Isometries to Analyze the Stability of Norm-Based Functional Equations in p-Uniformly Convex Spaces

Abstract

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, $\parallel \eta \left(gh\right)\parallel =\parallel \eta \left(g\right)+\eta \left(h\right)\parallel$ and $\parallel \eta \left(g{h}^{-1}\right)\parallel =\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \mathrm{for}\mathrm{every}g,h\in G,$ where $(G,·)$ denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function $\eta :G\to B$ is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where $p\ge 1$. This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form.

1. Introduction to Stability Analysis and Historical Context

Stability analysis is crucial for understanding the stability of FEs under perturbations. Stability is commonly applied in mathematical analysis to describe the characteristic that small variations in a function’s initial conditions result in small changes in its output, ensuring the reliability and consistency of mathematical models. This concept, initially introduced by Hyers and later developed by Ulam, Rassias, and others, is essential in various applications, from differential equations to dynamical systems. In this article, we explore the generalized stability of norm-additive FEs within arbitrary groups. We build upon previous research in the field by examining the specific conditions under which these equations demonstrate Hyers–Ulam stability.

S.M. Ulam [1] became the first mathematician who suggested the problem regarding the stability of functional equations. Ulam’s problems may have been specified as follows:

Problem 1. 

Consier a group A and a metric group B with the metric $d(·,·)$. For given $ϵ>0$, does there exist $\delta >0$ such that for a mapping $\eta :A\to B$ satisfying the condition

$$\begin{array}{c}\hfill d\left(\eta \right(gh),\eta (g\left)\eta \right(h\left)\right)<\delta \mathit{for}\mathit{every}g,h\in A,\end{array}$$

(1)

there exists a homomorphism mapping $\mu :A\to B$ such that $d\left(\eta \right(g),\mu (g\left)\right)<ϵ$ for every $g\in A$?

D.H. Hyers [2] provided a partial answer to Ulam’s question 1 when both A and B are Banach spaces with assumption that $ϵ=\delta >0$. The result is as follows:

Theorem 1 

([2]). Consider Banach spaces A and B, and let $ϵ>0$. Assume that the function $\eta :A\to B$ fulfills the following condition:

$$\begin{array}{c}\hfill \parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel \le ϵ\end{array}$$

for all $g,h\in A$. Then, there exists a mapping $\mu :A\to B$ such that

$$\begin{array}{c}\hfill \mu \left(g\right)=\underset{n\to \infty }{lim}\frac{\eta \left(2^{n}g\right)}{2^n}.\end{array}$$

Moreover, function $\mu :A\to B$ is unique and satisfying

$$\begin{array}{c}\hfill \parallel \eta \left(g\right)-\mu \left(g\right)\parallel \le ϵforallg\in A.\end{array}$$

Many authors, including Hyers [3], Skof [4], Forti [5,6], Moszner [7,8], Tabor et al. [9], Volkmann et al. [10], Dong [11], and Gilányi et al. [12], have conducted extensive research on the theory of stability.

Rassias [13] demonstrated generalized stability results by considering a Cauchy difference. The proposed findings are as follows:

Theorem 2. 

Suppose that A and B be Banach spaces, and there exist a positive constant λ and a parameter $0\le q<1$. Suppose that the mapping $\eta \left(tx\right)$ from A to B, is continuous with respect to t for each fixed $x\in A$. Suppose that

$$\begin{array}{c}\hfill \parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel \le \lambda (\parallel g{\parallel }^q+\parallel h{\parallel }^q),\end{array}$$

for every $g,h\in A$; then, there exist a unique additive mapping $\mu :A\to B$ that fulfills the condition

$$\begin{array}{c}\hfill \parallel \eta \left(g\right)-\mu \left(g\right)\parallel \le \frac{2\lambda }{2-2^q}{\parallel g\parallel }^q,\end{array}$$

for every $g\in A$.

P. Šemrl [14] tried to generalize the results of Rassias for $q=1$ in a different approach. Specifically, he investigated an inequality comparable to the results of Theorem 2 having a finite arbitrary number of variables.

Theorem 3. 

Suppose $\lambda >0$ and let $\eta :\mathbb{R}\to \mathbb{R}$ be a continuous mapping that satisfies

$$\left|\eta \left(\sum _{i=1}^{\ell }{x}_i\right)-\sum _{i=1}^{\ell }\eta \left(x_i\right)\right|\le \lambda \sum _{i=1}^{\ell }\left|x_i\right|,x_1,\dots ,x_{\ell }\in \mathbb{R},$$

for every $\ell \in \mathbb{N}$. Then $\chi :\mathbb{R}\to \mathbb{R}$ is an additive mapping such that

$$\left|\eta \right(x)-\chi (x\left)\right|\le \theta \left|x\right|$$

for all $x\in \mathbb{R}$.

Our suggested study compares Hyers’s and Rassias’s results on the stability of FEs with current developments using groups and p-uniformly convex spaces. These proposed results extend the foregoing approaches to mappings from groups to p-uniformly convex spaces, applying a suprema-based condition over differences, which leads to a novel approach for studying the stability of such functions. The presented results not only extend Hyers’s and Rassias’s stability results by extending their relevancy to arbitrary groups but also modify the conditions whereby stability can be guaranteed, taking into account hyperstability. This improves understanding of the dynamics of normed spaces, FEs, and stability analysis.

2. Norm-Additive Functional Equations and Further Developments

Several well-known researchers have collaborated significantly on the subsequent norm-additive FE:

$$\begin{array}{c}\hfill \parallel \eta (g+h)\parallel =\parallel \eta \left(g\right)+\eta \left(h\right)\parallel .\end{array}$$

(2)

See, for instance, R. Ger [15,16,17], J. Dhombres [18], and J. Aczél and J. Dhombres [19]. If a function $\eta$ fulfills Equation (2), we refer to it as a norm-additive (norm-based) functional equation. M. Hosszu’s research article [20] encouraged the research of this FE, which focuses on the equation $e\left(x\right)\eta (g+h)=\eta \left(g\right)+\eta \left(h\right)$.

The norm-preserving functional Equation (2) pertains to the stability analysis of functional equations. Essentially, the function $\eta$, when evaluated at the sum of two elements g and h from a group, is expected to have a norm that is equivalent to the sum of the norms of the function evaluated at each element separately. This criterion is crucial for comprehending the behavior of the function when summing inputs, which is significant in various real-world scenarios where linearity or approximate linearity is presumed. In this setting, stability analysis investigates how accurately a function adheres to the norm-additive condition. In this study, we derive upper bounds on $\parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel$ to identify the optimal additive function described by the FE.

Fischer and Muszély [21] proposed the FE (2) for the Hilbert spaces. The FE (2) is significant for its characterization of convex spaces, as given in [15]. The solution $\eta$ to the FE proposed by Fischer–Muszély becomes additive for strictly convex spaces.

