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Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

Author(s): Serguei I. Iakovlev | Valentina Iakovleva

Journal: Opuscula Mathematica
ISSN 1232-9274

Volume: 33;
Issue: 1;
Start page: 81;
Date: 2013;
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Keywords: Steklov's smoothing operator | spectrum | eigenvalues | eigenfunctions | mixed-type differential-difference equations | initial function | method of steps | countably normed space | transformation group | generator

It is shown that any \(\mu \in \mathbb{C}\) is an infinite multiplicity eigenvalue of the Steklov smoothing operator \(S_h\) acting on the space \(L^1_{loc}(\mathbb{R})\). For \(\mu \neq 0\) the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.
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