Asymptotically regular mappings in modular function spaces

Abstract

Let ρ be a modular function satisfying a ∆2-type condition and Lρ the corresponding modular space. The main result in this paper states that if C is a ρ-bounded and ρ-a.e sequentially compact subset of Lρ and T : C → C is an asymptotically regular mapping such that lim inf n→∞ [Tn] < 2, where |S| denotes the Lipschitz constant of S, then T has a fixed point. We show that the estimate lim inf n→∞ [Tn] < 2 cannot be, in general, improved.