Stability and asymptotic behavior of the systems with delay and bounded nonlinearity

This paper has a strong motivation in the models occurring in the problem of PIO (Pilot In-the-Loop Oscillations) where any pilot model contains at least one pure delay; at the same time in the so-called PIO II category all dynamics are assumed linear except the position and rate limiters whose models include the saturation nonlinearity - a typical bounded nonlinear function. Interesting enough, an equally motivating application arises from neural networks dynamics where both delays and bounded sigmoids are to be met. In this paper we start from some classical results for integral equations concerning boundedness and asymptotic behavior, improve the results of their application to systems with lumped parameters (i.e. described by ordinary differential equations) and extend them to an interesting class of time delay systems whose structure incorporates the systems described by input/output delay-differential equations of higher order.