Orthogonal Stability and Solution of a Three-Variable Functional Equation in Extended Banach Spaces

Abstract

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-$\beta$-Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases.

1. Introduction

In the present era, the domain of functional Equations (FEs) represents a continuously expanding area within mathematics, holding significant implications across various applications. The relatively recent emergence of functional Equation (FE) theory has led to the creation of potent tools within modern mathematics. FEs encompass a traditional mathematical discipline encompassing diverse algebraic, analytic, order-theoretic, and topological exploration avenues. Concurrently, numerous mathematical concepts from various disciplines have become fundamental to the underpinnings of FEs. This framework is progressively being employed to scrutinize challenges in unrelated fields like mathematical analysis, combinatorics, biology, behavioral and social sciences, and engineering.

Within the topic of FEs, several sectors cover a broad range of research areas; one of them is investigating the stability of FEs. In 1940, Ulam [1] initiated research into the stability of FEs, introducing the question: “Under which conditions is it possible for a solution to a slightly altered equation closely approximates the solution of the original equation?”. Subsequent investigations have built upon this inquiry. Hyers [2] provided a solution to Ulam’s stability problem for Banach spaces in 1941. Aoki [3] introduced additive mappings to this concept, and Rassias [4] extended the Hyers theorem for additive maps in 1978. In subsequent years, numerous mathematicians [5,6,7,8,9] have extensively explored stability studies. FEs can manifest in Banach spaces, Hilbert spaces, and other function spaces to examine operators’ and functionals’ stability and continuity traits. Refs. [10,11,12,13,14,15,16,17,18,19] would be beneficial to understand the diverse stability issues linked to FEs.

In an arbitrary real Banach space, multiple concepts of orthogonality are generalizations of inner product spaces. These concepts hold intrinsic geometric significance. The general solution of the Cauchy FE $f(u+v)=f\left(u\right)+f\left(v\right),u\perp v$, was found by Gudder and Strawther [20] in 1975. They also established an abstract orthogonality relation ⊥ via five axioms. Rätz [21] gave an innovative concept of orthogonality in 1985 by applying stricter restrictions than Gudder and Strawther.

Let us suppose we have a complex normed space denoted as $(Z,\parallel ·\parallel )$, $Z^k$ ($Z\oplus Z\oplus \dots \oplus Z$) represents a linear space comprising k-tuples, expressed as $(u_1,u_2,\dots ,u_k)$ where $u_1,u_2,\dots ,u_k\in Z$, and the zero element of $(Z,\parallel ·\parallel )$ is symbolized by 0. Additionally, ${\Omega }_k$ represents a permutation group of k symbols. Initially, we will focus on understanding terminology, defining key concepts, establishing notation, and understanding common attributes within the subject matter. Then, we are conveying these definitions in this paper as follows:

Definition 1 

([22]). A sequence ${\parallel ·\parallel }_k$ of norms on $\{Z^k:k\in \mathbb{N}\}$ is called a multi norm if, for each $u\in Z$, ${\left|\right|u\left|\right|}_1=\left|\right|u\left|\right|$ and satisfies the following axioms with $k\ge 2$, $\sigma \in {\Omega }_k,u_1,\dots ,u_k\in Z$, ${\alpha }_1,\dots ,{\alpha }_k\in C$

1. 

$\left|\right|(u_{\sigma \left(1\right)},\dots ,u_{\sigma \left(k\right)}){\left|\right|}_k$=$\left|\right|(u_1,\dots ,u_k){\left|\right|}_k$;

2. 

$\left|\right|({\alpha }_1{u}_1,\dots ,{\alpha }_k{u}_k){\left|\right|}_k\le {max}_{i\in N_k}\left\{\right|{\alpha }_i\left|\right\}\left|\right|(u_1,\dots ,u_k){\left|\right|}_k$;

3. 

$\left|\right|(u_1,\dots ,u_{k-1},0){\left|\right|}_k=\left|\right|(u_1,\dots ,u_{k-1}){\left|\right|}_{k-1}$;

4. 

$\left|\right|(u_1,\dots ,u_{k-1},u_{k-1}){\left|\right|}_k=\left|\right|(u_1,\dots ,u_{k-1}){\left|\right|}_{k-1}$.

Hence, the space $(Z^k,\parallel ·{\parallel }_k:k\in \mathbb{N})$ is called multi-normed space. Also, $(Z^k,\parallel ·{\parallel }_k:k\in \mathbb{N})$ satisfies the subsequent two characteristics:

(M1) 

${\left|\right|u,\dots ,u\left|\right|}_k=\left|\right|u\left|\right|$;

(M2) 

${\max }_{i\in N_k}\left|\right|u_i\left|\right|\le \left|\right|u_1,\dots ,u_k{\left|\right|}_k\le {\sum }_{i=1}^k\left|\right|u_i\left|\right|\le k{max}_{i\in N_k}\left|\right|u_i\left|\right|$.

Lemma 1 

([22]). Let $(u_1,\dots ,u_k)\in Z^k$ and $k\in \mathbb{N}$. For each $j\in \{1,2,\dots k\}$, let ${\left\{{u_n}^j\right\}}_{n\in \mathbb{N}}$ be a sequence in Z such that ${lim}_{n\to \infty }{u_n}^j=u_j$. Then, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}(u_n^1-w_1,\dots ,u_n^k-w_k)=(u_1-w_1,\dots ,u_k-w_k)\forall (w_1,\dots ,w_k)\in Z^k.\end{array}$$

Definition 2 

([22]). A sequence $\left\{u_n\right\}$ in Z is called a multi-null sequence if for each $ϵ>0$ there exists $n_0\in \mathbb{N}$ in such a manner

$$\begin{array}{c}\hfill \underset{k\in \mathbb{N}}{sup}\left|\right|u_n,\dots ,u_{n+k-1}{\left|\right|}_k<ϵ,(n\ge n_0).\end{array}$$

Definition 3 

([21]). A real-valued function $\parallel ·\parallel$ is called a quasi-β-norm on a linear space U over a field $\mathcal{K}$ if it satisfies the following axioms:

1. 

$\left|\right|u\left|\right|\ge 0$ for all $u\in U$ and $\left|\right|u\left|\right|=0$ if and only if $u=0$;

2. 

$\left|\right|\lambda u\left|\right|={\left|\lambda \right|}^{\beta }.\left|\right|u\left|\right|$ for all $\lambda \in \mathcal{K}$;

3. 

There is a constant $K\ge 1$ such that

$$\left|\right|u_1+u_2\left|\right|\le K\left(\right||u_1\left|\right|+\left|\right|u_2\left|\right|)$$

for all $u_1,u_2\in U.$

The pair $(U,\parallel ·\parallel )$ is referred to as a quasi-β-normed space, where the function $\parallel ·\parallel$ is termed a quasi-β-norm on U. The smallest possible constant K that satisfies the corresponding inequality is identified as the modulus of concavity. A quasi-β-Banach space is defined as a quasi-β-normed space that is also complete. Moreover, if a quasi-β-norm $\parallel ·\parallel$ satisfies the inequality $\parallel u_1+u_2{ \parallel }^p\le \parallel u_1{\parallel }^p + {\parallel u_2\parallel }^p$ for all $u_1,u_2\in U$ and for some $0<p\le 1$, it is referred to as a $(\beta ,p)$ norm. In this context, a quasi-β-Banach space equipped with such a norm is called a $(\beta ,p)$-Banach space.

Definition 4 

([21]). Let U be a real vector space with $dim\left(U\right)\ge 2$, equipped with a binary relation ⊥ that satisfies the following conditions:

$\left(O1\right)$

Homogeneity: If $u,vs.\in U$ and $u\perp v$, then $\alpha u\perp \beta v$ for any scalars $\alpha ,\beta \in \mathbb{R}$.

$\left(O2\right)$

Independence: If $u,vs.\in U\setminus \left\{0\right\}$ and $u\perp v$, then u and v are linearly independent.

$\left(O3\right)$

Zero orthogonality: $u\perp 0$ and $0\perp u$ for all $u\in U$.

$\left(O4\right)$

Thales’ property: Given $u\in U$ and a two-dimensional subspace $U_0\subseteq U,$ for any $\lambda \in {\mathbb{R}}^{+}$, there exists $u_0\in U_0$ such that $u\perp u_0$ and $u+u_0\perp \lambda u-u_0$.

The structure $(U,\perp )$ is then referred to as an orthogonality space.

The FE

$$\begin{array}{c}\hfill f(2x-y)+f(2x+y)=12f\left(x\right)+2f(x-y)+2f(x+y)\end{array}$$

where $x,y$ are mutually orthogonal was given by Jun and Kim [23]. Additionally, they found the solution and stability. The above FE is called a cubic FE because its solution is $f\left(x\right)=a{x}^3$, where a is arbitrary constant. In this article, inspired by the vision and guidance of various researchers, we describe and find the solution of the following new three-dimensional orthogonal cubic FE

$$\begin{array}{ccc}\hfill f(2u+3v+4w)& =& 3f(u+3v+4w)+f(-u+3v+4w)+2f(u+3v)\hfill \\ & +& 2f(u+4w)-6f(u-3v)-6f(u-4w)\hfill \\ & -& 3f(3v+4w)+16[f(u-{\displaystyle \frac{3}{2}}v)+f(u-2w)]\hfill \\ & -& 18f\left(u\right)-6f\left(3v\right)-6f\left(4w\right),\hfill \end{array}$$

where $u,v,w$ are mutually orthogonal. We also investigate the stability of the above cubic equation in multi-Banach space and quasi $(\beta ,p)$ normed space. In the last segment, we also provide the case of instability.

A cubic relationship among three variables is the basis of a three-dimensional functional cubic equation, which finds application in several domains. It simulates nonlinear dynamics in physics, including chaotic behavior in fluid flow. Engineers use these formulas in structural analysis to forecast material stress and deformation. They optimize economic resource allocation by simulating production functions with decreasing or growing returns. They are also used in computer graphics to provide smooth 3D surfaces for animation and computer-aided design.

For notational convenience, we represent the cubic FE with a difference operator $\Delta f(u,v,w)$

$$\begin{array}{ccc}\hfill \Delta f(u,v,w)& =& f(2u+3v+4w)-3f(u+3v+4w)-f(-u+3v+4w)\hfill \\ & -& 2f(u+3v)-2f(u+4w)+6f(u-3v)+6f(u-4w)\hfill \\ & +& 3f(3v+4w)-16[f(u-{\displaystyle \frac{3}{2}}v)+f(u-2w)]+18f\left(u\right)\hfill \\ & +& 6f\left(3v\right)+6f\left(4w\right).\hfill \end{array}$$

Throughout this article, U denotes an orthogonality space, V is a Banach space, W is an orthogonal normed space, $(Z^k:k\in N)$ is a multi-Banach space where Z is also a Banach space, and X is a $(\beta ,p)$-Banach space.

2. General Solution

In this section, we find the solution of the given FE. We define a function $f:U\to V$ as orthogonally cubic if it satisfies any of the following FEs:

$$\begin{array}{c}\hfill f(2x-y)+f(2x+y)=2f(x+y)+12f\left(x\right)+2f(x-y)\end{array}$$

where $x,y$ are mutually orthogonal and

$$\begin{array}{ccc}\hfill f(2u+3v+4w)& =& 3f(u+3v+4w)+f(-u+3v+4w)+2f(u+3v)\hfill \\ \hfill & +& 2f(u+4w)-6f(u-3v)-6f(u-4w)\hfill \\ \hfill & -& 3f(3v+4w)+16[f(u-{\displaystyle \frac{3}{2}}v)+f(u-2w)]\hfill \\ \hfill & -& 18f\left(u\right)-6f\left(3v\right)-6f\left(4w\right),\hfill \end{array}$$

where $u,v,w$ are mutually orthogonal.

Theorem 1. 

An odd function $f:U\to V$ fulfills the FE

$$\begin{array}{c}\hfill f(2x+y)-2f(x+y)=2f(x-y)-f(2x-y)+12f\left(x\right)\end{array}$$

(1)

where $x,y$ are mutually orthogonal if and only if $f:U\to V$ fulfills the FE

$$\begin{array}{ccc}\hfill f(2u+3v+4w)& =& 3f(u+3v+4w)+f(-u+3v+4w)+2f(u+3v)\hfill \\ \hfill & +& 2f(u+4w)-6f(u-3v)-6f(u-4w)\hfill \\ \hfill & -& 3f(3v+4w)+16[f(u-{\displaystyle \frac{3}{2}}v)+f(u-2w)]\hfill \\ & -& 18f\left(u\right)-6f\left(3v\right)-6f\left(4w\right)\hfill \end{array}$$

(2)

where $u,v,w$ are mutually orthogonal and $x,y,u,v,w\in U.$

Proof. 

Let $f:U\to V$ satisfy the FE (1). Taking $(x,y)=(0,0)$ in (1), we have $f\left(0\right)=0$. Switching $(x,y)$ with $(x,0),(x,x)$ and $(x,2x)$, respectively, in (1), we obtain

$$\begin{array}{c}\hfill f\left(2x\right)=2^{3}f\left(x\right),f\left(3x\right)=3^{3}f\left(x\right),f\left(4x\right)=4^{3}f\left(x\right).\end{array}$$

(3)

Since we know that (1) is cubic FE and its solution is $f\left(x\right)=a{x}^3$ then for any positive integer c we have

$$\begin{array}{c}\hfill f\left(cx\right)=c^{3}f\left(x\right).\end{array}$$

(4)

Replacing $(x,y)$ with $(u,3v+4w)$ in (1), we have

$$\begin{array}{ccc}\hfill f(2u+3v+4w)& -& 2f(u+3v+4w)=f(-2u+3v+4w)\hfill \\ & +& 2f(u-3v-4w)+12f\left(u\right).\hfill \end{array}$$

(5)

Again, switching $(x,y)$ with $(3v+4w,-2u)$ in (1), we obtain

$$\begin{array}{ccc}\hfill 4f(-u+3v+4w)& +& 4f(u+3v+4w)-f(2u+3v+4w)\hfill \\ & -& 6f(3v+4w)=f(-2u+3v+4w).\hfill \end{array}$$

(6)

Substituting (6) in (5), we obtain

$$\begin{array}{ccc}\hfill f(2u+3v+4w)-3f(u+3v+4w)& =& f(-u+3v+4w)+6f\left(u\right)\hfill \\ & -& 3f(3v+4w).\hfill \end{array}$$

(7)

Changing $(x,y)$ to $(3v,2u)$ in (1), we have

$$\begin{array}{c}\hfill 4f(u+3v)-4f(u-3v)-6f\left(3v\right)=f(2u+3v)-f(2u-3v).\end{array}$$

(8)

Changing $(x,y)$ to $(4w,2u)$ in (1), we have

$$\begin{array}{c}\hfill 4f(u+4w)-4f(u-4w)-6f\left(4w\right)=f(2u+4w)-f(2u-4w).\end{array}$$

(9)

Adding (8) and (9), we have

$$\begin{array}{ccc}\hfill & & 4f(u+3v)-4f(u-3v)+4f(u+4w)-4f(u-4w)-6f\left(4w\right)\hfill \\ \hfill & & -6f\left(3v\right)-f(2u+3v)+f(2u-3v)-f(2u+4w)\hfill \\ & & +f(2u-4w)=0.\hfill \end{array}$$

(10)

Adding (7) and (10), we obtain

$$\begin{array}{ccc}\hfill & & f(2u+3v+4w)=3f(u+3v+4w)+f(-u+3v+4w)\hfill \\ \hfill & & -3f(3v+4w)+6f\left(u\right)+4f(u+3v)-4f(u-3v)+4f(u+4w)\hfill \\ \hfill & & -4f(u-4w)-6f\left(4w\right)-6f\left(3v\right)-f(2u+3v)+f(2u-3v)\hfill \\ & & -f(2u+4w)+f(2u-4w).\hfill \end{array}$$

(11)

Changing $(x,y)$ to $(-u,3v)$ in (1), we obtain

$$\begin{array}{ccc}\hfill -f(2u+3v)& =& f(2u-3v)-2f(u-3v)\hfill \\ & & -2f(u+3v)-12f\left(u\right).\hfill \end{array}$$

(12)

Changing $(x,y)$ to $(-u,4w)$ in (1), we have

$$\begin{array}{ccc}\hfill -f(2u+4w)& =& f(2u-4w)-2f(u-4w)\hfill \\ & & -2f(u+4w)-12f\left(u\right).\hfill \end{array}$$

(13)

Adding (12) and (13), we obtain

$$\begin{array}{ccc}\hfill & & -f(2u+3v)-f(2u+4w)=f(2u-3v)-2f(u-3v)-2f(u+3v)\hfill \\ & & +f(2u-4w)-2f(u-4w)-2f(u+4w)-24f\left(u\right).\hfill \end{array}$$

(14)

Substituting the value of $-f(2u+3v)-f(2u+4w)$ from (14) in (11) and using (3), we have

$$\begin{array}{ccc}\hfill f(2u+3v+4w)& =& 3f(u+3v+4w)+f(-u+3v+4w)+2f(u+3v)\hfill \\ \hfill & +& 2f(u+4w)-6f(u-3v)-6f(u-4w)\hfill \\ \hfill & -& 3f(3v+4w)+16[f(u-{\displaystyle \frac{3}{2}}v)+f(u-2w)]\hfill \\ & -& 18f\left(u\right)-6f\left(3v\right)-6f\left(4w\right).\hfill \end{array}$$

(15)

Conversely, suppose $f:U\to V$ satisfied the FE (2). Replacing $(u,3v,4w)$ by $(x,0,0),(0,x,0)$ and $(0,0,x)$, respectively, in (15), we obtain

$$\begin{array}{c}\hfill f\left(2x\right)=2^{3}f\left(x\right),f\left({\displaystyle \frac{1}{2}}x\right)={\displaystyle \frac{1}{8}}f\left(x\right),f\left({\displaystyle \frac{1}{2}}x\right)={\displaystyle \frac{1}{8}}f\left(x\right).\end{array}$$

(16)

Replacing $(u,3v,4w)$ with $(x,-y,0)$ in (2) and using (16), we obtain

$$\begin{array}{ccc}\hfill f(2x+y)-2f(2x-y)& =& 5f(x+y)-7f(x-y)-6f\left(x\right)\hfill \\ & -& 9f\left(y\right).\hfill \end{array}$$

(17)

Again, replacing $(u,3v,4w)$ with $(x,y,0)$ in (2) and using (16), we obtain

$$\begin{array}{ccc}\hfill f(2x-y)-2f(2x+y)& =& -7f(x+y)+5f(x-y)-6f\left(x\right)\hfill \\ & +& 9f\left(y\right).\hfill \end{array}$$

(18)

Adding Equations (17) and (18), we obtain Equation (1). □

3. Stability Results in Multi-Banach Space

A multi-Banach space generalizes the concept of Banach spaces by integrating multiple Banach spaces, each with its norm. This approach is crucial for understanding tensor products and complicated operators in functional analysis. Multi-Banach spaces are used in data science to improve optimization algorithms and facilitate the study of multi-dimensional data. Additionally, they are essential to mathematical physics in areas like field theory and quantum mechanics, where systems with numerous interacting Banach spaces are involved. Multi-Banach spaces offer a flexible and reliable framework for challenging issues in various mathematical and scientific fields.

Here, we use the direct and fixed-point techniques to determine the orthogonal stability of the given cubic FE.

Theorem 2. 

Consider an odd function $f:W\to Z$, which holds the following inequality

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|\Delta f(u_1,v_1,w_1),\dots ,\Delta f(u_k,v_k,w_k){\left|\right|}_k\le \alpha \end{array}$$

(19)

for all $u_1,\dots ,u_k,v_1,\dots ,v_k,w_1,\dots ,w_k\in W$, $\alpha \in [0,\infty )$ and $u_i,v_i,w_i$ are mutually orthogonal for all $i\in \{1,\dots ,k\}$.Then, there exists exactly one orthogonal cubic mapping $O_c:W\to Z$ such that

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|f\left(u_1\right)-O_c\left(u_1\right),\dots ,f\left(u_k\right)-O_c\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{\alpha }{7}}.\end{array}$$

(20)

Proof. 

Putting $v_1=\dots =v_k=w_1=\dots =w_k=0$ in inequality (19), we obtain

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|f\left(2u_1\right)-8f\left(u_1\right),\dots ,f\left(2u_k\right)-8f\left(u_k\right){\left|\right|}_k\le \alpha .\end{array}$$

(21)

Dividing both sides by 8,

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2u_1\right)}{8}}-f\left(u_1\right),\dots ,{\displaystyle \frac{f\left(2u_k\right)}{8}}-f\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{\alpha }{8}}.\end{array}$$

(22)

Now, replacing $u_1,\dots ,u_k$ with $2^n{u}_1,\dots ,2^n{u}_k$ in the inequality (22) and dividing both sides by $8^n$

$$\begin{array}{ccc}\hfill & & \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n+1}{u}_1\right)}{8^{n+1}}}-{\displaystyle \frac{f\left(2^n{u}_1\right)}{8^n}},\dots ,{\displaystyle \frac{f\left(2^{n+1}{u}_k\right)}{8^{n+1}}}-{\displaystyle \frac{f\left(2^n{u}_k\right)}{8^n}}{\left|\right|}_k\hfill \\ & & \le {\displaystyle \frac{\alpha }{8^{n+1}}}.\hfill \end{array}$$

(23)

Now, with the help of inequality (23) and applying mathematical induction, we observe that

$$\begin{array}{ccc}\hfill & & \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n+m}{u}_1\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^n{u}_1\right)}{8^n}},\dots ,{\displaystyle \frac{f\left(2^{n+m}{u}_k\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^n{u}_k\right)}{8^n}}{\left|\right|}_k\hfill \\ & & \le \sum _{i=n}^{n+m-1}{\displaystyle \frac{\alpha }{8^{i+1}}}.\hfill \end{array}$$

(24)

Now, fix an element $u\in W$. Also, replace $u_1,\dots ,u_k$ with $u,2u,\dots 2^{k-1}u$, and using the definition of multi-norm, we obtain

$$\begin{array}{ccc}& & \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n+m}u\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}},\dots ,{\displaystyle \frac{f\left(2^{n+m+k-1}u\right)}{8^{n+m+k-1}}}-{\displaystyle \frac{f\left(2^{n+k-1}u\right)}{8^{n+k-1}}}{\left|\right|}_k\hfill \\ & & =\underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n+m}u\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}},\dots ,{\displaystyle \frac{1}{8^{k-1}}}\left({\displaystyle \frac{f\left(2^{n+m}\left(2^{k-1}u\right)\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^{n+k-1}u\right)}{8^n}}\right){\left|\right|}_k\hfill \\ & & \le \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n+m}u\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}},\dots ,\left({\displaystyle \frac{f\left(2^{n+m}\left(2^{k-1}u\right)\right)}{8^{n+m}}}-{\displaystyle \frac{f\left(2^{n+k-1}u\right)}{8^n}}\right){\left|\right|}_k\hfill \\ & & \le \alpha \sum _{i=n}^{n+m-1}{8}^{-i-1}.\hfill \end{array}$$

Hence, by this result the sequence $\left\{{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}\right\}$ is Cauchy, since Z is Banach space; therefore, the sequence $\left\{{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}\right\}$ is convergent. Taking

$$\begin{array}{c}\hfill O_c\left(u\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}\forall u\in W.\end{array}$$

Therefore, for each $ϵ>0,$ there is $n_{k_0}$ such that

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}-O_c\left(u\right),\dots ,{\displaystyle \frac{f\left(2^{n+k-1}u\right)}{8^{n+k-1}}}-O_c\left(u\right){\left|\right|}_k<ϵ\end{array}$$

for all $n\ge n_{k_0}$. Now, by using the characteristic (M2) of multi-norm, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\left|\right|{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}-O_c\left(u\right)\left|\right|=0.\end{array}$$

(25)

Now, by substituting $n=0$ in (24), we obtain

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|{\displaystyle \frac{f\left(2^m{u}_1\right)}{8^m}}-f\left(u_1\right),\dots ,{\displaystyle \frac{f\left(2^m{u}_k\right)}{8^m}}-f\left(u_k\right){\left|\right|}_k\le \alpha \sum _{i=0}^{m-1}{8}^{-i-1}.\end{array}$$

Taking $m\to \infty$ and using Lemma 1, and equality (25), we obtain

$$\begin{array}{c}\hfill \underset{k\in N}{sup}\left|\right|O_c\left(u_1\right)-f\left(u_1\right),\dots ,O_c\left(u_k\right)-f\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{\alpha }{7}}.\end{array}$$

Letting $u,v,w$ be mutually orthogonal such that using the definition of orthogonal space we have that $2^{n}u,2^{n}v,2^{n}w$ are also mutually orthogonal. Taking $u_1=\dots =u_k=2u,v_1=v_2=\dots =v_k=2v,w_1=\dots =w_k=2w$ in inequality (19) and dividing both sides by $8^n$ and using the (M1) property of multi-norms, the authors obtain

$$\begin{array}{ccc}& & \left|\right|{\displaystyle \frac{f\left(2^n(2u+3v+4w)\right)}{8^n}}-3{\displaystyle \frac{f\left(2^n(u+3v+4w)\right)}{8^n}}-{\displaystyle \frac{f\left(2^n(-u+3v+4w)\right)}{8^n}}\hfill \\ & & -2{\displaystyle \frac{f\left(2^n(u+3v)\right)}{8^n}}-2{\displaystyle \frac{f\left(2^n(u+4w)\right)}{8^n}}+6{\displaystyle \frac{f\left(2^n(u-3v)\right)}{8^n}}+6{\displaystyle \frac{f\left(2^n(u-4w)\right)}{8^n}}\hfill \\ & & +3{\displaystyle \frac{f\left(2^n(3v+4w)\right)}{8^n}}-16[{\displaystyle \frac{f\left(2^n(u-\frac{3}{2}v)\right)}{8^n}}+{\displaystyle \frac{f\left(2^n(u-2w)\right)}{8^n}}]+18{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}\hfill \\ & & +6{\displaystyle \frac{f(3.2^{n}v)}{8^n}}+6{\displaystyle \frac{f(4.2^{n}w)}{8^n}}\left|\right|\le {\displaystyle \frac{\alpha }{8^{n+1}}}.\hfill \end{array}$$

Now, taking $n\to \infty$, we obtain

$$\begin{array}{ccc}& & O_c(2u+3v+4w)-3O_c(u+3v+4w)-O_c(-u+3v+4w)\hfill \\ & & -2O_c(u+3v)-2O_c(u+4w)+6O_c(u-3v)+6O_c(u-4w)\hfill \\ & & +3O_c(3v+4w)-16[O_c(u+{\displaystyle \frac{3}{2}}v)+O_c(u-2w)]+18O_c\left(u\right)\hfill \\ & & +6O_c\left(3v\right)+6O_c\left(4w\right)=0.\hfill \end{array}$$

Therefore, $O_c$ is a cubic orthogonal mapping. Consider other cubic mapping, $O_c^{\prime }$, which satisfies inequality (20), in order to demonstrate the uniqueness of $O_c$, so

$$\begin{array}{ccc}\hfill \left|\right|O_c\left(u\right)-O_c^{\prime }\left(u\right)\left|\right|& =& {\displaystyle \frac{1}{8^n}}\left|\right|O_c\left(2^{n}u\right)-O_c^{\prime }\left(2^{n}u\right)\left|\right|\hfill \\ & \le & {\displaystyle \frac{1}{8^n}}\left|\right|O_c\left(2^{n}u\right)-f_c\left(2^{n}u\right)\left|\right|+{\displaystyle \frac{1}{8^n}}\left|\right|f_c\left(2^{n}u\right)-O_c^{\prime }\left(2^{n}u\right)\left|\right|\hfill \\ & \le & {\displaystyle \frac{1}{8^n}}{\displaystyle \frac{2\alpha }{8}}.\hfill \end{array}$$

Using $n\to \infty$, the authors obtain $O_c=O_c^{\prime }$. □

We will now use the fixed-point approach to demonstrate FE stability.

