Stability of the Fixed Point Property for Nonexpansive Mappings

Garcia-Falset, Jesús; Jiménez-Melado, Antonio; Llorens-Fuster, Enrique

doi:10.1007/978-94-017-1748-9_7

Cited by 21 publications

(24 citation statements)

References 19 publications

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“…From Theorems 2, 5, and the characterization of WNUS spaces given in [2], we obtain the following corollary. This corollary provides a sufficient condition for the FPP of a Banach space in terms of its modulus of u-convexity which generalizes the one given in Theorem 3, and-according to Theorem 2-presents a family of uniformly nonsquare Banach spaces for which the FPP holds.…”

Section: Proof Letmentioning

confidence: 84%

On the modulus of u‐convexity of Ji Gao

Mazcuñán-Navarro

2003

Abstract and Applied Analysis

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We consider the modulus of u-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus of u-convexity.Let X be a Banach space and let C be a nonempty subset of X. A mapping T : C → C is said to be nonexpansive wheneverfor all x, y ∈ C. A Banach space X has the weak fixed-point property (WFPP) (resp., fixed-point property (FPP)) if for each nonempty weakly compact convex (resp., bounded, closed, and convex) set C ⊂ X and each nonexpansive mappingIt is well known that the WFPP holds for Banach spaces with certain geometrical properties. Among such properties, weak normal structure is, maybe, the most widely studied (see [5, Chapter 3.2]). In order to give sufficient conditions for the WFPP or weak normal structure, different moduli of convexity of Banach spaces have been introduced by several authors (see [5, Chapter 4.5]).At the origin of these moduli is the classical modulus of convexity introduced by J.

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Section: Proof Letmentioning

confidence: 84%

On the modulus of u‐convexity of Ji Gao

Mazcuñán-Navarro

2003

Abstract and Applied Analysis

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“…The modulus r X has already been used by many authors, see e.g [20], [21], [22], [34], [35], [42]. There is a simple connection with the modulus of asymptotic smoothness.…”

Section: Main Result and Two Examplesmentioning

confidence: 99%

Smoothness, asymptotic smoothness and the Blum-Hanson property

2015

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Abstract. We isolate various sufficient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at infinity, or if X is uniformly Gâteaux smooth and embeds isometrically into a Banach space with a 1-unconditional finite-dimensional decomposition.

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“…However, to this day, it remains unknown whether reflexivity (or even super-reflexivity) alone is sufficient for the FPP. For much of what is currently known about the fixed point theory for nonexpansive mappings we refer to the Handbook [35], and especially in the survey articles [25] and [17]. Three papers (Browder [6], Göhde [30], and Kirk [34]), published by coincidence in 1965, provided the foundation for the theory.…”

Section: T X − T Y ≤ X − Ymentioning

confidence: 99%

Remarks on the stability of the fixed point property for nonexpansive mappings

2012

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It is shown that in many instances the fixed point property for nonexpansive mappings actually implies the fixed point property for a strictly larger family of mappings. This paper is largely expository, but some of the observations are not readily available, and some appear here for the first time. Several related open questions in are discussed. The emphasis is on accessible problems, especially those that require little background. The problems themselves have been given little thought and may be trivial or difficult.

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Stability of the Fixed Point Property for Nonexpansive Mappings

Cited by 21 publications

References 19 publications

On the modulus of u‐convexity of Ji Gao

On the modulus of u‐convexity of Ji Gao

Smoothness, asymptotic smoothness and the Blum-Hanson property

Remarks on the stability of the fixed point property for nonexpansive mappings

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