We show that the l 1 sum of the van Dulst space with itself possesses the fixed point property, although it fails most of the known conditions that imply (FPP). Mathematics Subject Classification: Primary, 46B20; 47H09.

Abstract. In this paper we give new estimates for the Lipschitz constants of n-periodic mappings in Hilbert spaces, in order to assure the existence of fixed points and retractions on the fixed point set.1. Introduction. In order to assure the existence of fixed points for a continuous mapping on Banach spaces, we need to impose some conditions on the mapping or on the Banach space. We will deal with k-Lipschitzian mappings: Definition 1.1. Let T : C → C be a mapping with C a nonempty, closed and convex subset of a Banach space X. T is called a Lipschitzian mapping if there is k > 0 such thatholds for any x, y ∈ C and we will write T ∈ L (k). If k 0 is the smallest number such that T ∈ L (k), we will write T ∈ L 0 (k 0 ). Definition 1.2. Let T : C → C where C is a nonempty, closed and convex subset of a Banach space X. If T n = I, T is called an n-periodic mapping.In 1981 K. Goebel and M. Koter, see [1,, proved the following theorem which shows that the condition of periodicity for nonexpansive mappings is very strong:2000 Mathematics Subject Classification. 47H10, 47H09.

In this work, we introduce the notion of cascading non-expansive mappings in the setting of CAT(0) spaces. This family of mappings properly contains the non-expansive maps, but it differs from other generalizations of this class of maps. Considering the concept of Δ-convergence in metric spaces, we prove a principle of demiclosedness for this type of mappings and a Δ-convergence theorem for a Mann iteration process defined using cascading operators.

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