ОБ АСИМПТОТИЧЕСКОЙ УСТОЙЧИВОСТИ РЕШЕНИЙ ЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА В СЛУЧАЕ НЕДИФФЕРЕНЦИРУЕМОСТИ СРЕЗАННЫХ ФУНКЦИЙ

Abstract

Sufficient conditions are established for the asymptotic stability of any solution of a linear second-order Volterra integro-differential equation in the case when the truncated functions may be non-differentiable at some points of the semiaxis. To do this, first, using a non-standard method of reduction to a system, the equation under consideration is reduced to an equivalent system, and the method of transforming equations and the method of cutting functions are applied to this system. The truncated kernel and free term are associated with the coefficient of the new unknown function using the inequality between the geometric and arithmetic means of two functions. Using the method of integral inequalities, sufficient conditions are established for the boundedness on the semiaxis of all solutions of the system. Then the method of squaring equations is used to the first equation of the system and the Lyusternik-Sobolev lemma is applied. Finally, based on a non-standard substitution, we complete the proof of the asymptotic stability of solutions to a given second-order linear integro-differential equation. An illustrative example is given.