Rotative Mappings and Mappings with Constant Displacement

Kaczor, Wiesława; Koter-Mórgowska, Małgorzata

doi:10.1007/978-94-017-1748-9_11

Cited by 1 publication

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“…In 1971 K. Goebel and E. Złotkiewicz [2] proved that if k < 2, then Fix(T ) = ∅ for 2-periodic and k-Lipschitzian mappings in general Banach spaces X, that is, γ X 2 ≥ 2. Furthermore, in 1986 M. Koter (see also [4]) proved that γ H 2 ≥ √ π 2 − 3 ≈ 2.6209 for Hilbert spaces H.…”

Section: Definition 12mentioning

confidence: 97%

Fixed points of periodic mappings in Hilbert spaces

García

Nathansky

2010

Annales UMCS, Mathematica

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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.

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Section: Definition 12mentioning

confidence: 97%