Fredholm formulae for kernels which are linear with respect to parameter

In this paper, we construct formulae, which are similar to the classical determinant formulae of Fredholm, for solving second-kind integral equations in $L_2(\mathbb R)$ with continuous on $\mathbb R^2$ Carleman kernels of the form $H(s,t)+\mu G(s,t)$, where $\mu$ is a complex parameter