Outstanding Development of the Quadratic Φ(μ₁,μ₂)-Functional Inequatities with 2k-Variables in Fuzzy Banach Space

Ly Van An

Faculty of Mathematics Teacher Education, Tay Ninh University, Tay Ninh, Vietnam.

DOI: 10.4236/oalib.1110592                

Abstract

In this paper, I work on expanding the Quadratic Φ(μ₁,μ₂)-function inequalities by relying on the general quadratic functional equation with 2k-variables on the fuzzy Banach space. That’s the main result of this.

Keywords

Generalized Quadratic Type Φ(μ₁, μ₂)-Functional Inequality, Generalized Quadratic Type Functional Equations, Fuzzy Banach Space, Fuzzy Normed Vector Spaces

1. Introduction

Let $X$ and $Y$ are fuzzy normed spaces on the same field $\mathbb{K}$ , and $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping. I use the notation N are the norm on $X$ and on $Y$ respectively. In this paper, I study the relationship between Quadratic-type functional equations and Quadratic $\varphi \left({\mu }_1,{\mu }_2\right)$ -function inequalities when $\left(X\mathrm{,}N\right)$ is a fuzzy normed space and $\left(Y\mathrm{,}N\right)$ is a fuzzy Banach space.

In fact, when $X$ is a fuzzy normed space and $Y$ is a fuzzy Banach space we solve and prove the Hyers-Ulam stability of the following relationship between quadratic $\varphi \left({\mu }_1,{\mu }_2\right)$ -function inequalities and quadratic-type functional equations:

$\begin{array}{l}N\left(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t\right)\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (1)

based on following Generalized Quadratic functional equations with 2k-variable

$f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)=2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)+2{\displaystyle \underset{k=1}{\overset{k}{\sum }}}f(\; y\; i\; )$

$A_0=\left\{h:ℝ\to ℝ:g\left({\mu }_1,{\mu }_2\right)=\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}<1,{\mu }_1,{\mu }_2\in ℝ\right\}$ .

Note that: With k is a positive integer and $h\in A_0$ .

The study of the functional equation stability originated from a question of S.M. Ulam [1] , concerning the stability of group homomorphisms. Let $\left(\mathbb{G}\mathrm{,}\ast \right)$ be a group and let $\left({\mathbb{G}}^{\prime }\mathrm{,}\circ \mathrm{,}d\right)$ be a metric group with metric $d\left(\cdot \mathrm{,}\cdot \right)$ . Geven $\epsilon >0$ , does there exist a $\delta >0$ such that if $f\mathrm{<mo>\; :\; </mo>}\mathbb{G}\to {\mathbb{G}}^{\prime }$ satisfy the condition $d\left(f\left(x\ast y\right)\mathrm{,}f\left(x\right)\circ f\left(y\right)\right)<\delta$ , for all $x\mathrm{,}y\in \mathbb{G}$ then there is a homomorphism $h\mathrm{<mo>\; :\; </mo>}\mathbb{G}\to {\mathbb{G}}^{\prime }$ with $d\left(f\left(x\right),h\left(x\right)\right)<\epsilon$ , for all $x\in \mathbb{G}$ , if the answer, is affirmative, we would say that equation of homomophism $h\left(x\ast y\right)=h\left(y\right)\circ h\left(y\right)$ is stable. The concept of stability for a functional equation arises when we replace a functional equation with an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how the solutions of the inequality differ from those of the given function equation.

Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Th.M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvrut [5] by replacing the unbounded Cauchy difference with a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensive.

Through the process of studying the works of mathematicians see ( [6] [7] [8] [9] [10] [11] ) in 2020, I set up a general quadratic equation with 2k-variables on the space Non-Archimedean Banach.

$f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{k=1}{\overset{k}{\sum }}}f\left(y_i\right)$ (2)

Next in 2020, I build quadratic inequalities on the application of groups and rings,

${\Vert f\left({\displaystyle \underset{j=1}{\overset{n}{\sum }}}{x}_j+\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\sum }}}{x}_{n+j}\right)+f\left({\displaystyle \underset{j=1}{\overset{n}{\sum }}}{x}_j-\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\sum }}}{x}_{n+j}\right)-2{\displaystyle \underset{j=1}{\overset{n}{\sum }}}f\left(x_j\right)-2{\displaystyle \underset{j=1}{\overset{n}{\sum }}}f\left(\frac{x_{n+j}}{n}\right)\Vert }_{\mathbb{Y}}\le \epsilon \mathrm{,}$ (3)

for all $\epsilon \ge 0$ and

${\Vert f\left({\displaystyle \underset{j=1}{\overset{n}{\prod }}}{x}_j+\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\prod }}}{x}_{n+j}\right)+f\left({\displaystyle \underset{j=1}{\overset{n}{\prod }}}{x}_j-\frac{1}{n}{\displaystyle \underset{j=1}{\overset{n}{\prod }}}{x}_{n+j}\right)-2{\displaystyle \underset{j=1}{\overset{n}{\prod }}}f\left(x_j\right)-2{\displaystyle \underset{j=1}{\overset{n}{\prod }}}f\left(\frac{x_{n+j}}{n}\right)\Vert }_{\mathbb{Y}}\le \delta \mathrm{,}$ (4)

for all $\delta \ge 0$ .

