In this paper we develop the method suggested by S. Pchlivan and M. A. Mamedov ("Statistical Cluster Points and Turnpike," submitted), where it was proved that under some conditions optimal paths have the same unique stationary limit point-stationary cluster point. This notion was introduced by J. A. Fridy (1993, Proc. Amer. Math. Sec. 118, 1187-1192) and it turns out to he a very useful and interesting tool in turnpike theory. The purpose of this paper is to avoid the convexity conditions. Here the turnpike theorem is proved under conditions that are quite different from those of Pehlivan and Mamedov and may be satisfied for the mappings with nonconvex images and for nonconcave functions in the definition of functionals. (C) 2001 Academic Press.