Asymptotic behavior of the solutions of the integro-differential equations on positive half-axis with non-difference kernel of a certain type (Q1899927)

The author considers the following integro-differential equation: \[ - {{d^2 y} \over {dx^2}}+ y= \int_0^\infty R(x- t) y(t) dt+ \int_0^\infty R_1 (x+ t) y(t) dt, \qquad x>0. \tag{1} \] Equations of this kind arise in various fields of physics and have many interesting applications. The author studies equation (1) under the following conditions: (i) \(R(x)\) is an even function with positive range; (ii) \(\int_{- \infty}^\infty R(x) e^{sx} dx<\infty\); (iii) \(\int_{-\infty}^\infty |R_1 (x) |e^{sx} dx\) \(<\infty\), \(0\leq s< s^*\), and discusses the asymptotic behavior of solutions of this problem.