New examples of subsets of c with the FPP and stability of the FPP in hyperconvex spaces

Abstract

The purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences c where we study non-weakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set $$M=[-1,1]^\mathbb {N}$$ M = [ - 1 , 1 ] N has the fixed point property for d-nonexpansive mappings where $$d(\cdot ,\cdot )$$ d ( · , · ) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.

Fixed Points and Common Fixed Points for Orbit-Nonexpansive Mappings in Metric Spaces

“…As it is the case for contractions and isometries, being nonexpansive for a mapping is a pure metric feature that does not call for the definition of any underlying linear structure: it only depends on the relationship between the distance of points and the distance of their images. Therefore, it is not surprising that many authors have studied some types of extensions when it comes to the existence of fixed points for nonexpansive operators on complete metric spaces (see, for instance, [7][8][9][15][16][17]21,24,29] and references therein).…”

“…4, we will focus on the condition of uniform relative normal structure, initially defined by Soardi [28] in the linear case, for obtaining fixed points for nonexpansive operators defined in abstract M Banach lattices, and also used by Lau [23] for the action of isometries. Once more, our framework will be a general metric space as in [8,17] and the interlaced condition will be used to obtain fixed points for a single mapping and for the action of a group likewise. Some other interesting papers studying fixed-point theorems under metric conditions involving some particular orbit features of a given operator are [25,26].…”

“…In fact, it was proved in [20] that convex compact in measure subsets of L 1 [0, 1] have τ -normal structure. Actually, it is possible to find closed convex bounded subsets of a Banach space failing to have weak normal structure (when the convexity structure is the family of all convex weakly compact subsets) but they still have metric normal structure [8,11].…”

“…A new property that formally weakens URNS was introduced in [8] and it was used to prove that, for some metric spaces, the original metric can be slightly altered such that the following property is still preserved. Definition 4.3.…”

Research on Large Field of View Mosaic Technology Based on Surface Structured Light

“…The 3D reconstruction technology of FPP plane structured light is to project a beam of sinusoidal structured light from the projector onto the surface of the measured object, and the structured light modulated and reflected by the measured object's 3D surface is imaged onto the target surface of CCD camera through the lens [3]. The curved sinusoidal structured light stripe is displayed on the computer screen, and the deformed sinusoidal stripe carries the height information of the object's surface.…”