R. Ger provided a resolution to the norm-additive functional Equation (2), which describes a mapping from an abelian group to a normed vector space. For instance, refer to [17]. It is evident that every solution to the norm-additive FE (2) must be an odd. However, the solutions to the following FE

$$\begin{array}{c}\hfill \parallel \eta (g-h)\parallel =\parallel \eta \left(g\right)-\eta \left(h\right)\parallel ,\end{array}$$

(3)

are far from being odd functions. The analysis of norm-additive FEs (2) and (3) demonstrates that, in general, the Equation (2) implies (3) but not the other way around. Equation (2) implies (3), and vice versa, assuming the mapping $\eta$ is an odd function. Tabor [9] proved that FE (2) is stable when the surjective function is assumed.

Tabor [9] analyzed the stability of Fischer and Muszély’s FE, which is as follows:

Theorem 4. 

Assume that $(G,+)$ is a group and A is a Banach space. Consider $\eta :G\to A$ as a surjective function such that

$$\begin{array}{c}\hfill |\parallel \eta (g+h)\parallel -\parallel \eta (g)+\eta (h)\parallel |\le ϵ,\end{array}$$

for all $g,h\in G$, then

$$\begin{array}{c}\hfill \parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel \le 13ϵ.\end{array}$$

It is significant that if G is an abelian (or more generally, amenable) group, then Hyers’ Theorem and Theorem 4 together imply the stability of the Fischer–Muszély functional equation within the class of surjective functions. A $\delta$-surjective mapping from a nonempty set A to a Banach space B can be characterized as for each $y\in B$, there exists an element $x\in A$ that satisfies $\parallel \eta \left(x\right)-y\parallel <\delta$.

Employing the notion of $\delta -$ surjective function, Sikorska [22] introduced the stability results of the norm-additive functional Equation (3) for an abelian group as detailed below:

Theorem 5. 

Assume that a group $(G,+)$ is an abelian and let A be a Banach space. Let $\eta :G\to A$ be a $\delta -$ surjective function under the assumption that if for all $g,h\in G$,

$$\begin{array}{c}\hfill |\parallel \eta (g-h)\parallel -\parallel \eta (g)-\eta (h)\parallel |\le ϵ,\end{array}$$

then

$$\begin{array}{c}\hfill \parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel \le 5ϵ+5\delta .\end{array}$$

Dong and Zheng [23] presented the following results to establish the stability analysis of an additive mapping for abelian group.

Theorem 6. 

Assume that a group $(G,+)$ is an abelian and let A be a Banach space. Suppose that $\eta :G\to A$ is a bijective mapping and $\mu :[0,+\infty )\times [0,+\infty )\to [0,+\infty )$ is a mapping such that

$$\begin{array}{c}\hfill |\parallel \eta (g)-\eta (h)\parallel -\parallel \eta (g-h)\parallel |\le \mu (\parallel \eta (g)-\eta (h)\parallel ,\parallel \eta (g-h)\parallel )\end{array}$$

then $\eta (g+h)=\eta \left(g\right)+\eta \left(h\right)$ for all $g,h\in G$.

For a comprehensive understanding of the established stability theory, it is suggested that readers consult the papers [24,25,26], along with the citations included within these works.

The primary motivation behind addressing the problem of generalized Hyers–Ulam stability of norm-additive FEs in nonabelian groups and p-uniformly convex space is to extend the understanding of stability properties in a broader class of functional equations and spaces by applying the results that were determined by L. Cheng et al. [27]. Our method leverages the findings of L. Cheng to obtain pertinent outcomes in a broader context through the extensive perturbation technique, constrained by the condition of integral convergence.

Traditional studies on the stability of FEs, including those by Hyers, Tabor, Sikorska, Rassias, and Dong, have primarily focused on specific types of functions and abelian groups. However, many real-world applications and theoretical problems involve noncommutative structures where these traditional results may not be directly applicable.

Our study builds upon the foundational work of Hyers and Ulam by considering norm-additive FEs in nonabelian groups and leveraging the properties of p-uniformly convex spaces. This approach allows us to generalize and strengthen the existing stability results, thereby addressing more complex and realistic conditions encountered in practice.

Our approach involves the application of Cheng’s results to derive the relevant findings in a more generalized context for the following generalized norm-additive functional equations:

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)\parallel =\parallel \eta \left(g\right)-\eta \left(h\right)\parallel ,g,h\in G,\end{array}$$

(4)

and

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)\parallel =\parallel \eta \left(g\right)+\eta \left(h\right)\parallel ,g,h\in G.\end{array}$$

(5)

Lindenstrauss and Szankowski [28] proposed the following results, which were utilized by Sarfraz et al. [29] to establish the stability of FEs (4) and (5) for an arbitrary group as opposed to a commutative group.

Theorem 7. 

Suppose that $\eta :A\to B$ is a surjection from a Banach space A onto a Banach space B and assume that $\eta \left(0\right)=0$. If for $\nu \ge 0$ the criterion

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\psi }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty \end{array}$$

holds, where

$$\begin{array}{c}\hfill {\psi }_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel g-h\parallel |:\parallel g-h\parallel \le \nu \mathit{or}\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \le \nu \},\end{array}$$

then $\chi :A\to B$ is a linear isometry such that

$$\begin{array}{c}\hfill \parallel \eta \left(g\right)-\chi \left(g\right)\parallel =o(\parallel g\parallel ),\mathit{where}\parallel g\parallel \to \infty .\end{array}$$

Let B be a Banach space. Then it is classified as a p-uniformly convex space for each $0<ϵ\le 2$ if there is a constant $M>0$ such that ${\delta }_B\left(ϵ\right)\ge M{ϵ}^p$, where ${\delta }_B\left(ϵ\right)$ represents the modulus of convexity, given that ${\delta }_B\left(ϵ\right)>0$.

The subsequent theorem, credited to L. Cheng et al. [27], is applied in the upcoming section:

Theorem 8. 

Let $\eta :A\to B$ be a mapping, where A is a Banach space and B is a p-uniformly convex space. Assume that $\eta \left(0\right)=0$ and the following conditions

$$\begin{array}{c}\hfill {ϵ}_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel g-h\parallel |:\parallel g-h\parallel \le \nu ,\nu \ge 0\},\end{array}$$

and

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{{ϵ}_{\eta }\left(\nu \right)}^{1/p}}{{\nu }^{1+1/p}}}d\nu <\infty \end{array}$$

hold. Then $\chi :A\to B$ is a linear isometry such that

$$\begin{array}{c}\hfill \parallel \eta \left(g\right)-\chi \left(g\right)\parallel =o(\parallel g\parallel ),\parallel g\parallel \to \infty .\end{array}$$

The mapping $\eta :G\to B$ is presumed to be surjective, significantly broadening the scope of our findings to encompass a wider range of theoretical and practical problems. The primary innovation of our methodology is the application of the integral condition suggested by L. Cheng et al. [27], which serves as an effective condition for proving stability through the existence of a linear isometry that closely approximates the surjective mapping. This technique enables us to build upon the results of J. Tabor and J. Sikorska, providing a more extensive insight into the stability characteristics of functional equations under more inclusive and realistic conditions.

