Approximate solution of a multivariable Cauchy-Jensen functional equation

1.   Introduction

Hyers-Ulam stability in functional equations relates to the characteristic where a function may not perfectly satisfy a given equation but remains close to the actual solution if the error is within a limited range. This concept of stability can be applied to many types of functional equations. The concept of Hyers-Ulam stability is named after the mathematicians David Hyers and Stanislaw Ulam, who both discovered it in the 1940s. In 1978, Rassias ^[1] published the result that stability can be obtained even after generalizing the error from a constant to a variable. The stability problem where the error is a variable is called the Hyers-Ulam-Rassias stability.

This theory extends beyond the Cauchy equation to various other types of functional equations. For instance, it can be considered in more complex forms of functional equations involving power functions, trigonometric functions, exponential functions, and more. Additionally, this concept can be effectively applied in other mathematical structures like differential equations or integral equations.

Hyers-Ulam stability serves as an important tool in mathematical analysis and applied mathematics, playing a significant role in finding approximate solutions. It helps in finding exact solutions in real-world problems where errors are allowed and is useful in providing approximate interpretations in complex systems.

We would like to introduce the definition of the Cauchy-Jensen equation and past work.

Definition 1. ^[2] A mapping $f:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ is called a Cauchy-Jensen mapping if $f$ satisfies the system of equations

In 2006, W. G. Park and J. H. Bae ^[2] obtained the general solution of the Cauchy-Jensen functional equation

and its stability. Subsequent papers have been published since 2007 by several authors ^[3,4,5].

In 2012, J. H. Bae and W. G. Park ^[6] introduced the following multivariable Cauchy-Jensen functional equation

where $n$ is an integer greater than $1$. In Theorem 2.2 of ^[6], we proved that the functional Eqs (1.1) and (1.2) are equivalent. The functional Eq (1.1) is the case where $n = 2$ in the function Eq (1.2).

In 2011, W. G. Park ^[7] investigated the approximate additive and Jensen and quadratic mappings in $2$-Banach spaces. In 2015, S. Yun ^[8] corrected the statements of results in ^[7] and proved the corrected theorems.

In this paper, we investigate the stability of the functional Eq (1.2) in Banach spaces, quasi-Banach spaces, and normed $2$-Banach spaces.

2.   Main results

Let ${\mathcal{X}}$ be a normed space and ${\mathcal{Y}}$ a Banach space.

Theorem 1. Let $r\in(0, 1)$, ${\varepsilon} > 0$, and ${\delta}\ge0$, and let $f:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ be a mapping satisfying $f(x, 0) = 0$ such that

for all $x, x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, then there exists a Cauchy-Jensen mapping $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ such that

for all $x, y\in{\mathcal{X}}$.

Proof. Let $g(x, y): = f(x, y)-f(0, y)$ for all $x, y\in{\mathcal{X}}$, then $g(x, 0) = g(0, y) = 0$ for all $x, y\in{\mathcal{X}}$. By (2.1), $g$ satisfies

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$. Letting $x_1 = {\cdots} = x_n = x$ in (2.3), we gain

for all $x, y_1, {\cdots}, y_n\in{\mathcal{X}}$. Putting $y_1 = y, y_2 = -y$, and $y_3 = {\cdots} = y_n = 0$ in (2.4), we get

for all $x, y\in{\mathcal{X}}$. Setting $y_1 = {\cdots} = y_n = y$ in (2.4), we have

for all $x, y\in{\mathcal{X}}$. By the above two inequalities, we obtain

for all $x, y\in{\mathcal{X}}$. Taking $y_1 = y, y_2 = -(n+1)y$ and $y_j = 0$ $(3\le j\le n)$ in (2.4), we gain

for all $x, y\in{\mathcal{X}}$. By (2.5) and the above inequality, we get

for all $x, y\in{\mathcal{X}}$. Replacing $y$ by $(n+1)y$ in (2.5), we have

for all $x, y\in{\mathcal{X}}$. By (2.6) and the above inequality, we obtain

for all $x, y\in{\mathcal{X}}$. Replacing $x$ by $n^kx$ and $y$ by $(n+1)^ky$ in the above inequality and dividing $n^{k+1}(n+1)^{k+1}$, we see that

for all $x, y\in{\mathcal{X}}$ and all nonnegative integers $k$. Thus, we have

for all integers $l, \, m\, (0\le l < m)$, and all $x, y\in{\mathcal{X}}$. Thus, the sequence $\big\{ \frac 1{n^k(n+1)^k}g\big(n^kx, (n+1)^ky\big)\big\}$ is a Cauchy sequence for all $x, \, y\in{\mathcal{X}}$. Since ${\mathcal{Y}}$ is complete, the sequence $\big\{ \frac 1{n^k(n+1)^k}g\big(n^kx, (n+1)^ky\big)\big\}$ converges for all $x, \, y\in{\mathcal{X}}$. Define $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ by

for all $x, y\in{\mathcal{X}}$. By (2.3), we have

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, and all nonnegative integers $k$. Letting $k\to{\infty}$ in the above inequality, we obtain that $F$ satisfies (1.2). By ^[6], $F$ is a Cauchy-Jensen mapping. Setting $l = 0$ and taking $m\to{\infty}$ in (2.7), one can obtain the inequality (2.2). \qed

