Classical and Fixed Point Approach to the Stability Analysis of a Bilateral Symmetric Additive Functional Equation in Fuzzy and Random Normed Spaces

Abstract

In this article, a new kind of bilateral symmetric additive type functional equation is introduced. One of the interesting characteristics of the equation is the fact that it is ideal for investigating the Ulam–Hyers stabilities in two prominent normed spaces, namely fuzzy and random normed spaces simultaneously. This article analyzes the proposed equation in both spaces. The solution of this equation exhibits the property of symmetry, that is, the left of the object becomes the right of the image, and vice versa. Additionally, the stability results of this functional equation are determined in fuzzy and random normed spaces using direct and fixed point methods.

1. Introduction

Stability is integral to the study of functional equations (FEs) as it serves as an effective and reliable tool for quantifying the error that arises when replacing functions that satisfy some equations only roughly with the exact solutions to those equations. Presently, an equation is said to be stable in some set of functions if any function from that set that approximates the equation is comparable to an exact solution of the equation. Many mathematicians have studied quite a few stability problems of diverse functional equations (radical, reciprocal, logarithmic, algebraic) over the last few decades (see [1,2,3,4,5]).

Ulam raised a significant question about the stability of group homomorphisms in 1940 [6]. In the subsequent year, Hyers [7] provided a solution to Ulam’s question for the Cauchy additive FE. Rassias enhanced Hyers’ result after more than two decades by generalizing it [8]. Later, G$\check{\mathrm{a}}$vrut$\check{\mathrm{a}}$ [9] developed Rassias’ result by permitting unbounded control functions. The term “generalized Hyers–Ulam stability” of functional equations is now alluded to the stability concept first put forth by Rassias and G$\check{\mathrm{a}}$vruta. For the detailed literature on the stability of functional equations, one can see [10,11].

Using the fundamental results of fixed-point theory, the fixed point method provides one of the effective techniques that can be used to investigate the Ulam stability of a FE. Some recent research and findings on fixed-point theory (FPT) can be seen in [12,13,14,15,16,17].

The Ulam stability of the Cauchy FE

$$\mathcal{G}({\mathcal{U}}_1+{\mathcal{U}}_2)=\mathcal{G}\left({\mathcal{U}}_1\right)+\mathcal{G}\left({\mathcal{U}}_2\right)$$

in random normed spaces was investigated by D. Mihet et al. [18].

H.M. Kim et al. [19] studied a modified and generalized version of the Cauchy additive FE

$$\mathcal{G}\left(\frac{{\mathcal{U}}_1-{\mathcal{U}}_2}{n}+{\mathcal{U}}_3\right)+\mathcal{G}\left(\frac{{\mathcal{U}}_2-{\mathcal{U}}_3}{n}+{\mathcal{U}}_1\right)+\mathcal{G}\left(\frac{{\mathcal{U}}_3-{\mathcal{U}}_1}{n}+{\mathcal{U}}_2\right)=\mathcal{G}({\mathcal{U}}_1+{\mathcal{U}}_2+{\mathcal{U}}_3)$$

and demonstrated the Ulam stability (US) for any fixed non-zero integer n in fuzzy Banach spaces (BSs). It is obvious that a function $\mathcal{G}$ satisfies the aforementioned equation if and only if $\mathcal{G}$ is additive. Consequently, the equation is referred to as the Cauchy additive functional equation, and its general solution is considered as the Cauchy additive function.

The Ulam–Hyers stability (UHS) of fixed point problems and the fixed points of mappings over a locally convex topological vector space were presented by Roy et al. [20]. Saha et al. [21] provided a solution using FPT to the stability problems in intuitionistic fuzzy BSs. Alanazi et al. [22] examined the fuzzy stability of a finite variable additive FE using direct and fixed point approaches. Bae et al. [23] demonstrated the stability problem and proposed a theory that symmetry is repeated self-similarity by simulating the well-known Cauchy and Jensen equations in two variables. Turab et al. [24] studied the applications of Banach limit in UHS. Other studies have examined the 2-normed spaces, Euler–Lagrange–Rassias quadratic FE, etc. [25,26,27,28].

Recently, Agilan et al. [29,30] studied the GUH stability of the following additive FE

$$\begin{array}{cc}\hfill & \zeta \left({\eta }^{\ell +\wp }\gamma +{\eta }^{\hslash }\kappa \right)+\zeta \left({\eta }^{\ell +\wp }\kappa +{\eta }^{\hslash }\mu \right)+{\eta }^{\ell +\wp }\zeta (\gamma -\kappa )+{\eta }^{\hslash }\zeta (\kappa -\mu )\hfill \\ \hfill & =2\left({\eta }^{\ell +\wp }\zeta \left(\gamma \right)+{\eta }^{\hslash }\zeta \left(\kappa \right)\right)\hfill \\ \hfill & \left(\varsigma +\wp +\hslash \right)\mathfrak{Z}\left(\varsigma \mathfrak{p}+\wp \mathfrak{q}+\hslash \mathfrak{r}\right)+\varsigma \mathfrak{Z}\left(\wp \hslash \left(\mathfrak{p}-\mathfrak{q}\right)\right)+\wp \mathfrak{Z}\left(\varsigma \hslash \left(\mathfrak{q}-\mathfrak{r}\right)\right)+\hslash \mathfrak{Z}\left(\varsigma \wp \left(\mathfrak{r}-\mathfrak{p}\right)\right)\hfill \\ \hfill & =\left(\varsigma +\wp +\hslash \right)\left(\varsigma \mathfrak{Z}\left(p\right)+\wp \mathfrak{Z}\left(\mathfrak{q}\right)+\hslash \mathfrak{Z}\left(r\right)\right)\hfill \end{array}$$

in BS, quasi-$\beta$-normed spaces, and IFNS using direct and FPT. Some of the newly developed concepts and applications in the field of fuzzy normed spaces have been studied in [31,32,33].

Motivated by the above fact, this article introduces a novel class of bilateral symmetric additive type functional equation described, as below. The generalized Ulam–Hyers stabilities are determined for various general control functions of the equation

$$\begin{gathered} \left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)+{\gamma }^{b+c}\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)+{\gamma }^a\mathcal{G}({\mathcal{U}}_2-{\mathcal{U}}_3) \\ =\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{G}\left({\mathcal{U}}_1\right)+\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{G}\left({\mathcal{U}}_2\right)+\left(1-2{\gamma }^a\right)\mathcal{G}\left({\mathcal{U}}_3\right) \end{gathered}$$

(1)

with ${\gamma }^a\ne 0,\pm 1$ and ${\gamma }^{b+c}\ne 0$ in fuzzy and random normed spaces using direct and fixed point techniques.

This is the first time in the literature that the generalized Ulam–Hyers stabilities for bilateral symmetric additive functional equations are analyzed using two distinct techniques in two different spaces at the same time. Consequently, the findings that will be discussed in the subsequent sections are both novel and essential to the study of functional equations. In Section 1, the fundamentals of fuzzy normed spaces and random normed spaces are discussed briefly with suitable examples. In the following section, the general solution of the bilateral symmetric additive type functional equation is derived. Section 3 covers the Ulam stability analysis of the new class of equation using the direct and fixed point methods in fuzzy normed spaces. In Section 4, the Ulam stability analysis is performed for the new equation in random normed spaces using direct and fixed point techniques.

1.1. Fundamentals of Fuzzy Normed Spaces

The fundamentals of fuzzy normed spaces are discussed based on the definitions provided in [34,35,36,37,38].

Definition 1.

Let ${\mathbb{F}}^{*}$ be a real linear space. A function $\mathcal{N}:{\mathbb{F}}^{*}\times \mathbb{R}\to [0,1]$ (the so-called fuzzy subset) is said to be a fuzzy norm on X if for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$ and all $s,t\in \mathbb{R}$,

$\mathrm{(𝒜}$₁)

$\mathcal{N}({\mathcal{U}}_1,c)=0$ for $c\le 0;$

$\mathrm{(𝒜}$₂)

$x=0$ if and only if $\mathcal{N}({\mathcal{U}}_1,c)=1$ for all $c>0;$

$\mathrm{(𝒜}$₃)

$\mathcal{N}(c{\mathcal{U}}_1,t)=\mathcal{N}\left({\mathcal{U}}_1,\frac{t}{\left|c\right|}\right)$ if $c\ne 0;$

$\mathrm{(𝒜}$₄)

$\mathcal{N}({\mathcal{U}}_1+{\mathcal{U}}_2,s+t)\ge min\{\mathcal{N}({\mathcal{U}}_1,s),\mathcal{N}({\mathcal{U}}_2,t)\};$

$\mathrm{(𝒜}$₅)

$\mathcal{N}({\mathcal{U}}_1,·)$ is a non-decreasing function on $\mathbb{R}$ and $li{m}_{t\to \infty }\mathcal{N}({\mathcal{U}}_1,t)=1;$

$\mathrm{(𝒜}$₆)

For ${\mathcal{U}}_1\ne 0,\mathcal{N}({\mathcal{U}}_1,·)$ is (upper semi) continuous on $\mathbb{R}$.

