# zbMATH — the first resource for mathematics

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 $0<\lambda_{1}\leq 1,\,\,\,0<\lambda_{2}\leq 1,\,\,\,\lambda_{i}=\int_{-\infty}^{+\infty}K_{i}(x)dx=2\int_{a}^{b}\frac{1}{s}d\sigma_{1}(s), \,\,i=1,2.$
##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 44A10 Laplace transform
Full Text: