Investigating the Hyers–Ulam Stability of the Generalized Drygas Functional Equation: New Results and Methods

Abstract

In this paper, we explore the Hyers–Ulam stability of a generalized Drygas functional equation, which extends the classical Drygas equation by incorporating additional parameters and conditions. Our investigation focuses on mappings from a real vector space into a Banach space and employs the fixed-point method to establish stability criteria. Our findings provide new insights into the conditions under which the generalized Drygas equation maintains stability, contributing to the broader understanding of functional equations in mathematical analysis. The results have implications for the study of functional equations and their applications in various mathematical contexts.

1. Introduction and Preliminaries

The stability problem of functional equations derives from a question of Ulam [1]. In 1941, Hyers [2] obtained well-known and interesting results for the Cauchy equations in Banach spaces. Later, very many generalizations of the Cauchy equations were obtained (see [3,4,5]). For papers concerned with various functional equations, see [6,7,8,9,10,11,12,13,14]. Here is some information related to diverse functional equations. This information can help in understanding the properties, solutions, and applications of these equations across different fields such as mathematics, physics, and engineering [15,16,17]. It offers insights into how these equations govern relationships between functions and variables [18,19]. Research on functional equations primarily focuses on several key aspects. Firstly, different types of functional equations [20,21] are analyzed, exploring their unique characteristics. Secondly, the investigation delves into relevant space structures [22,23,24], which play a fundamental role. Thirdly, applications [25,26,27] across various disciplines are examined, highlighting the practical significance of these equations.

In 1987, Drygas [28] considered the functional equation in quasi-inner products

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

whose solution is called a Drygas mapping. The general solution of the above functional equation was given by Ebanks, Kannappan, and Sahoo [29] as

$$f\left(x\right)=Q\left(x\right)+A\left(x\right),$$

where A is an additive mapping and Q is a quadratic mapping in quasi-inner products.

In 1992, the following question was raised in [30]: Is the functional equation

$$\begin{array}{c}\hfill f(ax+by+c)=Af\left(x\right)+Bf\left(y\right)+C,(a,b,A,B\ne 0)\end{array}$$

stable? This problem is of profound significance for the stability of functional equations.

In this paper, we discuss the Hyers–Ulam stability of the functional equation

$$\begin{array}{c}\hfill f(ax+by)+f(ax-by)=Af\left(x\right)+Bf\left(y\right)+Df(-y),\end{array}$$

(1)

in Banach space, where $a,b,A,B,D\ne 0$ and $a+b\ne 0$. We call the functional equation the generalized Drygas functional equation and its solution is called a generalized Drygas mapping. Recently, several mathematicians have investigated the problem of Hyers–Ulam stability of Drygas functional equations (see [31,32,33,34,35,36,37,38,39]) in some complete spaces. While fixed-point methods are indeed well established, our generalization introduces two key innovations: (i) the incorporation of new parameter $a,b,A,B,D\ne 0$, which was not considered in previous Drygas-type equations; (ii) the relaxation of $a+b\ne 0$ required in prior studies.

Throughout the paper, let $\mathbb{N}$ be the set of all positive integers, ${\mathbb{R}}_{+}$ denote the set of all positive reals, and $Y^X$ be the family of all mappings from X to Y. We will introduce tBrzdek’s fixed-point theorem to obtain the stability result for the generalized Drygas functional equation in Banach spaces.

Theorem 1

([40]). Let X be a nonempty set and $(Y,d)$ be a Banach space, and let $f_1,\cdots ,f_k:X\to X$ be mappings and $L_1,\cdots ,L_k:X\to {\mathbb{R}}_{+}$ be functions. Let $\mathrm{\Lambda }:{\mathbb{R}}_{+}^X\to {\mathbb{R}}_{+}^X$ be a linear operator defined by

$$\begin{array}{c}\hfill \mathrm{\Lambda }\delta \left(x\right):={\displaystyle \sum _{i=1}^k}{L}_i\left(x\right)\delta \left(f_i\left(x\right)\right),x\in X.\end{array}$$

Suppose that $J:Y^X\to Y^X$ is an operator which satisfies the following inequality:

$$\begin{array}{c}\hfill \parallel J\xi \left(x\right)-J\mu \left(x\right)\parallel \le {\displaystyle \sum _{i=1}^k}{L}_i\left(x\right)\parallel \xi \left(f_i\left(x\right)\right)-\mu \left(f_i\left(x\right)\right)\parallel ,\xi ,\mu \in Y^X,x\in X,\end{array}$$

$\epsilon :X\to {\mathbb{R}}_{+}$ is a function and $\phi :X\to Y$ is a mapping such that, for all $x\in X$,

$$\begin{array}{c}\hfill \parallel J\phi \left(x\right)-\phi \left(x\right)\parallel \le \epsilon \left(x\right)\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{\mathrm{\Lambda }}^n\epsilon \left(x\right):={\epsilon }^{*}\left(x\right)<\infty .\end{array}$$

Then, there exists a unique fixed point ϕ of J such that

$$\begin{array}{c}\hfill \parallel \phi \left(x\right)-\varphi \left(x\right)\parallel \le {\epsilon }^{*}\left(x\right)\end{array}$$

for all $x\in X$. Moreover, for every $x\in X$, the limit

$$\begin{array}{c}\hfill \varphi \left(x\right):=\underset{n\to \infty }{lim}{J}^n\phi \left(x\right)\end{array}$$

exists.