The pair $({\mathbb{F}}^{*},\mathcal{N})$ is called a fuzzy normed linear space. $\mathcal{N}({\mathbb{F}}^{*},t)$ may be considered as the truth-value of the statement “the norm of ${\mathcal{U}}_1$ is less than or equal to the real number t”.

Example 1.

Let $\left({\mathbb{F}}^{*},\left|\right|·\left|\right|\right)$ be a normed linear space. Then,

$$\mathcal{N}\left({\mathcal{U}}_1,t\right)=\left\{\begin{array}{c}\frac{t}{t+∥{\mathcal{U}}_1∥},t>0,{\mathcal{U}}_1\in {\mathbb{F}}^{*},\hfill \\ 0,t\le 0,{\mathcal{U}}_1\in {\mathbb{F}}^{*}\hfill \end{array}\right.$$

is a fuzzy norm on ${\mathbb{F}}^{*}$.

1.2. Fundamentals of Random Normed Spaces

Fundamentals of random normed spaces are described in [39,40,41,42,43].

Definition 2.

A random normed space is a triple $({\mathbb{F}}^{*},\mu ,T)$, where ${\mathbb{F}}^{*}$ is a vector space, T is a continuous t-norm and μ is a mapping from ${\mathbb{F}}^{*}$ into $D^{+}$ satisfying the following conditions:

(I) 

${\displaystyle {\mu }_{{\mathcal{U}}_1}\left(t\right)=\epsilon \left(t\right)}$ for all $t>0$ if and only if ${\mathcal{U}}_1=0$;

(II) 

${\displaystyle {\mu }_{\alpha {\mathcal{U}}_1}\left(t\right)={\mu }_{{\mathcal{U}}_1}(t/|\alpha \left|\right)}$ for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$, and $\alpha \in \mathbb{R}$ with $\alpha \ne 0$;

(III) 

${\displaystyle {\mu }_{{\mathcal{U}}_1+{\mathcal{U}}_2}(t+s)\ge T\left({\mu }_{{\mathcal{U}}_1}\left(t\right),{\mu }_{{\mathcal{U}}_2}\left(s\right)\right)}$ for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$ and $t,s\ge 0$.

Example 2.

Every normed space $({\mathbb{F}}^{*},||·|\left|\right)$ defines a random normed space $({\mathbb{F}}^{*},\mu ,T_M)$, where

$${\mu }_{(}t)=\frac{t}{t+\left|\right|{\mathcal{U}}_1\left|\right|}$$

and $T_M$ is the minimum t-norm.

2. General Solution of Bilateral Symmetric Additive Functional Equation

In this section, let us consider ${\mathbb{F}}^{*}$ and ${\mathbb{I}}^{*}$ to be real vector spaces.

Lemma 1.

If an odd mapping $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfies the FE

$$\mathcal{G}({\mathcal{U}}_1+{\mathcal{U}}_2)=\mathcal{G}\left({\mathcal{U}}_1\right)+\mathcal{G}\left({\mathcal{U}}_2\right)$$

(2)

then $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfies the FE (1) for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$.

Proof. 

If ${\mathcal{U}}_1={\mathcal{U}}_2=0$ in (2), then $\mathcal{G}\left(0\right)=0$. Replacing ${\mathcal{U}}_1$ by $-{\mathcal{U}}_2$ in (2), the following result is obtained: $\mathcal{G}(-{\mathcal{U}}_2)=-\mathcal{G}\left({\mathcal{U}}_2\right)$ for all ${\mathcal{U}}_2\in {\mathbb{F}}^{*}$. Replacing ${\mathcal{U}}_2$ by ${\mathcal{U}}_1$ in (2),

$$\mathcal{G}\left(2{\mathcal{U}}_1\right)=2\mathcal{G}\left({\mathcal{U}}_1\right)$$

(3)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$. By induction of n,

$$\mathcal{G}\left(n{\mathcal{U}}_1\right)=n\mathcal{G}\left({\mathcal{U}}_1\right)$$

(4)

Taking ${\mathcal{U}}_2={\gamma }^{b+c}{\mathcal{U}}_2$ in (2) and using (4),

$$\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)=\mathcal{G}\left({\mathcal{U}}_1\right)+{\gamma }^{b+c}\mathcal{G}\left({\mathcal{U}}_2\right)$$

(5)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$. Multiplying both sides by $\left({\gamma }^a+1\right)$ in (5),

$$\left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)=\left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1\right)+\left({\gamma }^a+1\right){\gamma }^{b+c}\mathcal{G}\left({\mathcal{U}}_2\right)$$

(6)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$. Taking ${\mathcal{U}}_1={\gamma }^{b+c}{\mathcal{U}}_2,{\mathcal{U}}_2={\mathcal{U}}_3$ in (2) and using (4),

$$\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)={\gamma }^{b+c}\mathcal{G}\left({\mathcal{U}}_2\right)+\mathcal{G}\left({\mathcal{U}}_3\right)$$

(7)

for all ${\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Multiplying both sides by $\left(1-{\gamma }^a\right)$ in (6),

$$\left(1-{\gamma }^a\right)\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)=\left(1-{\gamma }^a\right)\mathcal{G}{\gamma }^{b+c}\left({\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{G}\left({\mathcal{U}}_3\right)$$

(8)

for all ${\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Replacing ${\mathcal{U}}_2$ by $-{\mathcal{U}}_2$ in (2), and using $\mathcal{G}(-{\mathcal{U}}_2)=-\mathcal{G}\left({\mathcal{U}}_2\right)$, the following result is obtained:

$$\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)=\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{G}\left({\mathcal{U}}_2\right)$$

(9)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$. Multiplying both sides by ${\gamma }^{b+c}$ in (7),

$${\gamma }^{b+c}\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)={\gamma }^{b+c}\mathcal{G}\left({\mathcal{U}}_1\right)-{\gamma }^{b+c}\mathcal{G}\left({\mathcal{U}}_2\right)$$

(10)

for all ${\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Replacing $\left({\mathcal{U}}_1,{\mathcal{U}}_2\right)$ by $\left({\mathcal{U}}_2,-{\mathcal{U}}_3\right)$ in (2), and using $\mathcal{G}(-{\mathcal{U}}_3)=-\mathcal{G}\left({\mathcal{U}}_3\right)$,

$$\mathcal{G}({\mathcal{U}}_2-{\mathcal{U}}_3)=\mathcal{G}\left({\mathcal{U}}_2\right)-\mathcal{G}\left({\mathcal{U}}_3\right)$$

(11)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2\in {\mathbb{F}}^{*}$. Multiplying both sides by ${\gamma }^a$ in (7),

$${\gamma }^a\mathcal{G}({\mathcal{U}}_2-{\mathcal{U}}_3)={\gamma }^a\mathcal{G}\left({\mathcal{U}}_2\right)-{\gamma }^a\mathcal{G}\left({\mathcal{U}}_3\right)$$

(12)

for all ${\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Adding (6), (8), (10), and (12), Equation (1) can be derived. □

3. Classical Approach in Fuzzy Normed Spaces

For the rest of this article, ${\mathbb{F}}^{*},({\mathbb{I}}^{*},{\mathcal{N}}^{\prime })$ and $(Y,{\mathcal{N}}^{\prime })$ represent a linear vector space, fuzzy Banach space, and fuzzy normed space, respectively. The above notations are used in the following given mapping $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$

$$\begin{array}{cc}\hfill & \mathbb{D}\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)=\left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)+{\gamma }^{b+c}\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)\hfill \\ \hfill & +{\gamma }^{a}h({\mathcal{U}}_2-{\mathcal{U}}_3)-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{G}\left({\mathcal{U}}_1\right)-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{G}\left({\mathcal{U}}_2\right)-\left(1-2{\gamma }^a\right)\mathcal{G}\left({\mathcal{U}}_3\right).\hfill \end{array}$$

Theorem 1 discusses the stability results of the proposed functional equations in the fuzzy normed spaces using classical direct method. Theorem 2 discusses the stability results of the proposed functional equations in the fuzzy normed spaces using fixed point technique.

Theorem 1.