Next in 2021, Ly Van An construct the quadratic inequality functional inequalities in non-Archimedean Banach spaces and Banach spaces,

$\begin{array}{l}{\Vert F\left(\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+F\left(\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}-{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(\frac{x_{k+j}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(x_j\right)\Vert }_{X_2}\\ \le \Vert F\left(\frac{1}{k^2}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}+\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+F\left(\frac{1}{k^2}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}-\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-{\frac{2}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(\frac{x_{k+j}}{k}\right)-\frac{2}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(x_j\right)\Vert }_{X_2}\end{array}$ (5)

and

$\begin{array}{l}{\Vert F\left(\frac{1}{k^2}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}+\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+F\left(\frac{1}{k^2}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}-\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-\frac{2}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(\frac{x_{k+j}}{k}\right)-\frac{2}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(x_j\right)\Vert }_{X_2}\\ \le {\Vert F\left(\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+F\left(\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_{k+j}-{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(\frac{x_{k+1}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}F\left(x_j\right)\Vert }_{X_2}\mathrm{,}\end{array}$ (6)

Continuing into 2021, Ly Van An construct the quadratic inequality on γ-homogeneous complex Banach space,

$\begin{array}{l}{\Vert f\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+f\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k}-{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(\frac{x_{k+j}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)\Vert }_Y\\ \le {\Vert \beta \left(kf\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k^2}+\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+kf\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k^2}-\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(\frac{x_{k+j}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)\right)\Vert }_Y\end{array}$ (7)

and

$\begin{array}{l}{\Vert kf\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k^2}+\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+kf\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k^2}-\frac{1}{k}{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(\frac{x_{k+j}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)\Vert }_Y\\ \le {\Vert \beta \left(f\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)+f\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_{k+j}}{k}-{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(\frac{x_{k+j}}{k}\right)-2{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)\right)\Vert }_Y\end{array}$ (8)

Next in 2023, Ly Van An generalized stability of functional inequalities with 3k-variables associated for Jordan-von Neumann-type additive functional equation,

${\Vert {\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(y_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(z_j\right)\Vert }_Y\le {\Vert 2kf\left(\frac{{\displaystyle {\sum }_{j=1}^k{x}_j}+{\displaystyle {\sum }_{j=1}^k{y}_j}+{\displaystyle {\sum }_{j=1}^k{z}_j}}{2k}\right)\Vert }_Y\mathrm{,}$ (9)

and

${\Vert {\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(y_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(z_j\right)\Vert }_Y\le {\Vert f\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}{x}_j+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{y}_j+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{z}_j\right)\Vert }_Y\mathrm{,}$ (10)

final

${\Vert {\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(y_j\right)+2k{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(z_j\right)\Vert }_Y\le {\Vert 2kf\left(\frac{{\displaystyle {\sum }_{j=1}^k{x}_j}+{\displaystyle {\sum }_{j=1}^k{y}_j}}{2k}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{z}_j\right)\Vert }_Y\mathrm{.}$ (11)

Continuing into 2023, Ly Van An construct the broadly derivation on fuzzy Banach algebra involving functional equations and general Cauchy-Jensen functional inequalities,

$\Vert {\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(x_j\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}f\left(y_j\right)+f\left(2k{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{z}_j\right)\Vert \le \Vert 2kf\left({\displaystyle \underset{j=1}{\overset{k}{\sum }}}\frac{x_j+y_j}{2k}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{z}_j\right)\Vert$ (12)

The paper is organized as followings:

In section preliminary, we remind some basic notations in [12] - [18] such as Fuzzy normed spaces, Extended metric space theorem and solutions of the Jensen function equation.

Section 3: Setting up quadratic $\varphi \left({\mu }_1\mathrm{,}{\mu }_2\right)$ -function inequalities (1) based on quadratic Equation (2).

3.1: Condition for existence of solution of (1).

3.2: Establishing a solution for the quadratic $h\left({\mu }_1\mathrm{,}{\mu }_2\right)$ -function inequality (1). So that we solve and proved the Hyers-Ulam type stability for functional Equation (1) i.e. the functional equations with 2k-variables. Under suitable assumptions on spaces $X$ and $Y$ , we will prove that the mappings satisfying the functional Equations (1).

Thus, the results in this paper are generalization of those in [19] - [65] .

2. Preliminaries

2.1. Fuzzy Normed Spaces

Let X be a real vector space. Afunction $N\mathrm{<mo>\; :\; </mo>}X\times R\to \left[\mathrm{0,1}\right]$ is called a fuzzy norm on X if for all $x\mathrm{,}y\in X$ and all $s\mathrm{,}t\in ℝ$ ,

1) (N1) $N\left(x\mathrm{,}t\right)=0$ for $t\le 0$ ;

2) (N2) $x=0$ if and only if $N\left(x\mathrm{,}t\right)=1$ for all $t>0$ ;

3) (N3) $N\left(cx\mathrm{,}t\right)=N\left(x\mathrm{,}\frac{t}{\left|c\right|}\right)$ if $c\ne 0$ ;

4) (N4) $N\left(x+y,s+t\right)\ge \min \left\{N\left(x,s\right),N\left(y,t\right)\right\}$ ;

5) (N5) $N\left(x\mathrm{,}\cdot \right)$ is a non-decreasing function of $ℝ$ and ${\lim }_{t\to \infty }N\left(x\mathrm{,}t\right)=1$ ;

6) (N6) for $x\ne 0$ , $N\left(x\mathrm{,}\cdot \right)$ is continuous on $ℝ$ .

The pair $\left(X\mathrm{,}N\right)$ is called a fuzzy normed vector space:

1) Let $\left(X\mathrm{,}N\right)$ be a fuzzy normed vector space. A sequence $\left\{x_n\right\}$ in X is said to be convergent or converge if there exists an $x\in X$ such that ${\lim }_{n\to \infty }N\left(x_n-x\mathrm{,}t\right)=1$ for all $t>0$ . In this case, x is called the limit of the sequence $\left\{x_n\right\}$ and we denote it by $N-{\lim }_{n\to \infty }{x}_n=x$ .

2) Let $\left(X\mathrm{,}N\right)$ be a fuzzy normed vector space. A sequence $\left\{x_n\right\}$ in X is called Cauchy if for each $\epsilon >0$ and each $t>0$ there exists an $n_0\in N$ such that for all $n=n_0$ and all $p>0$ , we have $N\left(x_{n+p}-x_n,t\right)>1-\epsilon$ .

It is well-known that every convergent sequence in a fuzzy normedvector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping $f\mathrm{<mo>\; :\; </mo>}X\to Y$ between fuzzy normed vector spaces X and Y is continuous at a point $x_0\in X$ if for each sequence $\left\{x_n\right\}$ converging to $x_0$ in X, then the sequence $\left\{f\left(x_n\right)\right\}$ converges to $f\left(x_0\right)$ . If $f\mathrm{<mo>\; :\; </mo>}X\to Y$ is continuous at each $x\in X$ , then $f\mathrm{<mo>\; :\; </mo>}X\to Y$ is said to be continuous on X.