In ^[9,10,11], one can find the concept of quasi-Banach spaces.

Definition 2. Let ${\mathcal{X}}$ be a real vector space. A quasi-norm is a real-valued function on ${\mathcal{X}}$ satisfying the following:

(i) $\|x\|\ge0$ for all $x\in{\mathcal{X}}$, and $\|x\| = 0$ if, and only if, $x = 0$.

(ii) $\|{\lambda} x\| = |{\lambda}|\|x\|$ for all ${\lambda}\in{\mathbb{R}}$ and all $x\in{\mathcal{X}}$.

(iii) There is a constant $K\ge1$ such that $\|x+y\|\le K(\|x\|+\|y\|)$ for all $x, \, y\in{\mathcal{X}}$.

Definition 3. The pair $({\mathcal{X}}, \, \|{\cdot}\|)$ is called a quasi-normed space if $\|{\cdot}\|$ is a quasi-norm on ${\mathcal{X}}$. The smallest possible $K$ is called the modulus of concavity of $\|{\cdot}\|$. A quasi-Banach space is a complete quasi-normed space. A quasi-norm $\|{\cdot}\|$ is called a $p$-norm ($0 < p\le 1$) if

for all $x, \, y\in{\mathcal{X}}$. In this case, a quasi-Banach space is called a $p$-Banach space.

We will use the following lemma in the proof of the next theorem.

Lemma 1. ^[12] Let $0\le p\le1$ and let $x_1, x_2, {\cdots}, x_n$ be nonnegative real numbers, then

From now on, assume that ${\mathcal{X}}$ is a quasi-normed space with quasi-norm $\|{\cdot}\|$ and that ${\mathcal{Y}}$ is a $p$-Banach space with $p$-norm $\|{\cdot}\|_{\mathcal{Y}}$.

The following theorem proves the stability of the Eq $(1.2)$ in quasi-Banach spaces.

Theorem 2. Let $r\in(0, 1)$, ${\varepsilon} > 0$, and ${\delta}\ge0$, and let $f:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ be a mapping satisfying $f(x, 0) = 0$ such that

for all $x, x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, then there exists a Cauchy-Jensen mapping $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ such that

for all $x, y\in{\mathcal{X}}$.

Proof. Let $g(x, y): = f(x, y)-f(0, y)$ for all $x, y\in{\mathcal{X}}$, then $g(x, 0) = g(0, y) = 0$ for all $x, y\in{\mathcal{X}}$. By (2.8) and by using Lemma 1, $g$ satisfies

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$. Letting $x_1 = {\cdots} = x_n = x$ in (2.10), we gain

for all $x, y_1, {\cdots}, y_n\in{\mathcal{X}}$. Putting $y_1 = y, y_2 = -y$, and $y_3 = {\cdots} = y_n = 0$ in (2.11), we get

for all $x, y\in{\mathcal{X}}$. Setting $y_1 = {\cdots} = y_n = y$ in (2.11), we have

for all $x, y\in{\mathcal{X}}$. By the above two inequalities, we obtain

for all $x, y\in{\mathcal{X}}$. Taking $y_1 = y, y_2 = -(n+1)y$, and $y_3 = {\cdots} = y_n = 0$ in (2.11), we gain

for all $x, y\in{\mathcal{X}}$. By (2.12) and the above inequality, we get

for all $x, y\in{\mathcal{X}}$. Replacing $y$ by $(n+1)y$ in (2.12), we have

for all $x, y\in{\mathcal{X}}$. By (2.13) and the above inequality, we obtain

for all $x, y\in{\mathcal{X}}$. Replacing $x$ by $n^kx$ and $y$ by $(n+1)^ky$ in the above inequality and dividing $n^{(k+1)p}(n+1)^{(k+1)p}$, we see that