By using the fixed-point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [40,41,42,43]). To establish the stability of the multi-coefficient functional equation, we employ the fixed-point theorem as a fundamental tool. Specifically, by constructing an appropriate operator in a complete metric space, we demonstrate that the solution satisfies a contraction condition under given assumptions. This approach not only guarantees the existence and uniqueness of the solution but also ensures its Hyers–Ulam stability. The key innovation lies in the careful selection of control parameters that accommodate multiple coefficients, extending beyond classical cases. Compared to previous results, our method provides a unified framework for stability analysis, yielding explicit error bounds that refine existing estimates in the literature. We now cite the recent work of Girgin et al. [25] on Ulam–Hyers stability for Caputo-type fractional differential equations, emphasizing their use of fixed-point methods to derive stability criteria for nonlinear systems. Their approach aligns with our results but focuses on fractional operators, whereas our results address multi-parameter functional equations.

2. The Stability of the Generalized Drygas Function Equation

Theorem 2.

Suppose that X is a linear normed space, Y is a Banach space, and $\theta \ge 0$. Let $f:X\to Y$ be a mapping satisfying

$$\begin{array}{c}\hfill \parallel f(ax+by)+f(ax-by)-Af\left(x\right)-Bf\left(y\right)-Df(-y)\parallel \le \theta (\parallel x{\parallel }^p+\parallel y{\parallel }^p)\end{array}$$

(2)

for all $x,y\in X$. If p, a, b, A, B, and D satisfy $p>0$ and ${\displaystyle \frac{A}{{\left|2a\right|}^p}}+{\displaystyle \frac{B}{{\left|2b\right|}^p}}+{\displaystyle \frac{D}{{\left|2b\right|}^p}}<1$, then there exists a unique generalized Drygas mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\theta }{1-\left\{\frac{A}{{\left|2a\right|}^p}+\frac{B}{{\left|2b\right|}^p}+\frac{D}{{\left|2b\right|}^p}\right\}}}{\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p}}{\parallel x\parallel }^p\end{array}$$

for all $x\in X$.

Proof. 

Letting $x=y=0$ in (2), we obtain $f\left(0\right)=0$. Letting $y={\displaystyle \frac{a}{b}}x$ in (2), we obtain

$$\begin{array}{c}\hfill ∥f\left(x\right)-Af\left({\displaystyle \frac{1}{2a}}x\right)-Bf\left({\displaystyle \frac{1}{2b}}x\right)-Df\left(-{\displaystyle \frac{1}{2b}}x\right)∥\le \theta \left({\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{2^p{\left|ab\right|}^p}}{\parallel x\parallel }^p\right)\end{array}$$

(3)

for all $x\in X$.

Consider $J:Y^X\to Y^X$ and $\epsilon :X\to {\mathbb{R}}_{+}$ given as

$$\begin{array}{c}\hfill J\xi \left(x\right)=A\xi \left({\displaystyle \frac{1}{2a}}x\right)+B\xi \left({\displaystyle \frac{1}{2b}}x\right)+D\xi \left(-{\displaystyle \frac{1}{2b}}x\right)\end{array}$$

for all $x\in X,\xi \in Y^X$ and

$$\epsilon \left(x\right)=\theta {\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p}}{\parallel x\parallel }^p,x\in X.$$

The inequality in (3) becomes

$$\begin{array}{c}\hfill \parallel Jf\left(x\right)-f\left(x\right)\parallel \le \epsilon \left(x\right),\forall x\in X.\end{array}$$

For every $g,h\in Y^X$ and $x\in X$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel Jg\left(x\right)-Jh\left(x\right)\parallel =& ∥Ag\left({\displaystyle \frac{1}{2a}}x\right)+Bg\left({\displaystyle \frac{1}{2b}}x\right)+Dg\left({\displaystyle \frac{-1}{2b}}x\right)\hfill \\ \hfill & -Ah\left({\displaystyle \frac{1}{2a}}x\right)-Bh\left({\displaystyle \frac{1}{2b}}x\right)-Dh\left({\displaystyle \frac{-1}{2b}}x\right)∥\hfill \\ \hfill & \le \left|A\right|∥g\left({\displaystyle \frac{1}{2a}}x\right)-h\left({\displaystyle \frac{1}{2a}}x\right)∥\hfill \\ \hfill & +\left|B\right|∥g\left({\displaystyle \frac{1}{2b}}x\right)-h\left({\displaystyle \frac{1}{2b}}x\right)∥\hfill \\ \hfill & +\left|D\right|∥g\left({\displaystyle \frac{-1}{2b}}x\right)-h\left({\displaystyle \frac{-1}{2b}}x\right)∥.\hfill \end{array}\end{array}$$