Let $(G,·)$ be an arbitrary group and B is a real Banach space for mapping $\eta :G\to B$. Suppose that $\nu \ge 0$. Then, consider the following function defined based on the findings of Sarfraz et al. [29]:

$$\begin{array}{c}\hfill {\psi }_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(g\right)-\eta \left(h\right)\parallel |:\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \le \nu \mathrm{or}\parallel \eta \left(g{h}^{-1}\right)\parallel \le \nu \}.\end{array}$$

(6)

$$\begin{array}{c}\hfill {\overline{\psi }}_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(gh\right)\parallel -\parallel \eta \left(g\right)+\eta \left(h\right)\parallel |:\parallel \eta \left(g\right)+\eta \left(h\right)\parallel \le \nu \mathrm{or}\parallel \eta \left(gh\right)\parallel \le \nu \}.\end{array}$$

(7)

We apply the following Theorems 9–12 (Sarfraz et al. [29]) based on the defined functions presented in (6) and (7) to analyze the generalized norm-additive FEs (4) and (5), where the proposed domain for mapping $\eta$ is an arbitrary group, and for $p\ge 1$, p-uniform convex space is considered as codomain.

Theorem 9 

([29]). Let $(G,·)$ be an arbitrary group and let B be a real Banach space. Assume that a function $\eta :G\to B$ is surjective such that $\eta \left(e\right)=0$. If the condition ${\psi }_{\eta }$ in (6) and

$$\begin{array}{c}\hfill {\int }_1^{\infty }\frac{{\psi }_{\eta }\left(\nu \right)}{{\nu }^2}d\nu <\infty \end{array}$$

(8)

hold, then we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,g,h\in G.\end{array}$$

Theorem 10 

([29]). Let $(G,·)$ be an arbitrary group and let B be a real Banach space. Assume that a function $\eta :G\to B$ is surjective. If the condition $\overline{\psi }$ in (7) and

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\psi }}_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty \end{array}$$

(9)

hold, then we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,g,h\in G.\end{array}$$

Theorem 11 

([29]). Let $(G,·)$ be an arbitrary group and let B be a real Banach space. Assume that a function $\eta :G\to B$ is surjective such that $\eta \left(e\right)=0$. If the condition ${\psi }_{\eta }$ in (6) and

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\psi }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty \end{array}$$

hold, then we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,g,h\in G.\end{array}$$

Theorem 12 

([29]). Let $(G,·)$ be an arbitrary group and let B be a real Banach space. Assume that a function $\eta :G\to B$ is surjective. If the condition $\overline{\psi }$ in (7) and

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\psi }}_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty \end{array}$$

hold, then for every $g,h\in G$, we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),\end{array}$$

as $\parallel \eta \left(g\right)\parallel \to \infty$.

The primary benefits of our suggested research are as follows:

• Unlike previous studies confined to abelian groups or particular types of mappings, our investigation does not enforce these restrictions. By exploring noncommutative groups, our findings are relevant to a wider range of mathematical and practical challenges where the underlying algebraic structures are non-abelian.
• The assumption that the function $\eta :G\to B$ is surjective improves the application of our findings. Surjectivity ensures that the mapping spans the entire p-uniformly convex space B, which is crucial for establishing strong stability properties.
• Our approach involves the application of the large perturbation method in p-uniformly convex space. This method applies an effective method for analyzing stability integral conditions that account for large-scale perturbations. This is especially crucial in applications that deal with extensive structures, where asymptotic behaviors are of significant importance.
• Our findings present conditions under which the mappings not only approximate an additive functionn but also achieve this in a way that the discrepancy diminishes asymptotically as the norms of the variables grow.

Y. Sun et al. [30] first showed a stability analysis for $(\delta ,\theta ,ϵ)$-isometry from a Banach space E into a p-uniformly convex space F.

Theorem 13 

([30]). Suppose that a mapping η is a $(\delta ,\theta ,ϵ)$-isometry from a Banach space E into a p-uniformly convex space F for $p\ge 1$, then for $ϵ,\delta \ge 0$ there are constants ${\lambda }^{*}\left(ϵ\right)\ge 0$ and $\lambda (\delta ,\theta ,p)\ge 0$ with ${lim}_{\delta \to 0}\lambda (\delta ,\theta ,p)=0$ and ${lim}_{ϵ\to 0}{\lambda }^{*}\left(ϵ\right)=0$ and there exists a linear isometry $L:E\to F$ such that

$$\begin{array}{cc}\hfill \parallel \eta \left(g\right)-L\left(g\right)\parallel \le & \lambda (\delta ,\theta ,p)max\{\parallel g{\parallel }^{\theta },\parallel g{\parallel }^{1-(1-\theta )/p}\}\hfill \\ \hfill & +{\lambda }^{*}{\left(ϵ\right)max\{1,\parallel g\parallel }^{1-1/p}\},forallg\in A.\hfill \end{array}$$

Furthermore, Y. Sun et al. [30] examined an abelian group $(G,+)$ and a $(\delta ,\theta )$-surjective mapping to investigate the stability. The derived results are as follows.

Theorem 14. 

Let $(G,+)$ be an abelian group and F is a p-uniformly convex space for $p\ge 1$. Assume that a mapping $\eta :G\to B$ is a $(\delta ,\theta )$-surjective. If

$$\begin{array}{c}\hfill |\parallel \eta (g-h)\parallel -\parallel \eta (g)-\eta (h)\parallel |\le ϵ,\end{array}$$

for $0<\theta <1$, $ϵ,\delta \ge 0$, then there are two constants ${\lambda }^{*}\left(2ϵ\right)$ and $\lambda (\delta ,\theta ,p)$ such that

$$\begin{array}{cc}\hfill \parallel \eta (g+h)-\eta \left(g\right)-\eta \left(h\right)\parallel \le & \lambda (\delta ,\theta ,p)max\{\parallel \eta \left(h\right){\parallel }^{\theta },\parallel \eta \left(h\right){\parallel }^{1-(1-\theta )/p}\}\hfill \\ \hfill & +{\lambda }^{*}{\left(2ϵ\right)max\{1,\parallel \eta \left(h\right)\parallel }^{1-1/p}\}+ϵ,\mathit{for}\mathit{all}g,h\in G.\hfill \end{array}$$

Y. Sun et al. [30] presented $(\delta ,\theta ,ϵ)$-isometries, advancing the understanding of how near behaviors can contribute in the stability of mappings. Although their study mainly concentrates on abelian groups, our proposed approach can generalize the findings of Dong [23] and Y. Sun [30] to non-abelian contexts by employing the results of L. Cheng [27]. Our methodology extends Sun’s results to non-abelian groups using the large perturbation method, providing a more generalized framework for stability analysis.