for all $x, y\in{\mathcal{X}}$ and all nonnegative integers $k$. Thus, we have

for all integers $l, \, m\, (0\le l < m)$, and all $x, y\in{\mathcal{X}}$. Thus, the sequence $\big\{ \frac 1{n^k(n+1)^k}g(n^kx, (n+1)^ky)\big\}$ is a Cauchy sequence for all $x, \, y\in{\mathcal{X}}$. Since ${\mathcal{Y}}$ is complete, the sequence $\big\{ \frac 1{n^k(n+1)^k}g(n^kx, (n+1)^ky)\big\}$ converges for all $x, \, y\in{\mathcal{X}}$. Define $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ by

for all $x, y\in{\mathcal{X}}$. By (2.10), we have

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, and all nonnegative integers $k$. Letting $k\to{\infty}$ in the above inequality, we obtain that $F$ satisfies (1.2). By ^[6], $F$ is a Cauchy-Jensen mapping. Setting $l = 0$ and taking $m\to{\infty}$ in (2.14), one can obtain the inequality (2.9).

Taking $n = 2$ and ${\delta} = 0$ in Theorem 2, we obtain the following corollary.

Corollary 1. Let ${\varepsilon} > 0$ be fixed. Suppose that a mapping $f:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ is a mapping satisfying $f(x, 0) = 0$ such that

for all $x, y, z, w\in{\mathcal{X}}$, then there exists a unique bi-additive mapping $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ satisfying

for all $x, y\in{\mathcal{X}}$.

We introduce some definitions on $2$-Banach spaces ^[13,14,15].

Definition 4. Let ${\mathcal{X}}$ be a real vector space with $\dim{{\mathcal{X}}}\ge2$ and $\|{\cdot}, {\cdot}\|:{\mathcal{X}}^2\to{\mathbb{R}}$ be a function, then $({\mathcal{X}}, \|{\cdot}, {\cdot}\|)$ is called a linear $2$-normed space if the following conditions hold:

(a) $\|x, y\| = 0$ if, and only if, $x$ and $y$ are linearly dependent,

(b) $\|x, y\| = \|y, x\|$,

(c) $\|{\alpha} x, y\| = |{\alpha}|\|x, y\|$,

(d) $\|x, y+z\|\le\|x, y\|+\|x, z\|$ for all ${\alpha}\in{\mathbb{R}}$ and $x, y, z\in{\mathcal{X}}$. In this case, the function $\|{\cdot}, {\cdot}\|$ is called a $2$-norm on ${\mathcal{X}}$.

Definition 5. A sequence $\{x_n\}$ in a linear $2$-normed space ${\mathcal{X}}$ is called a Cauchy sequence if there are two linearly independent points $y, z\in{\mathcal{X}}$ such that for any ${\varepsilon} > 0$, there exists $N\in{\mathbb{N}}$ such that for all $m, n\ge N$, $\|x_m-x_n, y\| < {\varepsilon}$, and $\|x_m-x_n, z\| < {\varepsilon}$. For convenience, we will write $\lim_{m, n\to{\infty}}\|x_n-x_m, y\| = 0$ for a Cauchy sequence $\{x_n\}$.

Definition 6. Let $\{x_n\}$ be a sequence in a linear $2$-normed space ${\mathcal{X}}$. The sequence $\{x_n\}$ is said to convergent in ${\mathcal{X}}$ if there exits an element $x\in{\mathcal{X}}$ such that

for all $y\in{\mathcal{X}}$. In this case, we say that a sequence $\{x_n\}$ converges to the limit $x$, simply denoted by $\lim_{n\to{\infty}}x_n = x$.

Definition 7. A $2$-Banach space is defined to be a linear $2$-normed space in which every Cauchy sequence is convergent. $({\mathcal{X}}, \|{\cdot}\|, \|{\cdot}, {\cdot}\|)$ is called a normed $2$-Banach space if $({\mathcal{X}}, \|{\cdot}\|)$ is a normed space such that $({\mathcal{X}}, \|{\cdot}, {\cdot}\|)$ is a $2$-Banach space.

In the following lemma, we obtain some basic properties in a linear $2$-normed space, which will be used to prove the stability results.

Lemma 2. ^[7] Let $({\mathcal{X}}, \|{\cdot}, {\cdot}\|)$ be a linear $2$-normed space and $x\in{\mathcal{X}}$.

(a) If $\|x, y\| = 0$ for all $y\in{\mathcal{X}}$, then $x = 0$.

(b) $\big|\|x, z\|-\|y, z\|\big|\le\|x-y, z\|$ for all $x, y, z\in{\mathcal{X}}$.

(c) If a sequence $\{x_n\}$ is convergent in ${\mathcal{X}}$, then $\lim_{n\to{\infty}}\|x_n, y\| = \|\lim_{n\to{\infty}}x_n, y\|$ for all $y\in{\mathcal{X}}$.

