Nonstandard Methods in Fixed Point Theory

Abstract

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Uniform asymptotic normal structure, the uniform semi‐Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

“…We say that X has asymptotic normal structure (with respect to the weak topology) [4], AN S (respectively, w-AN S) for short, if for each bounded closed (weakly compact) and convex subset C of X consisting of more than one point and each asymptotically regular sequence {x n } in C, there is a point x ∈ C such that lim inf n x − x n < diam (C) (see also [1,2,7,8,19,20,26,30,36]). …”

“…For more information about the connections between the above mentioned geometric properties of Banach spaces (and other ones) see [1,2,3,13,14,18,19,20,27,29,33,34,35,37,38,39,40].…”

Uniform asymptotic normal structure, the uniform semi‐Opialproperty, and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part II

“…The asymptotic center of {x n } in C [13] is the set Ac (C, {x n }) = {x ∈ C : r a (x, {x n }) = r (C, {x n })} . For more details see [1], [16] and [17].…”

“…In a Banach space (X, · ) we denote by κ 0 (X) the infimum of the numbers κ (C) where C is a closed, convex, bounded and nonempty subset of X. It is known [2] , [16] (1) and 0 (X) < 1 if and only κ 0 (X) > 1 [12]. Therefore κ 0 (X) ≤ √ 2 [2].…”

Fixed point property and normal structure for Banach spaces associated to locally compact groups

“…In [14] Kirk, extending Browder's Theorem, showed that a weakly compact convex subset of a Banach space with normal structure has the fpp. In [2] Alspach exhibited a weakly compact convex subset K of the Lebesgue space L 1 [0,1] and an isometry T : K → K without a fixed point, proving, thereby, that the space L 1 [0, 1] does not have the fpp. However, in [19], Maurey, using the techniques of ultraproducts, showed that reflexive subspaces of L 1 [0, 1], as well as the sequence space c 0 , have the fpp.…”