Let X be an algebra and $\left(X\mathrm{,}N\right)$ a fuzzy normed space.

1) The fuzzy normed space $\left(X\mathrm{,}N\right)$ is called a fuzzy normed algebra if $N\left(xy\mathrm{,}st\right)\ge N\left(x\mathrm{,}s\right)\cdot N\left(y\mathrm{,}t\right)$ , for all $x\mathrm{,}y\in X$ and all positive real numbers s and t.

2) A complete fuzzy normed algebra is called a fuzzy Banach algebra.

Let $\left(X\mathrm{,}{N}_X\right)$ and $\left(Y\mathrm{,}N\right)$ be fuzzy normed algebras. Then a multiplicative $ℝ$ -linear mapping $H\mathrm{<mo>\; :\; </mo>}\left(X\mathrm{,}{N}_X\right)\to \left(Y\mathrm{,}N\right)$ is called a fuzzy algebra homomorphism. Example:

Let $\left(X\mathrm{,}\Vert \cdot \Vert \right)$ be a normed algebra. Let $N\left(x,t\right)=\{\begin{array}{ll}\frac{t}{t+\Vert x\Vert }\hfill & t>0\hfill \\ 0\hfill & t\le 0x\in X\hfill \end{array}$ . Then $N\left(x\mathrm{,}t\right)$ is a fuzzy norm on X and $\left(X\mathrm{,}N\left(x\mathrm{,}t\right)\right)$ is a fuzzy normed algebra. Let $\left(X\mathrm{,}{N}_X\right)$ and $\left(Y\mathrm{,}N\right)$ be fuzzy normed algebras. Then a multiplicative $ℝ$ -linear mapping $H\mathrm{<mo>\; :\; </mo>}\left(X\mathrm{,}{N}_X\right)\to \left(Y\mathrm{,}N\right)$ is called a fuzzy algebra homomorphism.

2.2. Extended Metric Space Theorem

Theorem 1. Let $\left(X\mathrm{,}d\right)$ be a complete generalized metric space and let $J\mathrm{<mo>\; :\; </mo>}X\to X$ be a strictly contractive mapping with Lipschitz constant $L<1$ . Then for each given element $x\in X$ , either $d\left(J^n\mathrm{,}{J}^{n+1}\right)=\infty$ , for all nonnegative integers n or there exists a positive integer $n_0$ such that

1) $d\left(J^n\mathrm{,}{J}^{n+1}\right)<\infty$ , $\forall n\ge n_0$ ;

2) The sequence $\left\{J^{n}x\right\}$ converges to a fixed point $y^{\mathrm{<mo>\; *\; </mo>}}$ of J;

3) $y^{\mathrm{<mo>\; *\; </mo>}}$ is the unique fixed point of J in the set $Y=\left\{y\in X|d\left(J^n,J^{n+1}\right)<\infty \right\}$ ;

4) $d\left(y\mathrm{,}{y}^{\mathrm{<mo>\; *\; </mo>}}\right)\le \frac{1}{1-l}d\left(y\mathrm{,}Jy\right)$ $\forall y\in Y$ .

2.3. Solutions of the Equation

The functional equation $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$ is called the Qquadratic equation. In particular, every solution of the quadratic equation is said to be a quadratic mapping.

2.4. Solutions of the Inequalities

The solution of the quadratic function inequalities is called the quadratic mapping.

3. Setting up Quadratic $\left({\mu }_1,{\mu }_2\right)$ -Function Inequalities (1) Based on Quadratic Equation (2)

3.1. Condition for Existence of Solution of (1)

In this section, assume that $X$ and $Y$ be a fuzzy normed vector spaces Under this setting, we can show that the mappings satisfying (1) is quadratic and $h\in A$ .

Lemma 2. Suppose that $\left(Y\mathrm{,}N\right)$ be a fuzzy normed vector space and let $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping and it satisfies the functional inequality

$\begin{array}{l}N\left(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t\right)\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (13)

For all $x_i\mathrm{,}{y}_i\in X\mathrm{,}i=1\to k$ and all $t>0$ then f is quadratic.

Proof. I replacing $\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)$ by $\left(\mathrm{0,}\cdots \mathrm{,0,0,}\cdots \mathrm{,0}\right)$ in (13), we have

$N\left(-3k{\mu }_{1}f\left(0\right)\mathrm{,}t\right)\ge N\left(\mathrm{0,}t\right)=1$ (14)

Thus $f\left(0\right)=0$ .

Next I replacing $\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)$ by $\left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)$ in (13), we have

$1\le N\left({\mu }_1\left(f\left(2kx\right)-4kf\left(x\right)\mathrm{,}t\right)\right)$ (15)

So

$f\left(2kx\right)=4kf\left(x\right)$ (16)

For all $x\in X$ .

Now I consider $G\mathrm{<mo>\; :\; </mo>}X\to Y$ . That

$\begin{array}{l}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right)\\ =2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\end{array}$ . (17)

It follows from (13) and (14)

$\begin{array}{l}N\left(\frac{1}{2k}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),t\right)\\ \le \min \left(N\left({\mu }_{1}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),t\right),N\left({\mu }_{2}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),t\right)\right)\end{array}$ . (18)

Next I put $v=2t$ (18) I have

$\begin{array}{l}N\left(G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),v\right)\\ \le \min \left(N\left({\mu }_{1}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),\frac{v}{2}\right),N\left({\mu }_{2}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),\frac{v}{2}\right)\right)\\ =\min \left(N\left(\frac{1}{2k}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),\frac{v}{4k\left|{\mu }_1\right|}\right),N\left(\frac{1}{2k}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),\frac{v}{4k\left|{\mu }_2\right|}\right)\right)\\ \le N\left(G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right),\frac{1}{4k}h\left({\mu }_1,{\mu }_2\right)v\right)\\ \le N\left(G\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)\mathrm{,}h\left({\mu }_1\mathrm{,}{\mu }_2\right)v\right)\end{array}$ (19)

for all $v>0$ . By (N₅) and (N₆) I have

$\begin{array}{l}G\left(x_1,\cdots ,x_k,y_1,\cdots ,y_k\right)\\ =f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)=2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)+2{\displaystyle \underset{k=1}{\overset{k}{\sum }}}f\left(y_i\right)\end{array}$ (20)

for all $x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\in X$ , since $h\left({\mu }_1\mathrm{,}{\mu }_2\right)\in A_0$ .

