Opuscula Math. 33, no. 1 (2013), 81-98
http://dx.doi.org/10.7494/OpMath.2013.33.1.81

Opuscula Mathematica

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

Serguei I. Iakovlev
Valentina Iakovleva

Abstract. 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.

Keywords: Steklov's smoothing operator, spectrum, eigenvalues, eigenfunctions, mixed-type differential-difference equations, initial function, method of steps, countably normed space, transformation group, generator.

Mathematics Subject Classification: 47A75, 34K99.

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