Approximate Bi-Affine Mappings

Abstract

In this paper, we introduce a multi-variable bi-affine functional equation of the form $f\left({\sum }_{i=1}^m{\alpha }_i{x}_i,{\sum }_{j=1}^n{\beta }_j{y}_j\right)={\sum }_{i=1}^m{\sum }_{j=1}^n{\alpha }_i{\beta }_{j}f(x_i,y_j)$, where m and n are integers and $m,n\ge 2$ and ${\alpha }_i,{\beta }_j$ are nonzero scalars. We investigate the Hyers–Ulam stability of this functional equation in Banach spaces using the direct method. The results obtained in this paper can be regarded as a generalization of stability results for the classical bi-Jensen functional equation and its multi-variable mean-type variants.

1. Introduction

The stability theory of functional equations investigates whether a mathematical object satisfying a functional equation approximately is close to an exact solution. This fundamental problem was first posed by S. M. Ulam [1] (origins of the stability problem posed in 1940) concerning group homomorphisms and was subsequently solved by D. H. Hyers [2] for the Cauchy additive functional equation in Banach spaces. Since these pioneering works, the theory has been extensively developed by mathematicians such as T. Aoki [3] and Th. M. Rassias [4], who introduced generalized bounds involving sums of powers of norms, and was further refined by Găvruta [5] and others. In the context of multi-variable mappings, significant progress has been made in analyzing Jensen-type equations. Park and Bae [6] derived the general solution and stability for the Cauchy–Jensen functional equation and later extended these results to the multi-variable Cauchy–Jensen functional equation. Research has also expanded into two-variable mappings; the authors [7,8] previously established the solution and stability for the bi-Jensen functional equation and, more recently, for the multi-variable bi-Jensen functional equation. In this paper, we generalize these findings by introducing a weighted multi-variable functional equation, referred to as the bi-affine functional equation. Unlike previous bi-Jensen models that typically utilize uniform weights, our proposed equation incorporates the arbitrarily fixed nonzero scalars ${\alpha }_i$ and ${\beta }_j$ as follows:

$$f\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)=\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}f(x_i,y_j)$$

(1)

where m and n are integers with $m,n\ge 2$. From [9,10,11,12,13,14,15,16], one can find interesting results on multi-additive mappings, multi-quadratic mappings, multi-Jensen mappings, multi-Cauchy–Jensen mappings, multi-Cauchy-Jensen-quadratic mappings and general linear mappings, etc. They contain stability results for equations closely related to Equation (1). The primary objective of this paper is to investigate the Hyers–Ulam stability of functional Equation (1) in Banach spaces by employing the direct method. Furthermore, we provide a definition for bi-affine mappings and analyze their properties in relation to the generalized weights. The results presented herein can be regarded as a broad generalization of earlier stability results concerning bi-Jensen functional equations and their mean-type variants.

2. General Solution and Hyers–Ulam Stability

We define the constants $\alpha$ and $\beta$ as follows:

$$\alpha :=\sum _{i=1}^m{\alpha }_i,\beta :=\sum _{j=1}^n{\beta }_j$$

where we assume that $\alpha \ne 0$ and $\beta \ne 0$.

Definition 1.

Let $\mathbf{a}=({\alpha }_1,\dots ,{\alpha }_m)$ and $\mathbf{b}=({\beta }_1,\dots ,{\beta }_n)$ be fixed scalar vectors. For real vector spaces $X,Y$ and Z, a mapping $f:X\times Y\to Z$ is called an a-b-bi-affine mapping (or simply a bi-affine mapping) if it satisfies functional Equation (1).

Theorem 1.

Let $X,Y$ and Z be real vector spaces and let $f:X\times Y\to Z$ be a mapping. Define three mappings $A:X\to Z$, $A^{\prime }:Y\to Z$, and $B:X\times Y\to Z$ by $A\left(x\right):=f(x,0)-f(0,0)$, $A^{\prime }\left(y\right):=f(0,y)-f(0,0)$, and $B(x,y):=f(x,y)-A\left(x\right)-A^{\prime }\left(y\right)-f(0,0)$ for all $(x,y)\in X\times Y$. Assume that $A,A^{\prime }$ and B satisfy

$$A\left({\alpha }_{i}x\right)={\alpha }_{i}A\left(x\right),A^{\prime }\left({\beta }_{j}y\right)={\beta }_j{A}^{\prime }\left(y\right),B({\alpha }_{i}x,y)={\alpha }_{i}B(x,y),B(x,{\beta }_{j}y)={\beta }_{j}B(x,y)$$

(2)

for all $(x,y)\in X\times Y$ and all $(i,j)\in \{1,\dots ,m\}\times \{1,\dots ,n\}$. Then, f satisfies Equation (1) if, and only if, $f,A,A^{\prime }$ and B satisfy the following conditions:

(a)

A and $A^{\prime }$ are additive and B is bi-additive.

(b)

If A is a non-zero mapping, then $\beta =1$.

(c)

If $A^{\prime }$ is a non-zero mapping, then $\alpha =1$.

(d)

If $f(0,0)\ne 0$, then $\alpha \beta =1$.

Proof. 

Let f satisfy the functional Equation (1) and let $c:=f(0,0)$. Setting $x_1=\dots =x_m=y_1=\dots =y_n=0$ in (1), we have $c=\alpha \beta c$, and so $(1-\alpha \beta )c=0$. Thus, if $\alpha \beta \ne 1$, c must be 0. That is, the condition (d) holds.

