TY  - JOUR
ID  - Iakovlev2013
LB  - Iakovlev2013
AU  - Iakovlev, Serguei I.
AU  - Iakovleva, Valentina
TI  - Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type
ST  - Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2013
VL  - 33
IS  - 1
SP  - 81
EP  - 98
DO  - http://dx.doi.org/10.7494/OpMath.2013.33.1.81
UR  - http://dx.doi.org/10.7494/OpMath.2013.33.1.81
AB  - 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.
KW  - Steklov's smoothing operator
KW  - spectrum
KW  - eigenvalues
KW  - eigenfunctions
KW  - mixed-type differential-difference equations
KW  - initial function
KW  - method of steps
KW  - countably normed space
KW  - transformation group
KW  - generator
SN  - 1232-9274
ER  -