Theorem 3. 

Suppose $\varphi :W^{3k}\to [0,\infty )$ is a mapping such that

$$\begin{array}{c}\hfill \varphi (2u_1,\dots ,2u_k,2v_1,\dots ,2v_k,2w_1,\dots ,2w_k)\le \alpha \varphi (u_1,\dots ,u_k,v_1,\dots ,v_k,w_1,\dots ,w_k),\end{array}$$

for all $u_1,\dots ,u_k,v_1,\dots ,v_k,w_1,\dots ,w_k\in W$ and $0<\alpha <8$. Suppose that $f:W\to Z$ is a function holding

$$\begin{array}{c}\hfill \left|\right|\Delta f(u_1,v_1,w_1),\dots ,\Delta f(u_k,v_k,w_k){\left|\right|}_k\le \varphi (u_1,\dots ,u_k,v_1,\dots ,v_k,w_1,\dots ,w_k)\end{array}$$

(26)

where $u_i,v_i,w_i$ are mutually orthogonal for all $i\in \{1,\dots ,k\}$. Then, there is an orthogonal cubic mapping $O_c:W\to Z$, which satisfies the following inequality

$$\begin{array}{c}\hfill \left|\right|f\left(u_1\right)-O_c\left(u_1\right),\dots ,f\left(u_k\right)-O_c\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{1}{(8-\alpha )}}\varphi (u_1,\dots ,u_k,0,\dots ,0).\end{array}$$

(27)

Proof. 

Let $H=\{f:W\to Z\}$ and define a generalized metric d on H with

$$\begin{array}{ccc}\hfill d(g,h)& =& \inf \{\beta \in (0,\infty ):\left|\right|g\left(u_1\right)-h\left(u_1\right),\dots ,g\left(u_k\right)-h\left(u_k\right){\left|\right|}_k\hfill \\ & \le & \beta \psi (u_1,\dots ,u_k),\forall u_1,\dots u_k\in W\},\hfill \end{array}$$

where $\psi (u_1,\dots u_k)=\varphi (u_1,\dots ,u_k,0,\dots ,0)$. Subsequently, proving that d is a complete generalized metric on H is simple. We have considered a mapping $Mg\left(u\right)={\displaystyle \frac{1}{8}}g\left(2u\right)$ and proved that mapping M is strictly contractive. Consider an arbitrary constant $\beta \in (0,\infty )$ such that $d(g,h)<\beta$ and $g,h\in H$. Now, it is easy to see that

$$\begin{array}{c}\hfill \left|\right|g\left(u_1\right)-h\left(u_1\right),\dots ,g\left(u_k\right)-h\left(u_k\right){\left|\right|}_k\le \beta \psi (u_1,\dots ,u_k).\end{array}$$

Therefore,

$$\begin{array}{ccc}& & \left|\right|Mg\left(u_1\right)-Mh\left(u_1\right),\dots ,Mg\left(u_k\right)-Mh\left(u_k\right){\left|\right|}_k\hfill \\ & & =\left|\right|{\displaystyle \frac{1}{8}}g\left(2u_1\right)-{\displaystyle \frac{1}{8}}h\left(2u_1\right),\dots ,{\displaystyle \frac{1}{8}}g\left(2u_k\right)-{\displaystyle \frac{1}{8}}h\left(2u_k\right){\left|\right|}_k\hfill \\ & & \le {\displaystyle \frac{1}{8}}\alpha \beta \psi (u_1,\dots ,u_k).\hfill \end{array}$$

So, we have $d(Mg,Mh)\le {\displaystyle \frac{1}{8}}\alpha d(g,h)$ for all $g,h\in H$. Taking $v_1=\dots =v_k=w_1=\dots =w_k=0$ in (26), we have

$$\begin{array}{c}\hfill \left|\right|f\left(2u_1\right)-8f\left(u_1\right),\dots ,f\left(2u_k\right)-8f\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{1}{8}}\psi (u_1,u_2,\dots ,u_k).\end{array}$$

(28)

It follows from (28) that $d(Mf,f)\le {\displaystyle \frac{1}{8}}$. By applying the fixed-point Theorem, we conclude that a fixed point M exists, i.e., a mapping $O_c:W\to Z$, having $O_c\left(2u\right)=8O_c\left(u\right)$. It follows from the condition $d(M^{n}f,O_c)\to 0$ that

$$\begin{array}{c}\hfill O_c\left(u\right)=\underset{n\to \infty }{lim}\left(M^{n}f\right)\left(u\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{f\left(2^{n}u\right)}{8^n}}.\end{array}$$

Moreover, $d(f,O_c)\le {\displaystyle \frac{8}{8-\alpha }}d(Mf,f)$ implies the inequality

$$\begin{array}{c}\hfill d(f,O_c)\le {\displaystyle \frac{1}{8-\alpha }}.\end{array}$$

Therefore, inequality (27) is satisfied. Let $u,v,w$ be mutually orthogonal such that $2^{n}u, 2^{n}v$, $2^{n}w$ are also mutually orthogonal and take $u_1=\dots u_k=2^{n}u,v_1=\dots =v_k=2^{n}v,w_1=\dots =w_k=2^{n}w$ in (26) and multiplying both side by $8^{-n}$; we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{8^n}}\left|\right|\Delta f(2^{n}u,2^{n}v,2^{n}w)\left|\right|& \le & {\displaystyle \frac{1}{8^n}}\varphi (2^{n}u,\dots ,2^{n}u,2^{n}v,\dots ,2^{n}v,2^{n}w,\dots ,2^{n}w)\hfill \\ & \le & {\left({\displaystyle \frac{\alpha }{8}}\right)}^n\varphi (u,\dots ,u,v,\dots ,v,w,\dots ,w).\hfill \end{array}$$

We conclude that $O_c$ is an orthogonal cubic mapping by taking the limit as $n\to \infty$. $O_c$ is a unique mapping since $O_c$ is the unique fixed point of M that satisfies the condition that there is a positive integer $\gamma$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u_1\right)-O_c\left(u_1\right),\dots ,f\left(u_k\right)-O_c\left(u_k\right){\left|\right|}_k\le \gamma \psi (u_1,\dots ,u_k).\end{array}$$

□

Corollary 1. 