Putting $y_1=\dots =y_n=0$ in (1), we have

$$A\left(\sum _{i=1}^m{\alpha }_i{x}_i\right)+c=\beta \sum _{i=1}^m{\alpha }_i(A\left(x_i\right)+c)=\beta \sum _{i=1}^m{\alpha }_{i}A\left(x_i\right)+\alpha \beta c$$

for all $x_1,\dots ,x_m\in X$. Since $c=\alpha \beta c$, this reduces to

$$A\left(\sum _{i=1}^m{\alpha }_i{x}_i\right)=\beta \sum _{i=1}^m{\alpha }_{i}A\left(x_i\right)$$

(3)

for all $x_1,\dots ,x_m\in X$.

If $\beta \ne 1$, setting $x_2=\dots =x_m=0$ in Equation (3) gives $A\left({\alpha }_1{x}_1\right)=\beta {\alpha }_{1}A\left(x_1\right)$. By the assumption (2), we also know $A\left({\alpha }_1{x}_1\right)={\alpha }_{1}A\left(x_1\right)$. Thus, ${\alpha }_{1}A\left(x_1\right)=\beta {\alpha }_{1}A\left(x_1\right)$, which implies $(1-\beta ){\alpha }_{1}A\left(x_1\right)=0$. Since ${\alpha }_1\ne 0$ and $\beta \ne 1$, we must have $A\left(x\right)=0$ for all $x\in X$. Hence, condition (b) holds.

If $\beta =1$, Equation (3) becomes $A\left({\sum }_{i=1}^m{\alpha }_i{x}_i\right)={\sum }_{i=1}^m{\alpha }_{i}A\left(x_i\right)$ for all $x_1,\dots ,x_m\in X$. Setting $x_3=\dots =x_m=0$, we have $A({\alpha }_1{x}_1+{\alpha }_2{x}_2)={\alpha }_{1}A\left(x_1\right)+{\alpha }_{2}A\left(x_2\right)$ for all $x_1,x_2\in X$. By the assumption (2), we obtain $A({\alpha }_1{x}_1+{\alpha }_2{x}_2)=A\left({\alpha }_1{x}_1\right)+A\left({\alpha }_2{x}_2\right)$ for all $x_1,x_2\in X$. This shows that A is additive.

By applying a similar argument, we can deduce that if $\alpha \ne 1$, then $A^{\prime }\left(y\right)=0$ for all $y\in Y$, and if $\alpha =1$, then $A^{\prime }$ is additive. Note that condition (c) holds.

By the definition of B, we have $B(x,0)=0$ and $B(0,y)=0$. Since f and c independently satisfy Equation (1), we have

$$\begin{array}{cc}\hfill B(\sum _{i=1}^m& {\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j)=f\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)-A\left(\sum _{i=1}^m{\alpha }_i{x}_i\right)-A^{\prime }\left(\sum _{j=1}^n{\beta }_j{y}_j\right)-c\hfill \\ \hfill & =\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}f(x_i,y_j)-\sum _{i=1}^m{\alpha }_{1}A\left(x_i\right)-\sum _{j=1}^n{\beta }_j{A}^{\prime }\left(y_j\right)-\alpha \beta c\hfill \\ \hfill & =\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}f(x_i,y_j)-\beta \sum _{i=1}^m{\alpha }_{1}A\left(x_i\right)-\alpha \sum _{j=1}^n{\beta }_j{A}^{\prime }\left(y_j\right)-\alpha \beta c=\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}B(x_i,y_j)\hfill \end{array}$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. Setting $y_1=y$ and $y_2=\dots =y_n=0$, we obtain

$$B\left(\sum _{i=1}^m{\alpha }_i{x}_i,{\beta }_{1}y\right)={\beta }_1\sum _{i=1}^m{\alpha }_{i}B(x_i,y)$$

for all $x_1,\dots ,x_m\in X$ and all $y\in Y$. By the assumption (2), we get ${\beta }_{1}B({\sum }_{i=1}^m{\alpha }_i{x}_i,y)={\beta }_1{\sum }_{i=1}^m{\alpha }_{i}B(x_i,y)$ for all $x_1,\dots ,x_m\in X$ and all $y\in Y$. Canceling ${\beta }_1$ yields $B({\sum }_{i=1}^m{\alpha }_i{x}_i,y)={\sum }_{i=1}^m{\alpha }_{i}B(x_i,y)$ for all $x_1,\dots ,x_m\in X$ and all $y\in Y$. Setting $x_3=\dots =x_m=0$ and applying (2), it readily follows that $B(x_1+x_2,y)=B(x_1,y)+B(x_2,y)$ for all $x_1,x_2\in X$ and all $y\in Y$. Through symmetric reasoning for the second variable, B is proven to be bi-additive. Therefore condition (a) holds.

Conversely, assume that $f,A,A^{\prime }$ and B satisfy conditions (a), (b), (c), and (d). By condition (a), we have

$$\begin{array}{cc}\hfill f\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)& =B\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)+A\left(\sum _{i=1}^m{\alpha }_i{x}_i\right)+A^{\prime }\left(\sum _{j=1}^n{\beta }_j{y}_j\right)+c\hfill \\ \hfill & =\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}B(x_i,y_j)+\sum _{i=1}^m{\alpha }_{i}A\left(x_i\right)+\sum _{j=1}^n{\beta }_j{A}^{\prime }\left(y_j\right)+c\hfill \end{array}$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. By conditions (b), (c), and (d), we can rewrite the single-variable and constant terms as double sums exactly matching the right-hand side of (1). Thus, f satisfies functional Equation (1). □

Remark 1.

If $\beta =1$, then $A\left({\alpha }_{i}x\right)={\alpha }_{i}A\left(x\right)$ for all $x\in X$ and all $i\in \{1,\dots ,m\}$. If $\alpha =1$, then $A^{\prime }\left({\beta }_{j}y\right)={\beta }_j{A}^{\prime }\left(y\right)$ for all $y\in Y$ and all $j\in \{1,\dots ,n\}$.

