On periodic solutions of differential equations with piecewise constant argument

The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367–383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646–663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall–Bellman Lemma for functions with piecewise constant and retarded–advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results.