Volume 3, Issue 2

This article is a survey of the recent results that concern real functions (from into ℝ) and whose solutions or statements involve the use of set theory. The choice of the topics follows the author's personal interest in the subject, and there are probably some important results in this area that did not make it to this survey. Most of the results presented here are left without proofs.

In the present paper we discuss some problems formulated in Ciesielski [J. Appl. Anal. 3: 143–190, 1997].

We discuss some set–theoretical questions concerning the notion of sup-measurability of functions of two variables and the existence and uniqueness of solutions of ordinary differential equations.

The gradient method is considered in Hilbert spaces. Earlier results on linear convergence are extended to systems of equations with certain non-differentiable operators. The method includes the approximate solution of elliptic systems of quasilinear boundary value problems.

In this paper, we prove a main result about Ky Fan minimax inequality in any compact H -space. Then, we apply this result to fixed point problem, intersection problem of sets with convex sections, minimax inequality of the Von Neumann type, and variational inequality.

The existence of a Walras equilibrium is proved in the framework of a finitely economy with production. The theorem assumes the production set correspondence to be closed–valued and absolutely continuous with respect to the “weighting” finitely additive measure. The result is obtained by using the Kakutani's fixed point theorem and some convergence results stated for multifunctions.

We present an adaptation of the (by now classical) parallel projection method for finding a point in the nonempty intersection of a finite number of closed convex sets in a Hilbert space. The adaptation consists of controlling at each iteration step whether or not some condition is fulfilled; if not, the adapted next iteration point is determined such that its position with respect to the intersection is better than the usual next iteration point. This may improve the speed of convergence.