The fixed point property for nonexpansive type mappings in nonreflexive Banach spaces
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The fixed point property for nonexpansive type mappings in nonreflexive Banach spaces Rahul Shukla1 Received: 28 July 2020 / Accepted: 30 September 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract In this paper, we present the fixed point property for nonexpansive type mappings in Banach spaces endowed with near-infinity concentrated norms. We also obtain a stability result. Finally, we present a nontrivial example to show the usefulness of these results. Keywords Fixed point · Nonexpansive mapping · Banach space Mathematics Subject Classification Primary 47H10 · 47H09
1 Introduction Let (E, ‖.‖) be a Banach space and K a nonempty subset of E . A mapping T ∶ K → K is said to be nonexpansive if ‖T(u) − T(v)‖ ≤ ‖u − v‖ for all u, v ∈ K . The Banach space (E, ‖.‖) is said to satisfy fixed point property (in short, FPP) for nonexpansive mappings if every nonexpansive self-mapping defined on a closed bounded convex subset K of E has a fixed point. Many reflexive Banach spaces enjoy the FPP for nonexpansive mappings, for instance, Hilbert spaces, uniformly convex Banach spaces, Banach spaces with normal structure etc. Further, reflexive Banach spaces with many other geometric properties are known to imply the FPP for nonexpansive mappings (uniformly Kadec-Klee property, uniform Opial property, existence of monotone unconditional basis etc.) [8]. However, the sequence Banach space 𝓁1 with norm ‖.‖1 is a classical example of a nonreflexive Banach space that fails to have the FPP for nonexpansive mappings [8]. For a long time, it was an open question whether all Banach spaces with FPP for nonexpansive mappings are reflexive. In 2008, P. K. Lin [11] answered this question in negative by giving a nonreflexive Banach space with the FPP for nonexpansive mappings. In fact, Lin considered the sequence space 𝓁1 endowed with a norm equivalent to the usual one. Many authors extended Lin’s results in different directions [3, 9, 10, 12, 13]. Recently, Castillo-Sántos et al. [4] defined the concept of a near-infinity concentrated norm on a Banach space with a boundedly complete Schauder basis. They proved that a Banach space * Rahul Shukla [email protected]; [email protected] 1
Department of Mathematics and Applied Mathematics, Kingsway Campus, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa
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endowed with this norm enjoys the FPP for nonexpansive mappings. They also enlarged the class of renorms on 𝓁1 satisfying the FPP for nonexpansive mappings. These results emphasize the interrelation between both renorming theory and metric fixed point theory. Earlier, both theories have been usually studied as independent topics. In this paper, we continue this direction of research and study new connections between renorming theory and metric fixed point theory in some nonreflexive Banach spaces for nonexpansive type mappings. On the other hand a number of nonlinear mappings have been appeared in literature as generalizations and extensions of n
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