Electron. J. Diff. Eqns., Vol. 1999(1999), No. 44, pp. 116.
Fredholm linear operators associated with ordinary differential
equations on noncompact intervals
Mariella Cecchi, Massimo Furi, Mauro Marini, & Maria Patrizia Pera
Abstract:
In the noncompact interval
we consider a linear
problem of the form Lx=y,
,
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
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.
Submitted March 9, 1999. Published October 31, 1999.
Math Subject Classifications: 34B05, 47A53.
Key Words: Fredholm operators, noncompact intervals.
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Mariella Cecchi
Dipartimento di Ingegneria Elettronica
Universitá di Firenze
Via S. Marta, 3  50139 Firenze, Italy
email address: cecchi@diefi.die.unifi.it 

Massimo Furi
Dipartimento di Matematica Applicata
Universitá di Firenze
Via S. Marta, 3  50139 Firenze, Italy
email address: furi@dma.unifi.it 

Mauro Marini
Dipartimento di Ingegneria Elettronica
Universitá di Firenze
Via S. Marta, 3  50139 Firenze, Italy
email address: marini@ingfi1.ing.unifi.it 

Maria Patrizia Pera
Dipartimento di Matematica Applicata
Universitá di Firenze
Via S. Marta, 3  50139 Firenze, Italy
email address: pera@dma.unifi.it 
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