Fredholm linear operators associated with ordinary differential equations on noncompact intervals

Cecchi, Mariella ; Furi, Massimo ; Marini, Mauro ; Pera, Maria Patrizia

Electronic journal of differential equations, Tome 1999 (1999)

Résumé

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 Frechet 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 $L_{loc}^{1}(J,{\Bbb R}^{n})$ 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.

Détail

Classification : 34B05, 47A53
Keywords: Fredholm operators, noncompact intervals