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On a conservative integral equation with two kernels

Authors:

L G Arabadzhyan

Institution:

(1) Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia, Armenia

Abstract:

We study the solvability of the integral equation

$f(x) = g(x) + \smallint _0^\infty T_1 (x - t)f(t)dt + \smallint _{ - \infty }^0 T_2 (x - t)f(t)dt, x \in \mathbb{R}$

, wheref∈L ₁ ^loc(ℝ) is the unknown function andg,T ₁, andT ₂ are given functions satisfying the conditions

$g \in L_1 (\mathbb{R}), 0 \leqslant T_j \in L_1 (\mathbb{R}), \smallint _{ - \infty }^\infty T_j (t)dt = 1, j = 1,2$

. Most attention is paid to the nontrivial solvability of the homogeneous equation

$s(x) = \smallint _0^\infty T_1 (x - t)s(t) dt + \smallint _{ - \infty }^0 T_2 (x - t)s(t) dt, x \in \mathbb{R}$

. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 323–331, September, 1997. Translated by M. A. Shishkova

Keywords:

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