Hence f is quadratic mapping as we expected. □

3.2. Establishing a Solution for the Quadratic $h\left({\mu }_1,{\mu }_2\right)$ -Function Inequality (1)

In this section, assume that $\left(X\mathrm{,}N\right)$ is a fuzzy normed space and $\left(Y\mathrm{,}N\right)$ is a fuzzy Banach space. Under this setting, we can show that the mappings satisfying (1) is quadratic and $h\in A_0$ .

Theorem 3. Let $\psi \mathrm{<mo>\; :\; </mo>}{X}^{2k}\to \left[\mathrm{0,}\infty \right)$ be a function such that there exists an $L<\frac{1}{2k}$ ,

$\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)\le 4kL\psi \left(\frac{x_1}{2k}\mathrm{,}\cdots \mathrm{,}\frac{x_1}{2k}\mathrm{,}\frac{y_1}{2k}\mathrm{,}\cdots \mathrm{,}\frac{y_k}{2k}\right)$ (21)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ .

Let $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping satisfying

$\begin{array}{l}\min (N(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)\\ -{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t),\frac{t}{t+\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)})\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (22)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ . Then

$A\left(x\right)=N-\underset{n\to \infty }{\lim }\frac{1}{{\left(4k\right)}^n}f\left({\left(2k\right)}^{n}x\right)$ (23)

exists each $x\in X$ and defines a quadratic mapping $A\mathrm{<mo>\; :\; </mo>}X\to Y$ such that

$N\left(f\left(x\right)-A\left(x\right)\mathrm{,}t\right)\ge \frac{4k\left|{\mu }_1\right|\left(1-L\right)t}{4k\left|{\mu }_1\right|\left(1-L\right)t+\psi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}$ (24)

for all $x\in X$ and $t>0$ .

Proof. I replacing $\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)$ by $\left(\mathrm{0,}\cdots \mathrm{,0,0,}\cdots \mathrm{,0}\right)$ in (22), I have

$N\left(-3k{\mu }_{1}f\left(0\right)\mathrm{,}t\right)\ge \frac{t}{t+\phi \left(\mathrm{0,}\cdots \mathrm{,0,0,}\cdots \mathrm{,0}\right)}=1$ (25)

Thus $f\left(0\right)=0$ .

Next I replacing $\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)$ by $\left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)$ in (22), we get

$\begin{array}{c}\frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\le N\left({\mu }_1\left(f\left(2kx\right)-4kf\left(x\right)\right)\mathrm{,}t\right)\\ \le N\left(f\left(x\right)-\frac{1}{4k}f\left(2kx\right)\mathrm{,}\frac{t}{4k\left|{\mu }_1\right|}\right)\end{array}$ (26)

for all $x\in X$ . Now we consider the set $\mathbb{M}\mathrm{<mo>\; :\; </mo>}=\left\{h\mathrm{<mo>\; :\; </mo>}X\to Y\right\}$ , and introduce the generalized metric on $\mathbb{M}$ as follows:

$\begin{array}{l}d\left(g,h\right):=\inf \{\beta \in {ℝ}_{+}:N\left(g\left(x\right)-h\left(x\right),\beta t\right)\begin{array}{c}\\ \end{array}\\ \ge \frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\mathrm{,}\forall x\in X\mathrm{,}\forall t>0\}\mathrm{,}\end{array}$ (27)

where, as usual, $\inf \varphi =+\infty$ . That has been proven by mathematicians $\left(\mathbb{M}\mathrm{,}d\right)$ is complete (see [47] ).

Now we cosider the linear mapping $T\mathrm{<mo>\; :\; </mo>}\mathbb{M}\to \mathbb{M}$ such that $Tg\left(x\right):=\frac{1}{4k}g\left(2kx\right)$ , for all $x\in X$ . Let $g\mathrm{,}h\in \mathbb{M}$ be given such that $d\left(g\mathrm{,}h\right)=\epsilon$ then $N\left(g\left(x\right)-h\left(x\right)\mathrm{,}\epsilon t\right)\ge \frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\mathrm{,}\forall x\in X\mathrm{,}\forall t>0.$ Hence

$\begin{array}{c}N\left(g\left(x\right)-h\left(x\right)\mathrm{,}\epsilon Lt\right)=N\left(\frac{1}{4k}g\left(2kx\right)-\frac{1}{4k}h\left(2kx\right)\mathrm{,}L\epsilon t\right)\\ =N\left(g\left(2kx\right)-h\left(2kx\right)\mathrm{,4}L\epsilon t\right)\\ \ge \frac{4Lt}{4Lt+\phi \left(2kx\mathrm{,}\cdots \mathrm{,2}kx\mathrm{,2}kx\mathrm{,}\cdots \mathrm{,2}kx\right)}\\ \ge \frac{4Lt}{4Lt+4L\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\\ \mathrm{<mo>\; =\; </mo>}\frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\mathrm{,}\forall x\in X\mathrm{,}\forall t>0.\end{array}$ (28)

So $d\left(g\mathrm{,}h\right)=\epsilon$ implies that $d\left(Tg\mathrm{,}Th\right)\le L\cdot \epsilon$ . This means that $d\left(Tg\mathrm{,}Th\right)\le Ld\left(g\mathrm{,}h\right)$ , for all $g\mathrm{,}h\in \mathbb{M}$ . It folows from (38) that

$\frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\le N\left(f\left(x\right)-\frac{1}{4k}f\left(2kx\right)\mathrm{,}\frac{t}{4k\left|{\mu }_1\right|}\right)$ (29)

for all $x\in \mathbb{X}$ . So $d\left(f\mathrm{,}Tf\right)\le \frac{1}{4k\left|{\mu }_1\right|}$ . By Theorem 1, there exists a mapping $A\mathrm{<mo>\; :\; </mo>}X\to Y$ satisfying the following:

1) A is a fixed point of T, i.e.,

$A\left(2kx\right)=4kA\left(x\right)$ (30)

for all $x\in X$ . The mapping A is a unique fixed point T in the set $\mathbb{Q}=\left\{g\in \mathbb{M}:d\left(f,g\right)<\infty \right\}$ . This implies that A is a unique mapping satisfying (38) such that there exists a $\beta \in \left(\mathrm{0,}\infty \right)$ satisfying

$N\left(f\left(x\right)-A\left(x\right)\mathrm{,}\beta t\right)\ge \frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\mathrm{,}\forall x\in X\mathrm{.}$

2) $d\left(T^{l}f\mathrm{,}H\right)\to 0$ as $l\to \infty$ . This implies equality

$N-\underset{l\to \infty }{\lim }\frac{1}{{\left(4k\right)}^l}f\left({\left(2k\right)}^{l}x\right)=A\left(x\right)$ , for all $x\in X$ .

