zbMATH — the first resource for mathematics

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
Full Text: DOI
[1] Banaś, J., On modulus of noncompact convexity and its properties, Can. math. bull., 30, 186-192, (1987) · Zbl 0585.46011
[2] Van, Dulst D.; Sims, B., Proc. Conf. on Banach Space Theory and its Applications, Bucharest, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type KK, (1981) · Zbl 0512.46015
[3] Goebel, K., Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compos. math., 22, 269-274, (1970) · Zbl 0202.12802
[4] Goebel, K.; Sekowski, T., The modulus of noncompact convexity, Ann. univ. marie Curie-skłodowska, 38, 41-48, (1984), (Sect. A) · Zbl 0607.46011
[5] Gossez, J.P.; Lami Dozo, E., Some geometric properties related to the fixed point property for nonexpansive mappings, Pacif. J. math., 40, 565-573, (1972) · Zbl 0223.47025
[6] Huff, R., Banach spaces which are nearly uniformly convex, Rocky mount. J. math., 10, 743-749, (1980) · Zbl 0505.46011
[7] \scJimenez M.A. & \scLlorens F.E., On the stability of the fixed point property for nonexpansive mappings (preprint).
[8] Karlovitz, L.A., Existence of fixed points for nonexpansive mappings in a space without normal structure, Pacif. J. math., 66, 153-159, (1976) · Zbl 0349.47043
[9] Kirk, W.A., A fixed point theorem for mappings which do not increase distances, Am. math. mon., 72, 1004-1006, (1965) · Zbl 0141.32402
[10] \scKuczumow T. & \scPrus S., Compact asymptotic centers and fixed points of multivalued nonexpansive mappings (preprint).
[11] Lin, P.K., Unconditional bases and fixed points of nonexpansive mappings, Pacif. J. math., 116, 69-76, (1985) · Zbl 0566.47038
[12] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. am. math. soc., 73, 591-597, (1967) · Zbl 0179.19902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.