On solution of a two kernel equation represented by exponents

The integral equation with two kernels
$$f(x)=g(x)+\int_0^\infty K_1(x-t)f(t)\,dt+\int_{-\infty}^0K_2(x-t)f(t)\,dt,\quad-\infty<x<+\infty,$$
where the kernel functions $K_{1,2}(x)\in L$, is considered on the whole line. The present paper is devoted to solvability of the equation, investigation of properties of solutions and description of their structure. It is assumed that the kernel functions $K_m\ge0$ are even and represented by exponentials as a mixture of the two-sided Laplace distributions:
$$K_m(x)=\int_a^be^{-|x|s}\,d\sigma_m(s)\ge0,\quad m=1,2.$$
Here $\sigma_{1,2}$ are nondecreasing functions on $(a,b)\subset(0,\infty)$ such that
$$0<\lambda_1\le1,\ \ 0<\lambda_2<1,\quad\text{где}\quad\lambda_i=\int_{-\infty}^\infty K_i(x)\,dx=2\int_a^b\frac1s\,d\sigma_i(s),\ \ i=1,2.$$