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.