Solutions of integro-differential equations on the half-axis with rapidly decreasing non-difference kernels (Q1607589)

The author studies the set of all solutions of the integro-differential equation \[ -\frac{d^2 y}{dx^2} + y = \int^{\infty}_0 R_1 (x-t)y(t) dt + \int^{\infty}_0 R_2 (x+t)y(t) dt, \quad x>0 \] under various assumptions on the kernels \(R_1 (x)\) and \(R_2 (x)\). With the introduction of a large parameter \(\nu\), he is able to define the Cauchy problem and boundary value problems and also obtains a convenient algorithm for calculating the approximate solutions of these two problems.