On an estimate of solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case on the half-line

Abstract

We obtain sufficient conditions for a certain estimate to hold on the half-line for all solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case. We present some corollaries concerning the absolute integrability of powers of the solutions on the half-line and the convergence of solutions to zero (including the exponential and power-law convergence) as the independent variable tends to infinity. Illustrative examples are given.