The stability of a biadditive functional equation via a new direct method

TL;DR

This paper introduces a new direct method for proving the stability of biadditive functional equations, simplifying the process and demonstrating its effectiveness across different mathematical contexts.

Contribution

The paper presents a novel direct approach for establishing the stability of complex functional equations, improving upon traditional methods in simplicity and efficiency.

Findings

The direct method effectively proves stability of the given functional equations.

The approach outperforms traditional proof techniques in simplicity and efficacy.

Application to various equations confirms the method's broad validity.

Abstract

Addressing stability in functional equations is a critical task with broad implications across mathematics and its applications. In this paper, we present a novel direct method for proving the stability of the following equation, \begin{eqnarray*} f(x,y)=\alpha f(f_1(x,y))+\beta f(f_2(x,y)) \end{eqnarray*} subjecting to certain constraints on the constants $\alpha$ and $\beta$, as well as the functions $f_1(x,y)$ and $f_2(x,y)$. We introduce the direct method to prove the stability of the following functional equation and inequality, \begin{eqnarray*} \|f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\| \leq \|x\|^p\|y\|^p\|z\|^p\|w\|^p; \end{eqnarray*} \begin{eqnarray*} \begin{split} \;&\|f\left(x+y,z-w\right)+af\left(\frac{x-y}{a},z+w\right)-2f(x,z)+2f(y,w) \|\\ \;&\leq \left\| \rho\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)\right\|. \end{split} \end{eqnarray*} This technique efficiently confirms stability, outperforming traditional proof methods in simplicity and efficacy. The application of our direct approach to various equations solidifies its validity, ensuring stable behavior across different mathematical contexts.