In this paper, G signifies any given group, and e symbolizes the identity element of G. Since G is an arbitrary group, the group operation is consistently represented by multiplication, denoted as $gh\in G$ for every $g,h\in G$.

3. Main Results

Assume mapping $\eta$ from group $(G,·)$ to a real Banach space B and define ${\mu }_{\eta }\left(\nu \right)$ and ${\overline{\mu }}_{\eta }\left(\nu \right)$ for $\nu \ge 0$ such that

$$\begin{array}{c}\hfill {\mu }_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(g\right)-\eta \left(h\right)\parallel |:\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \le \nu ,g,h\in G\},\end{array}$$

(10)

$$\begin{array}{c}\hfill {\overline{\mu }}_{\eta }\left(\nu \right)=sup\left\{\right|\parallel \eta \left(gh\right)\parallel -\parallel \eta \left(g\right)+\eta \left(h\right)\parallel |:\parallel \eta \left(g\right)+\eta \left(h\right)\parallel \le \nu ,g,h\in G\}.\end{array}$$

(11)

Before proceeding to the main theorems, we first specify that the condition (8) presented in Theorem 9 is equivalent to the condition described below:

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty ,\end{array}$$

(12)

and condition (9) from Theorem 10 is also equivalent to the following condition

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\mu }}_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty .\end{array}$$

(13)

Lemma 1. 

Assume that $(G,·)$ is an arbitrary group and let B be a real Banach space. Suppose $\eta :G\to B$ is a surjective function. Given the function ${\mu }_{\eta }$ defined in (10), assume that condition (12) holds. Then, for every $g,h\in G$, the subsequent asymptotic behavior is observed:

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\mathit{as}\parallel \eta \left(g\right)\parallel \to \infty .\end{array}$$

Proof. 

Assuming that condition (12) is satisfied, our first step is to verify the existence of a constant $\lambda >0$ such that $2(\nu -{\mu }_{\eta }\left(\nu \right))>\nu$ for every $\lambda <\nu$. In cases where such a $\lambda$ does not exist, for any $m>0$, we can find $m<{\nu }_m$ such that ${\mu }_{\eta }\left({\nu }_m\right)\ge {\displaystyle \frac{{\nu }_m}{2}}$. This leads to the following result:

$$\begin{array}{c}\hfill \underset{{\nu }_m}{\overset{2{\nu }_m}{\int }}{\displaystyle \frac{{\mu }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu \ge \underset{{\nu }_m}{\overset{2{\nu }_m}{\int }}{\displaystyle \frac{{\mu }_{\eta }\left({\nu }_m\right)}{{\nu }^2}}d\nu ={\displaystyle \frac{{\mu }_{\eta }\left({\nu }_m\right)}{2{\nu }_m}}\ge {\displaystyle \frac{1}{4}}.\end{array}$$

This statement presents a contradiction. Let us assume that $\parallel \eta \left(g{h}^{-1}\right)\parallel \le \nu$ and if $\parallel \eta \left(g\right)-\eta \left(h\right)\parallel >\lambda$, we have

$$\begin{array}{cc}\hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel & <2(\parallel \eta \left(g\right)-\eta \left(h\right)\parallel -{\mu }_{\eta }(\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \left)\right)\hfill \\ \hfill & \le 2\parallel \eta \left(g{h}^{-1}\right)\parallel \hfill \\ \hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel & \le 2\nu ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill |\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(g\right)-\eta \left(h\right)\parallel |\le {\mu }_{\eta }\left(2\nu \right).\end{array}$$

(14)

If $\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \le \lambda$, then we obtain

$$\begin{array}{c}\hfill |\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(g\right)-\eta \left(h\right)\parallel |\le {\mu }_{\eta }\left(\lambda \right).\end{array}$$

(15)

Likewise, suppose that $\parallel \eta \left(g\right)-\eta \left(h\right)\parallel \le \nu$, then the following holds:

$$\begin{array}{c}\hfill |\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(g\right)-\eta \left(h\right)\parallel |\le {\mu }_{\eta }\left(\nu \right).\end{array}$$

(16)

The combination of the inequalities specified in (14)–(16) and the expression ${\psi }_{\eta }\left(\nu \right)$ from (6) yields

$$\begin{array}{c}\hfill {\psi }_{\eta }\left(\nu \right)\le max\{{\mu }_{\eta }\left(\lambda \right),{\mu }_{\eta }\left(2\nu \right)\}.\end{array}$$

So, we can conclude

$$\begin{array}{cc}\hfill \underset{\lambda }{\overset{\infty }{\int }}{\displaystyle \frac{{\psi }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu & \le \underset{\lambda }{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }\left(2\nu \right)}{{\nu }^2}}d\nu =2\underset{2\lambda }{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu <\infty ,\hfill \\ \hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\psi }_{\eta }\left(\nu \right)}{{\nu }^2}}d\nu & <\infty .\hfill \end{array}$$

Hence, by utilizing Theorem 9, we can obtain the following result:

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty .\end{array}$$

□

Similarly, it can be demonstrated that the condition (9) stated in Theorem 10 is equivalent to condition (13).

Example 1. 

Consider the non-abelian group $G=GL(2,\mathbb{R})$ which defined as the group of invertible $2\times 2$ matrices over the real numbers, and let B be the space of all such matrices equipped with the Frobenius norm ${\parallel ·\parallel }_F$ (the space of matrices equipped with this norm is a Banach space). The function $\eta :G\to B$ is defined by

$$\eta \left(A\right)=A^{-1}$$

for all $A\in G$. We verify the lemma using this mapping η.

Consider two matrices $A,H\in GL(2,\mathbb{R})$ for some $\nu >0$; we have

$${\mu }_{\eta }\left(\nu \right)=sup\left\{|\parallel H{A}^{-1}{\parallel }_F-\parallel A^{-1}-H^{-1}{\parallel }_F|:{\parallel A^{-1}-H^{-1}\parallel }_F\le \nu \right\},$$

Assume $\parallel A^{-1}-H^{-1}{\parallel }_F\le \nu$. The Frobenius norm of the difference of inverses is bounded by

$$\parallel A^{-1}-H^{-1}{\parallel }_F \le \parallel A^{-1}{\parallel }_F+{\parallel H^{-1}\parallel }_F.$$

Since $G=GL(2,\mathbb{R})$, $\parallel A^{-1}{\parallel }_F$ and $\parallel H^{-1}{\parallel }_F$ are finite for all $A,H\in G$. Thus, we can find some constant value $\lambda >0$ such that

$${\mu }_{\eta }\left(\nu \right)\le \lambda .$$

Therefore,

$${\int }_1^{\infty }\frac{{\mu }_{\eta }\left(\nu \right)}{{\nu }^2}d\nu \le \lambda {\int }_1^{\infty }\frac{1}{{\nu }^2}d\nu =\lambda {\left[-\frac{1}{\nu }\right]}_1^{\infty }=\lambda .$$

Hence, the integral condition is satisfied.