So, J satisfies Inequality (2) with $f_1\left(x\right)={\displaystyle \frac{1}{2a}}x, f_2\left(x\right)={\displaystyle \frac{1}{2b}}x, f_3\left(x\right)={\displaystyle \frac{-1}{2b}}x$, $L_1\left(x\right)=\left|A\right|, L_2\left(x\right)=\left|B\right|,$ and $L_3\left(x\right)=\left|D\right|$. By (2), the operator $\mathrm{\Lambda }:{\mathbb{R}}_{+}^X\to {\mathbb{R}}_{+}^X$ is given by

$$\begin{array}{c}\hfill \mathrm{\Lambda }\eta \left(x\right)=\left|A\right|\eta \left({\displaystyle \frac{1}{2a}}x\right)+\left|B\right|\eta \left({\displaystyle \frac{1}{2b}}x\right)+\left|D\right|\eta \left({\displaystyle \frac{-1}{2b}}x\right)\end{array}$$

for all $x\in X$. In particular,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \mathrm{\Lambda }\epsilon \left(x\right)=\left|A\right|\epsilon \left({\displaystyle \frac{x}{2a}}\right)+\left|B\right|\epsilon \left({\displaystyle \frac{x}{2b}}\right)+\left|D\right|\epsilon \left({\displaystyle \frac{-x}{2b}}\right)\hfill \\ \hfill & =\left|A\right|\theta {\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p{\left|2a\right|}^p}}{\parallel x\parallel }^p+\left|B\right|\theta {\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p{\left|2b\right|}^p}}{\parallel x\parallel }^p+\left|D\right|\theta {\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p{\left|2b\right|}^p}}{\parallel x\parallel }^p\hfill \\ \hfill & =\left\{{\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}\right\}\epsilon \left(x\right).\hfill \end{array}\end{array}$$

Since $\mathrm{\Lambda }$ is linear, we can obtain

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}^n\epsilon \left(x\right)={\left\{{\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}\right\}}^n\epsilon \left(x\right),x\in X,n\in {\mathbb{N}}_0.\end{array}$$

Since ${\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}<1$, the series ${\sum }_{n=0}^{\infty }{\mathrm{\Lambda }}^n\epsilon \left(x\right)$ is convergent for every $x\in X$ and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\epsilon }^{*}\left(x\right)=\sum _{n=0}^{\infty }{\mathrm{\Lambda }}^n\epsilon \left(x\right)=\sum _{n=0}^{\infty }{\left({\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}\right)}^n\epsilon \left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{1-\left\{\frac{\left|A\right|}{{\left|2a\right|}^p}+\frac{\left|B\right|}{{\left|2b\right|}^p}+\frac{\left|D\right|}{{\left|2b\right|}^p}\right\}}}\epsilon \left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{1-\left\{\frac{\left|A\right|}{{\left|2a\right|}^p}+\frac{\left|B\right|}{{\left|2b\right|}^p}+\frac{\left|D\right|}{{\left|2b\right|}^p}\right\}}}\theta {\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p}}{\parallel x\parallel }^p,x\in X\hfill \end{array}\end{array}$$

By Theorem 1, there exists a mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & k\left(x\right)=\underset{n\to \infty }{lim}{J}^{n}f\left(x\right)\hfill \\ \hfill & k\left(x\right)=\left|A\right|k\left({\displaystyle \frac{x}{2a}}\right)+\left|B\right|k\left({\displaystyle \frac{x}{2b}}\right)+\left|D\right|k\left({\displaystyle \frac{-x}{2b}}\right)\hfill \\ \hfill & \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\theta }{1-\left\{\frac{\left|A\right|}{{\left|2a\right|}^p}+\frac{\left|B\right|}{{\left|2b\right|}^p}+\frac{\left|D\right|}{{\left|2b\right|}^p}\right\}}}{\displaystyle \frac{{\left|a\right|}^p+{\left|b\right|}^p}{{\left|2a\right|}^p{\left|b\right|}^p}}{\parallel x\parallel }^p.\hfill \end{array}\end{array}$$

Next, we prove that k satisfies the generalized Drygas equation. Letting $x={\displaystyle \frac{x}{a}}$ and $y={\displaystyle \frac{y}{b}}$ in (2), we have

$$\begin{array}{c}\hfill ∥f(x+y)+f(x-y)-Af\left({\displaystyle \frac{x}{a}}\right)-Bf\left({\displaystyle \frac{y}{b}}\right)-Df\left(-{\displaystyle \frac{y}{b}}\right)∥\le \theta \left({∥{\displaystyle \frac{x}{a}}∥}^p+{∥{\displaystyle \frac{y}{b}}∥}^p\right)\end{array}$$

