On a Homogeneous Integral Equation with Two Kernels

The present paper is devoted to the finding conditions of nontrivial (non-zero) solvability of some classes of equations of the form S(x)=∫0∞T1(x−t)S(t)dt+∫−∞0T2(x−t)S(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\left( x \right) = \int_0^\infty {{T_1}\left( {x - t} \right)S\left( t \right)} dt + \int_{ - \infty }^0 {{T_2}\left( {x - t} \right)S\left( t \right)} dt$$\end{document} , x ∈ R, with respect to unknown function S. The asymptotic behavior of the solution S is also studied.