On the asymptotic stability of solutions of a linear Volterra integro-differential equation of the third order with nonsmooth cut functions

Sufficient conditions for the asymptotic stability of solutions to a third-order linear integro-differential equation of Volterra type on the half-axis are established. To do this, first the given equation, using a non-standard substitution, is reduced to an equivalent system consisting of one second-order differential equation and one first-order Volterra integro-differential equation. Then the method of squaring equations and the method of cutting functions are developed for this system, and in contrast to previous studies, transformations are made that cover the case when the cut terms of the kernels and free terms can be non-differentiable at discrete points of the semi-axis. After applying the integral inequality method, the Lyusternik-Sobolev lemma is used. An illustrative example is constructed.