Stability of systems of general functional equations in the compact-open topology

TL;DR

This paper establishes a stability result for general systems of functional equations in the compact-open topology, showing that approximately satisfying solutions are close to exact solutions under certain conditions.

Contribution

It introduces a broad concept of functional equations and proves their stability using nonstandard analysis, extending classical stability results to more general systems.

Findings

Stability holds for systems with modest continuity and boundedness conditions.

Approximate solutions are close to true solutions on large compact sets.

The proof employs an 'almost-near' principle via nonstandard analysis.

Abstract

We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as various types of functional equations and recursion formulas occurring in mathematical analysis or combinatorics, respectively, become special cases (of systems) of such equations. Assuming that $X$ is a locally compact and $Y$ is a completely regular topological space, we show that systems of such functional equations, with parameters satisfying rather a modest continuity condition, are stable in the following intuitive sense: Every $k$-tuple of ``sufficiently continuous,'' ``reasonably bounded'' functions $X \to Y$ satisfying the given system with a ``sufficient precision'' on a ``big enough'' compact set is already ``arbitrarily close'' on an ``arbitrarily big'' compact set to a $k$-tuple of continuous functions solving the system. The result is derived as a consequence of certain intuitively appealing ``almost-near'' principle using the relation of infinitesimal nearness formulated in terms of nonstandard analysis.