The super fixed point property for asymptotically nonexpansive mappings

<p>Kirk introduced the notion of pointwise eventually asymptotically non-expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically non expansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings when ever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {T<sup>n</sup>x} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map.</p>

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The fixed-point property in Banach spaces containing a copy ofc0

Abstract and Applied Analysis ◽

10.1155/s1085337503203055 ◽

2003 ◽

Vol 2003 (3) ◽

pp. 183-192

Author(s):

Maria A. Japón Pineda

Keyword(s):

Banach Space ◽

Fixed Point ◽

Banach Spaces ◽

Weak Topology ◽

Nonexpansive Mappings ◽

Fixed Point Property ◽

Point Property ◽

Asymptotically Nonexpansive ◽

Locally Convex Topology ◽

Locally Convex

We prove that every Banach space containing an isomorphic copy ofc0fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy ofc0is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.

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THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000440 ◽

2013 ◽

Vol 89 (1) ◽

pp. 79-91 ◽

Cited By ~ 1

Author(s):

ANDRZEJ WIŚNICKI

Keyword(s):

Fixed Point ◽

Nonexpansive Mappings ◽

Fixed Point Property ◽

Point Property ◽

Direct Sums ◽

Definitive Answer ◽

Direct Sum ◽

Asymptotically Nonexpansive ◽

Weak Fixed Point Property ◽

Monotone Norm

AbstractWe show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

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Towards the fixed point property for superreflexive spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700019900 ◽

2001 ◽

Vol 64 (3) ◽

pp. 435-444 ◽

Cited By ~ 4

Author(s):

Andrzej Wiśnicki

Keyword(s):

Banach Space ◽

Fixed Point ◽

Unit Sphere ◽

Convex Set ◽

Nonexpansive Mappings ◽

Fixed Point Property ◽

Point Property ◽

Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

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On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

Fixed Point Theory and Applications ◽

10.1155/2010/596952 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

Jingxin Zhang ◽

Yunan Cui

Keyword(s):

Fixed Point ◽

Nonexpansive Mappings ◽

Fixed Point Property ◽

Point Property ◽

Geometric Constants

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Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2012-230 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 3

Author(s):

Withun Phuengrattana ◽

Suthep Suantai

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Nonexpansive Mappings ◽

Asymptotically Nonexpansive Mappings ◽

Asymptotically Nonexpansive

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Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

Abstract and Applied Analysis ◽

10.1155/2014/809528 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Yuanheng Wang

Keyword(s):

Banach Space ◽

Fixed Point ◽

Real Banach Space ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Common Fixed Points ◽

Uniformly Gâteaux Differentiable Norm ◽

Asymptotically Nonexpansive Mappings ◽

Asymptotically Nonexpansive ◽

Differentiable Norm

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

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