Banach spaces with the uniform Opial property. (English) Zbl 0786.46023

The Hausdorff measure of noncompactness \(\chi(A)\) of a bounded subset \(A\) of a Banach space \(X\) is defined as the infimum of all \(r>0\) such that \(A\) can be covered by a finite union of balls of radius \(r\). For \(\varepsilon\in[0,1]\), the modulus of non compact convexity [introduced by K. Goebel and T. Sekowski, Ann. Univ. Marie Curie- Skłodowska, Sect. A 38, 41-48 (1984; Zbl 0607.46011)] is defined by \(\Delta(\varepsilon)= \inf(1-\inf \{\| x\|\): \(x\in A\})\), where the first infimum is taken over all closed convex subsets \(A\) of the unit ball of \(X\) with \(\chi(A)\geq\varepsilon\). Let \(\Delta(1-)= \lim_{\varepsilon\to 1-} \Delta(\varepsilon)\). If \(\Delta(1-)>0\) then \(X\) is reflexive (Theorem 1.2) and if \(\Delta(1-)=1\) then \(X\) is said to satisfy property (L). The Banach space \(X\) has the uniform Opial property provided for every \(c>0\) there is an \(r>0\) such that \(1+r\leq\liminf_ n \| x+x_ n\|\), for each \(x\in X\) with \(\| x\|\geq c\) and each sequence \((x_ n)\) in \(X\) with \(\text{w-lim } x_ n=0\) and \(\liminf_ n \| x_ n\|\geq 1\). The author proves (Theorem 1.3) that the space \(X\) has property (L) if and only if it is reflexive and has the uniform Opial property.
Property (L) is also relevant in fixed point theory. D. van Dulst and B. Sims, Lect. Notes Math. 991, 35-43 (1983; Zbl 0512.46015), proved that if \(X\) is nearly uniformly convex [a notion due to R. Huff, Rocky Mount. J. Math. 10, 743-749 (1980; Zbl 0505.46011)] then it has the fixed point property for nonexpansive mappings on bounded closed convex sets. The author proves that the same is true for \(X\) and \(X^*\), whenever \(X^*\) has property (L) (in this case \(X\) is necessarily reflexive).


46B20 Geometry and structure of normed linear spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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