3) $d\left(f\mathrm{,}A\right)\le \frac{1}{1-L}d\left(f\mathrm{,}Tf\right)$ , which implies the inequality.

4) $d\left(f\mathrm{,}A\right)\le \frac{1}{\left|4k\right|\left(1-L\right)}$ .

This implies that the inequality (24) holds.

By (22)

$\begin{array}{l}\min (N(\frac{1}{{\left(4k\right)}^n}(2kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)+2kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right),\frac{t}{{\left(4k\right)}^n}),\\ \frac{t}{t+\psi \left({\left(2k\right)}^n{x}_1,\cdots ,{\left(2k\right)}^n{x}_k,{\left(2k\right)}^n{y}_1,\cdots ,{\left(2k\right)}^n{y}_k\right)})\end{array}$

$\begin{array}{l}\le \min (N(\frac{{\mu }_1}{{\left(4k\right)}^n}(f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)+f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right)),\frac{t}{{\left(4k\right)}^n}),\end{array}$

$\begin{array}{l}N(\frac{{\mu }_2}{{\left(4k\right)}^n}(4kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_{i}2k\right)\right)+f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right)),\frac{t}{{\left(4k\right)}^n}))\end{array}$ (31)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ and for all $n\in \mathbb{N}$ . So

$\begin{array}{l}\min (N(\frac{1}{{\left(4k\right)}^n}(2kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)+2kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right),t),\frac{{\left(4k\right)}^{k}t}{{\left(4k\right)}^{k}t+{\left(4k\right)}^k\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)})\end{array}$

$\begin{array}{l}\le \min (N(\frac{{\mu }_1}{{\left(4k\right)}^n}(f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)+f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right)),t),\\ N(\frac{{\mu }_2}{{\left(4k\right)}^n}(4kf\left({\left(2k\right)}^{n-1}\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_{i}2k\right)\right)+f\left({\left(2k\right)}^n\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)\right)\\ -2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{x}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left({\left(2k\right)}^n{y}_i\right)),t))\end{array}$ (32)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ and for all $n\in \mathbb{N}$ . So since $\underset{n\to \infty }{\lim }\frac{{\left(4k\right)}^{n}t}{{\left(4k\right)}^{n}t+{\left(4k\right)}^n{L}^n\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\mathrm{,}{z}_1\mathrm{,}\cdots \mathrm{,}{z}_k\right)}=1$ , for all $x_j\mathrm{,}{y}_j\mathrm{,}{z}_j\in \mathbb{X}$ for all $j\to k$ , $\forall t>0$ , $q\in ℝ$ . So

$\begin{array}{l}N\left(2kA\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kA\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(y_i\right),t\right)\\ \le \min (N\left({\gamma }_1\left(A\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+A\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(y_i\right)\right),t\right),\\ N\left({\gamma }_2\left(4kA\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+A\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}A\left(y_i\right)\right),t\right))\end{array}$ (33)

So the mapping $A\mathrm{<mo>\; :\; </mo>}X\to X$ is a Quadratic mapping, as I desired. □

Theorem 4. Let $\psi \mathrm{<mo>\; :\; </mo>}{X}^{2k}\to \left[\mathrm{0,}\infty \right)$ be a function such that there exists an $L<\frac{1}{2k}$ ,

$\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)\le \frac{1}{4kL}\psi \left(2k{x}_1\mathrm{,}\cdots \mathrm{,2}k{x}_1\mathrm{,2}k{y}_1\mathrm{,}\cdots \mathrm{,2}k{y}_k\right)$ (34)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ .

Let $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping satisfying

$\begin{array}{l}\min (N(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)\\ -{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t),\frac{t}{t+\psi \left(x_1\mathrm{,}\cdots \mathrm{,}{x}_k\mathrm{,}{y}_1\mathrm{,}\cdots \mathrm{,}{y}_k\right)})\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (35)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ . Then

$A\left(x\right)=N-\underset{n\to \infty }{\lim }{\left(4k\right)}^{n}f\left(\frac{x}{{\left(2k\right)}^n}\right)$ (36)

exists each $x\in X$ and defines a quadratic mapping $A\mathrm{<mo>\; :\; </mo>}X\to Y$ such that

$N\left(f\left(x\right)-A\left(x\right)\mathrm{,}t\right)\ge \frac{4k\left|{\mu }_1\right|\left(1-L\right)t}{4k\left|{\mu }_1\right|\left(1-L\right)+L\psi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}$ (37)

for all $x\in X$ and $t>0$ .

Proof. Suppose that $\left(\mathbb{M}\mathrm{,}d\right)$ be the generalized metric space defined in the proof of theorem 3.