Example 1.

Consider the real vector space of $d\times d$ real matrices, denoted by $X=Y=Z=M_d\left(\mathbb{R}\right)$. We examine the case where the scalar constants satisfy $\alpha =\beta =1$. For instance, this corresponds to $m=2$ with ${\alpha }_1=1/3$ and ${\alpha }_2=2/3$, and $n=2$ with ${\beta }_1=1/2$ and ${\beta }_2=1/2$.

Let $f:M_d\left(\mathbb{R}\right)\times M_d\left(\mathbb{R}\right)\to M_d\left(\mathbb{R}\right)$ be a mapping defined by

$$f(L,M):=LM+L^T+\mathrm{trace}\left(M\right)I+C\mathrm{for}\mathrm{all}L,M\in M_d\left(\mathbb{R}\right),$$

where $L^T$ denotes the transpose of L, $\mathrm{trace}\left(M\right)$ is the trace of M, I is the identity matrix, and C is a fixed nonzero constant matrix.

Since f satisfies (1), according to Theorem 1, the general solution is of the form $f(L,M)=B(L,M)+A\left(L\right)+A^{\prime }\left(M\right)+C$, where $B(L,M)=LM$, $A\left(L\right)=L^T$ and $A^{\prime }\left(M\right)=\mathrm{trace}\left(M\right)$ for all $L,M\in M_d\left(\mathbb{R}\right)$.

This example comprehensively demonstrates that the generalized bi-affine functional equation covers not only polynomial-type mappings but also more general linear algebraic structures involving matrix operations, perfectly aligning with the necessary conditions $\left(\mathrm{b}\right)$, $\left(\mathrm{c}\right)$, and $\left(\mathrm{d}\right)$ of Theorem 1.

Let X and Y be vector spaces and $\mathcal{Z}$ be a Banach space. We define the difference mapping $Df:X^m\times Y^n\to \mathcal{Z}$ for a mapping $f:X\times Y\to \mathcal{Z}$ by

$$Df(x_1,\dots ,x_m,y_1,\dots ,y_n):=f\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)-\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}f(x_i,y_j)$$

for all $x_1,\dots ,x_m\in X$ and $y_1,\dots ,y_n\in Y$.

Theorem 2.

Let $\left|\alpha \beta \right|>1$ and let $X,Y$ be vector spaces and $\mathcal{Z}$ be a Banach space. Suppose that a mapping $f:X\times Y\to \mathcal{Z}$ satisfies the inequality

$$\parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n)\parallel \le \phi (x_1,\dots ,x_m,y_1,\dots ,y_n)$$

(4)

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$, where $\phi :X^m\times Y^n\to [0,\infty )$ is a given control function. If the function φ satisfies the convergence condition

$$\sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\phi ({\alpha }^k{x}_1,\dots ,{\alpha }^k{x}_m,{\beta }^k{y}_1,\dots ,{\beta }^k{y}_n)<\infty$$

(5)

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$, then there exists a unique mapping $F:X\times Y\to Z$ satisfying functional Equation (1) and the inequality

$$\parallel f(x,y)-F(x,y)\parallel \le \sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y),$$

(6)

where $\tilde{\phi }(x,y):=\phi (\underset{m}{\underbrace{x,\dots ,x}},\underset{n}{\underbrace{y,\dots ,y}})$ for all $(x,y)\in X\times Y$.

Proof. 

Putting $x_1=\dots =x_m=x$ and $y_1=\dots =y_n=y$ in (4), we obtain

$$\parallel f(\alpha x,\beta y)-\alpha \beta f(x,y)\parallel \le \tilde{\phi }(x,y)$$

for all $(x,y)\in X\times Y$. Dividing both sides by $\left|\alpha \beta \right|$, we have

$$∥{\displaystyle \frac{f(\alpha x,\beta y)}{\alpha \beta }}-f(x,y)∥\le {\displaystyle \frac{1}{\left|\alpha \beta \right|}}\tilde{\phi }(x,y)$$

for all $(x,y)\in X\times Y$. For $k\in \mathbb{N}$, replacing x with ${\alpha }^{k}x$ and y with ${\beta }^{k}y$ in the above inequality, we get

$$∥{\displaystyle \frac{f({\alpha }^{k+1}x,{\beta }^{k+1}y)}{\alpha \beta }}-f({\alpha }^{k}x,{\beta }^{k}y)∥\le {\displaystyle \frac{1}{\left|\alpha \beta \right|}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)$$

for all $x,y\in X$. Dividing by ${\left|\alpha \beta \right|}^k$ in the above inequality, we obtain

$$∥{\displaystyle \frac{f({\alpha }^{k+1}x,{\beta }^{k+1}y)}{{\left(\alpha \beta \right)}^{k+1}}}-{\displaystyle \frac{f({\alpha }^{k}x,{\beta }^{k}y)}{{\left(\alpha \beta \right)}^k}}∥\le {\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)$$

(7)

for all $(x,y)\in X\times Y$ and all $k\in \mathbb{N}$. Now, we construct a sequence $\left\{g_k\right\}$ defined by

$$g_k(x,y):={\displaystyle \frac{f\left({\alpha }^{k}x,{\beta }^{k}y\right)}{{\left(\alpha \beta \right)}^k}}$$

(8)

for all $(x,y)\in X\times Y$ and all $k\in \mathbb{N}$. For any integers $p>q\ge 0$, by the triangle inequality and (7), we have

$$\begin{array}{cc}\hfill \parallel g_p(x,y)-g_q& (x,y)\parallel =∥\sum _{k=q}^{p-1}\left[g_{k+1}(x,y)-g_k(x,y)\right]∥\hfill \\ \hfill & \le \sum _{k=q}^{p-1}\parallel g_{k+1}(x,y)-g_k(x,y)\parallel \le \sum _{k=q}^{p-1}{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }\left({\alpha }^{k}x,{\beta }^{k}y\right)\hfill \end{array}$$

