Piotr Maćkowiak

Nonautonomous superposition operators in the spaces of functions of bounded variation

The main goal of this paper is to give an answer to the main problem of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we give necessary and sufficient conditions which guarantee that nonautonomous superposition operators map that space into itself and are locally bounded. Moreover, special attention is drawn to nonautonomous superposition operators generated by locally bounded mappings as well as to superposition operators generated by functions with separable variables.

A counterexample to Ljamin’s theorem

One of the well-known results ensuring that a nonautonomous superposition operator maps the set of functions of one variable of bounded variation in the sense of Jordan into itself is the theorem by A. G. Ljamin. According to that theorem it suffices to consider the class of functions which are uniformly Lipschitz w.r.t. the second variable and of uniformly bounded variation w.r.t. the first variable. Unfortunately, Ljamin’s result is false. Here we deliver an example contradicting sufficiency of those conditions.

The existence of equilibrium in a simple exchange model

This paper gives a new proof of the existence of equilibrium in a simple model of an exchange economy. We first formulate and prove a simple combinatorial lemma and then we use it to prove the existence of equilibrium. The combinatorial lemma allows us to derive an algorithm for the computation of equilibria. Though the existence theorem is formulated for functions defined on open simplices, it is equivalent to the Brouwer fixed point theorem.MSC: Primary 91B02; secondary 91B50; 54H25.