Let $f:W\to Z$ be a mapping that satisfies

$$\begin{array}{ccc}\hfill \left|\right|\Delta f(u_1,v_1,w_1),\dots ,\Delta f(u_k,v_k,w_k){\left|\right|}_k& \le & \alpha \left(\right|\left|u_1\right|{|}^q+\dots \left|\right|u_k|{|}^q\hfill \\ & +& \left|\right|v_1{\left|\right|}^q+\dots +\left|\right|v_k{\left|\right|}^q\hfill \\ & +& \left|\right|w_1{\left|\right|}^q+\dots +\left|\right|w_k{\left|\right|}^q),\hfill \end{array}$$

where $u_i,v_i,w_i$ are mutually orthogonal for all $i\in \{1,2,\dots ,k\}$, $\alpha >0$ and $0<q<3$. Then, there is a cubic orthogonal mapping $O_c:W\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u_1\right)-O_c\left(u_1\right),\dots ,f\left(u_k\right)-O_c\left(u_k\right){\left|\right|}_k\le {\displaystyle \frac{\alpha }{8-2^q}}\left(\right||u_1{\left|\right|}^q+\dots \left|\right|u_k{\left|\right|}^q).\end{array}$$

4. Stability Results in $(\mathit{\beta },\mathit{p})$-Banach Space

The triangle inequality is relaxed in a quasi-Banach space such that the norm satisfies $\left(\right||u+v||\le K(\parallel u\parallel + \left|\right|v\left|\right|\left)\right)$ for some constant $(K\ge 1)$. This generalizes Banach spaces. Quasi-Banach spaces are helpful in harmonic analysis to examine different function spaces and in functional analysis for interpolation theory because of their flexibility. Additionally, they are employed in probability theory to analyze stable distributions and spaces of random variables and in approximation theory for signal processing, where conventional norm characteristics might not hold. A brief overview of the quasi-$\beta$-normed space is given before we go into our findings.

Definition 5. 

The function $\psi :U\to X$ is said to be

1. 

A subadditive function if it satisfies the property

$$\begin{array}{c}\hfill \psi (u+v)\le \psi \left(u\right)+\psi \left(v\right).\end{array}$$

2. 

A contractively subadditive function if it satisfies the property

$$\begin{array}{c}\hfill \psi (u+v)\le L\left[\psi \right(u)+\psi (v\left)\right],\end{array}$$

3. 

An expansively superadditive function if it satisfies the property

$$\begin{array}{c}\hfill \psi (u+v)\le {\displaystyle \frac{1}{L}}[\psi \left(u\right)+\psi \left(v\right)],\end{array}$$

where $0<L<1$.

Theorem 4. 

Consider a function $f:U\to X$ for which there is a contractively subadditive mapping $\psi :U^3\to [0,\infty )$ such that

$$\begin{array}{c}\hfill {\left|\right|\Delta f(u,v,w)\left|\right|}_X\le \psi (u,v,w),u,v,w\in U,\end{array}$$

(29)

with $2^{(1-3\beta )}L<1$, where L is any constant, $u,v,w\in U$, and $u,v,w$ are mutually orthogonal. Then, there is one and only one orthogonal cubic mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right){\left|\right|}_X\le {\displaystyle \frac{K}{{(2^{3\beta p}-{\left(2L\right)}^p)}^{\frac{1}{p}}}}\psi (u,0,0),\end{array}$$

(30)

where K is the modulus concavity of ${\parallel ·\parallel }_X$.

Proof. 

Substituting $(u,v,w)=(u,0,0)$ in inequality (29), we have

$$\begin{array}{c}\hfill \left|\right|f\left(2u\right)-2^{3}f\left(u\right){\left|\right|}_X\le \psi (u,0,0).\end{array}$$

(31)

$$\begin{array}{c}\hfill \left|\right|{\displaystyle \frac{f\left(2u\right)}{8}}-f\left(u\right){\left|\right|}_X\le {\displaystyle \frac{1}{8^{\beta }}}\psi (u,0,0).\end{array}$$

(32)

Therefore, it follows from the last inequality with $2^{t}u$ in the place of u and using an iterative method that

$$\begin{array}{ccc}\hfill \left|\right|{\displaystyle \frac{f\left(2^{m+1}u\right)}{2^{3(m+1)}}}-{\displaystyle \frac{f\left(2^{l}u\right)}{2^{3l}}}{\left|\right|}_X^p& \le & \sum _{t=l}^m\left|\right|{\displaystyle \frac{f\left(2^{t+1}u\right)}{2^{3(t+1)}}}-{\displaystyle \frac{f\left(2^{t}u\right)}{2^{3t}}}{\left|\right|}_X^p\hfill \\ \hfill & \le & \sum _{t=l}^m{\displaystyle \frac{K^p}{2^{3(t+1)\beta p}}}{\psi }^p(2^{t}u,0,0)\hfill \\ & \le & {\displaystyle \frac{K^p{\psi }^p(u,0,0)}{2^{3p\beta }}}\sum _{t=l}^m{\left(2^{(1-3\beta )}L\right)}^{tp}\hfill \end{array}$$

(33)

where $m\ge l>0$. Since X is complete space, the Cauchy sequence $\left\{{\displaystyle \frac{f\left(2^{m}u\right)}{2^{3m}}}\right\}$ is convergent. Now, define a mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill O_c\left(u\right)=\underset{m\to \infty }{lim}{\displaystyle \frac{f\left(2^{m}u\right)}{2^{3m}}},\forall u\in U.\end{array}$$

(34)

Taking $l\to 0^{+}$ and $m\to \infty$ in (33), we obtain inequality (30). Now, we have to prove that $O_c$ is a cubic mapping, and using the inequality (29) and (33)

$$\begin{array}{ccc}\hfill {\left|\right|\Delta f(u,v,w)\left|\right|}_X^p& =& \underset{m\to \infty }{lim}\left|\right|{\displaystyle \frac{\Delta f(2^{m}u,2^{m}v,2^{m}w)}{2^{3m}}}{\left|\right|}_X^p\hfill \\ \hfill & \le & \underset{m\to \infty }{lim}{\displaystyle \frac{K^p}{2^{3m\beta p}}}{\psi }^p(2^{m}u,2^{m}v,2^{m}w)\hfill \\ \hfill & \le & \underset{m\to \infty }{lim}{\displaystyle \frac{K^p{\left(2L\right)}^{mp}}{2^{3m\beta p}}}{\psi }^p(u,v,w)\hfill \\ \hfill & \le & \underset{m\to \infty }{lim}{K}^p{\left(2^{(1-3\beta )}L\right)}^{mp}{\psi }^p(u,v,w)\hfill \\ & =& 0.\hfill \end{array}$$

(35)

Therefore, $O_c$ is a cubic mapping. Consider another cubic mapping $O_c^{\prime }$, which satisfies inequality (30) to demonstrate the uniqueness of $O_c$.

