Symmetric Quadratic Energy and Hyers-Ulam Stability in Extended $b$-Metric Spaces

Symmetric Quadratic Energy and Hyers-Ulam Stability

Abstract

This paper examines the Hyers--Ulam stability of a quadratic functional equation arising naturally in nonlinear analysis and energy modeling.
By employing both the classical direct method and a fixed point approach, we establish stability results within the framework of extended
$b$-metric spaces. The fixed point method provides a unified technique for proving the existence and uniqueness of an exact quadratic mapping
approximating any function that satisfies the associated inequality. Furthermore, we present an application to a quadratic energy model to
demonstrate the effectiveness of the proposed stability analysis. The results obtained extend and generalize several known stability theorems
for quadratic functional equations in various metric structures.