for all $x,y\in X$. Then, from the above inequality, we obtain

$$\begin{array}{c}\hfill \parallel f(x+y)+f(x-y)-2f\left(x\right)-f\left(y\right)-f(-y)\parallel \le 2\theta \left({∥{\displaystyle \frac{x}{a}}∥}^p+{∥{\displaystyle \frac{y}{b}}∥}^p\right),x,y\in X.\end{array}$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel Jf(x+y)+Jf(x-y)-2Jf\left(x\right)-Jf\left(y\right)-Jf(-y)\parallel \hfill \\ \hfill & \le \left|A\right|∥f\left({\displaystyle \frac{x+y}{2a}}\right)+f\left({\displaystyle \frac{x-y}{2a}}\right)-2f\left({\displaystyle \frac{x}{2a}}\right)-f\left({\displaystyle \frac{y}{2a}}\right)-f\left(-{\displaystyle \frac{y}{2a}}\right)∥\hfill \\ \hfill & +\left|B\right|∥f\left({\displaystyle \frac{x+y}{2b}}\right)+f\left({\displaystyle \frac{x-y}{2b}}\right)-2f\left({\displaystyle \frac{x}{2b}}\right)-f\left({\displaystyle \frac{y}{2b}}\right)-f\left(-{\displaystyle \frac{y}{2b}}\right)∥\hfill \\ \hfill & +\left|D\right|∥f\left(-{\displaystyle \frac{x+y}{2b}}\right)+f\left(-{\displaystyle \frac{x-y}{2b}}\right)-2f\left(-{\displaystyle \frac{x}{2b}}\right)-f\left(-{\displaystyle \frac{y}{2b}}\right)-f\left({\displaystyle \frac{y}{2b}}\right)∥\hfill \\ \hfill & \le 2\theta \left({\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}\right)\left({∥{\displaystyle \frac{x}{a}}∥}^p+{∥{\displaystyle \frac{y}{b}}∥}^p\right)\hfill \end{array}\end{array}$$

and so

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel J^{n}f(x+y)+J^{n}f(x-y)-2J^{n}f\left(x\right)-J^{n}f\left(y\right)-J^{n}f(-y)\parallel \hfill \\ \hfill & \le 2\theta {\left({\displaystyle \frac{\left|A\right|}{{\left|2a\right|}^p}}+{\displaystyle \frac{\left|B\right|}{{\left|2b\right|}^p}}+{\displaystyle \frac{\left|D\right|}{{\left|2b\right|}^p}}\right)}^n\left({∥{\displaystyle \frac{x}{a}}∥}^p+{∥{\displaystyle \frac{y}{b}}∥}^p\right)\hfill \end{array}\end{array}$$

for all $n\in {\mathbb{N}}_0$ and $x,y\in X$. Letting $n\to \infty$, we obtain

$$\begin{array}{c}\hfill k(x+y)+k(x-y)-2k\left(x\right)-k\left(y\right)-k(-y)=0\end{array}$$

for all $x,y\in X.$ □

Notion Forti’s [33] seminal work established stability criteria for the classical Drygas equation

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

using direct methods. Our Theorem 2 generalizes this by incorporating multi-parameter dependence $(a,b,A,B,D)$ and relaxes the condition $a+b\ne 0$ required in earlier studies. The fixed-point approach here unifies stability analysis for a broader class of equations, whereas Forti’s results were restricted to specific coefficient choices.

Theorem 3.

Suppose that X is a linear normed space, Y is a Banach space, and $\theta \ge 0$; let $f:X\to Y$ be a mapping satisfying (2) for all $x,y\in X$. If p, a, and A satisfy $p>0$ and ${2\left|a\right|}^p<\left|A\right|$, then there exists a unique generalized Drygas mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\theta }{\left|A\right|-{2\left|a\right|}^p}}{\parallel x\parallel }^p\end{array}$$

for all $x\in X$.

Proof. 

Letting $x=y=0$ in (2), we obtain $f\left(0\right)=0$. Letting $y=0$ in (2), we have

$$\begin{array}{c}\hfill \parallel f\left(x\right)-{\displaystyle \frac{2}{A}}f\left(ax\right)\parallel \le {\displaystyle \frac{\theta }{\left|A\right|}}{\parallel x\parallel }^p,x\in X.\end{array}$$

Consider the mapping $J:Y^X\to Y^X$ such that

$$\begin{array}{c}\hfill J\xi \left(x\right)={\displaystyle \frac{2}{A}}\xi \left(ax\right),x\in X,\xi \in Y^X.\end{array}$$

And letting $\epsilon \left(x\right)={\displaystyle \frac{\theta }{\left|A\right|}}{\parallel x\parallel }^p$, we have

$$\begin{array}{c}\hfill \parallel Jf\left(x\right)-f\left(x\right)\parallel \le \epsilon \left(x\right),x\in X.\end{array}$$

For every $\xi ,\mu \in Y^X$,

$$\begin{array}{c}\hfill \parallel J\xi \left(x\right)-J\mu \left(x\right)\parallel ={\displaystyle \frac{2}{\left|A\right|}}\parallel \xi \left(x\right)-\mu \left(x\right)\parallel .\end{array}$$

Thus, J satisfies Inequality (2) with $f_1\left(x\right)=2ax$ and $L_1\left(x\right)=\left|A\right|$. Next, we define $\mathrm{\Lambda }:{\mathbb{R}}_{+}^X\to {\mathbb{R}}_{+}^X$ by

$$\begin{array}{c}\hfill \mathrm{\Lambda }\eta \left(x\right)={\displaystyle \frac{2}{\left|A\right|}}\eta \left(ax\right),x\in X,\eta \in {\mathbb{R}}_{+}^X.\end{array}$$