Let $\mathcal{E}\in \{-1,1\}$ be fixed and let $\mathbb{K}:{\mathbb{F}}^{*}3\to {\mathbb{I}}^{*}$ be a mapping such that for some $\mathcal{D}$ with ${\displaystyle 0<{\left(\frac{\mathcal{D}}{2}\right)}^{\mathcal{E}}<1}$

$${\mathcal{N}}^{\prime }\left(\mathbb{K}\left({\mathcal{B}}^{\mathcal{E}}{\mathcal{U}}_1,{\mathcal{B}}^{\mathcal{E}}{\mathcal{U}}_1,{\mathcal{B}}^{\mathcal{E}}{\mathcal{U}}_1,\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left({\mathcal{D}}^{\mathcal{E}}\mathbb{K}\left({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1\right),\mathcal{R}\right)$$

(13)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$, $\mathcal{D}>0$, and

$$\underset{k\to \infty }{lim}{\mathcal{N}}^{\prime }\left(\mathbb{K}\left({\mathcal{B}}^{\mathcal{E}k}{\mathcal{U}}_1,{\mathcal{B}}^{\mathcal{E}k}{\mathcal{U}}_2,{\mathcal{B}}^{\mathcal{E}k}{\mathcal{U}}_3\right),{\mathcal{B}}^{\mathcal{E}k}\mathcal{R}\right)=1$$

(14)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Let a function $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfy the inequality

$$\mathcal{N}\left(D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)$$

(15)

for all $\mathcal{R}>0$ and all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Then, the limit

$$\mathcal{A}\left({\mathcal{U}}_1\right)=N-\underset{k\to \infty }{lim}\frac{\mathcal{G}\left({\mathcal{B}}^{\mathcal{E}k}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{\mathcal{E}k}}$$

(16)

exists for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and the mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ is a unique additive mapping such that

$$\mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),2\mathcal{R}|\mathcal{B}-\mathcal{D}|\right)$$

(17)

with $\mathcal{B}=\left(1+{\gamma }^{b+c}\right)$.

Proof. 

Replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1)$ in (15),

$$\mathcal{N}\left(2\mathcal{G}\left(\mathcal{B}{\mathcal{U}}_1\right)-2\mathcal{B}\mathcal{G}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)$$

(18)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Replacing ${\mathcal{U}}_1$ by ${\mathcal{B}}^k{\mathcal{U}}_1$ in (18), the following result is obtained:

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+1}{\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right),\frac{r}{2\mathcal{B}}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1),\mathcal{R}\right)$$

(19)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Using (13), $\left({\mathcal{A}}_3\right)$ in (19),

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+1}{\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right),\frac{r}{2\mathcal{B}}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{\mathcal{R}}{{\mathcal{D}}^k}\right)$$

(20)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. It is easy to verify from (20) that

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},\frac{\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^k}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{\mathcal{R}}{{\mathcal{D}}^k}\right)$$

(21)

holds for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Replacing $\mathcal{R}$ by ${\mathcal{D}}^k\mathcal{R}$ in (21),

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},\frac{{\mathcal{D}}^k\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^k}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)$$

(22)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. It is obvious that

$$\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\mathcal{G}\left({\mathcal{U}}_1\right)=\sum _{i=0}^{k-1}\left[\frac{\mathcal{G}\left({\mathcal{B}}^{i+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(i+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^i{\mathcal{U}}_1\right)}{{\mathcal{B}}^i}\right]$$

(23)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$. From Equations (22) and (23),

$$\begin{array}{c}\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\mathcal{G}\left({\mathcal{U}}_1\right),\sum _{i=0}^{k-1}\frac{{\mathcal{D}}^i\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^i}\right)\hfill \\ \ge min\bigcup _{i=0}^{k-1}\left\{\frac{\mathcal{G}\left({\mathcal{B}}^{i+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(i+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^i{\mathcal{U}}_1\right)}{{\mathcal{B}}^i},\frac{{\mathcal{D}}^i\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^i}\right\}\hfill \\ \ge min\bigcup _{i=0}^{k-1}\left\{{\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)\right\}\hfill \\ \ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)\hfill \end{array}$$

(24)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Replacing ${\mathcal{U}}_1$ by ${\mathcal{B}}^m{\mathcal{U}}_1$ in (24) and using (13), $\left({\mathcal{A}}_3\right)$,

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+m}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+m)}}-\frac{\mathcal{G}\left({\mathcal{B}}^m{\mathcal{U}}_1\right)}{{\mathcal{B}}^m},\sum _{i=0}^{k-1}\frac{{\mathcal{D}}^i\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^{(i+m)}}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{\mathcal{R}}{{\mathcal{D}}^m}\right)$$

(25)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$ and all $m,k\ge 0$. Replacing $\mathcal{R}$ by ${\mathcal{D}}^m\mathcal{R}$ in (25),

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+m}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+m)}}-\frac{\mathcal{G}\left({\mathcal{B}}^m{\mathcal{U}}_1\right)}{{\mathcal{B}}^m},\sum _{i=m}^{m+k-1}\frac{{\mathcal{D}}^i\mathcal{R}}{2\mathcal{B}·{\mathcal{B}}^i}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)$$

(26)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$ and all $m,k\ge 0$. Using $\left({\mathcal{A}}_3\right)$ in (26),

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^{k+m}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+m)}}-\frac{\mathcal{G}\left({\mathcal{B}}^m{\mathcal{U}}_1\right)}{{\mathcal{B}}^m},\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{r}{{\sum }_{i=m}^{m+k-1}\frac{{\mathcal{D}}^i}{2\mathcal{B}·{\mathcal{B}}^i}}\right)$$

(27)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$ and all $m,k\ge 0$. Since $0<\mathcal{D}<\mathcal{B}$ and ${\displaystyle \sum _{i=0}^k{\left(\frac{d}{\mathcal{B}}\right)}^i<\infty }$, the Cauchy criterion for convergence and $\left({\mathcal{A}}_5\right)$ implies that ${\displaystyle \left\{\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}\right\}}$ is a Cauchy sequence in $({\mathbb{I}}^{*},{\mathcal{N}}^{\prime })$. Since $({\mathbb{I}}^{*},{\mathcal{N}}^{\prime })$ is a fuzzy Banach space, this sequence converges to some point $\mathcal{A}\left({\mathcal{U}}_1\right)\in {\mathbb{I}}^{*}$. So, one can define the mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ by

$$\mathcal{A}\left({\mathcal{U}}_1\right)=N-\underset{k\to \infty }{lim}\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}.$ Letting $m=0$ in (27),

$$\mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\mathcal{G}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{\mathcal{R}}{{\sum }_{i=0}^{k-1}\frac{{\mathcal{D}}^i}{2\mathcal{B}·{\mathcal{B}}^i}}\right)$$

(28)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Letting $k\to \infty$ in (28) and using $\left({\mathcal{A}}_6\right)$, the following result is obtained:

$$\mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),2\mathcal{R}(\mathcal{B}-\mathcal{D})\right)$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. To prove that $\mathcal{A}$ satisfies Equation (1), we have to replace $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3)$ in (15), respectively,

$$\mathcal{N}\left(\frac{1}{{\mathcal{B}}^k}D\mathcal{G}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3),{\mathcal{B}}^k\mathcal{R}\right)$$

(29)

for all $\mathcal{R}>0$ and all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Now,

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right.\hfill \\ \hfill & +{\gamma }^{b+c}\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)+{\gamma }^a\mathcal{G}({\mathcal{U}}_2-{\mathcal{U}}_3)\hfill \\ \hfill & \left.-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{G}\left({\mathcal{U}}_1\right)-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{G}\left({\mathcal{U}}_2\right)-\left(1-2{\gamma }^a\right)\mathcal{G}\left({\mathcal{U}}_3\right),\mathcal{R}\right)\hfill \\ \hfill & \ge min\left\{\mathcal{N}\left(\left(\left({\gamma }^a+1\right)\mathcal{A}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)\right)-\left(\left({\gamma }^a+1\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)\right)\right),\frac{\mathcal{R}}{8}\right)\right.,\hfill \\ \hfill & \mathcal{N}\left(\left(\left(1-{\gamma }^a\right)\mathcal{A}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right)-\left(\left(\left(1-{\gamma }^a\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right)\right)\right),\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left({\gamma }^{b+c}\mathcal{A}({\mathcal{U}}_1-{\mathcal{U}}_2)+\left({\gamma }^{b+c}\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_1-{\mathcal{U}}_2)\right)\right),\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left({\gamma }^a\mathcal{A}({\mathcal{U}}_2-{\mathcal{U}}_3)+\left({\gamma }^a\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_2-{\mathcal{U}}_3)\right),\right)\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left(-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{A}\left({\mathcal{U}}_1\right)+\left(-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\mathcal{U}}_1\right)\right)\right),\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left(-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{A}\left({\mathcal{U}}_2\right)+\left(-\left({\gamma }^a+{\gamma }^{b+c}\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_2)\right)\right),\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left(-\left(1-2{\gamma }^a\right)\mathcal{A}\left({\mathcal{U}}_3\right)+\left(-\left(1-2{\gamma }^a\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_3)\right)\right),\frac{\mathcal{R}}{8}\right),\hfill \\ \hfill & \mathcal{N}\left(\left(\left({\gamma }^a+1\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)\right)+\left(\left(1-{\gamma }^a\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right)\right)\right)\right.\hfill \\ \hfill & +\left({\gamma }^{b+c}\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_1-{\mathcal{U}}_2)\right)\right)+\left({\gamma }^a\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_2-{\mathcal{U}}_3)\right)\right)\hfill \\ \hfill & -\left(\left({\gamma }^a+{\gamma }^{b+c}+1\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}\left({\mathcal{B}}^k\left({\mathcal{U}}_1\right)\right)\right)-\left(\left({\gamma }^a+{\gamma }^{b+c}\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_2)\right)\right)\hfill \\ \hfill & -\left.\left.\left(\left(1-2{\gamma }^a\right)\frac{1}{{\mathcal{B}}^k}\mathcal{G}({\mathcal{B}}^k\left({\mathcal{U}}_3)\right)\right),\frac{\mathcal{R}}{8}\right)\right\}\hfill \end{array}$$