$$\begin{array}{ccc}\hfill \left|\right|O_c-O_c^{\prime }{\left|\right|}_X& =& \underset{n\to \infty }{lim}{\displaystyle \frac{1}{2^{3m\beta }}}\left|\right|O_c\left(2^{m}u\right)-O_c^{\prime }\left(2^{m}u\right){\left|\right|}_X\hfill \\ & \le & \underset{n\to \infty }{lim}{\displaystyle \frac{K\psi (2^{m}u,0,0)}{2^{3m\beta }{(2^{3p\beta }-{\left(2L\right)}^p)}^{\frac{1}{p}}}}\hfill \\ & \le & \underset{m\to \infty }{lim}{\displaystyle \frac{K{2}^{{(1-3\beta )L)}^m}}{{(2^{3p\beta }-{\left(2L\right)}^p)}^{\frac{1}{p}}}}\psi (u,0,0)=0.\hfill \end{array}$$

Hence, $O_c=O_c^{\prime }$. Therefore, we have established the required result. □

Corollary 2. 

Suppose $\kappa >0$, $\beta =1$, and a mapping $f:U\to X$ is such that ${\left|\right|\Delta f(u,v,w)\left|\right|}_X\le \kappa$, where $u,v,w$ are mutually orthogonal. Then, there is one and only one orthogonal cubic mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right){\left|\right|}_X\le {\displaystyle \frac{K\kappa }{{(2^{3\beta p}-{\left(2L\right)}^p)}^{\frac{1}{p}}}}\forall u\in U.\end{array}$$

Theorem 5. 

Consider a function $f:U\to X$ for which there is an expansive superadditive function $\psi :U^3\to [0,\infty )$ satisfying

$$\begin{array}{c}\hfill {\left|\right|\Delta f(u,v,w)\left|\right|}_X\le \psi (u,v,w),\end{array}$$

with $2^{(3\beta -1)}L<1$, where L is any constant, $w,v,u\in U$, and $u,v,w$ are mutually orthogonal. Then, there is one and only one orthogonal cubic mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right){\left|\right|}_X\le {\displaystyle \frac{KL}{{(2^p-{\left(2^{3\beta }L\right)}^p)}^{\frac{1}{p}}}}\psi (u,0,0),\end{array}$$

where K is the modulus concavity of ${\parallel ·\parallel }_X$.

Proof. 

The proof is the same as the Theorem 4. □

Theorem 6. 

Consider a mapping $\varphi :U^3\to [0,\infty )$ in such a way that

$$\underset{m\to \infty }{lim}{\displaystyle \frac{1}{2^{3m}}}\varphi (2^{m}u,2^{m}v,2^{m}w)=0,$$

(36)

for all $u,v,w\in U$, $u,v,w$ are mutually orthogonal and

$$\begin{array}{c}\hfill \tilde{\varphi }\left(u\right)=\sum _{t=0}^{\infty }{\displaystyle \frac{K^p}{2^{3t\beta p}}}{\varphi }^p(2^{t}u,0,0)<\infty .\end{array}$$

(37)

Let there exist a mapping $f:U\to X$ such that

$$\begin{array}{c}\hfill {\left|\right|\Delta f(u,v,w)\left|\right|}_X\le \varphi (u,v,w),\end{array}$$

(38)

where $u,v,w$ are mutually orthogonal. Then, there is one and only one orthogonal cubic mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right){\left|\right|}_X\le {\displaystyle \frac{K}{2^{3\beta }}}{\left[\tilde{\varphi }\left(u\right)\right]}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

(39)

Proof. 

Substituting $(u,v,w)=(u,0,0)$ in (38), we have

$$\begin{array}{c}\hfill \left|\right|f\left(2u\right)-2^{3}f\left(u\right){\left|\right|}_X\le \varphi (u,0,0).\end{array}$$

(40)

Now,

$$\begin{array}{ccc}\hfill \left|\right|{\displaystyle \frac{f\left(2^{m+1}u\right)}{2^{3(m+1)}}}-{\displaystyle \frac{f\left(2^{l}u\right)}{2^{3l}}}{\left|\right|}_X^p& \le & \sum _{t=l}^m\left|\right|{\displaystyle \frac{f\left(2^{t+1}u\right)}{2^{3(t+1)}}}-{\displaystyle \frac{f\left(2^{t}u\right)}{2^{3t}}}{\left|\right|}_X^p\hfill \\ & \le & \sum _{t=l}^m{\displaystyle \frac{K^p}{2^{3(t+1)\beta p}}}{\psi }^p(2^{t}u,0,0)\hfill \end{array}$$

(41)

where $m\ge l>0.$ From inequalities (37) and (41), we can deduce that the sequence $\left\{{\displaystyle \frac{f\left(2^{m}u\right)}{2^{3m}}}\right\}$ is Cauchy. However, the space X is complete; therefore, the sequence $\left\{{\displaystyle \frac{f\left(2^{m}u\right)}{2^{3m}}}\right\}$ is convergent. Now, define a mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill O_c\left(u\right)=\underset{m\to \infty }{lim}{\displaystyle \frac{f\left(2^{m}u\right)}{2^{3m}}},\forall u\in U.\end{array}$$

(42)

Taking $l=0$ and $m\to \infty$ in (41), the authors obtain inequality (39). We must prove that $O_c$ is cubic mapping. The remaining arguments follow the proof of Theorem 4. □

Corollary 3. 

Let us consider two positive real numbers ${\alpha }_1$ and ${\alpha }_2$ with condition ${\alpha }_1+{\alpha }_2<3\beta$ or ${\alpha }_1+{\alpha }_2>3\beta$. Let there be a mapping $f:U\to X$ such that

$$\begin{array}{c}\hfill {\left|\right|\Delta f(u,v,w)\left|\right|}_X\le \prod _{t=1}^3\left|\right|u_t{\left|\right|}_X^{{\alpha }_1+{\alpha }_2}+\sum _{t=1}^3\left|\right|u_t{\left|\right|}_X^{3({\alpha }_1+{\alpha }_2)}\end{array}$$

where $u_1=u,u_2=v,u_3=w$. Then, there is one and only one orthogonal cubic mapping $O_c:U\to X$ such that

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right){\left|\right|}_X\le {\displaystyle \frac{{K\left|\right|u\left|\right|}_X^{3({\alpha }_1+{\alpha }_2)}}{|2^{3p\beta }-2^{3({\alpha }_1+{\alpha }_2)p}{|}^{\frac{1}{p}}}}.\end{array}$$

Nonstability Example for Special Case

Example 1. 