(9)

for all $(x,y)\in X\times Y$. By (5), the right-hand side of the above inequality tends to zero as $q\to \infty$. Thus, the sequence $\left\{g_k(x,y)\right\}$ is a Cauchy sequence in the Banach space $\mathcal{Z}$ for all $(x,y)\in X\times Y$. Therefore, we can define a mapping $F:X\times Y\to \mathcal{Z}$ by

$$F(x,y):=\underset{k\to \infty }{lim}{\displaystyle \frac{f({\alpha }^{k}x,{\beta }^{k}y)}{{\left(\alpha \beta \right)}^k}}$$

(10)

for all $(x,y)\in X\times Y$. By letting $q=0$ in the inequality (9), we obtain

$$∥g_p(x,y)-f(x,y)∥\le \sum _{k=0}^{p-1}{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)$$

for all $(x,y)\in X\times Y$ and all positive integers p. Taking the limit as $p\to \infty$ in the above inequality, and using the definition of $F(x,y)$ given in (10), we get

$$\parallel F(x,y)-f(x,y)\parallel =\underset{p\to \infty }{lim}\parallel g_p(x,y)-f(x,y)\parallel \le \sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)$$

for all $(x,y)\in X\times Y$.

By (10), we have

$$\begin{array}{cc}\hfill DF(x_1,\dots ,x_m,& y_1,\dots ,y_n)=F\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)-\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}F(x_i,y_j)\hfill \\ \hfill & =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\left(\alpha \beta \right)}^k}}\left[f\left({\alpha }^k\sum _{i=1}^m{\alpha }_i{x}_i,{\beta }^k\sum _{j=1}^n{\beta }_j{y}_j\right)-\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}f({\alpha }^k{x}_i,{\beta }^k{y}_j)\right]\hfill \end{array}$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. Thus, taking the norm on both sides, we obtain

$$\parallel DF(x_1,\dots ,x_m,y_1,\dots ,y_n)\parallel =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^k}}∥Df({\alpha }^k{x}_1,\dots ,{\alpha }^k{x}_m,{\beta }^k{y}_1,\dots ,{\beta }^k{y}_n)∥$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. By the inequality (4), we have

$$\parallel DF(x_1,\dots ,x_m,y_1,\dots ,y_n)\parallel \le \underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^k}}\phi ({\alpha }^k{x}_1,\dots ,{\alpha }^k{x}_m,{\beta }^k{y}_1,\dots ,{\beta }^k{y}_n)$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. By the condition (5), we have $DF(x_1,\dots ,x_m,y_1,\dots ,y_n)=0$ for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. That is, the mapping F satisfies functional Equation (1).

To prove the uniqueness of the mapping F, let $F^{\prime }:X\times Y\to \mathcal{Z}$ be another mapping satisfying the functional Equation (1) and the inequality (6). Note that

$$F\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)=\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_{j}F(x_i,y_j)$$

and

$$F^{\prime }\left(\sum _{i=1}^m{\alpha }_i{x}_i,\sum _{j=1}^n{\beta }_j{y}_j\right)=\sum _{i=1}^m\sum _{j=1}^n{\alpha }_i{\beta }_j{F}^{\prime }(x_i,y_j)$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. Putting $x_1=\dots =x_m=x$ and $y_1=\dots =y_n=y$ into the above two equations, we have $F(\alpha x,\beta y)=\alpha \beta F(x,y)$ and $F^{\prime }(\alpha x,\beta y)=\alpha \beta F^{\prime }(x,y)$ for all $(x,y)\in X\times Y$. Thus, we have

$$F(x,y)={\displaystyle \frac{1}{{\left(\alpha \beta \right)}^k}}F\left({\alpha }^{k}x,{\beta }^{k}y\right)\mathrm{and}{F}^{\prime }(x,y)={\displaystyle \frac{1}{{\left(\alpha \beta \right)}^k}}{F}^{\prime }\left({\alpha }^{k}x,{\beta }^{k}y\right)$$

for all $(x,y)\in X\times Y$ and all positive integers k. It follows from the inequality (6) that

$$\begin{array}{cc}\hfill \parallel F(x,& y)-F^{\prime }(x,y)\parallel =∥{\displaystyle \frac{1}{{\left(\alpha \beta \right)}^k}}F\left({\alpha }^{k}x,{\beta }^{k}y\right)-{\displaystyle \frac{1}{{\left(\alpha \beta \right)}^k}}{F}^{\prime }\left({\alpha }^{k}x,{\beta }^{k}y\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{{\left|\alpha \beta \right|}^k}}\left(\parallel F\left({\alpha }^{k}x,{\beta }^{k}y\right)-f\left({\alpha }^{k}x,{\beta }^{k}y\right)\parallel +\parallel f\left({\alpha }^{k}x,{\beta }^{k}y\right)-F^{\prime }\left({\alpha }^{k}x,{\beta }^{k}y\right)\parallel \right)\hfill \\ \hfill & \le {\displaystyle \frac{2}{{\left|\alpha \beta \right|}^k}}\sum _{j=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{j+1}}}\tilde{\phi }\left({\alpha }^{k+j}x,{\beta }^{k+j}y\right)=2\sum _{j=k}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{j+1}}}\tilde{\phi }\left({\alpha }^{j}x,{\beta }^{j}y\right)\to 0\mathrm{as}k\to \infty \hfill \end{array}$$

for all $(x,y)\in X\times Y$. □

Corollary 1.