Now, we verify the equation

$$\parallel \eta \left(A{H}^{-1}\right)-\eta \left(A\right)+\eta \left(H\right){\parallel }_F=o(\parallel \eta \left(A\right){\parallel }_F),\mathrm{as}\parallel \eta \left(A\right){\parallel }_F\to \infty .$$

First, note that

$$\parallel \eta \left(A{H}^{-1}\right)-\eta \left(A\right)+\eta \left(H\right){\parallel }_F={\parallel {\left(A{H}^{-1}\right)}^{-1}-A^{-1}+H^{-1}\parallel }_F.$$

Using the Frobenius norm properties, we know

$$\parallel {\left(A{H}^{-1}\right)}^{-1}-A^{-1}+H^{-1}{\parallel }_F\le \parallel {\left(A{H}^{-1}\right)}^{-1}{\parallel }_F+\parallel A^{-1}{\parallel }_F+{\parallel H^{-1}\parallel }_F.$$

As ${\parallel \eta \left(A\right)\parallel }_F\to \infty$, $\parallel A^{-1}{\parallel }_F$ and $\parallel H^{-1}{\parallel }_F$ must grow without bound because A and H are invertible matrices, and their inverses’ norms are related to the original matrices’ norms.

Thus, we see that

$$\parallel {\left(A{H}^{-1}\right)}^{-1}-A^{-1}+H^{-1}{\parallel }_F=o(\parallel A^{-1}{\parallel }_F)=o(\parallel \eta \left(A\right){\parallel }_F).$$

Hence,

$$\parallel \eta \left(A{H}^{-1}\right)-\eta \left(A\right)+\eta \left(H\right){\parallel }_F=o(\parallel \eta \left(A\right){\parallel }_F),\mathit{as}\parallel \eta \left(A\right){\parallel }_F\to \infty .$$

Theorem 15. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space B such that $\eta \left(e\right)=0$. Define the function ${\mu }_{\eta }$ as in (10).

If the integral condition

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{{\mu }_{\eta }\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty ,p\ge 1\end{array}$$

(17)

holds, then for all $g,h\in G$, we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty .\end{array}$$

Proof. 

Assume that condition (17) holds. Consider an element $h\in B$ and define a set-valued mapping $H_h:B\to 2^B$ by

$$H_h\left(r\right)=\{\eta \left(b_r{h}^{-1}\right)+\eta \left(h\right):b_r\in {\eta }^{-1}\left(r\right)\},\mathrm{for}\mathrm{all}r\in B.$$

For two fixed arbitrary elements $r,s\in B$, we can find two elements $z_r\in H_h\left(r\right)$ and $z_s\in H_h\left(s\right)$ such that there exist $b_r\in {\eta }^{-1}\left(r\right)$ and $b_s\in {\eta }^{-1}\left(s\right)$ such that we can write $z_r=\eta \left(b_r{h}^{-1}\right)+\eta \left(h\right)$ and $z_s=\eta \left(b_s{h}^{-1}\right)+\eta \left(h\right)$, then we can obtain

$$\begin{array}{cc}\hfill |\parallel z_r-z_s\parallel -\parallel r-s\parallel |& =|\parallel \eta \left(b_r{h}^{-1}\right)-\eta \left(b_s{h}^{-1}\right)\parallel -\parallel r-s\parallel |\hfill \\ \hfill & =|\parallel \eta \left(b_r{h}^{-1}\right)-\eta \left(b_s{h}^{-1}\right)\parallel -\parallel \eta \left(b_r{b_s}^{-1}\right)\parallel \hfill \\ \hfill & +\parallel \eta \left(b_r{b_s}^{-1}\right)\parallel -\parallel r-s\parallel |\hfill \\ \hfill & \le |\parallel \eta \left(b_r{h}^{-1}\right)-\eta \left(b_s{h}^{-1}\right)\parallel -\parallel \eta \left(b_r{b_s}^{-1}\right)\parallel |\hfill \\ \hfill & +|\parallel \eta (b_r{b_s}^{-1})\parallel -\parallel r-s\parallel |\hfill \\ \hfill \parallel z_r-z_s\parallel -\parallel r-s\parallel & \le {\mu }_{\eta }\{\parallel z_r-z_s\parallel +\parallel r-s\parallel \}.\hfill \end{array}$$

(18)

Our first step is to verify the existence of a constant $\lambda \left(\mu \right)>0$ such that $2(\nu -{\mu }_{\eta }\left(\nu \right))>\nu$ for every $\lambda \left(\mu \right)<\nu$. For any $m>0$, we can find $m<{\nu }_m$ such that ${\mu }_{\eta }\left({\nu }_m\right)\ge {\displaystyle \frac{{\nu }_m}{2}}$. This gives us the following results:

$$\underset{{\nu }_m}{\overset{2{\nu }_m}{\int }}{\displaystyle \frac{{\mu }_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu \ge \underset{{\nu }_m}{\overset{2{\nu }_m}{\int }}{\displaystyle \frac{{\mu }_{\eta }{\left({\nu }_m\right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu ={\mu }_{\eta }{\left({\nu }_m\right)}^{\frac{1}{p}}{\displaystyle \frac{p(2^{\frac{1}{p}}-1)}{2^{\frac{1}{p}}{\nu }_m^{\frac{1}{p}}}}\ge {\nu }_m^{\frac{1}{p}}{\displaystyle \frac{p(2^{\frac{1}{p}}-1)}{2^{\frac{1}{p}}{\nu }_m^{\frac{1}{p}}}}\ge {\displaystyle \frac{p(2^{\frac{1}{p}}-1)}{2^{\frac{1}{p}}}}>0,$$

which specifies the contradiction to the given condition (17). Further, assume that $\parallel r-s\parallel \le \nu$, then we can obtain

$$\begin{array}{c}\hfill \parallel \eta \left(b_r{b_s}^{-1}\right)\parallel \le \parallel r-s\parallel +{\mu }_{\eta }(\parallel r-s\parallel )\le \nu +{\mu }_{\eta }\left(\nu \right).\end{array}$$

(19)

If $\parallel z_r-z_s\parallel >\lambda \left(\mu \right)$, then we can conclude

$$\begin{array}{c}\hfill \parallel z_r-z_s\parallel <2\{\parallel z_r-z_s\parallel -{\mu }_{\eta }(\parallel z_r-z_s\parallel )\le 2\parallel \eta \left(b_r{b_s}^{-1}\right)\parallel \le 2(\nu +{\mu }_{\eta }\left(\nu \right)).\end{array}$$

(20)

If $\parallel z_r-z_s\parallel \le \lambda \left(\mu \right)$, then inequality (18) gives

$$\begin{array}{c}\hfill |\parallel z_r-z_s\parallel -\parallel r-s\parallel |\le {\mu }_{\eta }\left(\lambda \left(\mu \right)\right)+{\mu }_{\eta }\left(\nu \right).\end{array}$$

(21)

