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Zbl 0914.45003
On a conservative integral equation with two kernels.
(English. Russian original)
[J] Math. Notes 62, No.3, 271-277 (1997); translation from Mat. Zametki 62, No.3, 323-331 (1997). ISSN 0001-4346; ISSN 1573-8876/e

Summary: We study the solvability of the integral equation $$f(x)= g(x)+ \int_0^\infty T_1(x-t)f(t)dt+ \int_{-\infty}^0 T_2(x-t)f(t)dt, \qquad x\in\bbfR,$$ where $f\in L_1^{\text{loc}} (\bbfR)$ is the unknown function and $g$, $T_1$, and $T_2$ are given functions satisfying the conditions $$g\in L_1(\bbfR), \quad 0\leq T_j\in L_1(\bbfR), \qquad \int_{-\infty}^\infty T_j(t)dt=1, \quad j=1,2.$$ Most attention is paid to the nontrivial solvability of the homogeneous equation $$s(x)= \int_0^\infty T_1(x-t) s(t)dt+ \int_{-\infty}^0 T_2(x-t)s(t)dt, \quad x\in\bbfR.$$
MSC 2000:
*45E10 Integral equations of the convolution type

Keywords: conservative integral equation; two kernels; local integrability; nontrivial solvability

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