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THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS $R(a, X)$

Part of: Nonlinear operators and their properties Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  28 June 2013

ANDRZEJ WIŚNICKI
ANDRZEJ WIŚNICKI*
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland
*
[email protected]
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Abstract

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We 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.

Keywords

nonexpansive mappingasymptotically nonexpansive mappingfixed pointdirect sum

MSC classification

Secondary: 47H10: Fixed-point theorems 46B20: Geometry and structure of normed linear spaces 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 79 - 91
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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