(30)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Using (29) and $\left({\mathcal{A}}_5\right)$ in (30), the following result is derived:

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\left({\gamma }^a+1\right)\mathcal{A}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{A}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right.\hfill \\ \hfill & +{\gamma }^{b+c}\mathcal{A}({\mathcal{U}}_1-{\mathcal{U}}_2)+{\gamma }^a\mathcal{A}({\mathcal{U}}_2-{\mathcal{U}}_3)\hfill \\ \hfill & \left.-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{A}\left({\mathcal{U}}_1\right)-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{A}\left({\mathcal{U}}_2\right)-\left(1-2{\gamma }^a\right)\mathcal{A}\left({\mathcal{U}}_3\right),\mathcal{R}\right)\hfill \\ \hfill & \ge min\left\{1,1,1,1,1,1,1{\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3),{\mathcal{B}}^k\mathcal{R}\right)\right\}\hfill \\ \hfill & \ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3),{\mathcal{B}}^k\mathcal{R}\right)\hfill \end{array}$$

(31)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Letting $k\to \infty$ in (31) and using (14),

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\left({\gamma }^a+1\right)\mathcal{A}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{A}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)\right.\hfill \\ \hfill & +{\gamma }^{b+c}\mathcal{A}({\mathcal{U}}_1-{\mathcal{U}}_2)+{\gamma }^a\mathcal{A}({\mathcal{U}}_2-{\mathcal{U}}_3)\hfill \\ \hfill & \left.-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{A}\left({\mathcal{U}}_1\right)-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{A}\left({\mathcal{U}}_2\right)-\left(1-2{\gamma }^a\right)\mathcal{A}\left({\mathcal{U}}_3\right),\mathcal{R}\right)=1\hfill \end{array}$$

(32)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Using $\left({\mathcal{A}}_2\right)$ in the above inequality gives

$$\begin{array}{cc}\hfill & \left({\gamma }^a+1\right)\mathcal{A}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{A}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)+{\gamma }^{b+c}\mathcal{A}({\mathcal{U}}_1-{\mathcal{U}}_2)+{\gamma }^a\mathcal{A}({\mathcal{U}}_2-{\mathcal{U}}_3)\hfill \\ \hfill & =\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{A}\left({\mathcal{U}}_1\right)+\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{A}\left({\mathcal{U}}_2\right)+\left(1-2{\gamma }^a\right)\mathcal{A}\left({\mathcal{U}}_3\right)\hfill \end{array}$$

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Hence, $\mathcal{A}$ satisfies the additive functional Equation (1). In order to prove $\mathcal{A}\left({\mathcal{U}}_1\right)$ is unique, let ${\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right)$ be another additive functional equation satisfying (1) and (17). Hence,

$$\begin{array}{cc}\hfill N(\mathcal{A}\left({\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right),\mathcal{R})& =\mathcal{N}\left(\frac{\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\frac{{\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},\mathcal{R}\right)\hfill \\ \hfill & \ge min\{\mathcal{N}\left(\frac{\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},\frac{\mathcal{R}}{2}\right),\hfill \\ \hfill & \mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\frac{{\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},\frac{\mathcal{R}}{2}\right)\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1),\frac{2\mathcal{R}{\mathcal{B}}^k(\mathcal{B}-\mathcal{D})}{2}\right)\hfill \\ \hfill & \ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{2\mathcal{R}{\mathcal{B}}^k(\mathcal{B}-\mathcal{D})}{2{\mathcal{D}}^k}\right)\hfill \end{array}$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$. Since

$$\underset{k\to \infty }{lim}\frac{2\mathcal{R}{\mathcal{B}}^k(\mathcal{B}-\mathcal{D})}{2{\mathcal{D}}^k}=\infty ,$$

$$\underset{k\to \infty }{lim}{\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\frac{2\mathcal{R}{\mathcal{B}}^k(\mathcal{B}-\mathcal{D})}{2{\mathcal{D}}^k}\right)=1.$$

Thus,

$$N(\mathcal{A}\left({\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right),\mathcal{R})=1$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$, which implies that $\mathcal{A}\left({\mathcal{U}}_1\right)={\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right)$. Therefore, $\mathcal{A}\left({\mathcal{U}}_1\right)$ is unique. □

Corollary 1.

Suppose that a function $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfies the inequality

$$\begin{array}{cc}\hfill N\left(D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)& \ge \left\{\begin{array}{cc}{\mathcal{N}}^{\prime }\left(\Psi ,\mathcal{R}\right),\hfill & \\ {\mathcal{N}}^{\prime }\left(\Psi \sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,\mathcal{R}\right),\hfill & s\ne 1;\hfill \\ {\mathcal{N}}^{\prime }\left(\Psi \prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,\mathcal{R}\right),\hfill & s\ne \frac{1}{3};\hfill \\ {\mathcal{N}}^{\prime }\left(\Psi \left(\prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+\sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right),\mathcal{R}\right),\hfill & s\ne \frac{1}{3};\hfill \end{array}\right.\hfill \end{array}$$

(33)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$, where $\Psi ,s$ are constants with $\Psi >0$, then there exists a unique additive mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ such that

$$\begin{array}{c}\hfill N\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge \left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left(\Psi ,2|\mathcal{B}-1|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(3\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2|\mathcal{B}-{\mathcal{B}}^s|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2|\mathcal{B}-{\mathcal{B}}^{3s}|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(4\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2|\mathcal{B}-{\mathcal{B}}^{3s}|\mathcal{R}\right)}\hfill \end{array}\right.\end{array}$$

(34)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$.

Fixed Point Approach in Fuzzy Normed Spaces

Let us assume that ${\delta }_i$ is a constant such that

$${\delta }_i=\left\{\begin{array}{ccc}\mathcal{B}\hfill & if& i=0,\hfill \\ \frac{1}{\mathcal{B}}\hfill & if& i=1\hfill \end{array}\right.$$

and $\Omega$ is the set such that

$$\Omega =\left\{g|g:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*},g\left(0\right)=0\right\}.$$

Theorem 2.

Let $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ be a mapping for which there exists a function $\mathbb{K}:{\mathbb{F}}^{*3}\to {\mathbb{I}}^{*}$ with the condition

$$\underset{k\to \infty }{lim}{\mathcal{N}}^{\prime }\left(\mathbb{K}\left({\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3\right),{\delta }_i^k\mathcal{R}\right)=1forall{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*},\mathcal{R}>0$$

(35)

and satisfying the functional inequality

$$\mathcal{N}\left(D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)forall{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*},\mathcal{R}>0.$$

(36)

If there exists $L=L\left(i\right)$ such that the function ${\mathcal{U}}_1\to \mathcal{E}\left({\mathcal{U}}_1\right)=\frac{1}{2}\mathbb{K}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}\right)$ has the property

$${\mathcal{N}}^{\prime }\left(L\frac{1}{{\delta }_i}\mathcal{E}\left({\delta }_i{\mathcal{U}}_1\right),\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{R}>0.$$

(37)

Then, there exists a unique additive function $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfying the functional Equation (1) and

$$\mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-Q\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\frac{L^{1-i}}{1-L}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{R}>0,$$

(38)

with $\mathcal{B}=\left(1+{\gamma }^{b+c}\right)$.

Proof. 

