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In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form  x ″( t ) +  px ″( t  − 1) =  qx ([ t ]) +  f ( t ), where [⋅] denotes the greatest integer function,  p  and  q  are nonzero real or complex constants, and  f ( t ) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments