It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y has the asymptotic (P) property (which is weaker than the condition WCS(Y) > 1), then X ⊕ Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-1-8,
author = {Stanis\l aw Prus and Andrzej Wi\'snicki},
title = {On the fixed point property in direct sums of Banach spaces with strictly monotone norms},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {87-99},
zbl = {1147.47038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-1-8}
}
Stanisław Prus; Andrzej Wiśnicki. On the fixed point property in direct sums of Banach spaces with strictly monotone norms. Studia Mathematica, Tome 187 (2008) pp. 87-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-1-8/