Assume d be a general metric on $\Omega ,$ such that

$$d(g,h)=inf\left\{K\in (0,\infty )|\mathcal{N}\left(g\left({\mathcal{U}}_1\right)-h\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(K\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{R}>0\right\}.$$

It is apparent that $(\Omega ,d)$ is complete. Let $T:\Omega \to \Omega$ by ${\displaystyle Tg\left({\mathcal{U}}_1\right)=\frac{1}{{\delta }_i}g\left({\delta }_i{\mathcal{U}}_1\right),}$ for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}.$ For $g,h\in \Omega$, $d(g,h)\le K$

$$\begin{array}{cc}\hfill \Rightarrow & \mathcal{N}\left(g\left({\mathcal{U}}_1\right)-h\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(K\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill \Rightarrow & \mathcal{N}\left(\frac{g\left({\delta }_i{\mathcal{U}}_1\right)}{{\delta }_i}-\frac{h\left({\delta }_i{\mathcal{U}}_1\right)}{{\delta }_i},\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\frac{K}{{\delta }_i}\mathcal{E}\left({\delta }_i{\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill \Rightarrow & \mathcal{N}\left(Tg\left({\mathcal{U}}_1\right)-Th\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(KL\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill \Rightarrow & d\left(Tg\left({\mathcal{U}}_1\right),Th\left({\mathcal{U}}_1\right)\right)\le KL\hfill \\ \hfill \Rightarrow & d\left(Tg,Th\right)\le Ld(g,h)\hfill \end{array}$$

(39)

for all $g,h\in \Omega .$ Therefore, T is a strictly contractive mapping on $\Omega$ with Lipschitz constant $L.$ Replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1)$ in (36),

$$\begin{array}{cc}\hfill & \mathcal{N}\left(2\mathcal{G}\left(\mathcal{B}{\mathcal{U}}_1\right)-2\mathcal{B}\mathcal{G}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)\hfill \end{array}$$

(40)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$, $\mathcal{R}>0.$ Using (F3) in (40),

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\frac{1}{2\mathcal{B}}\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1),\mathcal{R}\right)\hfill \end{array}$$

(41)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$, $\mathcal{R}>0$ with the help of (37) when $i=0$, it follows from (41),

$$\begin{array}{cc}\hfill \Rightarrow & \mathcal{N}\left(\frac{\mathcal{G}\left({\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(L\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill \Rightarrow & d(T\mathcal{G},\mathcal{G})\le L=L^1=L^{1-i}\hfill \end{array}$$

(42)

Replacing ${\mathcal{U}}_1$ by $\frac{{\mathcal{U}}_1}{\mathcal{B}}$ in (40),

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{B}\mathcal{G}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}}\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\frac{1}{2}\mathbb{K}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}\right),\mathcal{R}\right)\hfill \end{array}$$

(43)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$, $\mathcal{R}>0$ with the help of (37) when $i=1$, it follows from (43) that

$$\begin{array}{cc}\hfill \Rightarrow & \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{B}\mathcal{G}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}}\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill \Rightarrow & d(\mathcal{G},T\mathcal{G})\le 1=L^0=L^{1-i}\hfill \end{array}$$

(44)

Then, from (42) and (44), it can be concluded that

$$\begin{array}{cc}\hfill & d(\mathcal{G},T\mathcal{G})\le L^{1-i}<\infty \hfill \end{array}$$

From the fixed point alternative in both cases, it follows that a fixed point $\mathcal{A}$ of T exists in $\Omega$ such that

$$\begin{array}{c}\hfill \mathcal{A}\left({\mathcal{U}}_1\right)=N-\underset{k\to \infty }{\overset{}{lim}}\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k},forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{R}>0.\end{array}$$

(45)

Replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\delta }_i{\mathcal{U}}_1,{\delta }_i{\mathcal{U}}_2,{\delta }_i{\mathcal{U}}_3)$ in (36), the following equation is obtained:

$$\mathcal{N}\left(\frac{1}{{\delta }_i^k}D\mathcal{G}({\delta }_i{\mathcal{U}}_1,{\delta }_i{\mathcal{U}}_2,{\delta }_i{\mathcal{U}}_3),\mathcal{R}\right)\ge N^{\prime }\left(\mathbb{K}({\delta }_i{\mathcal{U}}_1,{\delta }_i{\mathcal{U}}_2,{\delta }_i{\mathcal{U}}_3),{\delta }_i^k\mathcal{R}\right)$$

(46)

for all $\mathcal{R}>0$ and all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Using the same procedure as in Theorem 13, it can be proved that the function $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfies the FE (1).

Using fixed point alternative, since $\mathcal{A}$ is a unique fixed point of T in the set

$$\Delta =\left\{\mathcal{G}\in \Omega \left|d\right(\mathcal{G},\mathcal{A})<\infty \right\},$$

C is a unique function such that

$$\begin{array}{cc}\hfill & \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(K\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \end{array}$$

(47)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$, $\mathcal{R}>0$ and $K>0.$ The following equation is obtained using the fixed point alternative again:

$$\begin{array}{cc}\hfill & d(\mathcal{G},\mathcal{A})\le \frac{1}{1-L}d(\mathcal{G},T\mathcal{G})\hfill \\ \hfill \Rightarrow & d(\mathcal{G},\mathcal{A})\le \frac{L^{1-i}}{1-L}\hfill \\ \hfill \Rightarrow & \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\frac{L^{1-i}}{1-L}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),\hfill \end{array}$$

(48)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and $\mathcal{R}>0$. □

Corollary 2.

Let a function $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfy the inequality

$$\begin{array}{cc}\hfill N\left(D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3),\mathcal{R}\right)& \ge \left\{\begin{array}{cc}{\mathcal{N}}^{\prime }\left(\Psi ,\mathcal{R}\right),\hfill & \\ {\mathcal{N}}^{\prime }\left(\Psi \sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,\mathcal{R}\right),\hfill & s\ne 1;\hfill \\ {\mathcal{N}}^{\prime }\left(\Psi \prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,\mathcal{R}\right),\hfill & s\ne \frac{1}{3};\hfill \\ {\mathcal{N}}^{\prime }\left(\Psi \left(\prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+\sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right),\mathcal{R}\right),\hfill & s\ne \frac{1}{3};\hfill \end{array}\right.\hfill \end{array}$$

(49)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$, where $\Psi ,s$ are constants with $\Psi >0$. Then, there exists a unique additive mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ such that

$$\begin{array}{c}\hfill N\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\ge \left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left(\Psi ,2|\mathcal{B}-1|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(3\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2|\mathcal{B}-{\mathcal{B}}^s|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2|\mathcal{B}-{\mathcal{B}}^{3s}|\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(4\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2|\mathcal{B}-{\mathcal{B}}^{3s}|\mathcal{R}\right)}\hfill \end{array}\right.\end{array}$$

(50)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{R}>0$.

Proof. 

Setting

$$\mathbb{K}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)=\left\{\begin{array}{c}{\displaystyle \Psi ,}\hfill \\ {\displaystyle \Psi \sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,}\hfill \\ {\displaystyle \Psi \prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,}\hfill \\ {\displaystyle \Psi \left(\prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+\sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right).}\hfill \end{array}\right.$$

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Then,

$$\begin{array}{cc}\hfill & {\mathcal{N}}^{\prime }\left(\mathbb{K}({\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3),{\delta }_i^k\mathcal{R}\right)\hfill \\ \hfill & =\left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left(\Psi ,{\delta }_i^k\mathcal{R}\right)}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\Psi \sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,{\delta }_i^{(1-s)k}\mathcal{R}\right)}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\Psi \prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s,{\delta }_i^{(1-3s)k}\mathcal{R}\right)}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\Psi \left(\prod _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+\sum _{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right),{\delta }_i^{(1-3s)k}\mathcal{R}\right)}\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left\{\begin{array}{c}\to 1\mathrm{as}k\to \infty ,\hfill \\ \to 1\mathrm{as}k\to \infty ,\hfill \\ \to 1\mathrm{as}k\to \infty ,\hfill \\ \to 1\mathrm{as}k\to \infty .\hfill \end{array}\right.\hfill \end{array}$$

Thus, (35) holds. However, $\mathcal{E}\left({\mathcal{U}}_1\right)=\frac{1}{2}\mathbb{K}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}\right)$ has the property

$${\mathcal{N}}^{\prime }\left(L\frac{1}{{\delta }_i}\mathcal{E}\left({\delta }_i{\mathcal{U}}_1\right),\mathcal{R}\right)\ge {\mathcal{N}}^{\prime }\left(\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{R}>0.$$

Hence,

$$\begin{array}{c}\hfill {\mathcal{N}}^{\prime }\left(\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\mathbb{K}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}\right),2\mathcal{R}\right)=\left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left(\Psi ,2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{3\Psi }{{\mathcal{B}}^s}\left|\right|{\mathcal{U}}_1{\left|\right|}^s,2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{\Psi }{{\mathcal{B}}^{3s}}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{4\Psi }{{\mathcal{B}}^{3s}}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2\mathcal{R}\right).}\hfill \end{array}\right.\end{array}$$

Now,

$$\begin{array}{cc}\hfill {\mathcal{N}}^{\prime }\left(\frac{1}{{\delta }_i}\mathcal{E}\left({\delta }_i{\mathcal{U}}_1\right),\mathcal{R}\right)=\left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left(\frac{\Psi }{{\delta }_i},2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{\Psi }{{\delta }_i}\left(\frac{n}{{\mathcal{B}}^s}\right)\left|\right|{\delta }_i{\mathcal{U}}_1{\left|\right|}^s,2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{\Psi }{{\delta }_i}\left(\frac{1}{{\mathcal{B}}^{3s}}\right)\left|\right|{\delta }_i{\mathcal{U}}_1{\left|\right|}^{3s},2\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left(\frac{\Psi }{{\delta }_i}\left(\frac{4}{{\mathcal{B}}^{3s}}\right)\left|\right|{\delta }_i{\mathcal{U}}_1{\left|\right|}^{3s},2\mathcal{R}\right)}\hfill \end{array}\right.& =\left\{\begin{array}{c}{\displaystyle {\mathcal{N}}^{\prime }\left({\delta }_i^{-1}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left({\delta }_i^{s-1}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left({\delta }_i^{3s-1}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right),}\hfill \\ {\displaystyle {\mathcal{N}}^{\prime }\left({\delta }_i^{3s-1}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right).}\hfill \end{array}\right.\hfill \end{array}$$

