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A note on properties that imply the fixed point property. (English) Zbl 1147.46011
There are many geometric properties of a Banach space that imply that it possesses the weak fixed point property for nonexpansive mappings. In this article, the authors consider the relationships between some of these geometric properties: the Schur property, Opial’s condition, weak orthogonality, \(k\)-uniform rotundity, property \((M)\), and several others. For example, the authors note that a Banach space \(X\) has the Schur property if and only if it satisfies Opial’s condition and \(R(X)=1\) (where \(R(X) \overset{\text{def}}{=} \sup\{\liminf_n \| x_n -x\| : x_n\overset{w}{} x, \| x_n\| \leq 1, \| x\|\leq 1\}\) is a parameter introduced by J. García-Falset in [Houston J. Math. 20, No. 3, 495–506 (1994; Zbl 0816.47062)]. The authors also prove that, if the Jordan-von Neumann constant of a Banach space \(X\) is less than \((1 + \sqrt{3})/2\), then \(X\) has uniform normal structure.

MSC:
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
Citations:
Zbl 0816.47062
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References:
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