Let p be a real number such that $p>0$ and satisfies $max\{1,|\alpha {|}^p,\left|\beta {|}^p\right\}<\left|\alpha \beta \right|$. Let $X,Y$ be normed vector spaces and $\mathcal{Z}$ be a Banach space and suppose that a mapping $f:X\times Y\to \mathcal{Z}$ satisfies $f(x,0)=0$ for all $x\in X$ and the inequality

$$∥Df(x_1,\dots ,x_m,y_1,\dots ,y_n)∥\le \epsilon +\delta \left(\sum _{i=1}^m\parallel x_i{\parallel }^p+\sum _{j=1}^n{\parallel y_j\parallel }^p\right)$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$, where $\epsilon \ge 0$ and $\delta \ge 0$ are constants. Then there exists a unique mapping $F:X\times Y\to \mathcal{Z}$ satisfying the functional Equation (1) and the inequality

$$\parallel f(x,y)-F(x,y)\parallel \le {\displaystyle \frac{\epsilon }{\left|\alpha \beta \right|-1}}+\delta \left({\displaystyle \frac{{m\parallel x\parallel }^p}{\left|\alpha \beta \right|-{\left|\alpha \right|}^p}}+{\displaystyle \frac{{n\parallel y\parallel }^p}{\left|\alpha \beta \right|-{\left|\beta \right|}^p}}\right)$$

(11)

for all $(x,y)\in X\times Y$.

Proof. 

Let $\phi (x_1,\dots ,x_m,y_1,\dots ,y_n):=\epsilon +\delta ({\sum }_{i=1}^m\parallel x_i{\parallel }^p+{\sum }_{j=1}^n\parallel y_j{\parallel }^p)$ for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. For $x_1=\dots =x_m=x$ and $y_1=\dots =y_n=y$, we define

$$\tilde{\phi }(x,y):=\epsilon +{\delta (m\parallel x\parallel }^p+{n\parallel y\parallel }^p)$$

for all $(x,y)\in X\times Y$. Since $max\{1,|\alpha {|}^p,\left|\beta {|}^p\right\}<\left|\alpha \beta \right|$, we have that $\phi$ satisfies (5) as follows:

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }& {\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^k{x}_1,\dots ,{\alpha }^k{x}_m,{\beta }^k{y}_1,\dots ,{\beta }^k{y}_n)\hfill \\ \hfill & =\sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\left[\epsilon +\delta \left(\sum _{i=1}^m\parallel {\alpha }^k{x}_i{\parallel }^p+\sum _{j=1}^n{\parallel {\beta }^k{y}_j\parallel }^p\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{\left|\alpha \beta \right|}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{1}{\left|\alpha \beta \right|}}\right)}^k+{\displaystyle \frac{\delta }{\left|\alpha \beta \right|}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{{\left|\alpha \right|}^p}{\left|\alpha \beta \right|}}\right)}^k\sum _{i=1}^m\parallel x_i{\parallel }^p+{\displaystyle \frac{\delta }{\left|\alpha \beta \right|}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{{\left|\beta \right|}^p}{\left|\alpha \beta \right|}}\right)}^k\sum _{j=1}^n{\parallel y_j\parallel }^p\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{\left|\alpha \beta \right|}}·{\displaystyle \frac{1}{1-\frac{1}{\left|\alpha \beta \right|}}}+{\displaystyle \frac{\delta }{\left|\alpha \beta \right|}}·{\displaystyle \frac{1}{1-\frac{{\left|\alpha \right|}^p}{\left|\alpha \beta \right|}}}\sum _{i=1}^m\parallel x_i{\parallel }^p+{\displaystyle \frac{\delta }{\left|\alpha \beta \right|}}·{\displaystyle \frac{1}{1-\frac{{\left|\beta \right|}^p}{\left|\alpha \beta \right|}}}\sum _{j=1}^n{\parallel y_j\parallel }^p\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{\left|\alpha \beta \right|-1}}+\delta \left({\displaystyle \frac{1}{\left|\alpha \beta \right|-{\left|\alpha \right|}^p}}\sum _{i=1}^m\parallel x_i{\parallel }^p+{\displaystyle \frac{1}{\left|\alpha \beta \right|-{\left|\beta \right|}^p}}\sum _{j=1}^n{\parallel y_j\parallel }^p\right)<\infty \hfill \end{array}$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$.

By Theorem 2, there exists a unique mapping $F:X\times Y\to \mathcal{Z}$ satisfying the functional Equation (1). The inequality (11) holds as follows:

$$\begin{array}{cc}\hfill \parallel f(x,y)-F(x,y)\parallel & \le \sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }(\underset{m}{\underbrace{{\alpha }^{k}x,\dots ,{\alpha }^{k}x}},\underset{n}{\underbrace{{\beta }^{k}y,\dots {\beta }^{k}y}})\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{\left|\alpha \beta \right|-1}}+\delta \left({\displaystyle \frac{{m\parallel x\parallel }^p}{\left|\alpha \beta \right|-{\left|\alpha \right|}^p}}+{\displaystyle \frac{{n\parallel y\parallel }^p}{\left|\alpha \beta \right|-{\left|\beta \right|}^p}}\right)\hfill \end{array}$$

for all $(x,y)\in X\times Y$. □

Example 2.

Let $X=Y=\mathcal{Z}=\mathbb{R}$. We consider the bi-affine functional equation with $m=n=2$ and ${\alpha }_1={\alpha }_2={\beta }_1={\beta }_2=2$. Then $\alpha =4$, $\beta =4$ and $\left|\alpha \beta \right|=16$.

