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On solution of an equation with two kernels represented by exponents. (Russian. English summary) Zbl 1249.45003
Author’s abstract: The integral equation with two kernels \(\displaystyle f(x)=g(x)+\int_{0}^{\infty}K_{1}(x-t)f(t)dt+\int_{-\infty}^{0}K_{2}(x-t)f(t)dt,\) \(-\infty<x<+\infty,\) where the kernel functions \(K_{1},K_{2}(x)\in L_{1}(-\infty,+\infty),\) is considered on the whole line. The present paper is devoted to the solvability of the equation, an investigation of properties of the solutions and a description of their structure. It is assumed that the kernel functions \( K_{m} \geq 0\) are even and represented by exponentials as a mixture of the two-sided Laplace distributions: \(\displaystyle K_{m}(x)=\int_{a}^{b}e^{-|x|s}d\sigma_{m}(s)\geq 0,\,\,\,m=1,2. \) Here \(\sigma_{1},\) \(\sigma_{2}\) are nondecreasing functions on \((a,b)\subset(0,\infty)\) such that \[ 0<\lambda_{1}\leq 1,\,\,\,0<\lambda_{2}\leq 1,\,\,\,\lambda_{i}=\int_{-\infty}^{+\infty}K_{i}(x)dx=2\int_{a}^{b}\frac{1}{s}d\sigma_{1}(s), \,\,i=1,2. \]
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
44A10 Laplace transform
Full Text: MNR