ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 1. ABSTRACT Let ρ be a modular function satisfying a Δ 2 -type condition and L ρ the cor- responding modular space. The main result in this paper states that if C is a ρ-bounded and ρ-a.e sequentially compact subset of L ρ and T : C → C is an asymptotically regular mapping such that lim inf n→∞ |T n | < 2, where |S | denotes the Lipschitz constant of S , then T has a fixed point. We show that the estimate lim inf n→∞ |T n | < 2 cannot be, in general, improved. 1991 Mathematics subject classification: Primary 46E30; Secondary 47H09, 47H10. Key Words: asymptotically regular mappings, fixed point, modular functions, Opial property. The first author is partially supported by PB-96-1338-C01-C02 and PAI-FMQ-0127. 1