A converse of the Banach contraction principle for partial metric spaces and the continuum hypothesis

A version of the Bessaga inverse of the Banach contraction principle for partial metric spaces is presented. Equivalence of that version and the continuum hypothesis is shown as well

Compactness in normed spaces: a unified approach through semi-norms

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,\mathbb{R})$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known compactness criteria (including the Arzel\`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation and provide a full characterization of linear integral operators acting from the space of functions of bounded Jordan variation to the space of functions of bounded Schramm variation in therms of their generating kernels.Comment: Th version accepted for publication in Topological Methods in Nonlinear Analysi

Integral operators in the spaces of functions of bounded Schramm variation

In this paper we provide a full characterization of linear integral operators acting from the space of functions of bounded Jordan variation to the space of functions of bounded Schramm variation in terms of their generating kernels.Comment: This paper grew out of the unpublished part of the paper by the same authors, Compactness in normed spaces: a unified approach through semi-norms, arXiv:2111.10547v

Adaptive Rolling Plans Are Good

Here we prove the goodness property of adaptive rolling plans in a multisector optimal growth model under decreasing returns in deterministic environment. Goodness is achieved as a result of fast convergence (at an asymptotically geometric rate) of the rolling plan to balanced growth path. Further on, while searching for goodness, we give a new proof of strong concavity of an indirect utility function – this result is achieved just with help of some elementary matrix algebra and differential calculus

The existence of equilibrium without fixed-point arguments

This paper gives a proof of the existence of general equilibrium without the use of a fixed point theorem. Unlike other results of this type, the conditions we use do not imply that the set of equilibrium prices is convex. We use an assumption on the excess demand correspondence that is related to, but weaker than, the weak axiom of revealed preference (WARP). The proof is carried out for compact and convex valued upper hemicontinuous excess demand correspondences satisfying this WARP-related condition and some other standard conditions. We also provide an algorithm for finding equilibrium prices

Some remarks on lower hemicontinuity of convex multivalued mappings

For a multifunction a condition sufficient for lower hemicontinuity is presented. It is shown that under convexity of graph it is possible for a multifunction to be not continuous only when a special representation of points of its domain is not feasible

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

Some remarks on lower hemicontinuity of convex multivalued mappings

For a multifunction a condition sufficient for lower hemicontinuity is presented. It is shown that under convexity of graph it is possible for a multifunction to be not continuous only when a special representation of points of its domain is not feasible

Uniform boundedness of feasible per capita output streams under convex technology and non-stationary labor

This paper shows that under classical assumptions on technological mapping and presence of an indispensable production factor there is a bound on long-term per capita production. The bound does not depend on initial state of economy. It is shown that all feasible processes converge uniformly over every bounded set of initial inputs p.c. to some set (dependent on technology)