From now on, let ${\mathcal{X}}$ be a normed space and ${\mathcal{Y}}$ a normed $2$-Banach space. The following theorem proves the stability of the Eq $(1.2)$ in normed $2$-Banach spaces.

Theorem 3. Let $r\in(0, 1)$, ${\varepsilon} > 0$, ${\delta}$, and $\eta\ge0$, and let $f:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ be a mapping satisfying $f(x, 0) = 0$ such that

for all $x, x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, and all $z\in{\mathcal{Y}}$, then there exists a Cauchy-Jensen mapping $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ such that

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$.

Proof. Let $g(x, y): = f(x, y)-f(0, y)$ for all $x, y\in{\mathcal{X}}$, then $g(x, 0) = g(0, y) = 0$ for all $x, y\in{\mathcal{X}}$. By (2.15), $g$ satisfies

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, and all $z\in{\mathcal{Y}}$. Letting $x_1 = {\cdots} = x_n = x$ in (2.17), we gain

for all $x, y_1, {\cdots}, y_n\in{\mathcal{X}}$, and all $z\in{\mathcal{Y}}$. Putting $y_1 = y, y_2 = -y$, and $y_3 = {\cdots} = y_n = 0$ in (2.18), we get

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. Setting $y_1 = {\cdots} = y_n = y$ in (2.18), we have

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. By the above two inequalities, we obtain

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. Taking $y_1 = y, y_2 = -(n+1)y$ and $y_3 = \cdots = y_n = 0$ in (2.18), we gain

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. By (2.19) and the above inequality, we get

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. Replacing $y$ by $(n+1)y$ in (2.19), we have

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. By (2.20) and the above inequality, we obtain

for all $x, y\in{\mathcal{X}}$ and all $z\in{\mathcal{Y}}$. Replacing $x$ by $n^kx$ and $y$ by $(n+1)^ky$ in the above inequality and dividing $n^{k+1}(n+1)^{k+1}$, we see that

for all $x, y\in{\mathcal{X}}$, all $z\in{\mathcal{Y}}$, and all nonnegative integers $k$. Thus, we have

for all integers $l, \, m\, (0\le l < m)$, all $x, y\in{\mathcal{X}}$, and all $z\in{\mathcal{Y}}$. Thus, the sequence $\big\{ \frac 1{n^k(n+1)^k}g\big(n^kx, (n+1)^ky\big)\big\}$ is a Cauchy sequence in $({\mathcal{Y}}, \|{\cdot}, {\cdot}\|)$ for all $x, \, y\in{\mathcal{X}}$. Since $({\mathcal{Y}}, \|{\cdot}, {\cdot}\|)$ is complete, the sequence $\big\{ \frac 1{n^k(n+1)^k}g(n^kx, (n+1)^ky)\big\}$ converges for all $x, \, y\in{\mathcal{X}}$.

Define $F:{\mathcal{X}}{\times}{\mathcal{X}}\to{\mathcal{Y}}$ by

for all $x, y\in{\mathcal{X}}$. By (2.17), we have

for all $x_1, {\cdots}, x_n, y_1, {\cdots}, y_n\in{\mathcal{X}}$, all $z\in{\mathcal{Y}}$, and all nonnegative integers $k$. Letting $k\to{\infty}$ in the above inequality and using Lemma 2, we obtain that $F$ satisfies (1.2). By ^[6], $F$ is a Cauchy-Jensen mapping. Setting $l = 0$ and taking $m\to{\infty}$ in (2.21), using Lemma 2, one can obtain the inequality (2.16).

3.   Conclusions

This paper deals with the stability of the Cauchy-Jensen functional equation, presenting connections to three different types of spaces: normed spaces, quasi-normed spaces, and normed 2-normed spaces. These spaces differ in the criteria used to measure the distance and magnitude of the function. A norm provide functions that measure the length or size of vectors, which play an important role in many mathematical analyses and optimization problems. Quasi-normed spaces have less stringent conditions and do not have to satisfy all the properties of a norm, allowing for analysis in more general situations. A 2-normed space is a space whose norms are defined specifically by Euclidean norms (or 2-norms), which are a generalization of distances and angles in vector spaces.

By proving the stability of the Cauchy-Jensen functional equation on these three spaces, the paper shows how the equation can be applied in these different contexts. In other words, the paper shows that the Cauchy-Jensen equation maintains consistent properties in these different types of spaces, and gives a set of conditions that guarantee the existence of the solution of the equation in each space. This suggests that the equation is a powerful tool with a wide range of applications across a variety of mathematical structures.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors state no conflict of interest.