Asymptotic normal structure and the semi-opial property

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

“…The condition w-AN (X) > 1 implies the weak fixed point property for nonexpansive mappings as a consequence of the Baillon-Schöne-berg theorem (see also [7]). …”

“…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]). …”

“…Recall that a Banach space is said to have the semi-Opial (weak semiOpial) property [8,25], SO (w-SO) for short, if for each bounded nonconstant asymptotically regular sequence {x n } (with a weakly compact convex hull), there exists a subsequence {x n i } , weakly convergent to x, such that…”

“…A proof similar to the above one was used in [8] to prove that every Banach space with the SO property is reflexive.…”

“…James [5]. This is essentially the space which has been discussed in various places in the literature, e.g., [1,2,4,5,7,8,10,15,16,19,20,21,22,23,25,26,28,39].…”