1.1 Introduction

Tracing research focused on friction it seems that the model introduced by Coulomb is still one of the most appropriate (perhaps with slight modifications) and therefore it will be further used in our considerations. It appears that friction causes not only problems in industrial equipments but also there are some not yet satisfactorily solved problems of mathematical nature. Although since the Coulomb s work published in 1781, a great amount of papers devoted to friction phenomena have appeared, the problem remains still open. We do not review the state of the art in this field but we only indicate some research directions oriented on friction. The oldest approach is based on an experiment and modelling of friction (see, for instance, works [Dowell and Schwartz (1983b)], [Tomlinson and Chen (1996)] and [Hess and Soom (1990)]). Some papers are devoted to numerical investigations of systems with friction ([Stewart (1996)], [Stewart and Trinkle (1997)]). Analytical and asymptotical approaches to analyse the friction phenomena are sometime also applied ([Capecchi and Vestroni (1995)] and [Altpeter et. al.(1998)]).

Stick-slip and chaotic phenomena caused by friction have been analysed by [Pratt and Williams (1981)], [Awrejcewicz and Delfs (1990a)], [Awrejcewicz and Delfs (1990b)], [Stelter (1992)], [Ionescu and Paumier (1993)], [Baumberger et. al. (1995)], [Shaw (1985)] and [Madan (1993)]. More theoretically oriented research has been carried out by [Deimling (1992)], [Deimling and Szil gyi (1994)] and [Fe?kan (1997)]. For instance it is possible to generalize classical Conley index theory to nonsmooth system and use it to prove...