From (38),

Axiom: 1 $L={\mathcal{B}}^{-1}$ for $s=0$ if $i=0$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{{\mathcal{B}}^{-1}}{1-{\mathcal{B}}^{-1}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{\Psi }{2(\mathcal{B}-1)}\left|\right|{\mathcal{U}}_1{\left|\right|}^s,\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2(\mathcal{B}-1)\mathcal{R}\right).\hfill \end{array}$$

Axiom: 2 $L={\mathcal{B}}^3$ for $s=0$ if $i=1$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{1}{1-{\mathcal{B}}^3}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{\Psi }{2(1-\mathcal{B})}\left|\right|{\mathcal{U}}_1{\left|\right|}^s,\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2(1-\mathcal{B})\mathcal{R}\right).\hfill \end{array}$$

Axiom: 3 $L={\mathcal{B}}^{s-1}$ for $s>3$ if $i=0$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{{\mathcal{B}}^{s-1}}{1-{\mathcal{B}}^{s-1}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{3\Psi }{2(\mathcal{B}-{\mathcal{B}}^s)}\left|\right|{\mathcal{U}}_1{\left|\right|}^s,\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(3\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2(\mathcal{B}-{\mathcal{B}}^s)\mathcal{R}\right).\hfill \end{array}$$

Axiom: 4 $L={\mathcal{B}}^{1-s}$ for $s<1$ if $i=1$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{1}{1-{\mathcal{B}}^{1-s}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{3\Psi }{2({\mathcal{B}}^s-\mathcal{B})}\left|\right|{\mathcal{U}}_1{\left|\right|}^s,\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(3\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s,2({\mathcal{B}}^s-\mathcal{B})\mathcal{R}\right).\hfill \end{array}$$

Axiom: 5 $L={\mathcal{B}}^{3s-1}$ for $s>\frac{1}{3}$ if $i=0$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{{\mathcal{B}}^{3s-1}}{1-{\mathcal{B}}^{3s-1}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{\Psi }{2(\mathcal{B}-{\mathcal{B}}^{3s})}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2(\mathcal{B}-{\mathcal{B}}^{3s})\mathcal{R}\right).\hfill \end{array}$$

Axiom: 6 $L={\mathcal{B}}^{1-3s}$ for $s<\frac{1}{3}$ if $i=1$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{1}{1-{\mathcal{B}}^{1-3s}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{\Psi }{2({\mathcal{B}}^{3s}-\mathcal{B})}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2({\mathcal{B}}^{3s}-\mathcal{B})\mathcal{R}\right).\hfill \end{array}$$

Axiom: 7 $L={\mathcal{B}}^{3s-1}$ for $s>\frac{1}{3}$ if $i=0$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{{\mathcal{B}}^{3s-1}}{1-{\mathcal{B}}^{3s-1}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{4\Psi }{2(\mathcal{B}-{\mathcal{B}}^{3s})}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(4\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2(\mathcal{B}-{\mathcal{B}}^{3s})\mathcal{R}\right).\hfill \end{array}$$

Axiom: 8 $L={\mathcal{B}}^{1-3s}$ for $s<\frac{1}{3}$ if $i=1$

$$\begin{array}{cc}\hfill \mathcal{N}\left(\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right),\mathcal{R}\right)& \ge {\mathcal{N}}^{\prime }\left(\frac{1}{1-{\mathcal{B}}^{1-3s}}\mathcal{E}\left({\mathcal{U}}_1\right),\mathcal{R}\right)\hfill \\ \hfill & ={\mathcal{N}}^{\prime }\left(\frac{4\Psi }{2({\mathcal{B}}^{3s}-\mathcal{B})}\left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},\mathcal{R}\right)={\mathcal{N}}^{\prime }\left(4\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s},2({\mathcal{B}}^{3s}-\mathcal{B})\mathcal{R}\right).\hfill \end{array}$$

□

4. Classical Approach in Random Normed Spaces

Let ${\mathbb{F}}^{*}$ be a linear space and $({\mathbb{I}}^{*},\mu ,T)$ be a complete random normed space. A mapping is defined as $D\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ by

$$\begin{array}{cc}\hfill & \mathbb{D}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)=\left({\gamma }^a+1\right)\mathcal{G}\left({\mathcal{U}}_1+{\gamma }^{b+c}{\mathcal{U}}_2\right)+\left(1-{\gamma }^a\right)\mathcal{G}\left({\gamma }^{b+c}{\mathcal{U}}_2+{\mathcal{U}}_3\right)+{\gamma }^{b+c}\mathcal{G}({\mathcal{U}}_1-{\mathcal{U}}_2)\hfill \\ \hfill & +{\gamma }^{a}h({\mathcal{U}}_2-{\mathcal{U}}_3)-\left({\gamma }^a+{\gamma }^{b+c}+1\right)\mathcal{G}\left({\mathcal{U}}_1\right)-\left({\gamma }^a+{\gamma }^{b+c}\right)\mathcal{G}\left({\mathcal{U}}_2\right)-\left(1-2{\gamma }^a\right)\mathcal{G}\left({\mathcal{U}}_3\right)\hfill \end{array}$$

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$.

Theorem 3 discusses the stability results of the proposed functional equations in the random normed spaces using the classical direct method. Theorem 4 discusses the stability results of the proposed functional equations in the random normed spaces using the fixed point technique.

Theorem 3.

Let $j=\pm 1$. Let $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ be an odd mapping for which there exists a function $\eta :{\mathbb{F}}^{*}n\to D^{+}$ with the condition

$$\begin{array}{c}\hfill {\displaystyle \underset{k\to \infty }{lim}{T}_{i=0}^{\infty }\left({\eta }_{{\mathcal{B}}^{(k+i)j}{\mathcal{U}}_1,{\mathcal{B}}^{(k+i)j}{\mathcal{U}}_2,{\mathcal{B}}^{(k+i)j}{\mathcal{U}}_3}\left({\mathcal{B}}^{(k+i+1)j}\mathcal{Q}\right)\right)=1}\\ \hfill {\displaystyle =\underset{k\to \infty }{lim}{\eta }_{{\mathcal{B}}^{kj}{\mathcal{U}}_1,{\mathcal{B}}^{kj}{\mathcal{U}}_2,{\mathcal{B}}^{kj}{\mathcal{U}}_3}\left({\mathcal{B}}^{kj}\mathcal{Q}\right)}\end{array}$$

(51)

such that the functional inequality with $\mathcal{G}\left(0\right)=0$ such that

$${\displaystyle {\mu }_{D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)}\left(\mathcal{Q}\right)\ge {\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3}\left(\mathcal{Q}\right)}$$

(52)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Then, there exists a unique additive mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfying the functional Equation (1) and

$${\displaystyle {\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)\ge T_{i=0}^{\infty }\left({\eta }_{{\mathcal{B}}^{(i+1)j}{\mathcal{U}}_1,{\mathcal{B}}^{(i+1)j}{\mathcal{U}}_1,{\mathcal{B}}^{(i+1)j}{\mathcal{U}}_1}\left({\mathcal{B}}^{(i+1)j}\mathcal{Q}\right)\right)}$$

(53)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. The mapping $\mathcal{A}\left({\mathcal{U}}_1\right)$ is defined by

$${\displaystyle {\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)=\underset{k\to \infty }{lim}{\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^{kj}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{kj}}}\left(\mathcal{Q}\right)}$$

(54)

with $\mathcal{B}=\left(1+{\gamma }^{b+c}\right)$ for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$.

Proof. 