Since $\mathrm{\Lambda }$ is linear, we obtain

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}^n\epsilon \left(x\right)={\left({\displaystyle \frac{{2\left|a\right|}^p}{\left|A\right|}}\right)}^n\epsilon \left(x\right)\end{array}$$

and

$$\begin{array}{c}\hfill {\epsilon }^{*}\left(x\right)=\sum _{n=0}^{\infty }\left({\displaystyle \frac{{2\left|a\right|}^p}{\left|A\right|}}\right)\epsilon \left(x\right)={\displaystyle \frac{\theta }{\left|A\right|-{2\left|a\right|}^p}}{\parallel x\parallel }^p.\end{array}$$

The rest of the proof is similar to the proof of Theorem 2. □

Notion Jung and Sahoo [35] proved stability for the Drygas equation in Banach spaces under the assumption that $f\left(0\right)=0$. Theorem 3 extends their framework by introducing scaling parameters $(a,A)$ and deriving explicit error bounds

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\theta }{\left|A\right|-{2\left|a\right|}^p}}{\parallel x\parallel }^p\end{array}$$

which were absent in their work. The condition ${2\left|a\right|}^p<\left|A\right|$ in our paper refines their stability threshold.

Theorem 4.

Suppose that X is a linear normed space, Y is a Banach space, and $\theta \ge 0$; let $f:X\to Y$ be a mapping satisfying (2) for all $x,y\in X$. If p, a, and A satisfy $p>0$ and ${2\left|a\right|}^p<\left|A\right|$, then there exists a unique generalized Drygas mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{2\theta }{\left|A\right|-{2\left|a\right|}^p}}{\parallel x\parallel }^p\end{array}$$

for all $x\in X$.

Proof. 

Let $f_e:X\to Y$ and $f_o:X\to Y$ be the even and odd part of function f, respectively. That is, $f\left(x\right)=f_e+f_o$. We can see that $f\left(0\right)=f_e\left(0\right)=f_o\left(x\right)=0$. It is easy to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel f_j(x+y)+f_j(x-y)-2f_j\left(x\right)-f_j\left(y\right)-f_j(-y)\parallel \hfill \\ \hfill & \le 2\theta \left({∥{\displaystyle \frac{x}{a}}∥}^p+{∥{\displaystyle \frac{y}{b}}∥}^p\right),x,y\in X,j=e,o.\hfill \end{array}\end{array}$$

Similar to the proof of Theorem 3, we can obtain

$$\begin{array}{c}\hfill k_j\left(x\right)=\underset{n\to \infty }{lim}{J}^n{f}_j\left(x\right),k_j={\displaystyle \frac{2}{\left|A\right|}}k\left(ax\right),x\in X,j=e,o.\end{array}$$

Thus, $k=k_e+k_o$ satisfies the generalized Drygas equation and

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{2\theta }{\left|A\right|-{2\left|a\right|}^p}}{\parallel x\parallel }^p\end{array}$$

for all $x\in X.$ □

3. The Stability of the Generalized Drygas Function Inequation

Park [44] defined additive $\rho$-functional inequalities and proved the Hyers–Ulam stability of the additive $\rho$-functional inequalities in Banach spaces using a direct method. The paper innovatively studies two additive $\rho$-functional inequalities, deeply exploring their relationships with additive mappings. By constructing lemmas, it proves that mappings satisfying specific inequalities are additive. Moreover, it demonstrates the Hyers–Ulam stability of related inequalities and equations in complex Banach spaces, enriching the theoretical achievements in this field and providing new ideas and methods for subsequent research. The paper mainly conducts research in complex Banach spaces, with relatively few studies on extensions to other spaces. Future research could consider generalizing the relevant conclusions to broader types of spaces, such as non-Banach spaces, to further expand the scope of application of the theory. Meanwhile, it can explore the applications of additive $\rho$ -functional inequalities and equations in other branches of mathematics or practical problems.

In 2016, Choi [45], etc., studied the additive $\rho -$functional inequalities in normed spaces. By using the fixed-point method, they proved the Hyers–Ulam stability of the following two additive $\rho -$functional inequalities:

$$\begin{array}{c}\hfill |f(x+y)-f\left(x\right)-f\left(y\right)|\le |\rho (2f({\displaystyle \frac{x+y}{2}})-f\left(x\right)-f\left(y\right))|\end{array}$$

and

$$\begin{array}{c}\hfill |(2f({\displaystyle \frac{x+y}{2}})-f\left(x\right)-f\left(y\right))|\le |\rho (f(x+y)-f\left(x\right)-f\left(y\right))|\end{array}$$