(1) 

The case where the direct method succeeds (i.e., a stable result is obtained)

Given a constant $\delta >0$, let us define a mapping $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by

$$f(x,y):=xy+\delta \left(\right|x|+|y\left|\right)$$

for all $x,y\in \mathbb{R}$. Then, we have

$$\begin{array}{cc}\hfill Df(x_1,x_2,y_1,y_2)& =f(2x_1+2x_2,2y_1+2y_2)-\sum _{i=1}^2\sum _{j=1}^{2}4f(x_i,y_j)\hfill \\ & =\delta \left(\right|2x_1+2x_2|+|2y_1+2y_2\left|\right)-\sum _{i=1}^2\sum _{j=1}^{2}4\delta \left(\right|x_i|+|y_j\left|\right)\hfill \end{array}$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. By applying the triangle inequality, we obtain

$$|Df(x_1,x_2,y_1,y_2)|\le 10\delta (|x_1|+|x_2|+|y_1|+|y_2\left|\right)$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. This matches the condition of Corollary 1 with $p=1$. The convergence condition is strictly satisfied since the ratio $r={\displaystyle \frac{4^1}{16}}={\displaystyle \frac{1}{4}}<1$. By Corollary 1, there exists a unique exact bi-affine mapping $F(x,y)=xy$, and the actual error $\delta \left(\right|x|+|y\left|\right)$ is safely bounded by the theoretical limit ${\displaystyle \frac{5\delta }{3}}\left(\right|x|+\left|y\right|)$.

(2) 

The case where the direct method fails (i.e., it is impossible to determine stability using the direct method)

Given a constant $\delta >0$, let us define another mapping $g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ with a quadratic perturbation ($p=2$) by $g(x,y):=xy+\delta (x^2+y^2)$ for all $x,y\in \mathbb{R}$. Then we obtain that

$$\begin{array}{cc}\hfill Dg(x_1,x_2,y_1,y_2)=& g(2x_1+2x_2,2y_1+2y_2)-\sum _{i=1}^2\sum _{j=1}^{2}4g(x_i,y_j)\hfill \\ & =\delta ({(2x_1+2x_2)}^2+{(2y_1+2y_2)}^2)-4\delta \sum _{i=1}^2\sum _{j=1}^2(x_i^2+y_j^2)\hfill \\ & =4\delta (x_1^2+2x_1{x}_2+x_2^2+y_1^2+2y_1{y}_2+y_2^2)-8\delta (x_1^2+x_2^2+y_1^2+y_2^2)\hfill \\ & =-4\delta \left[{(x_1-x_2)}^2+{(y_1-y_2)}^2\right]\hfill \end{array}$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. Using the basic inequality ${(a-b)}^2\le 2a^2+2b^2$ for real numbers a and b, we have

$$|Dg(x_1,x_2,y_1,y_2)|\le 4\delta (2x_1^2+2x_2^2+2y_1^2+2y_2^2)=8\delta (x_1^2+x_2^2+y_1^2+y_2^2)$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. This exactly matches the bounding condition for $p=2$. If we attempt to apply the direct method by constructing the sequence $\left\{g_k\right\}$ given by (8), we obtain that

$$g_k(x,y)={\displaystyle \frac{{16}^{k}xy+{16}^k\delta (x^2+y^2)}{{16}^k}}=xy+\delta (x^2+y^2)=g(x,y)$$

for all $x,y\in \mathbb{R}$ and all $k\in \mathbb{N}$. The sequence $\left\{g_k(x,y)\right\}$ is perfectly constant for all k, meaning its limit is $g(x,y)$ itself. However, $g(x,y)$ is clearly not a solution to the bi-affine Equation (1) due to the presence of the quadratic term. This demonstrates that for $p=2$, the direct method fails to construct an exact bi-affine mapping, and thus the Hyers–Ulam stability does not hold in this case.

3. Hyers–Ulam Stability in 2-Banach Spaces

We generalize Theorem 2 to the context of 2-Banach spaces. Let $\mathcal{Z}$ be a 2-Banach space equipped with a 2-norm $\parallel ·,·\parallel$.

Definition 2

([17]). For $l\in \mathbb{N}$, a 2-normed space $(\mathcal{Z},\parallel ·,·\parallel )$ is said to satisfy property $\left(K_l\right)$ if there exist linearly independent elements $a_1,a_2,\dots ,a_l$ in $\mathcal{Z}$ such that for every sequence $\left\{z_k\right\}$ in $\mathcal{Z}$ and $z\in \mathcal{Z}$, the condition

$$\underset{k\to \infty }{lim}\parallel z-z_k,a_i\parallel =0\mathrm{for}\mathrm{all}i=1,\dots ,l$$

implies that

$$\underset{k\to \infty }{lim}\parallel z-z_k,a\parallel =0\mathrm{for}\mathrm{all}a\in \mathcal{Z}.$$

Property $\left(K_l\right)$ is indeed an additional topological restriction and is not automatically satisfied by all arbitrary 2-Banach spaces. However, it is satisfied by most common and widely studied finite-dimensional 2-Banach spaces, including ${\mathbb{R}}^2$ equipped with the standard 2-norm.

Theorem 3

([17]). Let $(\mathcal{Z},\parallel ·,·\parallel )$ be a 2-normed space. Then, the following are equivalent: I. $\mathcal{Z}$ satisfies property $\left(K_l\right)$ for some $l\in \mathbb{N}$; II. there exists a linearly independent set $\{a_1,a_2,\dots ,a_l\}$, such that the topology of the norm defined by

$${\parallel z\parallel }_{\infty }:=sup\{\parallel z,a_1\parallel ,\parallel z,a_2\parallel ,\dots ,\parallel z,a_l\parallel \}\mathrm{for}\mathrm{all}z\in \mathcal{Z}$$

matches with the topology of $(\mathcal{Z},\parallel ·,·\parallel )$.

