Approximate solutions of quasiequilibrium problems in Banach spaces

TL;DR

This paper critically examines a recent claim about quasiequilibrium problems in Banach spaces, providing a counterexample to disprove it and offering a new lemma to establish approximate solutions.

Contribution

It corrects a previous incorrect result and introduces a generalized lemma to prove the existence of approximate solutions in quasiequilibrium problems.

Findings

Counterexample disproves a recent existence claim.

Generalized lemma on continuous '-minimizers.

Establishment of approximate solutions for quasiequilibrium problems.

Abstract

In this note we show that a recent existence result on quasiequilibrium problems, which seems to improve deeply some well-known results, is not correct. We exhibit a counterexample and we furnish a generalization of a lemma about continuous "-minimizers of quasiconvex functions depending on a parameter. This allows to establish an existence result of approximate solutions of quasiequilibrium problems.