For any choice of the mapping ${\chi }_h$ of $H_h$, concluding (20) and (21) for $\nu \ge 0$, it can be concluded that

$$\begin{array}{c}\hfill {ϵ}_{{\chi }_h}\left(\nu \right)\le max\{{\mu }_{\eta }\left(2(\nu +{\mu }_{\eta }\left(\nu \right))\right)+{\mu }_{\eta }\left(\nu \right),{\mu }_{\eta }\left(\lambda \left(\mu \right)\right)+{\mu }_{\eta }\left(\nu \right)\}.\end{array}$$

(22)

It is clear that $\nu >\lambda \left(\mu \right)$, then we have $2(\nu -{\mu }_{\eta }\left(\nu \right))>\nu$, which implies that $2{\mu }_{\eta }\left(\nu \right)>\nu$, so we obtain

$$\begin{array}{cc}\hfill \underset{\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{{ϵ}_{{\chi }_h}\left(\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu & \le \underset{\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{{\mu }_{\eta }\left(2(\nu +{\mu }_{\eta }\left(\nu \right))\right)+{\mu }_{\eta }\left(\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu \hfill \\ \hfill & \le \underset{\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{2{\mu }_{\eta }\left(4\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu \hfill \\ \hfill & \le 2^{\frac{1}{p}}\underset{\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{{\mu }_{\eta }\left(4\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu \hfill \\ \hfill & =2^{\frac{3}{p}}\underset{4\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{{\mu }_{\eta }\left(\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty \hfill \\ \hfill \underset{\lambda \left(\mu \right)}{\overset{\infty }{\int }}{\displaystyle \frac{{\left\{{ϵ}_{{\chi }_h}\left(\nu \right)\right\}}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu & <\infty .\hfill \end{array}$$

According to the application of Theorem 8 proposed by Cheng et al. [27], a linear isometry $L_{{\chi }_h}:B\to B$ exists such that

$$\begin{array}{c}\hfill \parallel {\chi }_h\left(r\right)-L_{{\chi }_h}\left(r\right)\parallel =o(\parallel r\parallel ),\parallel r\parallel \to \infty .\end{array}$$

(23)

Consider an alternative mapping $\rho :B\to B$ for choosing $H_h$. As a result, the inequality (21) transforms into:

$$\begin{array}{c}\hfill \parallel {\rho }_h\left(r\right)-{\chi }_h\left(r\right)\parallel \le {\mu }_{\eta }\left(\lambda \left(\mu \right)\right)+{\mu }_{\eta }\left(0\right),\mathrm{for}\mathrm{all}r\in B.\end{array}$$

Then, we can obtain

$$\begin{array}{cc}\hfill \parallel L_{{\rho }_h}\left(r\right)-L_{{\chi }_h}\left(r\right)\parallel & \le \parallel L_{{\rho }_h}\left(r\right)-{\rho }_h\left(r\right)\parallel +\parallel {\rho }_h\left(r\right)-{\chi }_h\left(r\right)\parallel +\parallel {\chi }_h\left(r\right)-L_{{\chi }_h}\left(r\right)\parallel \hfill \\ \hfill \parallel L_{{\rho }_h}\left(r\right)-L_{{\chi }_h}\left(r\right)\parallel & \le 2o(\parallel r\parallel )+{\mu }_{\eta }\left(\lambda \left(\mu \right)\right)+{\mu }_{\eta }\left(0\right).\hfill \end{array}$$

From $\parallel r\parallel \to \infty$ we can get that $L_{{\rho }_h}\left(r\right)=L_{{\chi }_h}\left(r\right)$, therefore, by using $r=\eta \left(g\right)$ in (23) and denoting $L_{{\chi }_h}$ by $L_h$, we have

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)+\eta \left(h\right)-L_h\left(\eta \left(g\right)\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty .\end{array}$$

(24)

Consider two arbitrary elements $h_1,h_2\in G$, then we want to determine the relation with $L_{h_1}$ and $L_{h_2}$ using (18), (20), (21) and (24) as follows:

$$\begin{array}{cc}\hfill \parallel L_{h_1}\left(\eta \left(g\right)\right)-L_{h_2}\left(\eta \left(g\right)\right)\parallel \le & \parallel L_{h_1}\left(\eta \left(g\right)\right)-(\eta \left({g{h}_1}^{-1}\right)+\eta \left(h_1\right))\parallel \hfill \\ \hfill & +\parallel \eta \left({g{h}_1}^{-1}\right)+\eta \left(h_1\right)-(\eta \left({g{h}_2}^{-1}\right)+\eta \left(h_2\right))\parallel \hfill \\ \hfill & +\parallel \eta \left({g{h}_2}^{-1}\right)+\eta \left(h_2\right)-L_{h_2}\left(\eta \left(g\right)\right)\parallel \hfill \\ \hfill \le & 2o(\parallel \eta \left(g\right)\parallel )+\parallel \eta \left({g{h}_1}^{-1}\right)-\eta \left({g{h}_2}^{-1}\right)\parallel \hfill \\ \hfill & -\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel +2\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel ,\hfill \\ \hfill \le & 2o(\parallel \eta \left(g\right)\parallel )+2\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel \hfill \\ \hfill & +max\{{\mu }_{\eta }\left(2\right(\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel +{\mu }_{\eta }(\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel \left)\right))\hfill \\ \hfill & +{\mu }_{\eta }(\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel ),{\mu }_{\eta }\left(\lambda \left(\mu \right)\right)+{\mu }_{\eta }(\parallel \eta \left(h_1\right)-\eta \left(h_2\right)\parallel \left)\right\},\hfill \end{array}$$

that gives $L_{h_1}=L_{h_2}$ because $\parallel \eta \left(g\right)\parallel \to \infty$. Setting $h=e$ in (24), we get that

$$\begin{array}{cc}\hfill \parallel \eta \left(g\right)+\eta \left(e\right)-L_e\left(\eta \left(g\right)\right)\parallel & =o(\parallel \eta (g)\parallel ),\hfill \\ \hfill \parallel \eta \left(g\right)-L_e\left(\eta \left(g\right)\right)\parallel & =o(\parallel \eta (g)\parallel ).\hfill \end{array}$$

Since $\parallel \eta \left(g\right)\parallel \to \infty$, therefore, $L_e\left(\eta \left(g\right)\right)=\eta \left(g\right)$ gives that $L_e$ is an identity mapping. Also, identify the function $L_e$ as $L_h=I$ for all $g\in G$. Since $L_h=I$, $h\in G$, then Equation (24) gives

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\mathrm{where}\parallel \eta \left(g\right)\parallel \to \infty ,\end{array}$$

which is the required proof. □

Based on the findings presented by Hanche-Olsen [31] as well as the works of Clarkson [32] and Lovaglia [33] regarding p-uniformly convex spaces, consider the space $B=L^p,l^p$, where $1<p<\infty$. Based on these results, we can deduce the subsequent corollary.

