The Small Parameter and Differential Equations with Piecewise Constant Argument

Abstract

The problem of the existence of periodic solutions is one of the most interesting topics for applications. The method of small parameter is introduced by Poincaré [267] to investigate the problem, and it has been developed by many authors (see, for example, [218, 273], and the references cited therein). This method remains as one of the most effective methods for this problem and it is important that the results obtained in this field can be extended to the bifurcation theory [48, 233]. In this chapter, we investigate the existence and stability of periodic solutions of quasilinear system with piecewise constant an argument and a small parameter in noncritical and critical cases. Theorems on continuous dependence of solutions with respect to initial conditions and parameters, and an analogue of the Gronwall-Bellman lemma are also proved. Examples illustrating the obtained results are constructed as well. In the first section of this chapter we consider a simpler problem of a non-critical case with the delay argument-function β (t) In the last section critical case is considered as well as the argument-function γ(t) of the mixed, advanced-delayed, type is used for the system.