Volume 7, Issue 2

Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K . We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function ƒ on X . We prove (under certain assumptions on ƒ) that the iterates of a generic mapping in these spaces converge strongly to a retraction onto F .

In [Evans, Humke and O'Malley, J. Appl. Anal. 6: 1–16, 2000], the present authors and Richard O'Malley showed that in order for a function be universally polygonally approximable it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ – (1 – ε ) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, , such that each E i is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, ƒ, whose set of points of non-quasicontinuity is precisely E . Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of F σ sets.

We investigate the issue of uniqueness and nonuniqueness of minimizers for the approximation of variational problems. We show that when the continuous problem does not admit a minimizer its approximation by finite elements may lead to several discrete minimizers.

The canonical solutions of the truncated Hamburger moment problem (both in the classical and degenerate cases) are found. The Nevanlinna theorem which provides the noncanonical solutions of the truncated Hamburger problem is also rederived in the framework of the operator approach.

We show that for a σ -finite diffused Borel measure in a nondiscrete locally bounded topological group there is a meager set whose complement is of measure zero.

In this article we investigate the pointwise, discrete and transfinite convergences in the classes of real functions defined on topological spaces which are upper and lower quasicontinuous at each point.

A cardinal related to compositions of Sierpiński-Zygmund functions will be considered. A combinatorial characterization of the cardinal is given and is used to answer some questions of K. Ciesielski and T. Natkaniec. It is shown that the bounding number of the continuum may be strictly smaller than continuum.

We consider the numerical solution of quasilinear elliptic Neumann problems. The basic difficulty is the non-injectivity of the operator, which can be overcome by suitable factorization. We extend the gradient-finite element method (GFEM), introduced earlier by the authors for Dirichlet problems, to the Neumann problem. The algorithm is constructed and its convergence is proved.

The aim of the paper is to characterize those sets of points at which sequence of real functions from a given class F converges as well as sets of points of convergence to infinity of such sequences. As F we consider quasi-continuous functions and some other subclasses of Baire measurable functions.

Each statistic, which is pairwise sufficient and (in a natural sense) countably complete, is a minimal pairwise sufficient statistic. The Basu theorem for pairwise sufficient statistic is also obtained.