Corollary 1. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space $B=L^p,l^p$ such that $\eta \left(e\right)=0$ Consider defined function ${\mu }_{\eta }$ from (10). If the integral condition

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{{\mu }_{\eta }\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty ,1<p<\infty ,\end{array}$$

holds, then we have

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,\mathit{for}\mathit{all}g,h\in G.\end{array}$$

Example 2. 

Assume that $G=\mathbb{Z}$ is the group of integers under addition, and let $B={\ell }^p\left(\mathbb{Z}\right)$ be the space of p-summable sequences for some $p\ge 1$. Define the mapping $\eta :\mathbb{Z}\to {\ell }^p\left(\mathbb{Z}\right)$ by

$$\eta \left(n\right)=\left\{\begin{array}{cc}(n,0,0,\dots )\hfill & \mathit{if}n\ge 0,\hfill \\ (0,0,\dots ,-n,0,0,\dots )\hfill & \mathit{if}n<0,\hfill \end{array}\right.$$

where n is placed at the first position for $n\ge 0$ and at the $\left|n\right|$-th position for $n<0$. This mapping is surjective and $\eta \left(0\right)=0$. We define ${\mu }_{\eta }\left(\nu \right)$ as

$${\mu }_{\eta }\left(\nu \right)=sup\left\{\left|{\parallel \eta (m-n)\parallel }_p-{\parallel \eta \left(m\right)-\eta \left(n\right)\parallel }_p\right|:{\parallel \eta \left(m\right)-\eta \left(n\right)\parallel }_p\le \nu ,m,n\in \mathbb{Z}\right\}.$$

For any $m,n\in \mathbb{Z}$, we have:

$${\parallel \eta \left(m\right)-\eta \left(n\right)\parallel }_p=|m-n|,$$

since the sequence will have a single nonzero element, which is $m-n$.

Thus,

$${\parallel \eta (m-n)\parallel }_p=|m-n|,$$

and it follows that

$${\mu }_{\eta }\left(\nu \right)=sup\left\{\left||m-n|-|m-n|\right|:|m-n|\le \nu \right\}=0.$$

Since ${\mu }_{\eta }\left(\nu \right)=0$, the integral condition holds trivially. First, note that $\eta (m-n)=\eta \left(m\right)-\eta \left(n\right)$, so

$${\parallel \eta (m-n)-\eta \left(m\right)+\eta \left(n\right)\parallel }_p={\parallel (\eta \left(m\right)-\eta \left(n\right))-\eta \left(m\right)+\eta \left(n\right)\parallel }_p=0.$$

Therefore,

$${\parallel \eta (m-n)-\eta \left(m\right)+\eta \left(n\right)\parallel }_p={o(\parallel \eta \left(m\right)\parallel }_p{)\mathit{as}\parallel \eta \left(m\right)\parallel }_p\to \infty .$$

Next, by utilizing Theorem 15, we establish Theorem 16. To accomplish this, we present a new generalized condition for the function ${\overline{\mu }}_{\eta }$, which incorporates the generalized form of the functional Equation (5) to obtain the following FE

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,\end{array}$$

by using an appropriate restriction on surjective mapping.

Theorem 16. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space B. Define the function ${\overline{\mu }}_{\eta }$ as in (11).

If the integral condition

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\mu }}_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}},d\nu <\infty ,p\ge 1\end{array}$$

(25)

holds, then we obtain

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,\mathit{for}\mathit{all}g,h\in G.\end{array}$$

Proof. 

Assume that the condition (25) holds, then we have a constant $\lambda \left(\overline{\mu }\right)>0$ such that $2(\nu -{\overline{\mu }}_{\eta }\left(\nu \right))>\nu$ for every $\lambda \left(\overline{\mu }\right)<\nu$. Suppose that $\parallel \eta \left(gh\right)\parallel \le \nu$ and if $\parallel \eta \left(g\right)+\eta \left(h\right)\parallel >\lambda \left(\overline{\mu }\right)$, then we obtain

$$\parallel \eta \left(g\right)+\eta \left(h\right)\parallel <2(\parallel \eta \left(g\right)+\eta \left(h\right)\parallel -{\overline{\mu }}_{\eta }(\parallel \eta \left(g\right)+\eta \left(h\right)\parallel \left)\right)\le 2\parallel \eta \left(gh\right)\parallel \le 2\nu .$$

So, we can obtain

$$\begin{array}{c}\hfill \parallel \eta \left(g\right)+\eta \left(h\right)\parallel \le max\{2\parallel \eta \left(gh\right)\parallel ,\lambda \left(\overline{\mu }\right)\}.\end{array}$$

(26)

Setting $h=g^{-1}$ in (26), for all $h\in G$, we get

$$\begin{array}{c}\hfill \parallel \eta \left(h\right)+\eta \left(h^{-1}\right)\parallel \le max\{2\parallel \eta \left(e\right)\parallel ,\lambda \left(\overline{\mu }\right)\}\equiv \beta .\end{array}$$

(27)

Consider

$$\begin{array}{cc}\hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel & =\parallel \eta \left(h^{-1}\right)-\eta \left(h^{-1}\right)+\eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel \hfill \\ \hfill & \le \parallel \eta \left(h^{-1}\right)+\eta \left(h\right)\parallel +\parallel \eta \left(g\right)+\eta \left(h^{-1}\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel \hfill \\ \hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel & \le |\parallel \eta \left(g\right)+\eta \left(h^{-1}\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel |+\beta .\hfill \end{array}$$

(28)

Again, consider

$$\begin{array}{cc}\hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel & =\parallel \eta \left(h^{-1}\right)-\eta \left(h^{-1}\right)+\eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel \hfill \\ \hfill & \ge \parallel \eta \left(g\right)+\eta \left(h^{-1}\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel -\parallel \eta \left(h^{-1}\right)+\eta \left(h\right)\parallel \hfill \\ \hfill \parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel & \ge -|\parallel \eta \left(g\right)+\eta \left(h^{-1}\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel |-\beta .\hfill \end{array}$$

(29)

From inequalities (28) and (29), we obtain that

$$\begin{array}{c}\hfill |\parallel \eta \left(g\right)-\eta \left(h\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel |\le |\parallel \eta \left(g\right)+\eta \left(h^{-1}\right)\parallel -\parallel \eta \left(g{h}^{-1}\right)\parallel |+\beta .\end{array}$$

(30)

Also,

$$\begin{array}{cc}\hfill \parallel \eta \left(h^{-1}\right)+\eta \left(g\right)\parallel & \le \parallel \eta \left(g\right)-\eta \left(h\right)\parallel +\parallel \eta \left(h\right)+\eta \left(h^{-1}\right)\parallel \hfill \\ \hfill \parallel \eta \left(h^{-1}\right)+\eta \left(g\right)\parallel & \le \parallel \eta \left(g\right)-\eta \left(h\right)\parallel +\beta .\hfill \end{array}$$

(31)

By (30) and (31), for every $\nu >0$, we can express ${\overline{\mu }}_{\eta }$ and ${\mu }_{\eta }$ as follows:

$$\begin{array}{c}\hfill {\mu }_{\eta }\left(\nu \right)\le {\overline{\mu }}_{\eta }(\nu +\beta )+\beta .\end{array}$$

(32)

Applying (25) in (32), we have

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty .\end{array}$$

□

Again, we can deduce the following corollary if we consider the p-uniformly convex spaces $B=L^p,l^p$, where $1<p<\infty$.

