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Convolution equation with a completely monotonic kernel on the half-line. (English. Russian original) Zbl 0873.45002
Sb. Math. 187, No. 10, 1465-1485 (1996); translation from Mat. Sb. 187, No. 10, 53-72 (1996).
The Wiener-Hopf (convolution) integral equation \[ f(x)=g(x)+\int_0^\infty K(x-y)f(y)dy\tag{1} \] and the related factorization problem are considered for the kernels \[ K(\pm x)=\int\limits_a^b e^{-xp}d\sigma_\pm(p),\quad\mu\equiv\sum\limits_\pm\int\limits_a^b\frac{1}{p}d\sigma_\pm<+\infty \] where \(\sigma_\pm(p)\) are non-decreasing functions continuous from the left. Equation (1) is tightly connected with the nonlinear Ambartsumyan equation \[ \varphi_\pm(p)=1+\varphi_\pm(p)\int\limits_a^b\frac{\varphi_\pm(q)}{p+q}d\rho_\pm(q), \] appearing in the problem of radiation transfer. Since \(\int\limits_{-\infty}^\infty|K(x)|dx\leq\mu\), (equality holds if \(K(x)\geq 0\)) equation (1) is uniquely solvable for \(\mu<1\) in different functional spaces. The authors consider mostly the so-called supercritical case \(\mu>1\) and apply the factorization method.

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G10 Integral operators
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