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On a conservative integral equation with two kernels. (English. Russian original) Zbl 0914.45003
Math. Notes 62, No. 3, 271-277 (1997); translation from Mat. Zametki 62, No. 3, 323-331 (1997).
Summary: We study the solvability of the integral equation \[ f(x)= g(x)+ \int_0^\infty T_1(x-t)f(t)dt+ \int_{-\infty}^0 T_2(x-t)f(t)dt, \qquad x\in\mathbb{R}, \] where \(f\in L_1^{\text{loc}} (\mathbb{R})\) is the unknown function and \(g\), \(T_1\), and \(T_2\) are given functions satisfying the conditions \[ g\in L_1(\mathbb{R}), \quad 0\leq T_j\in L_1(\mathbb{R}), \qquad \int_{-\infty}^\infty T_j(t)dt=1, \quad j=1,2. \] Most attention is paid to the nontrivial solvability of the homogeneous equation \[ s(x)= \int_0^\infty T_1(x-t) s(t)dt+ \int_{-\infty}^0 T_2(x-t)s(t)dt, \quad x\in\mathbb{R}. \]

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text: DOI
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