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On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces

Kozlowski W. M.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

113--136

Let C be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space X. We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings Tt: C → C. Each of Tt: is assumed to be pointwise Lipschitzian, that is, there exists a family of functions αt: C → [0, ∞) such that ||Tt(x) — Tt(y)\\ ≤ αt:(x) || - y|| for x,y € C. The paper demonstrates how the weak compactness of C plays an essential role in proving the weak convergence of these processes to common fixed points.

On weak compactness and lower closure results for Pettis integrable (multi) functions

Balder E. J., Sambucini A. R.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 1

53-61

In [4, 5, 7] an abstract, versatile approach was given to sequential weak compactness and lower closure results for scalarly integrable functions and multifunctions. Its main tool is an abstract version of the Komlos theorem, which applies to scalarly integrable functions. Here it is shown that this same approach also applies to Pettis integrable multifunctions, because the abstract Komlos theorem can easily be extended so as to apply to generalized Pettis integrable functions. Some results in the literature are thus unified.

Weak sequential completeness and compactness in topological function spaces

Nowak M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

155-165

Let (E, r) be a Hausdorff locally convex-solid function space (over a cr-finite measure space) and let E* stand for its topological dual. It is proved that the space (E, r) is weakly sequentially complete if and only if r is a c-Lebesgue and cr-Levy topology. In particular, a characterization of weak sequential completeness of Or-licz spaces L* in terms of Orlicz functions is given. Moreover, it is proved that the Eberlein-Smulian type theorem remains valid for a locally convex space (E, o~(E, E*)). A characterization of conditional and relative weak compactness in (E, r) is obtained.