Set-Valued Analysis 6: 277–302, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 277 Lyapunov Sequences and a Turnpike Theorem without Convexity Z. DZALILOV and A. M. RUBINOV ⋆ School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat 3353, Victoria, Australia P. E. KLOEDEN Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany (Received: 11 March 1998; in final form: 25 September 1998) Abstract. A turnpike theorem is presented for a class of nonautonomous nonconvex difference inclusions defined by positively homogeneous increasing setvalued mappings. The proof involves a new concept of Lyapunov sequences based on Minkowski gauges of certain normal sets. Mathematics Subject Classification (1991): 90A16. Key words: nonautonomous difference inclusions, positively homogeneous increasing set-valued mappings, turnpike theorem, normal and stable sets, Minkowski gauge. 1. Introduction The dynamics of economical systems typically involves a large number of trajec- tories of which but a small proportion are of special interest, such as those that are optimal with respect to some prescribed optimality criterion or are efficient in some sense. Classical turnpike theorems (see, for example, [1, 5–11, 14–16]) describe the relationship of such trajectories with certain special states or trajectories of the system and are usually stated in terms of autonomous systems. Moreover, their proofs typically require convexity assumptions on the optimality criterion and the mapping generating the dynamics. In this paper we avoid the use of convexity assumptions by considering efficient rather than optimal trajectories for discrete-time set-valued dynamical systems that are assumed for greater generality to be nonautonomous, that is to have the form x t +1 ∈ a t (x t ) for set-valued mappings a t which are positively homogeneous and monotonic rather than convex. We say that a trajectory is efficient if each iterate is a weak Pareto optimal point (with respect to the ordering induced on the state space by some given cone) of the corresponding cocycle mapping generated by ⋆ Partly supported by the Australian Research Council grant A69701407 and the University of Ballarat Special Research Grant.