Generalized Quadratic Functional Equation and Its Stability over Non-Archimedean Normed Space

Abstract

A functional equation is one of the most important and fascinating areas of mathematics, which involves simple algebraic manipulations and can lead to a variety of interesting results. In recent decades, numerous authors have studied different types of functional equation and its stability, such as Hyers-Ulam Stability, Hyers-Ulam-Rassias Stability, and generalized Hyers-Ulam Stability. The stability of functional equations and mixedtype functional equations has been extensively explored by numerous researchers across various spaces, yielding intriguing results primarily in the classical (Archimedean) setting. In recent years, attention has shifted towards investigating the Hyers-Ulam stability (HUS) of generalized Quadratic functional equations in non-Archimedean normed spaces. This article demonstrates the Hyers-Ulam Stability (HUS) of Quadratic functional equations. 𝑔(3𝑥 − 𝑦) + 𝑔(𝑥 + 3𝑦) = 10𝑔(𝑥) + 10𝑔(𝑦), 𝑔(𝜐𝑥 − 𝑦) + 𝑔(𝑥 + 𝜐𝑦) = (𝜐 2 + 1)𝑔(𝑥) + (𝜐 2 + 1)𝑔(𝑦), for any integer 𝜐 ≠ 0, in NAN space by using the direct method. Also, we have given some suitable counterexamples.