Assume $j=1$. Setting $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)=({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1)$ in (52),

$${\displaystyle {\mu }_{2\mathcal{G}\left({\mathcal{U}}_1\right)-2\mathcal{B}\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)\ge {\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1}\left(\mathcal{Q}\right)}$$

(55)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. It follows from (55) and $\left(RN2\right)$,

$${\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)\ge {\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1}\left(2\mathcal{B}\mathcal{Q}\right)}$$

(56)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Replacing ${\mathcal{U}}_1$ by ${\mathcal{B}}^k{\mathcal{U}}_1$ in (56),

$${\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^{k+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}}\left(\mathcal{Q}\right)\ge {\eta }_{{\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_1}\left(2{\mathcal{B}}^{(k+1)}t\right)}$$

(57)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. It can be observed that

$${\displaystyle \frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\mathcal{G}\left({\mathcal{U}}_1\right)=\sum _{i=0}^{k-1}\frac{\mathcal{G}\left({\mathcal{B}}^{i+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(i+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^i{\mathcal{U}}_1\right)}{{\mathcal{B}}^i}}$$

(58)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$. From Equations (57) and (58),

$$\begin{array}{cc}\hfill {\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)}& {\displaystyle ={\mu }_{{\sum }_{i=0}^{k-1}\frac{\mathcal{G}\left({\mathcal{B}}^{i+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(i+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^i{\mathcal{U}}_1\right)}{{\mathcal{B}}^i}}\left(\mathcal{Q}\right)}\hfill \\ \hfill & {\displaystyle \ge T_{i=0}^{k-1}{\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^{i+1}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(i+1)}}-\frac{\mathcal{G}\left({\mathcal{B}}^i{\mathcal{U}}_1\right)}{{\mathcal{B}}^i}}\left(\mathcal{Q}\right)}\hfill \\ \hfill & {\displaystyle \ge T_{i=0}^{k-1}{\eta }_{{\mathcal{B}}^i{\mathcal{U}}_1,{\mathcal{B}}^i{\mathcal{U}}_1,{\mathcal{B}}^i{\mathcal{U}}_1}\left(2{\mathcal{B}}^{(i+1)}\mathcal{Q}\right)}\hfill \end{array}$$

(59)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. To prove the convergence of the following sequence

$$\left\{\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}\right\},$$

${\mathcal{U}}_1$ is replaced by ${\mathcal{B}}^m{\mathcal{U}}_1$ in (59).

$$\begin{array}{cc}\hfill {\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^{k+m}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{(k+m)}}-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)}& {\displaystyle \ge T_{i=0}^{k-1}{\eta }_{{\mathcal{B}}^{i+m}{\mathcal{U}}_1,{\mathcal{B}}^{i+m}{\mathcal{U}}_1,{\mathcal{B}}^{i+m}{\mathcal{U}}_1}\left(2{\mathcal{B}}^{(i+m+1)}\mathcal{Q}\right)}\hfill \\ \hfill & {\displaystyle =T_{i=m}^{m+n-1}{\eta }_{{\mathcal{B}}^i{\mathcal{U}}_1,{\mathcal{B}}^i{\mathcal{U}}_1,{\mathcal{B}}^i{\mathcal{U}}_1}\left(2{\mathcal{B}}^{(i+1)}{\mathcal{U}}_1\right)}\hfill \\ \hfill & \to 0asm\to \infty \hfill \end{array}$$

(60)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Thus, ${\displaystyle \left\{\frac{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}{{\mathcal{B}}^k}\right\}}$ is a Cauchy sequence. Since ${\mathbb{F}}^{*}$ is complete, there exists a mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$, defined as

$${\displaystyle {\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)=\underset{k\to \infty }{lim}{\mu }_{\frac{\mathcal{G}\left({\mathcal{B}}^{kj}{\mathcal{U}}_1\right)}{{\mathcal{B}}^{kj}}}\left(\mathcal{Q}\right)}$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Letting $m=0$ and $k\to \infty$ in (60), we arrive at (53) for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Now, to show that $\mathcal{A}$ satisfies (1), replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3)$, we have

$$\begin{array}{cc}\hfill {\displaystyle {\mu }_{\frac{1}{{\mathcal{B}}^k}D\mathcal{G}({\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3)}\left(\mathcal{Q}\right)}& {\displaystyle \ge {\eta }_{{\mathcal{B}}^k{\mathcal{U}}_1,{\mathcal{B}}^k{\mathcal{U}}_2,{\mathcal{B}}^k{\mathcal{U}}_3}\left({\mathcal{B}}^{k}t\right)}\hfill \\ \hfill & {\displaystyle =T_{i=m}^{m+k-1}\left({\eta }_{{\mathcal{B}}^{i+1}{\mathcal{U}}_1,{\mathcal{B}}^{i+1}{\mathcal{U}}_1,{\mathcal{B}}^{i+1}{\mathcal{U}}_1}\right)\left({\mathcal{B}}^{(i+m+1)}t\right)}\hfill \end{array}$$

(61)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Taking $k\to \infty$ on both sides, we find that $\mathcal{A}$ satisfies (1) for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$. Therefore, the mapping $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ is additive.

To demonstrate the uniqueness of the additive function $\mathcal{A}$ subject to (54), suppose there exists an additive function ${\mathcal{A}}^{\prime }$ that satisfies (53) and (54). Since $\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)={\mathcal{B}}^k\mathcal{A}\left({\mathcal{U}}_1\right)$ and ${\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)={\mathcal{B}}^k{\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right)$ for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $k\in \mathbb{N}$, it follows from (54) that

$$\begin{array}{cc}\hfill {\displaystyle {\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{U}}_1\right)}\left(2\mathcal{Q}\right)}& {\displaystyle ={\mu }_{\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}\left(2{\mathcal{B}}^k\mathcal{Q}\right)}\hfill \\ \hfill & {\displaystyle ={\mu }_{\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)-\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)+\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}\left(2{\mathcal{B}}^k\mathcal{Q}\right)}\hfill \\ \hfill & \ge T\left({\displaystyle {\mu }_{\mathcal{A}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)-\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}\left({\mathcal{B}}^k\mathcal{Q}\right),{\mu }_{\mathcal{G}\left({\mathcal{B}}^k{\mathcal{U}}_1\right)-{\mathcal{A}}^{\prime }\left({\mathcal{B}}^k{\mathcal{U}}_1\right)}\left({\mathcal{B}}^k\mathcal{Q}\right)}\right)\hfill \\ \hfill & =T\left({\displaystyle T_{i=0}^{\infty }\left({\eta }_{{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1,{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1,{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1}\right)\left(2{\mathcal{B}}^{(i+k+1)}\mathcal{Q}\right),}\right.\hfill \\ \hfill & \left.{\displaystyle T_{i=0}^{\infty }\left({\eta }_{{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1,{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1,{\mathcal{B}}^{(i+k+1)}{\mathcal{U}}_1}\right)\left(2{\mathcal{B}}^{(i+k+1)}\mathcal{Q}\right)}\right)\hfill \\ \hfill & \to 0ask\to \infty \hfill \end{array}$$

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Hence, $\mathcal{A}$ is unique. □

Corollary 3.

Let Ψ and s be nonnegative real numbers. Let an additive function $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfy the inequality

$$\begin{array}{cc}\hfill {\mu }_{D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)}\left(\mathcal{Q}\right)& \ge \left\{\begin{array}{cc}{\eta }_{\Psi }\left(\mathcal{Q}\right),\hfill & \\ {\eta }_{\Psi {\sum }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),\hfill & s\ne 3;\hfill \\ {\eta }_{\Psi {\prod }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),\hfill & s\ne \frac{1}{3};\hfill \\ {\eta }_{\Psi \left({\prod }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+{\sum }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right)}\left(\mathcal{Q}\right),\hfill & s\ne \frac{1}{3};\hfill \end{array}\right.\hfill \end{array}$$

(62)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Then, there exists a unique additive function $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ such that

$${\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)\le \left\{\begin{array}{c}{\displaystyle {\eta }_{\frac{\Psi }{2|\mathcal{B}-1|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{n\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s}{2|\mathcal{B}-{\mathcal{B}}^s|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s}}{2|\mathcal{B}-{\mathcal{B}}^{3s}|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s}}{2}\left(\frac{4}{|\mathcal{B}-{\mathcal{B}}^{3s}|}\right)}\left(\mathcal{Q}\right),}\hfill \end{array}\right.$$

(63)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$.

Fixed Point Approach in Random Normed Spaces

Theorem 4.

Let $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ be a mapping for which there exists a function $\eta :{\mathbb{F}}^{*3}\to D^{+}$ with the condition

$$\underset{k\to \infty }{lim}{\eta }_{{\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3}\left({\delta }_i^{k}t\right)=1,forall{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathcal{U}}_1,\mathcal{Q}>0$$

(64)

and satisfying the functional inequality

$${\mu }_{D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)}\left(\mathcal{Q}\right)\ge {\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3}\left(\mathcal{Q}\right),forall{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathcal{U}}_1,\mathcal{Q}>0.$$

(65)

If there exists $L=L\left(i\right)$ such that the function

$${\mathcal{U}}_1\to \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})={\eta }_{\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}}\left(2\mathcal{Q}\right),$$

has the property

$$\mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\le L\frac{1}{{\delta }_i}\mathcal{E}\left({\delta }_i{\mathcal{U}}_1,\mathcal{Q}\right),forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0.$$

(66)

Then, there exists a unique additive function $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfying the functional Equation (1) and

$${\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\frac{L^{1-i}}{1-L}\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q}),forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0,$$

(67)

with $\mathcal{B}=\left(1+{\gamma }^{b+c}\right)$.

Proof. 

