Fixed points of periodic mappings in Hilbert spaces

Abstract

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.

Hölder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

“…We start with the following variant of a well known result (see, e.g., [2, Prop. 1.10], [16,Lemma 2.1]). Lemma 1.…”

“…[8,9]) proved several structural results concerning uniformly Lipschitzian mappings but many questions remain open. In [16], Pérez García and Fetter Nathansky gave conditions under which Fix T is a Hölder continuous retract and applied them to the study of n-periodic mappings in Hilbert spaces.…”