Theorem 4

([17]). Let $(\mathcal{Z},\parallel ·,·\parallel )$ be a 2-normed space and ${\parallel ·\parallel }_{\infty }$ be the derived norm defined by ${\parallel z\parallel }_{\infty }:={max}_{1\le k\le l}\parallel z,a_k\parallel$. Then, the convergence of sequences in $(\mathcal{Z},\parallel ·,·\parallel )$ and $(\mathcal{Z},\parallel ·{\parallel }_{\infty })$ is equivalent if, and only if, both spaces have the same topology.

Theorem 5.

Let $X,Y$ be the vector spaces and $\mathcal{Z}$ be a 2-Banach space satisfying property $\left(K_l\right)$ for some $l\in \mathbb{N}$. Let $\{a_1,a_2,\dots ,a_l\}$ be a linearly independent set in $\mathcal{Z}$. Let $M:={max}_{1\le k\le l}{\parallel a_k\parallel }_{\infty }$. Let $\left|\alpha \beta \right|>1$ and $\phi :X^m\times Y^n\to [0,\infty )$ be a control function satisfying the convergence condition (5). Suppose that a mapping $f:X\times Y\to \mathcal{Z}$ satisfies

$$\parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n),z\parallel \le \phi (x_1,\dots ,x_m,y_1,\dots ,y_n){\parallel z\parallel }_{\infty }$$

(12)

for all $x_1,\dots ,x_m\in X$, all $y_1,\dots ,y_n\in Y$, and all $z\in \mathcal{Z}$.

Then there exists a unique bi-affine mapping $F:X\times Y\to \mathcal{Z}$ satisfying functional Equation (1) and the inequality

$${\parallel f(x,y)-F(x,y)\parallel }_{\infty }\le M\sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\phi }({\alpha }^{k}x,{\beta }^{k}y)$$

(13)

for all $(x,y)\in X\times Y$.

Proof. 

Since $\mathcal{Z}$ satisfies Property $\left(K_l\right)$, it follows from Theorem 3 that $\mathcal{Z}$ is normable and the topology induced by the 2-norm $\parallel ·,·\parallel$ coincides with the topology of the derived norm ${\parallel ·\parallel }_{\infty }$.

By substituting $z=a_k$ ($k=1,\dots ,l$) into (12), we have

$$\parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n),a_k\parallel \le \phi (x_1,\dots ,x_m,y_1,\dots ,y_n){\parallel a_k\parallel }_{\infty }$$

for all $x_1,\dots ,x_m\in X$, all $y_1,\dots ,y_n\in Y$, and all $k=1,\dots ,l$. Taking the maximum over $k=1,\dots ,l$, we obtain an inequality for the derived norm:

$$\begin{array}{cc}\hfill & \parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n){\parallel }_{\infty }=\underset{1\le k\le l}{max}\parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n),a_k\parallel \hfill \\ \hfill & \le \phi (x_1,\dots ,x_m,y_1,\dots ,y_n)\underset{1\le k\le l}{max}{\parallel a_k\parallel }_{\infty }=M\phi (x_1,\dots ,x_m,y_1,\dots ,y_n)\hfill \end{array}$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. Let $\Phi (x_1,\dots ,x_m,y_1,\dots ,y_n):=M\phi (x_1,\dots ,x_m,y_1,\dots ,y_n)$ for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. The condition (12) in the 2-Banach space is thus reduced to the following norm inequality:

$$\parallel Df(x_1,\dots ,x_m,y_1,\dots ,y_n){\parallel }_{\infty }\le \Phi (x_1,\dots ,x_m,y_1,\dots ,y_n)$$

for all $x_1,\dots ,x_m\in X$ and all $y_1,\dots ,y_n\in Y$. According to Theorem 4, the convergence of sequences in $(\mathcal{Z},\parallel ·,·\parallel )$ is equivalent to that in $(\mathcal{Z},\parallel ·{\parallel }_{\infty })$. Therefore, we can apply the same reasoning as in the proof of Theorem 2, regarding $(\mathcal{Z},\parallel ·{\parallel }_{\infty })$ as a Banach space. By setting $x_1=\dots =x_m=x$ and $y_1=\dots =y_n=y$, and dividing by $\left|\alpha \beta \right|$, we construct the Cauchy sequence $\left\{g_k\right\}$ defined by (8). The sequence $\left\{g_k\right\}$ converges to a unique bi-affine mapping F. The error bound is derived as

$${\parallel f(x,y)-F(x,y)\parallel }_{\infty }\le \sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\tilde{\Phi }({\alpha }^{k}x,{\beta }^{k}y)$$

for all $(x,y)\in X\times Y$. Substituting $\tilde{\Phi }=M\tilde{\phi }$, we obtain the result (13). □

Example 3.

Let $\mathcal{X}=\mathcal{Y}=\mathbb{R}$ and $\mathcal{Z}={\mathbb{R}}^2$. We define the 2-norm on ${\mathbb{R}}^2$ by

$$\parallel u,v\parallel :=|u_1{v}_2-u_2{v}_1|$$

for $u=(u_1,u_2)$ and $v=(v_1,v_2)$ in ${\mathbb{R}}^2$. Let $e_1=(1,0)$ and $e_2=(0,1)$. The derived norm is given by

$${\parallel z\parallel }_{\infty }=max\{\parallel z,e_1\parallel ,\parallel z,e_2\parallel \}=max\left\{\right|z_2|,|z_1\left|\right\}$$

for $z=(z_1,z_2)\in {\mathbb{R}}^2$. The derived norm coincides with the standard ${\ell }_{\infty }$-norm on ${\mathbb{R}}^2$. Here, $M=max\{\parallel e_1{\parallel }_{\infty },\parallel e_2{\parallel }_{\infty }\}=1$.

