The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings

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Journal of Fixed Point Theory and Applications

The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings Torrey M. Gallagher, Maria Jap´on

and Chris Lennard

Abstract. Let C be a convex subset of a Banach space X and let T be a mapping from C into C. Fix α = (α1 , α2 , . . . , αn ) a multi-index in Rn such that αi ≥ 0 (1 ≤ i ≤ n), n 1, α1 , αn > 0, and consider i=1 αi =  i the mapping Tα : C → C given by Tα = n i=1 αi T . Every fixed point of T is a fixed point for Tα but the converse does not hold in general. In this paper we study necessary and sufficient conditions to assure the existence of fixed points for T in terms of the existence of fixed points of Tα and the behaviour of the T -orbits of the points in the domain of T . As a consequence, we prove that the fixed point property for nonexpansive mappings and the fixed point property for mean nonexpansive mappings are equivalent conditions when the involved mappings are affine. Some extensions for more general classes of mappings are also achieved. Mathematics Subject Classification. 46B03, 47H09, 47H10. Keywords. Fixed point property, nonexpansive mappings, mean nonexpansive mappings, metric spaces, Banach spaces.

1. Introduction and Preliminaries Let (C, d) be a metric space. A mapping T : C → C is said to be Lipschitzian with constant k > 0 if d(T x, T y) ≤ kd(x, y), for every x, y ∈ C. The existence of a unique fixed point is well known when the metric space is complete and k < 1. This is Banach’s Contraction Theorem. The mapping T is said to be uniformly Lipschitzian if every iterate T n is Lipschitzian with the same uniform constant k > 0, and T is said to be nonexpansive when k = 1. The existence of fixed points for nonexpansive and uniformly Lipschitzian mappings has been extensively studied (see for instance the books Maria Jap´ on is partially supported by MICINN, Grant PGC2018-098474-B-C21 and Andalusian Regional Government Grant FQM-127. 0123456789().: V,-vol

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[2,14,16,20,28] and the references therein). In general, given a class of mappings A, we say that a subset C has the fixed point property (FPP) for the class A if every mapping T ∈ A with T : C → C has a fixed point. We recall that closed convex bounded sets of reflexive Banach spaces with normal structure or more generally, convex weakly compact sets of Banach spaces with some “nice” geometric property have the FPP for nonexpansive mappings [14,20,30,31]. By contrast, for every convex bounded set C of an infinitedimensional Banach space which is not norm-compact and for every k > 1, there exists a Lipschitzian mapping T : C → C with constant k, which fails to have a fixed point [24]. Regarding uniformly Lipschitzian mappings, some positive results have been achieved for some closed convex bounded sets of certain Banach spaces when the constant k is less than a certain coefficient depending on the geometry of the space (see for instance [16, Section 8]). In this article we will deal with a class of mappings which is strongly rela