PERIODIC-SOLUTIONS OF DRY FRICTION PROBLEMS

Dry friction problems lead to discontinuous differential equations. e. g. to x'' + alpha x' + mu sgn x' + beta(2)x = phi(t). where sgn gamma= gamma \gamma \ for gamma not equal 0 and sgn(0) = [- 1.1]. We study ex istence of omega-periodic solutions of (1) in case phi is omega-period ic. Results for alpha > 0 are given in the book ''Multivalued Differen tial Equations'' (K. Deimling: De Gruyter 1992), and preliminary ones for alpha = 0 are contained in K. Deimling ''Multivalued differential equations and dry friction problems'' (Proc. Conf. Differential & Dela y Equations. World Sci. Publ. 1992). Based on the latter and considera ble additional analysis, we give a complete description of the resonan t case alpha = 0, beta = 1, phi(t) = sin I. In particular, it turned o ut that for mu epsilon (pi 4.1) there is a unique globally asymptotica lly stable 2 pi-periodic solution x(mu). which necessarily has deadzon es (i.e. x(mu)(t) equivalent to c in certain intervals). In addition, the nonresonant case is solved by means of degree theory for multivalu ed maps, since in this situation a priori bounds can be found easily.