Fredholm Linear Operators Associated with Ordinary Differential Equations on Noncompact Intervals

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 the locally summable funtions on $J$, 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.