Solvability of convolution type integro-differential equations on ℝ

Abstract

The paper looks for the solutions of integro-differential equations of the form $$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $$ in the class of functions which are absolutely continuous and of slow growth on ℝ. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L1 (ℝ), 0 ≤ k ∈ L1 (ℝ), ∫ℝk(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ℝ. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.