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Ufimskii Matematicheskii Zhurnal, 2011, Volume 3, Issue 4, Pages 28–38 (Mi ufa115)  

On solution of a two kernel equation represented by exponents

A. G. Barseghyan

Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia
References:
Abstract: The integral equation with two kernels
$$ f(x)=g(x)+\int_0^\infty K_1(x-t)f(t)\,dt+\int_{-\infty}^0K_2(x-t)f(t)\,dt,\quad-\infty<x<+\infty, $$
where the kernel functions $K_{1,2}(x)\in L$, is considered on the whole line. The present paper is devoted to solvability of the equation, investigation of properties of solutions and description of their structure. It is assumed that the kernel functions $K_m\ge0$ are even and represented by exponentials as a mixture of the two-sided Laplace distributions:
$$ K_m(x)=\int_a^be^{-|x|s}\,d\sigma_m(s)\ge0,\quad m=1,2. $$
Here $\sigma_{1,2}$ are nondecreasing functions on $(a,b)\subset(0,\infty)$ such that
$$ 0<\lambda_1\le1,\ \ 0<\lambda_2<1,\quad\text{где}\quad\lambda_i=\int_{-\infty}^\infty K_i(x)\,dx=2\int_a^b\frac1s\,d\sigma_i(s),\ \ i=1,2. $$
Keywords: the basic solution, Ambartsumian equation, Laplace transform, system of integral equations.
Received: 10.09.2011
Bibliographic databases:
Document Type: Article
UDC: 517.968.2
Language: Russian
Citation: A. G. Barseghyan, “On solution of a two kernel equation represented by exponents”, Ufa Math. J., 3:4 (2011)
Citation in format AMSBIB
\Bibitem{Bar11}
\by A.~G.~Barseghyan
\paper On solution of a~two kernel equation represented by exponents
\jour Ufa Math. J.
\yr 2011
\vol 3
\issue 4
\mathnet{http://mi.mathnet.ru//eng/ufa115}
\zmath{https://zbmath.org/?q=an:1249.45003}
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