Submitted March 9, 1999. Published October 31, 1999.
Math Subject Classifications: 34B05, 47A53.
Key Words: Fredholm operators; noncompact intervals.