×

Fredholm linear operators associated with ordinary differential equations on noncompact intervals. (English) Zbl 0942.34013

In the noncompact interval \(J= [a,\infty)\) the authors consider a linear problem of the form \(L x=y, x\in S\), where \(L\) is a first-order differential operator, \(y\) a locally summable function in \(J\), and \(S\) a subspace of the FrĂ©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, the authors 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 is reduced to the task of finding linear functionals defined in a convenient subspace of \(L_{\text{loc}}^1(J,\mathbb{R}^n)\) whose “kernel intersection” coincides with \(L(S)\). The authors 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 the results are given.

MSC:

34B05 Linear boundary value problems for ordinary differential equations
47A53 (Semi-) Fredholm operators; index theories
PDFBibTeX XMLCite
Full Text: EuDML EMIS