From (35) I have

$\frac{t}{t+\phi \left(x\mathrm{,}\cdots \mathrm{,}x\mathrm{,}x\mathrm{,}\cdots \mathrm{,}x\right)}\le N\left(f\left(x\right)-4kf\left(\frac{x}{{\left(2k\right)}^n}\right)\mathrm{,}\frac{Lt}{4k\left|{\mu }_1\right|}\right)$ (38)

for all $x\in X$ , and for all $t\mathrm{<mo>\; >\; </mo>0.}$

Now we cosider the linear mapping $T\mathrm{<mo>\; :\; </mo>}\mathbb{M}\to \mathbb{M}$ such that $Tg\left(x\right)\mathrm{<mo>\; :\; </mo>}=4kg\left(\frac{x}{2k}\right)$ , for all $x\in X$ . So $d\left(f\mathrm{,}Tf\right)\le \frac{L}{4k\left|{\mu }_1\right|}$ . Thus $d\left(f\mathrm{,}A\right)\le \frac{L}{4k\left|{\mu }_1\right|\left(1-L\right)}\mathrm{.}$ which implies that the inequality (37) Satisfied. The rest of the proof is similar to the proof of Theorem 3. □

From the above theorems we have the following corollary:

Corollary 1. Suppose $\theta \ge 0$ and let p be a real number with $0<p<2$ . Let $X$ be a normed vector space with norm $\Vert \cdot \Vert$ Let $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping satisfying

$\min (N(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)$

$\begin{array}{l}-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t),\frac{t}{t+\theta \left({\displaystyle {\sum }_{i=1}^k{\Vert x_i\Vert }^p}+{\displaystyle {\sum }_{i=1}^k{\Vert y_i\Vert }^p}\right)})\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (39)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ . Then

$A\left(x\right)=N-\underset{n\to \infty }{\lim }\frac{1}{{\left(4k\right)}^n}f\left({\left(2k\right)}^{n}x\right)$ (40)

exists each $x\in X$ and defines a quadratic mapping $A\mathrm{<mo>\; :\; </mo>}X\to Y$ such that

$N\left(f\left(x\right)-A\left(x\right),t\right)\ge \frac{\left|{\mu }_1\right|\left(4k-{\left(2k\right)}^p\right)t}{4k\left|{\mu }_1\right|\left(4k-{\left(2k\right)}^p\right)t+\theta {\displaystyle {\sum }_{i=1}^k{\Vert 2k{x}_i\Vert }^p}}$ (41)

for all $x\in X$ and $t>0$ .

Corollary 2. Suppose $\theta \ge 0$ and let p be a real number with $p>2$ .Let $X$ be a normed vector space with norm $\Vert \cdot \Vert$ Let $f\mathrm{<mo>\; :\; </mo>}X\to Y$ be a mapping satisfying

$\begin{array}{l}\min (N(2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+2kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}-{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)\\ -{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right),t),\frac{t}{t+\theta \left({\displaystyle {\sum }_{i=1}^k{\Vert x_i\Vert }^p}+{\displaystyle {\sum }_{i=1}^k{\Vert y_i\Vert }^p}\right)})\\ \le \min (N\left({\mu }_1\left(f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right),\\ N\left({\mu }_2\left(4kf\left(\frac{{\displaystyle {\sum }_{i=1}^k{x}_i}+{\displaystyle {\sum }_{i=1}^k{y}_i}}{2k}\right)+f\left({\displaystyle \underset{i=1}{\overset{k}{\sum }}}{x}_i-{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{y}_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(x_i\right)-2{\displaystyle \underset{i=1}{\overset{k}{\sum }}}f\left(y_i\right)\right),t\right))\end{array}$ (42)

for all $x_j\mathrm{,}{y}_j\in X$ for $j=1\to k$ , for all $t>0$ . Then

$A\left(x\right)=N-\underset{n\to \infty }{\lim }{\left(4k\right)}^{n}f\left(\frac{1}{{\left(2k\right)}^n}x\right)$ (43)

exists each $x\in X$ and defines a quadratic mapping $A\mathrm{<mo>\; :\; </mo>}X\to Y$ such that

$N\left(f\left(x\right)-A\left(x\right),t\right)\ge \frac{4k\left|{\mu }_1\right|\left(4k-{\left(2k\right)}^p\right)t}{4k\left|{\mu }_1\right|\left(4k-{\left(2k\right)}^p\right)t+\theta {\displaystyle {\sum }_{i=1}^k{\Vert 4k{x}_i\Vert }^p}}$ (44)

for all $x\in X$ and $t>0$ .

4. Conclusion

In this paper, I construct the $\varphi \left({\mu }_1\mathrm{,}{\mu }_2\right)$ -function inequality on fuzzy space, which is a great idea for the field of functional equations. Then I show how to find their solutions in spaces constructed by Mathematicians.

