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Mat. Zametki, 2007, Volume 81, Issue 5, Pages 693–702 (Mi mz3712)  

This article is cited in 1 scientific paper (total in 1 paper)

On the Convolution Equation with Positive Kernel Expressed via an Alternating Measure

B. N. Enginbarian

Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: We consider the integral convolution equation on the half-line or on a finite interval with kernel
$$ K(x-t)=\int_a^be^{-|x-t|s} d\sigma(s) $$
with an alternating measure $d\sigma$ under the conditions
$$ K(x)>0, \quad \int_a^b\frac{1}{s} |d\sigma(s)|<+\infty, \quad \int_{-\infty}^\infty K(x) dx=2\int_a^b\frac{1}{s} d\sigma(s)\le1. $$
The solution of the nonlinear Ambartsumyan equation
$$ \varphi(s)=1+\varphi(s)\int_a^b\frac{\varphi(p)}{s+p} d\sigma(p), $$
is constructed; it can be effectively used for solving the original convolution equation.

Keywords: integral convolution equation, nonlinear Ambartsumyan equation, alternating measure, Wiener–Hopf operator, nonlinear factorization equation, Volterra equation

DOI: https://doi.org/10.4213/mzm3712

Full text: PDF file (472 kB)
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English version:
Mathematical Notes, 2007, 81:5, 620–627

Bibliographic databases:

UDC: 517.968.4
Received: 26.12.2005
Revised: 28.09.2006

Citation: B. N. Enginbarian, “On the Convolution Equation with Positive Kernel Expressed via an Alternating Measure”, Mat. Zametki, 81:5 (2007), 693–702; Math. Notes, 81:5 (2007), 620–627

Citation in format AMSBIB
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\by B.~N.~Enginbarian
\paper On the Convolution Equation with Positive Kernel Expressed via an Alternating Measure
\jour Mat. Zametki
\yr 2007
\vol 81
\issue 5
\pages 693--702
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\crossref{https://doi.org/10.4213/mzm3712}
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\zmath{https://zbmath.org/?q=an:1153.45003}
\elib{https://elibrary.ru/item.asp?id=9498097}
\transl
\jour Math. Notes
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\vol 81
\issue 5
\pages 620--627
\crossref{https://doi.org/10.1134/S0001434607050069}
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    This publication is cited in the following articles:
    1. Barsegyan A.G., “On the solution of the convolution equation with two kernels”, Differ. Equ., 48:5 (2012), 756–759  crossref  mathscinet  zmath  isi  elib  scopus
  • Математические заметки Mathematical Notes
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