We consider the functional Equation (1) with $m=n=2$ and scalars ${\alpha }_1={\alpha }_2=2$ and ${\beta }_1={\beta }_2=2$. Then $\alpha ={\sum }_{i=1}^2{\alpha }_i=4$ and $\beta ={\sum }_{j=1}^2{\beta }_j=4$, satisfying the convergence condition $\left|\alpha \beta \right|=16>1$.

For a given constant $\delta >0$, define a mapping $f:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^2$ by

$$f(x,y):=\left(xy+\delta sin(x+y),xy+\delta cos(x+y)\right)$$

for all $x,y\in \mathbb{R}$. If $D{f}_1(x_1,x_2,y_1,y_2)$ and $D{f}_2(x_1,x_2,y_1,y_2)$ are the first component of $Df(x_1,x_2,y_1,y_2)$ and the second component of $Df(x_1,x_2,y_1,y_2)$, respectively, for all $x_1,x_2,y_1,y_2\in \mathbb{R}$, then we have

$$\begin{array}{cc}\hfill D_{1}f(x_1,x_2,y_1,y_2)& =\delta (sin\left(2(x_1+x_2+y_1+y_2)\right)-4(sin(x_1+y_1)+sin(x_1+y_2)\hfill \\ \hfill & +sin(x_2+y_1)+sin(x_2+y_2)\left)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill D_{2}f(x_1,x_2,y_1,y_2)& =\delta (cos\left(2(x_1+x_2+y_1+y_2)\right)-4(cos(x_1+y_1)+cos(x_1+y_2)\hfill \\ \hfill & +cos(x_2+y_1)+cos(x_2+y_2)\left)\right)\hfill \end{array}$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. Therefore,

$$\parallel Df(x_1,x_2,y_1,y_2){\parallel }_{\infty }=max\left\{\right|D_{2}f(x_1,x_2,y_1,y_2)|,|D_{1}f(x_1,x_2,y_1,y_2)\left|\right\}\le 17\delta$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$.

For any $z=(z_1,z_2)\in {\mathbb{R}}^2$, the 2-norm satisfies

$$\parallel Df(x_1,x_2,y_1,y_2){,z\parallel \le 17\delta \parallel z\parallel }_{\infty }$$

for all $x_1,x_2,y_1,y_2\in \mathbb{R}$. Thus, we can define the constant control function $\phi (x_1,x_2,y_1,y_2):=17\delta$. The convergence condition (5) is satisfied as follows:

$$\sum _{k=0}^{\infty }{\displaystyle \frac{1}{{\left|\alpha \beta \right|}^{k+1}}}\phi ({\alpha }^k{x}_1,{\alpha }^k{x}_2,{\beta }^k{y}_1,{\beta }^k{y}_2)=\sum _{k=0}^{\infty }{\displaystyle \frac{17\delta }{{16}^{k+1}}}={\displaystyle \frac{17\delta }{15}}<\infty .$$

By Theorem 3, there exists a unique bi-affine mapping $F:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^2$ satisfying functional Equation (1), and the error bound is given by

$${\parallel f(x,y)-F(x,y)\parallel }_{\infty }\le M\sum _{k=0}^{\infty }{\displaystyle \frac{17\delta }{{16}^{k+1}}}={\displaystyle \frac{17\delta }{15}}$$

for all $(x,y)\in \mathbb{R}\times \mathbb{R}$. Indeed, the exact bi-affine mapping is $F:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^2$, given by $F(x,y)=(xy,xy)$ for all $x,y\in \mathbb{R}$.

Remark 2.

As noted by S. Yun [18], if the exponent of the norm ${\parallel z\parallel }_{\infty }$ in the control function in (12) is $r\ne 1$ (i.e., the bound is $\phi (x_1,\dots ,x_m,y_1,\dots ,y_n){\parallel z\parallel }_{\infty }^r$ with $r\ne 1$), then the mapping f must be strictly bi-affine (super-stability).

4. Conclusions

In this paper, we have introduced and investigated a generalized multi-variable bi-affine functional equation with arbitrary nonzero scalar weights. Our study provides a comprehensive analysis of the general solution and the Hyers–Ulam stability of this equation in various functional spaces. The key findings of this work are summarized as follows.

Firstly, we determined the explicit structure of the general solution (Theorem 1). We demonstrated that any mapping satisfying the proposed equation can be decomposed into a bi-additive mapping, two independent additive mappings, and a constant term. This characterization holds under specific parameter conditions (such as $\beta =1$, $\alpha =1$, or $\alpha \beta =1$) and clarifies the algebraic structure of bi-affine mappings.

Secondly, employing the direct method, we established the Hyers–Ulam stability of the functional equation in Banach spaces (Theorem 2). We derived an explicit error bound for the approximation when the perturbation is controlled by a generalized function $\phi$. Furthermore, through Corollary 1 and illustrative examples, we rigorously analyzed the stability behavior under polynomial-type perturbations (${\parallel x\parallel }^p+{\parallel y\parallel }^p$). Notably, we presented a counter-example (Example 2) to demonstrate that the stability does not hold in the quadratic case ($p=2$), thereby highlighting the boundary of stability for bi-affine mappings.

Finally, we successfully extended our investigation to the setting of 2-Banach spaces (Theorem 5). By utilizing the properties of the derived norm and Property $\left(K_l\right)$, we confirmed that the stability results obtained in standard Banach spaces can be naturally generalized to 2-Banach spaces.

To summarize, this study generalizes earlier results on bi-Jensen functional equations by incorporating arbitrary weights and provides a rigorous mathematical foundation for approximating multi-variable bi-affine mappings. These findings are expected to have applications in characterizing generalized linear structures in diverse function spaces.