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Ulam, S.M. (1960) A Collection of Mathematical Problems. Volume 8, Interscience Publishers, New York.
[2]	Hyers, D.H. (1941) On the Stability of the Functional Equation. Proceedings of the National Academy of the United States of America, 27, 222-224. https://doi.org/10.1073/pnas.27.4.222
[3]	Aoki, T. (1950) On the Stability of the Linear Transformation in Banach Space. Journal of the Mathematical Society of Japan, 2, 64-66. https://doi.org/10.2969/jmsj/00210064
[4]	Rassias, Th.M. (1978) On the Stability of the Linear Mapping in Banach Spaces. Proceedings of the American Mathematical Society, 72, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
[5]	Găvrut, P. (1994) A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. Journal of Mathematical Analysis and Applications, 184, 431-436. https://doi.org/10.1006/jmaa.1994.1211
[6]	Jun, K. and Kim, H. (2002) The Generalized Hyers-Ulam-Rassias Stability of a Cubic Functional Equation. Journal of Mathematical Analysis and Applications, 274, 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
[7]	Lee, S., Im, S. and Hwang, I. (2005) Quartic Functional Equations. Journal of Mathematical Analysis and Applications, 307, 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
[8]	Gilányi, A. (2001) Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequationes Mathematicae, 62, 303-309. https://doi.org/10.1007/PL00000156
[9]	Rätz, J. (2003) On Inequalities Associated with the Jordan-von Neumann Functional Equation. Aequationes Matheaticae, 66, 191-200. https://doi.org/10.1007/s00010-003-2684-8
[10]	Park, C., Cho, Y. and Han, M. (2007) Stability of Functional Inequalities Associated with Jordan-von Neumann Type Additive Functional Equations. Journal of Inequalities and Applications, 2007, Article ID: 041820. https://doi.org/10.1155/2007/41820
[11]	Fechner, W. (2006) Stability of a Functional Inequality Associated with the Jordan-Von Neumann Functional Equation. Aequationes Mathematicae, 71, 149-161. https://doi.org/10.1007/s00010-005-2775-9
[12]	Bag, T. and Samanta, S.K. (2003) Finite Dimensional Fuzzy Normed Linear Spaces. The Journal of Fuzzy Mathematics, 11, 687-705
[13]	Mirmostafaee, A.K. and Moslehian, M.S. (2008) Fuzzy Versions of Hyers-Ulam-Rassias Theorem. Fuzzy Sets and Systems, 159, 720-729. https://doi.org/10.1016/j.fss.2007.09.016
[14]	Mirmostafaee, A.K. and Moslehian, M.S. (2008) Fuzzy Approximately Cubic Mappings. Information Sciences, 178, 3791-3798. https://doi.org/10.1016/j.ins.2008.05.032
[15]	Mirmostafaee, A.K., Mirzavaziri, M. and Moslehian, M.S. (2008) Fuzzy Stability of the Jensen Functional Equation. Fuzzy Sets and Systems, 159, 730-738. https://doi.org/10.1016/j.fss.2007.07.011
[16]	Katsaras, A.K. (1984) Fuzzy Topological Vector Spaces II. Fuzzy Sets and Systems, 12, 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
[17]	Cădariu, L. andRadu, V. (2003) The Fixed Point Alternative and the Stability of Functional Equations. Fixed Point Theory, 4, 91-96.
[18]	Diaz, J. and Margolis, B. (1968) A Fixed Point Theorem of the Alternative for Contractions on a Generalized Complete Metric Space. Bulletin of the American Mathematical Society, 74, 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
[19]	Van An, L. (2020) Generalized Hyers-Ulam Type Stability of the Type Functional Equation with 2k-Variable in Non-Archimedean Space. International Journal of Mathematics Trends and Technology, 66, 134-147. https://doi.org/10.14445/22315373/IJMTT-V66I7P518
[20]	An, L.V. (2020) Generalized Hyers-Ulam Type Stability of the Additive Functional Equation Inequalities with 2n-Variables on an Approximate Group and Ring Homomorphism. Asia Mathematika, 4, 161-175.
[21]	An, L.V. (2021) Generalized Hyers-Ulam-Rassias Type Stability of the with 2k-Variable Quadratic Functional Inequalities in Non-Archimedean Banach Spaces and Banach Spaces. Asia Mathematika, 5, 69-83. https://doi.org/10.14445/22315373/IJMTT-V67I9P521
[22]	An, L.V. (2021) Generalized Hyers-Ulam Type Stability of the 2k-Variables Quadratic β-Functional Inequalities and Function in γ-Homogeneous Normed Space. International Journal of Mathematics and Its Applications, 9, 81-93.
[23]	An, L.V. (2023) Generalized Stability of Functional Inequalities with 3k-Variables Associated for Jordan-von Neumann-Type Additive Functional Equation. Open Access Library Journal, 10, e9681.
[24]	Park, C., Najati, A. and Jang, S. (2013) Fixed Points and Fuzzy Stability of an Additive-Quadratic Functional Equation. Journal of Computational Analysis and Applications, 15, 452-462.
[25]	Park, C. and Rassias, Th.M. (2007) Fixed Points and Generalized Hyers-Ulam Stability of Quadratic Functional Equations. Journal of Mathematical Inequalities, 1, 515-528. https://doi.org/10.7153/jmi-01-43
[26]	Radu, V. (2003) The Fixed Point Alternative and the Stability of Functional Equations. Fixed Point Theory, 4, 91-96.
[27]	Chang, I. and Lee, Y. (2013) Additive and Quadratic Type Functional Equation and Its Fuzzy Stability. Results in Mathematics, 63, 717-730. https://doi.org/10.1007/s00025-012-0229-y
[28]	Cheng, S.C. and Mordeson, J.M. (1994) Fuzzy Linear Operators and Fuzzy Normed Linear Spaces. Bulletin of the Calcutta Mathematical Society, 86, 429-436.
[29]	Rassias, T.M. (2000) On the Stability of Functional Equations in Banach Spaces. Journal of Mathematical Analysis and Applications, 251, 264-284. https://doi.org/10.1006/jmaa.2000.7046
[30]	Rassias, Th.M. (1990) Problem 16; 2, Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae, 39, 292-293, 309.
[31]	Gajda, Z. (1991) On Stability of Additive Mappings. International Journal of Mathematics and Mathematical Sciences, 14, 431-434. https://doi.org/10.1155/S016117129100056X
[32]	Rassias, Th.M. and Semrl, P. (1992) On the Behavior of Mappings Which Do Not Satisfy Hyers-Ulam Stability. Proceedings of the American Mathematical Society, 114, 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
[33]	An, L.V. (2021) Generalized Hyers-Ulam Stability of the Additive Functional Inequalities with 2n-Variables in Non-Archimedean Banach Space. Bulletin of Mathematics and Statistics Research, 9, 67-73.
