Numeric-analytic method for finding the periodic solutions of nonlinear differential-difference equations with impulses (Q1091777)

The aim of this paper is the study of periodic solutions of nonlinear systems of delay differential equations with impulses. In this type of equations a solution is a real function x(t) that satisfies \[ (1)\quad x'(t)=f(t,x(t),x(t-h))\text{ for all } t\neq t_ i,\quad x(t_ i+0)- x(t_ i-0)=I_ i(x(t_ i-0)) \] where h is a constant delay, \((t_ i)\) is a fixed set of discontinuity points and \(I_ i\) given functionals. Under the assumption that there exists a periodic solution of (1) and some Lipschitz conditions for f(t,x,y) and I(x) with suitable constants are fulfilled, the authors define an iterative scheme \(x_{m+1}=\phi (x_ m)\) which generalizes the classical Picard- Lindelöf method for ordinary differential equations. Then, they prove that \(\phi\) is contractive in a Bieleckij norm and thus \((x_ m)\) converges uniformly to the periodic solution. The paper ends with some remarks on the existence of periodic solutions of (1). It must be pointed out that a numerical implementation of this iterative method is not considered in this paper.