Approximately Quadratic Mappings on Restricted Domains

TL;DR

This paper introduces a generalized quadratic functional equation with parameters, explores its stability on restricted domains, and examines the asymptotic behavior of the solutions, extending classical quadratic functional analysis.

Contribution

It generalizes quadratic functional equations with parameters and investigates their stability and asymptotic properties on restricted domains, which is a novel extension.

Findings

The functional equation is quadratic when r, s are rational.

Stability results are established for the equation on restricted domains.

Asymptotic behavior of generalized quadratic mappings is characterized.

Abstract

In this paper, we introduce a generalized quadratic functional equation $f(rx + sy) = rf(x) + sf(y) - rsf(x - y)$ where $r, s$ are nonzero real numbers with $r + s = 1.$ We show that this functional equation is quadratic if $r, s$ are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.