[34]	An, L.V. (2023) Generalized Stability of Functional Inequalities with 3k-Variables Associated for Jordan-von Neumann-Type Additive Functional Equation. Open Access Library Journal, 10, e9681. https://www.scirp.org/journal/oalibj https://doi.org/10.4236/oalib.1109681
[35]	Lee, J.R., Park, C. and Shin, D.Y. (2014) Additive and Quadratic Functional in Equalities in Non-Archimedean Normed Spaces. International Journal of Mathematical Analysis, 8, 1233-1247. https://www.m-hikari.com https://doi.org/10.12988/ijma.2014.44113
[36]	Rassias, J.M. (1982) On Approximation of Approximately Linear Mappings by Linear Mappings. Journal of Functional Analysis, 46, 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
[37]	Acze’l, J. and Dhombres, J. (1989) Functional Equations in Several Variables, with Applications to Mathematics, Information Theory and to the Natural and Social Sciences. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge.
[38]	Cadariu, L. and Radu, V. (2003) Fixed Points and the Stability of Jensen’s Functional Equation. JIPAM. Journal of Inequalities in Pure and Applied Mathematics, 4, Article No. 4.
[39]	Chang, I.S., Eshaghi Gordji, M., Khodaei, H. and Kim, H.M. (2013) Nearly Quartic Mappings in β-Homogeneous F-Spaces. Results in Mathematics, 63, Article ID: 529541. https://doi.org/10.1007/s00025-011-0215-9
[40]	Cho, Y.J., Park, C.K. and Saadati, R. (2010) Functional Inequalities in Non-Archimedean Banach Spaces. Applied Mathematics Letters, 23, 1238-1242. https://doi.org/10.1016/j.aml.2010.06.005
[41]	Cholewa, P.W. (1984) Remarks on the Stability of Functional Equations. Aequationes Mathematicae, 27, 76-86. https://doi.org/10.1007/BF02192660
[42]	Ebadian, A., Ghobadipour, N., Rassias, T.M. and Eshaghi Gordji, M. (2011) Functional Inequalities Associated with Cauchy Additive Functional Equation in Non-Archimedean Spaces. Discrete Dynamics in Nature and Society, 2011, Article ID: 929824. https://doi.org/10.1155/2011/929824
[43]	Isac, G. and Rassias, T.M. (1993) On the Hyers-Ulam Stability of Additive Mappings. Journal of Approximation Theory, 72, 131-137. https://doi.org/10.1006/jath.1993.1010
[44]	Lakshmikantham, V., Leela, S. and Devi, J.V. (2009) Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge.
[45]	Lee, S.-B., Bae, J.-H. and Park, W.-G. (2014) On the Stability of an Additive Functional Inequality for the Fixed Point Alternative. Journal of Computational Analysis and Applications, 17, 361-371.
[46]	Lu, G. and Park, C.K. (2011) Hyers-Ulam Stability of Additive Set-Valued Functional Equations. Applied Mathematics Letters, 24, 1312-1316. https://doi.org/10.1016/j.aml.2011.02.024
[47]	Park, C.K., Cho, Y.S. and Han, M.-H. (2007) Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional Equations. Journal of Inequalities and Applications, 2007, Article ID: 041820. https://doi.org/10.1155/2007/41820
[48]	Rassias, T.M. (2000) Functional Equations and Inequalities. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht. https://doi.org/10.1007/978-94-011-4341-7
[49]	Rassias, T.M. (2000) On the Stability of Functional Equations and a Problem of Ulam. Acta Applicandae Mathematicae, 62, 23-130. https://doi.org/10.1023/A:1006499223572
[50]	Rassias, M. (2007) Refined Hyers-Ulam Approximation of Approximately Jensen Type Mappings. Bulletin des Sciences Mathématiques, 131, 89-98. https://doi.org/10.1016/j.bulsci.2006.03.011
[51]	Rassias, M. and Rassias, M.J. (2003) On the Ulam Stability of Jensen and Jensen Type Mappings on Restricted Domains. Journal of Mathematical Analysis and Applications, 281, 516-524. https://doi.org/10.1016/S0022-247X(03)00136-7
[52]	Jun, K.-W. and Lee, Y.-H. (2004) A Generalization of the Hyers-Ulam-Rassias Stability of the Pexiderized Quadratic Equations. Journal of Mathematical Analysis and Applications, 297, 70 86. https://doi.org/10.1016/j.jmaa.2004.04.009
[53]	Gilanyi, A. (2001) Eine zur Parallelogrammgleichung aquivalente Ungleichung. Aequationes Mathematicae, 62, 303-309. https://doi.org/10.1007/PL00000156
[54]	Bahyrycz, A. and Piszczek, M. (2014) Hyers Stability of the Jensen Function Equation. Acta Mathematica Hungarica, 142, 353-365. https://doi.org/10.1007/s10474-013-0347-3
[55]	Schwaiger, J. (2013) A System of Two Inhomogeneous Linear Functional Equations. Acta Mathematica Hungarica, 140, 377-406. https://doi.org/10.1007/s10474-013-0315-y
[56]	Maligranda, L. (2008) Tosio Aoki (1910-1989). In: International Symposium on Banach and Function Spaces (14/09/2006-17/09/2006), Yokohama Publishers, Yokohama, 1-23.
[57]	Najati, A. and Eskandani, G.Z. (2008) Stability of a Mixed Additive and Cubic Functional Equation in Quasi-Banach Spaces. Journal of Mathematical Analysis and Applications, 342, 1318-1331. https://doi.org/10.1016/j.jmaa.2007.12.039
[58]	Gilányi, A. (2002) On a Problem by K. Nikodem. Mathematical Inequalities & Applications, 5, 707-710. https://doi.org/10.7153/mia-05-71
[59]	Fechner, W. (2010) On Some Functional Inequalities Related to the Logarithmic Mean. Acta Mathematica Hungarica, 128, 36-45. https://doi.org/10.1007/s10474-010-9153-3
[60]	Park, C. (2014) Additive β-Functional Inequalities. Journal of Nonlinear Sciences and Applications, 7, 296-310. https://doi.org/10.22436/jnsa.007.05.02
[61]	Van An, L. (2019) Hyers-Ulam Stability of Functional Inequalities with Three Variable in Banach Spaces and non-Archimedean Banach Spaces. International Journal of Mathematical Analysis, 13, 519-537. https://doi.org/10.12988/ijma.2019.9954
[62]	Cho, Y.J., Park, C. and Saadati, R. (2010) Functional in Equalities in Non-Archimedean Normed Spaces. Applied Mathematics Letters, 23, 1238-1242. https://doi.org/10.1016/j.aml.2010.06.005
[63]	Aribou, Y. and Kabbaj, S. (2018) Generalized Functional in Inequalities in Non-Archimedean Normed Spaces. Applied Mathematics Letters, 2, 61-66.
[64]	An, L.V. (2020) Hyers-Ulam Stability Additive β-Functional Inequalities with Three Variable in Non-Archimedean Banach Space and Complex Banach Spaces. International Journal of Mathematical Analysis, 14, 219-239. https://doi.org/10.12988/ijma.2020.91169
[65]	An, L.V. (2023) Establish an Additive (s; t)-Function Inequlities Fixed Point Method and Direct Method with n-Variables Banach Space. IJRDO—Journal of Mathematics, 9, 1-13. https://doi.org/10.53555/m.v9i1.5515