Consider a function $\varphi :R\to R$ such that

$$\begin{array}{c}\hfill \varphi \left(u\right)=\left\{\begin{array}{cc}\theta u^3\hfill & \mathit{if}-1<u<1\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

(43)

Here, θ is a positive constant. Also, consider a mapping $f:R\to R$ defined as

$$\begin{array}{c}\hfill f\left(u\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{\varphi \left(2^{m}u\right)}{2^{3m}}}.\end{array}$$

(44)

Then, the mapping f fulfills the inequality

$$\begin{array}{c}\hfill \left|\right|\Delta f(u,v,w)\left|\right|\le {\displaystyle \frac{688}{7}}{\theta \left(\right|u|}^3+{\left|v\right|}^3+{\left|w\right|}^3)\end{array}$$

(45)

so that there does not exist any cubic mapping $O_c:R\to R$ which holding the inequality

$$\begin{array}{c}\hfill \left|\right|f\left(u\right)-O_c\left(u\right)\left|\right|\le {ϵ\left|u\right|}^3.\end{array}$$

(46)

where ϵ is a constant.

Proof. 

First, we prove f is bounded,

$$\begin{array}{c}\hfill \left|f\left(u\right)\right|=\sum _{m=0}^{\infty }{\displaystyle \frac{\left|\varphi \right(2^{m}u\left)\right|}{|2^{3m}|}}=\sum _{m=0}^{\infty }{\displaystyle \frac{\theta }{2^{3m}}}={\displaystyle \frac{8\theta }{7}}.\end{array}$$

Hence, f is bounded. The left side of (45) is less than ${\displaystyle \frac{44032}{7}}\theta$, if ${\left(\right|u|}^3+{\left|v\right|}^3+{\left|w\right|}^3)\ge {\displaystyle \frac{1}{2^3}}$. Now, we assume that

$$\begin{array}{c}\hfill {0<\left(\right|u|}^3+{\left|v\right|}^3+{\left|w\right|}^3)<{\displaystyle \frac{1}{2^3}}\end{array}$$

(47)

then there is an integer k such that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2^{3(k+2)}}}\le {\left(\right|u|}^3+{\left|v\right|}^3+{\left|w\right|}^3)<{\displaystyle \frac{1}{2^{3(k+1)}}},\end{array}$$

(48)

so that

$$2^k\left|u\right|<{\displaystyle \frac{1}{2}},2^k\left|v\right|<{\displaystyle \frac{1}{2}},2^k\left|w\right|<{\displaystyle \frac{1}{2}}$$

and

$$2^{m}u,2^{m}v,2^{m}w\in (-1,1).$$

For $m=0,1,2,\dots ,i-1$,

$$\begin{array}{ccc}& & \varphi \left(2^m(2u+3v+4w)\right)-3\varphi \left(2^m(u+3v+4w)\right)-\varphi \left(2^m(-u+3v+4w)\right)\hfill \\ \hfill & & -2\varphi \left(2^m(u+3v)\right)-2\varphi \left(2^m(u+4w)\right)+6\varphi \left(2^m(u-3v)\right)+6\varphi \left(2^m(u-4w)\right)\hfill \\ \hfill & & +3\varphi \left(2^m(3v+4w)\right)-16[\varphi \left(2^m(u+{\displaystyle \frac{3}{2}}v)\right)-\varphi \left(2^m(u-2w)\right)]+18\varphi \left(2^m\left(u\right)\right)\hfill \\ \hfill & & +6\varphi \left(2^m\left(3v\right)\right)+6\varphi \left(2^m\left(4w\right)\right)=0.\hfill \end{array}$$

Now,

$$\begin{array}{ccc}& & |\Delta f(u,v,w)|\le \sum _{m=i}^{\infty }{\displaystyle \frac{1}{2^{3m}}}|\Delta \varphi (2^{m}u,2^{m}v,2^{m}w)|\hfill \\ \hfill & & \le \sum _{m=i}^{\infty }{\displaystyle \frac{1}{2^{3m}}}|\varphi \left(2^m(2u+3v+4w)\right)-3\varphi \left(2^m(u+3v+4w)\right)\hfill \\ \hfill & & -\varphi \left(2^m(-u+3v+4w)\right)-2\varphi \left(2^m(u+3v)\right)-2\varphi \left(2^m(u+4w)\right)\hfill \\ \hfill & & +6\varphi \left(2^m(u-3v)\right)+6\varphi \left(2^m(u-4w)\right)+3\varphi \left(2^m(3v+4w)\right)\hfill \\ \hfill & & -16[\varphi \left(2^m(u-{\displaystyle \frac{3}{2}}v)\right)+\varphi \left(2^m(u-2w)\right)]+18\varphi \left(2^m\left(u\right)\right)\hfill \\ \hfill & & +6\varphi \left(2^m\left(3v\right)\right)+6\varphi \left(2^m\left(4w\right)\right)|\le \sum _{m=i}^{\infty }{\displaystyle \frac{1}{2^{3m}}}86\theta ={\displaystyle \frac{688}{7}}{\displaystyle \frac{1}{2^{3i}}}\theta \hfill \\ & & \le {\displaystyle \frac{44032}{7}}{\theta \left(\right|u|}^3+{\left|v\right|}^3+{\left|w\right|}^3).\hfill \end{array}$$

(49)

Assume that there is a contrary, cubic mapping $O_c:R\to R$, which satisfies all conditions. For every $u\in R$, f is continuous and bounded. Additionally, $O_c$ is continuous at the origin and bounded on any open interval containing the origin. Also, $O_c$ must be $O_c\left(u\right)=\lambda {\left|u\right|}^3$. Therefore, the authors have

$$\begin{array}{c}\hfill \left|f\left(u\right)\right|\le {(ϵ+|\lambda \left|\right)\left|u\right|}^3,u\in R.\end{array}$$

(50)

Let us choose a positive integer k such that

$$k\theta >ϵ+{\left|u\right|}^3,$$

if $u\in (0,{\displaystyle \frac{1}{2^{k-1}}}),$ then $2^{m}u\in (0,1)$ for any $m\in \{0,1,\dots ,k-1\}$; the authors obtain

$$\begin{array}{ccc}\hfill f\left(u\right)& =& \sum _{m=0}^{\infty }{\displaystyle \frac{\varphi \left(2^{m}u\right)}{2^{3m}}}\ge \sum _{m=0}^{k-1}{\displaystyle \frac{\theta {\left(2^{m}u\right)}^3}{2^{3m}}}\hfill \\ & =& {k\theta \left|u\right|}^3{>(ϵ+|\lambda \left|\right)\left|u\right|}^3,\hfill \end{array}$$

(51)

which contradicts, i.e., the FE is not stable. □

5. Conclusions

In this study, we introduced a new three-variable cubic functional equation and successfully derived its general solution. By employing both direct and fixed-point methods, we investigated the orthogonal stability of this equation within the frameworks of quasi-$\beta$-Banach spaces and multi-Banach spaces. Our results highlighted the stability properties in various extended Banach spaces and offered insights into mixed cases. Additionally, we provided an example demonstrating the absence of stability in a specific scenario, illustrating the nuanced nature of functional equations in these mathematical structures. The findings of this research contribute to the broader understanding of functional equations and their stability in multi-Banach and quasi-Banach spaces, offering new directions for future studies in this area.