(where $\rho <1$) in normed spaces. In 2023, Nawaz [46] and others analyzed the generalized Hyers–Ulam stability of the cubic and quartic $\rho$-functional inequalities in fuzzy matrices. They used the fixed-point method to study the following two functional inequalities:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left|f\right(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)-\hfill \\ \rho (4f(x+{\displaystyle \frac{y}{2}})+4(f(x-{\displaystyle \frac{y}{2}})-f(x+y)-f(x-y))-6f\left(x\right),r)|\ge {\displaystyle \frac{r}{r+\phi (x,y)}}\hfill \\ f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f\left(x\right)+6f\left(y\right)-\hfill \\ \rho (8f(x+{\displaystyle \frac{y}{2}})+8(f(x-{\displaystyle \frac{y}{2}})-2f(x+y)-2f(x-y))-12f\left(x\right)+3f\left(y\right),r)\ge {\displaystyle \frac{r}{r+\phi (x,y)}}\hfill \end{array}\right.,\end{array}$$

where $\rho \ne 2$ is a real number. The paper focuses on the stability of $\rho -$functional inequalities. It builds on the existing research of functional equation stability, with a continuous and expanded research topic, which promotes the development of the functional inequality stability theory. In the paper, detailed solutions and stability proofs are given for cubic and quartic $\rho -$functional inequalities. Studying the stability of $\rho -$functional inequalities in the context of fuzzy matrices expands the research scope of functional inequality stability. Compared with previous studies, there is certain innovation in research objects and application scenarios. The paper gives specific examples, such as defining a specific function $\psi$ to construct a function that satisfies the inequality, which enhances the persuasiveness of the theoretical results and builds a bridge between theory and practical applications.

Overall, the research on the stability of $\rho -$functional inequalities mainly focuses on using methods such as the fixed-point method to analyze and prove the stability of various types of $\rho -$functional inequalities under different spaces and conditions. Moreover, the research results have certain application prospects in fields such as numerical analysis, biology, and economics. However, there are still many issues to be further explored in this field—for example, for more complex forms of $\rho -$functional inequalities, as well as the research on stability in a wider range of spaces and practical application scenarios. In this section, we solve and investigate the generalized Drygas function inequation.

Theorem 5.

Let $\varphi (x,y):X^2\to [0,\infty )$ be a function with $\varphi (0,0)=0$ such that there exists an $L<1$ with

$$\begin{array}{c}\hfill \varphi (ax,ay)\le \left|a\right|L\varphi (x,y)\end{array}$$

for all $x,y\in X$. Suppose that X is a linear normed space, Y is a Banach space, and ρ is any real number; let $f:X\to Y$ be an odd mapping satisfying

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel f(ax+by)+f(ax-by)-Af\left(x\right)-Bf\left(y\right)-Df(-y)\parallel \hfill \\ \hfill & \le \parallel \rho \left(f\right(ax+by)-f(ax)-f(by\left)\right)\parallel +\varphi (x,y)\hfill \end{array}\end{array}$$

(4)

for all $x,y\in X$. If ρ, L, A, and a satisfy ${\displaystyle \frac{2\left|a\right|L}{\left|A\right|}}<1$, $\rho <1$, then there exists a unique additive mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\left|A\right|}{\left|A\right|-2L\left|a\right|}}\varphi (x,0)\end{array}$$

for all $x\in X$.

Proof. 

Letting $y=0$ in (4), we obtain

$$\begin{array}{c}\hfill \parallel f\left(x\right)-{\displaystyle \frac{2}{A}}f\left(ax\right)\parallel \le {\displaystyle \frac{1}{\left|A\right|}}\varphi (x,0),x\in X.\end{array}$$

Consider the mapping $J:Y^X\to Y^X$ such that

$$\begin{array}{c}\hfill J\xi \left(x\right)={\displaystyle \frac{2}{A}}\xi \left(ax\right),x\in X,\xi \in Y^X.\end{array}$$

Then, we have

$$\begin{array}{c}\hfill \parallel Jf\left(x\right)-f\left(x\right)\parallel \le {\displaystyle \frac{1}{\left|A\right|}}\varphi (x,0),x\in X.\end{array}$$

For every $\xi ,\mu \in Y^X$,

$$\begin{array}{c}\hfill \parallel Jg\left(x\right)-Jh\left(x\right)\parallel ={\displaystyle \frac{2}{\left|A\right|}}\parallel g\left(ax\right)-h\left(ax\right)\parallel .\end{array}$$

Thus, J satisfies Inequality (2) with $f_1\left(x\right)=2ax$ and $L_1\left(x\right)={\displaystyle \frac{2}{\left|A\right|}}$. Next, we define $\mathrm{\Lambda }:{\mathbb{R}}_{+}^X\to {\mathbb{R}}_{+}^X$ by

$$\begin{array}{c}\hfill \mathrm{\Lambda }\varphi (x,y)={\displaystyle \frac{2}{\left|A\right|}}\varphi (ax,ay)<{\displaystyle \frac{2\left|a\right|L}{\left|A\right|}}\varphi (x,y),x\in X,\varphi \in {\mathbb{R}}_{+}^X.\end{array}$$

Since $\mathrm{\Lambda }$ is linear, we obtain

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}^n\varphi (x,y)={\left({\displaystyle \frac{2\left|a\right|L}{\left|A\right|}}\right)}^n\varphi (x,y)\end{array}$$

and

$$\begin{array}{c}\hfill {\epsilon }^{*}(x,y)=\sum _{n=0}^{\infty }{\left({\displaystyle \frac{2\left|a\right|L}{\left|A\right|}}\right)}^n\varphi (x,y)={\displaystyle \frac{\left|A\right|}{\left|A\right|-2L\left|a\right|}}\varphi (x,y).\end{array}$$

