Chaplygin’s method for second-order neutral differential equations with piecewise constant deviating arguments

Abstract

The purpose of this paper is to extend Chaplygin’s theorem to second-order neutral differential equations with piecewise constant delay. We start with some auxiliary results concerning upper and lower solutions of second-order neutral differential equations. We then use these extended results to find bounds in terms of Chaplygin sequences for the solution of the addressed problem. These bounds, formed by the construction of upper and lower solutions, are shown to converge to the unique solution of the equation. Finally, we show that the error estimates obtained are sharper than those for ordinary and first-order neutral differential equations.