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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.

##### MSC:
 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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