By Theorem 1, there exists a mapping $k:X\to Y$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & k\left(x\right)=\underset{n\to \infty }{lim}{J}^{n}f\left(x\right)\hfill \\ \hfill & k\left(x\right)={\displaystyle \frac{2}{A}}k\left(ax\right)\hfill \\ \hfill & \parallel f\left(x\right)-k\left(x\right)\parallel \le {\displaystyle \frac{\left|A\right|}{\left|A\right|-2L\left|a\right|}}\varphi (x,0).\hfill \end{array}\end{array}$$

Next, we prove that k satisfies the generalized Drygas equation. Letting $x={\displaystyle \frac{x}{a}}$ and $y={\displaystyle \frac{y}{b}}$ in (4), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & ∥f(x+y)+f(x-y)-Af\left({\displaystyle \frac{x}{a}}\right)-Bf\left({\displaystyle \frac{y}{b}}\right)-Df\left(-{\displaystyle \frac{y}{b}}\right)∥\hfill \\ \hfill & \le \left|\rho \right|∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥+\varphi \left({\displaystyle \frac{x}{a}},{\displaystyle \frac{y}{b}}\right)\hfill \end{array}\end{array}$$

for all $x,y\in X$. Then, from the above inequality, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel f(x+y)+f(x-y)-2f\left(x\right)\parallel \hfill \\ \hfill & \le \left|\rho \right|∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥+\varphi \left({\displaystyle \frac{x}{a}},{\displaystyle \frac{y}{b}}\right)+\varphi \left({\displaystyle \frac{x}{a}},0\right)+\varphi \left(0,{\displaystyle \frac{y}{b}}\right)\hfill \end{array}\end{array}$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel Jf(x+y)+Jf(x-y)-2Jf\left(x\right)-Jf\left(y\right)-Jf(-y)\parallel \hfill \\ \hfill & ={\displaystyle \frac{2}{\left|A\right|}}∥f\left(a(x+y)\right)+f\left(a(x-y)\right)-2f\left(ax\right)-f\left(ay\right)-f\left(-ay\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{2}{\left|A\right|}}\left|\rho \right|\parallel f(x+{\displaystyle \frac{a}{b}}y)-f\left(x\right)-f({\displaystyle \frac{a}{b}}y)\parallel +{\displaystyle \frac{2}{\left|A\right|}}\varphi (x,y)+{\displaystyle \frac{2}{\left|A\right|}}\varphi (x,0)+{\displaystyle \frac{2}{\left|A\right|}}\varphi (0,y)\hfill \end{array}\end{array}$$

and so

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel k(x+y)+k(x-y)-k\left(x\right)-k\left(y\right)-k(-y)\parallel \hfill \\ \hfill & =\underset{n\to \infty }{lim}\parallel J^{n}f(x+y)+J^{n}f(x-y)-2J^{n}f\left(x\right)-J^{n}f\left(y\right)-J^{n}f(-y)\parallel \hfill \\ \hfill & \le \underset{n\to \infty }{lim}\parallel J^{n}f(x+{\displaystyle \frac{a}{b}}y)-J^{n}f\left(x\right)-J^{n}f({\displaystyle \frac{a}{b}}y)\parallel \hfill \\ \hfill & +\underset{n\to \infty }{lim}(J^n\varphi (x,y)+J^n\varphi (x,0)+J^n\varphi (0,y))\hfill \end{array}\end{array}$$

for all $n\in {\mathbb{N}}_0$ and $x,y\in X$. Letting $n\to \infty$, we obtain

$$\begin{array}{c}\hfill \parallel k(x+y)+k(x-y)-2k\left(x\right)\parallel \le \left|\rho \right|\parallel k(x+y)-k\left(x\right)-k\left(y\right)\parallel \end{array}$$

(5)

for all $x,y\in X.$ Letting $y=x$ in (5), we have

$$\begin{array}{c}\hfill \parallel k\left(2x\right)-2k\left(x\right)\parallel \le \left|\rho \right|\parallel k\left(2x\right)-2k\left(x\right)\parallel \end{array}$$

for all $x\in X$. Moreover, $k\left(2x\right)=2k\left(x\right)$ for all $x\in X$.

Letting $l=x+y,m=x-y$ in (5), we obtain

$$\begin{array}{c}\hfill ∥k\left(l\right)+k\left(m\right)-2k\left({\displaystyle \frac{l+m}{2}}\right)∥\le \left|\rho \right|∥k\left(l\right)-k\left({\displaystyle \frac{l+m}{2}}\right)-k\left({\displaystyle \frac{l-m}{2}}\right)∥\end{array}$$

(6)

for all $l,m\in X$. By (5) and (6),

$$\begin{array}{c}\hfill \parallel k\left(l\right)+k\left(m\right)-k(l+m)\parallel \le {\displaystyle \frac{1}{2}}|{\rho }_1{|}^2\parallel k\left(l\right)+k\left(m\right)-k(l+m)\parallel \end{array}$$

for all $l,m\in X$. Then, $k(l+m)=k\left(l\right)+k\left(m\right)$ for all $l,m\in X$. Therefore f is additive. □

Theorem 6.