Corollary 2. 

Let $(G,·)$ be an arbitrary group and suppose that $B=L^p$ or $B=l^p$ is a p-uniformly convex space with $1<p<\infty$. Assume $\eta :G\to B$ is a surjective mapping. If

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\mu }}_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty ,\end{array}$$

where ${\overline{\mu }}_{\eta }$ is defined in (11), then we have

$$\begin{array}{c}\hfill \parallel \eta \left(g{h}^{-1}\right)-\eta \left(g\right)+\eta \left(h\right)\parallel =o(\parallel \eta \left(g\right)\parallel ),g,h\in G,\end{array}$$

when $\parallel \eta \left(g\right)\parallel \to \infty$.

Remark 1. 

Theorem 16 implies Theorem 15 if a constant $\beta >0$ such that the integral condition involving ${\overline{\mu }}_{\eta }\left(\nu \right)$ can be bounded appropriately. Specifically, this holds if

$$\underset{1}{\overset{\infty }{\int }}\frac{{\overline{\mu }}_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}d\nu <\infty and{\beta }^{\frac{1}{p}}\underset{1}{\overset{\infty }{\int }}\frac{1}{{\nu }^{1+\frac{1}{p}}}d\nu <\infty .$$

Given the integral involving ${\overline{\mu }}_{\eta }\left(\nu \right)$ converges, the additional term involving β must also be bounded for the integral condition on ${\mu }_{\eta }\left(\nu \right)$ to hold.

By adhering to the approach detailed in Theorem 15 and defining the set-valued mapping $H_h$ from the Banach space B to $2^B$, with a particular emphasis on selecting a fixed element h from the noncommutative group G, we introduce Theorem 17 to demonstrate the stability of the functional Equation (5), which is as follows:

Theorem 17. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space B such that $\eta \left(e\right)=0$. Consider the function ${\mu }_{\eta }$ defined in (10). Assume that

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}},d\nu <\infty ,p\ge 1.\end{array}$$

Then, we have

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),\parallel \eta \left(g\right)\parallel \to \infty ,\mathit{for}\mathit{all}g,h\in G.\end{array}$$

We can deduce the following corollary by considering the p-uniformly convex spaces $B=L^p,l^p$, where $1<p<\infty$.

Corollary 3. 

Suppose $(G,·)$ is an arbitrary group and $B=L^p,l^p$. Let $\eta :G\to B$ be a surjective mapping having $\eta \left(e\right)=0$. Consider a function ${\mu }_{\eta }$ from (10), then we have

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),forallg,h\in G,\end{array}$$

as $\parallel \eta \left(g\right)\parallel \to \infty$, provided that

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\mu }_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty ,1<p<\infty .\end{array}$$

By specifying the mapping ${\overline{\mu }}_{\eta }\left(\nu \right)$ for $\nu \ge 0$ and adhering to the procedures described in Theorem 16, we now introduce Theorem 18 through the transformation of Theorem 17. This methodology leads to deriving the stability result for the FE (5). It can be stated as follows:

Theorem 18. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space B for $p\ge 1$. Suppose the following condition holds:

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\mu }}_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty .\end{array}$$

Consider the function ${\overline{\mu }}_{\eta }$ defined in (11). Then, we have

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),g,h\in G,\end{array}$$

as $\parallel \eta \left(g\right)\parallel \to \infty$.

Examine the p-uniformly convex spaces $B=L^p,l^p$, where $1<p<\infty$. Consequently, the following corollary can be deduced.

Corollary 4. 

Suppose that η is a surjective function from an arbitrary group $(G,·)$ to a p-uniformly convex space $B=L^p,l^p$. Consider a function ${\overline{\mu }}_{\eta }$ from (11) and if

$$\begin{array}{c}\hfill \underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{{\overline{\mu }}_{\eta }{\left(\nu \right)}^{\frac{1}{p}}}{{\nu }^{1+\frac{1}{p}}}}d\nu <\infty ,1<p<\infty ,\end{array}$$

then we have

$$\begin{array}{c}\hfill \parallel \eta \left(gh\right)-\eta \left(g\right)-\eta \left(h\right)\parallel =o(\parallel \eta (g)\parallel ),\mathit{for}\mathit{all}g,h\in G\end{array}$$

as $\parallel \eta \left(g\right)\parallel \to \infty$.

The investigation of p-uniformly convex spaces regarding the norm-based functional equations results in several significant open problems:

• While this study concentrated on norm-additive functional equations, it would be beneficial to analyze the stability for further FEs in p-uniformly convex domains. For example, can we obtain comparable stability results for quadratic, cubic, or other functional equations?
• The precise bounds of perturbations that assure stability in p-uniformly convex spaces remain an outstanding question. Further research could focus on improving these limits and understanding their relationship to space and group elements.
• Consider the analysis of the stability for FEs in higher-dimensional p-uniformly convex spaces. How do the stability characteristics change as the size of the space increases?
• Since this study focuses on arbitrarily (noncommutative) groups, it might be interesting to categorize specific types of noncommutative groups that exhibit distinguishing unique stability qualities for these functional equations.
• Consider the investigation of the implications of the stability results in p-uniformly convex spaces for similar problems in metric spaces. Can comparable stability results be obtained in the context of metric fixed point theory or other fields of analysis?

4. Summary and Future Directions

This research examined the generalization of the Hyers–Ulam stability of norm-additive FEs across arbitrary groups and p-uniformly convex spaces. Expanding on the contributions of Cheng [27] and Sarfraz [29], we utilized the perturbation method to determine the conditions that promote stability in surjective mappings from groups to p-uniformly convex space. Our findings broaden the scope of stability analysis to include non-abelian groups, addressing gaps in earlier studies by Tabor, Sikorska, Dong, and Sun, which mainly concentrated on abelian groups or specific types of mappings.

The methodology and findings presented enhance the domain of stability analysis by broadening traditional stability results to include noncommutative groups and p-uniformly convex spaces. Integral conditions and the surjectivity assumption are pivotal to our analysis, offering a solid framework for determining the Hyers–Ulam stability of norm-based FEs.

The proposed integral condition approach can be extended to other types of FEs beyond the norm-additive ones, for example, quadratic, cubic, exponential, and trigonometric functional equations. Future work could analyze the stability of different classes of FEs using similar methods. The concept of p-uniformly convex spaces has proven useful in establishing stability results.

The results indicate that the large perturbation method is an effective approach for analyzing stability in functional equations. It is recommended that researchers consider applying this technique to additional mathematical challenges where stability plays a crucial role, including differential equations and dynamical systems.