Fredholm Linear Operators Associated with Ordinary Differential Equations on Noncompact Intervals

Cecchi, Mariella
Furi, Massimo
Marini, Mauro
Pera, Maria Patrizia
Journal Title
Journal ISSN
Volume Title
Southwest Texas State University, Department of Mathematics
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 L1loc (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.
Fredholm operators, Noncompact intervals
Cecchi, M., Furi, M., Marini, M., & Pera, M. P. (1999). Fredholm linear operators associated with ordinary differential equations on noncompact intervals. <i>Electronic Journal of Differential Equations, 1999</i>(44), pp. 1-16.