Let $\psi :X^2\to [0,\infty )$ be a function such that there exists an $L<1$ with

$$\begin{array}{c}\hfill \psi \left(ax,ay\right)\le a^{2}L\psi (x,y)\end{array}$$

for all $x,y\in X$. Suppose that X is a linear normed space, Y is a Banach space, and ρ is any real number. Let $f:X\to Y$ be a even mapping with $f\left(0\right)=0$ satisfying

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel f(ax+by)+f(ax-by)-Af\left(x\right)-Bf\left(y\right)-Df(-y)\parallel \hfill \\ \hfill & \le \parallel \rho \left(f\right(ax+by)+f(ax-by)-2f(ax)-2f(by\left)\right)\parallel +\psi (x,y)\hfill \end{array}\end{array}$$

(7)

for all $x,y\in X$. If ρ, L, A, and a satisfy ${\displaystyle \frac{2a^{2}L}{\left|A\right|}}<1$, $\rho <1$, then there exists a unique quadratic mapping $H:X\to Y$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-H\left(x\right)\parallel \le {\displaystyle \frac{\left|A\right|}{\left|A\right|-2L{a}^2}}\psi (0,x)\end{array}$$

for all $x\in X$.

Proof. 

Letting $y=0$ in (7), we obtain

$$\begin{array}{c}\hfill \parallel f\left(x\right)-{\displaystyle \frac{2}{A}}f\left(ax\right)\parallel \le {\displaystyle \frac{1}{\left|A\right|}}\psi (x,0),x\in X.\end{array}$$

Similar to Theorem 5, we can prove that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & H\left(x\right)=\underset{n\to \infty }{lim}{J}^{n}f\left(x\right)\hfill \\ \hfill & H\left(x\right)={\displaystyle \frac{2}{A}}H\left(ax\right)\hfill \\ \hfill & \parallel f\left(x\right)-H\left(x\right)\parallel \le {\displaystyle \frac{\left|A\right|}{\left|A\right|-2L{a}^2}}\psi (x,0).\hfill \end{array}\end{array}$$

And

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel f(x+y)+f(x-y)-2f\left(x\right)-f\left(y\right)-f(-y)\parallel \hfill \\ \hfill & \le \parallel \rho \left(f\right(x+y)+f(x-y)-2f(x)-2f(y\left)\right)\parallel +\psi (x/a,y/b)+\psi (x/a,0)+\psi (0,y/b)\hfill \end{array}\end{array}$$

(8)

Next, that $H\left(x\right)$ is a quadratic function will be proved.

By the definition of $H\left(x\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel H(x+y)+H(x-y)-2H\left(x\right)-2H\left(y\right)\parallel \hfill \\ \hfill & \le \parallel \rho \left(H\right(x+y)+H(x-y)-2H(x)-2H(y\left)\right)\parallel \hfill \end{array}\end{array}$$

(9)

for all $x,y\in X$. So, $f(x+y)+f(x-y)=2f\left(x\right)+2f\left(y\right)$ for all $x,y\in X$. □

Notion Park’s [44] additive $\rho -$functional inequalities and Choi’s [45] fixed-point method inspired our treatment of inequalities (Theorems 5 and 6). However, our results address the generalized Drygas inequation with non-additive perturbations ($\rho \ne 1)$ and provide stability under weaker constraints (${\displaystyle \frac{2\left|a\right|L}{\left|A\right|}}<1$).

4. Conclusions

This paper focuses on the generalized Drygas functional equation and deeply explores its Hyers–Ulam stability in the context from a real vector space to a Banach space. A series of key results are obtained by using the fixed-point method. Regarding the stability of the generalized Drygas functional equation, under different conditions (such as the specific restrictions on $a,b,A,B,D$ in Theorems 2–4), the close relationship between the mapping satisfying the relevant inequality and the unique generalized Drygas mapping is clearly given, and the error estimate is accurately provided. This not only deepens the theoretical understanding of the properties of this equation but also provides an important methodological reference and theoretical basis for subsequent studies on the stability of such equations. In the study of the generalized Drygas functional inequality, it is successfully proven that there exist unique additive and quadratic mappings under specific conditions. These conclusions are of great significance in theoretical research fields such as mathematical analysis and function approximation. They provide new perspectives and methods for function classification and property characterization. In practical applications, such as in the design of optimization algorithms and signal processing, they provide powerful mathematical tools, which can effectively improve algorithm performance and optimize signal processing effects.

The research results of this paper expand the research boundaries of the stability theory of functional equations and open up new directions for the future development of this field. Follow up research can further explore the properties of the generalized Drygas functional equation and inequality in more complex spaces or under different types of operators. At the same time, it is necessary to strengthen the applied research in interdisciplinary fields, fully explore its potential value, and promote the coordinated development of related fields.