Let d be a general metric on $\Omega ,$ such that

$$d(g,h)=inf\left\{K\in (0,\infty )|{\mu }_{g\left({\mathcal{U}}_1\right)-h\left({\mathcal{U}}_1\right)}\left(K\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q}),{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0\right\}.$$

It is obvious that $(\Omega ,d)$ is complete. Define $T:\Omega \to \Omega$ by ${\displaystyle Tg\left({\mathcal{U}}_1\right)=\frac{1}{{\delta }_i}g\left({\delta }_i{\mathcal{U}}_1\right),}$ for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}.$ Now, for $g,h\in \Omega$, $d(g,h)\le K$

$$\begin{array}{cc}\hfill \Rightarrow & {\mu }_{g\left({\mathcal{U}}_1\right)-h\left({\mathcal{U}}_1\right)}\left(K\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\hfill \\ \hfill \Rightarrow & {\mu }_{\frac{g\left({\delta }_i{\mathcal{U}}_1\right)}{{\delta }_i}-\frac{h\left({\delta }_i{\mathcal{U}}_1\right)}{{\delta }_i}}\left(\frac{K\mathcal{Q}}{{\delta }_i}\right)\ge \mathcal{E}({\delta }_i{\mathcal{U}}_1,\mathcal{Q})\hfill \\ \hfill \Rightarrow & {\mu }_{Tg\left({\mathcal{U}}_1\right)-Th\left({\mathcal{U}}_1\right)}\left(\frac{Kt}{{\delta }_i}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\hfill \\ \hfill \Rightarrow & d\left(Tg\left({\mathcal{U}}_1\right),Th\left({\mathcal{U}}_1\right)\right)\le KL\hfill \\ \hfill \Rightarrow & d\left(Tg,Th\right)\le Ld(g,h)\hfill \end{array}$$

(68)

for all $g,h\in \Omega .$ Therefore, T is a strictly contractive mapping on $\Omega$ with Lipschitz constant $L.$ Replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1)$ in (65),

$$\begin{array}{cc}\hfill & {\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\frac{t}{\mathcal{B}}\right)\ge {\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_1,{\mathcal{U}}_1}\left(2\mathcal{Q}\right)}\hfill \end{array}$$

(69)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0$, with the help of (66) when $i=0$, it follows from (69) that

$$\begin{array}{cc}\hfill \Rightarrow & {\displaystyle {\mu }_{\frac{\mathcal{G}\left({\mathcal{U}}_1\right)}{\mathcal{B}}-\mathcal{G}\left({\mathcal{U}}_1\right)}\left(\frac{t}{{\mathcal{B}}^3}\right)\ge \mathcal{E}({\mathcal{U}}_1,t)}\hfill \\ \hfill \Rightarrow & d(T\mathcal{G},\mathcal{G})\le L=L^{1-0}<\infty .\hfill \end{array}$$

(70)

Replacing ${\mathcal{U}}_1$ by $\frac{{\mathcal{U}}_1}{\mathcal{B}}$ in (69),

$$\begin{array}{cc}\hfill & {\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{B}\mathcal{G}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}}\right)}\left(\mathcal{Q}\right)\ge {\eta }_{\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}},\frac{{\mathcal{U}}_1}{\mathcal{B}}}\left(2\mathcal{Q}\right)\hfill \end{array}$$

(71)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0$ with the help of (66) when $i=1$, it follows from (71),

$$\begin{array}{cc}\hfill \Rightarrow & {\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{B}\mathcal{G}\left(\frac{{\mathcal{U}}_1}{\mathcal{B}}\right)}\left(\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\hfill \\ \hfill \Rightarrow & d(\mathcal{G},T\mathcal{G})\le 1=L^0=L^{1-i}\hfill \end{array}$$

(72)

Then, from (70) and (72), it can be concluded that

$$\begin{array}{cc}\hfill & d(\mathcal{G},T\mathcal{G})\le L^0<\infty \hfill \end{array}$$

It can be deduced from the fixed point alternative in both cases that there is a fixed point $\mathcal{A}$ of T in $\Omega$ such that

$$\begin{array}{c}\hfill {\mu }_{\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)=\underset{k\to \infty }{\overset{}{lim}}\frac{{\mu }_{\mathcal{G}\left({\delta }_i^k{\mathcal{U}}_1\right)}}{{\delta }_i^k}\left(\mathcal{Q}\right),forall{\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0.\end{array}$$

(73)

Replacing $({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)$ by $({\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3)$ in (65),

$${\mu }_{\frac{1}{{\delta }_i^k}D\mathcal{G}({\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3)}\left(\mathcal{Q}\right)\ge {\eta }_{{\delta }_i^k{\mathcal{U}}_1,{\delta }_i^k{\mathcal{U}}_2,{\delta }_i^k{\mathcal{U}}_3}\left({\delta }_i^k\mathcal{Q}\right)$$

(74)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. By following the same procedure as in Theorem 3, it can be proved that $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfies the functional Equation (1).

By the fixed point method, since $\mathcal{A}$ is unique fixed point of T in the set

$$\Delta =\left\{\mathcal{G}\in \Omega \left|d\right(\mathcal{G},\mathcal{A})<\infty \right\},$$

$\mathcal{A}$ is a unique function such that

$$\begin{array}{cc}\hfill & {\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right)}\left(K\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\hfill \end{array}$$

(75)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*},\mathcal{Q}>0$ and $K>0.$ Using the fixed point alternative again, the following result is obtained:

$$\begin{array}{cc}\hfill & d(\mathcal{G},\mathcal{A})\le \frac{1}{1-L}d(\mathcal{G},T\mathcal{G})\hfill \\ \hfill \Rightarrow & d(\mathcal{G},\mathcal{A})\le \frac{L^{1-i}}{1-L}\hfill \\ \hfill \Rightarrow & {\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\frac{L^{1-i}}{1-L}\mathcal{Q}\right)\ge \mathcal{E}({\mathcal{U}}_1,\mathcal{Q})\hfill \end{array}$$

(76)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and $\mathcal{Q}>0$.

Corollary 4.

Let Ψ and s be nonnegative real numbers. Let a additive function $\mathcal{G}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ satisfy the inequality

$$\begin{array}{cc}\hfill {\mu }_{D\mathcal{G}({\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3)}\left(\mathcal{Q}\right)& \ge \left\{\begin{array}{cc}{\eta }_{\Psi }\left(\mathcal{Q}\right),\hfill & \\ {\eta }_{\Psi {\sum }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),\hfill & s\ne 3;\hfill \\ {\eta }_{\Psi {\prod }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),\hfill & s\ne \frac{1}{3};\hfill \\ {\eta }_{\Psi \left({\prod }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^s+{\sum }_{i=1}^3\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right)}\left(\mathcal{Q}\right),\hfill & s\ne \frac{1}{3};\hfill \end{array}\right.\hfill \end{array}$$

(77)

for all ${\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$. Then, there exists a unique additive function $\mathcal{A}:{\mathbb{F}}^{*}\to {\mathbb{I}}^{*}$ such that

$${\mu }_{\mathcal{G}\left({\mathcal{U}}_1\right)-\mathcal{A}\left({\mathcal{U}}_1\right)}\left(\mathcal{Q}\right)\le \left\{\begin{array}{c}{\displaystyle {\eta }_{\frac{\Psi }{2|\mathcal{B}-1|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{n\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^s}{2|\mathcal{B}-{\mathcal{B}}^s|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s}}{2|\mathcal{B}-{\mathcal{B}}^{3s}|}}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\frac{\Psi \left|\right|{\mathcal{U}}_1{\left|\right|}^{3s}}{2}\left(\frac{4}{|\mathcal{B}-{\mathcal{B}}^{3s}|}\right)}\left(\mathcal{Q}\right),}\hfill \end{array}\right.$$

(78)

for all ${\mathcal{U}}_1\in {\mathbb{F}}^{*}$ and all $\mathcal{Q}>0$.

Proof. 

Using

$${\eta }_{{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3}\left(\mathcal{Q}\right)=\left\{\begin{array}{c}{\displaystyle {\eta }_{\Psi }\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\Psi {\displaystyle \sum _{i=1}^3}\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\Psi {\displaystyle \prod _{i=1}^3}\left|\right|{\mathcal{U}}_i{\left|\right|}^s}\left(\mathcal{Q}\right),}\hfill \\ {\displaystyle {\eta }_{\Psi \left({\displaystyle \prod _{i=1}^3}\left|\right|{\mathcal{U}}_i{\left|\right|}^s+{\displaystyle \sum _{i=1}^3}\left|\right|{\mathcal{U}}_i{\left|\right|}^{3s}\right)}\left(\mathcal{Q}\right).}\hfill \end{array}\right.$$

□

5. Conclusions

In this article, a novel system of bilateral symmetry-type additive functional Equation (1) has been introduced. The general solution of the equation is derived and the Hyers–Ulam stability has been analyzed in fuzzy and random normed spaces simultaneously using direct and the fixed-point techniques. A few potential applications of this newly introduced equation and its stability analysis are also explored to help the readers appreciate and understand the significance of the functional equation. In the future, Hyers–Ulam stability for the same Equation (1) can be explored in other normed spaces such as L-Fuzzy normed spaces, Menger Probabilistic normed spaces, and non-Archimedian normed spaces. This is left as an open problem for future research work.