Fredholm Linear Operators Associated with Ordinary Differential Equations on Noncompact Intervals
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In the noncompact interval J = [α, ∞) we consider a linear problem of the form Lx = y, x ∈ S, where L is a first order differential operator, y a locally summable function in J, and S a subspace of the Fréchet 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 L1(loc) (J, Rn) 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.