# Ulam Stability of a General Linear Functional Equation in Modular Spaces

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- (i)
- ${\sum}_{j=1}^{n}{a}_{ij}{c}_{j}=1$for$i\in I$;
- (ii)
- ${\sum}_{i\in {I}_{0}}|{B}_{i}|{\omega}_{i}<\left|{\sum}_{i\in I}{B}_{i}\right|$;
- (iii)
- $\theta \left(\right)open="("\; close=")">\left(\right)open="("\; close=")">{\sum}_{j=1}^{n}{b}_{ij}{c}_{j}{x}_{n}$for$i\in {I}_{0}$and${x}_{1},\dots ,{x}_{n}\in X$.

**Definition**

**1.**

## 2. Preliminaries

**Definition**

**2.**

- M1.
- $\rho \left(x\right)=0$if and only if$x=0$;
- M2.
- $\rho \left(\alpha x\right)=\rho \left(x\right)$for every$\alpha \in \mathbb{K}$with$\left|\alpha \right|=1$;
- M3.
- $\rho (\alpha x+\beta y)\le \rho \left(x\right)+\rho \left(y\right)$for every$\alpha ,\beta \in {\mathbb{R}}_{+}$with$\alpha +\beta =1$.If we replace condition M3 with the following one:
- M4.
- $\rho (\alpha x+\beta y)\le \alpha \rho \left(x\right)+\beta \rho \left(y\right)$for every$\alpha ,\beta \in {\mathbb{R}}_{+}$with$\alpha +\beta =1$,

**Definition**

**3.**

**Remark**

**1.**

- (a)
- If $\rho $ is a modular on Y and $y\in Y$, then the function ${\mathbb{R}}_{+}\ni t\to \rho \left(ty\right)$ is non-decreasing, i.e., $\rho \left(ay\right)\le \rho \left(by\right)$ for every $a,b\in {\mathbb{R}}_{+}$ with $a<b$ (it is enough to take $y=0$ in M3).
- (b)
- For a convex modular $\rho $ on Y, we have $\rho \left(\alpha y\right)\le \left|\alpha \right|\rho \left(y\right)$ for all $y\in Y$ and $\alpha \in \mathbb{K}$ with $\left|\alpha \right|\le 1$ and, moreover,$$\rho \left(\right)open="("\; close=")">\sum _{j=1}^{n}{\alpha}_{j}{y}_{j}$$

**Definition**

**4.**

- (i)
- ${\left({y}_{n}\right)}_{n}$is ρ-convergent to a point$y\in Y$(which we denote by$y=\rho -{lim}_{n}{y}_{n}$), if$\rho ({y}_{n}-y)\to 0$as$n\to +\infty $;
- (ii)
- ${\left({y}_{n}\right)}_{n}$is ρ-Cauchy if for any$\u03f5>0$, we have$\rho ({y}_{n}-{y}_{m})<\u03f5$for sufficiently large$m,n\in \mathbb{N}$;
- (iii)
- ${Y}_{\rho}$is said to be ρ-complete if every ρ-Cauchy sequence in${Y}_{\rho}$is ρ-convergent.
- (iv)
- A subset$C\subset {Y}_{\rho}$is called ρ-closed if C contains every$x\in {Y}_{\rho}$such that there is a sequence${\left({x}_{n}\right)}_{n}$in C which is ρ-convergent to x.

**Definition**

**5.**

## 3. Stability of Equation (5)

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**1.**

**Remark**

**2.**

- (1)
- Every constant function $D:{X}^{n}\to Y$ satisfies condition (7).
- (2)
- If ${D}_{1},{D}_{2}:{X}^{n}\to Y$ satisfy (7), then so does the function ${\alpha}_{1}{D}_{1}+{\alpha}_{2}{D}_{2}$ for any fixed scalars ${\alpha}_{1}$, ${\alpha}_{2}$.
- (3)
- Consider the situation in Corollary 1 (i.e., when Equation (5) has the form (20)). Then, condition (7) has the form$$D({x}_{1}+{x}_{2},{x}_{1}+{x}_{2})-D({x}_{1},{x}_{1})-D({x}_{2},{x}_{2})=D(2{x}_{1},2{x}_{2})-2D({x}_{1},{x}_{2}).$$$$D({x}_{1},{x}_{2})=h({x}_{1}+{x}_{2})-h\left({x}_{1}\right)-h\left({x}_{2}\right),\phantom{\rule{1.em}{0ex}}{x}_{1},{x}_{2}\in X,$$

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Aboutaib, I.; Benzarouala, C.; Brzdęk, J.; Leśniak, Z.; Oubbi, L.
Ulam Stability of a General Linear Functional Equation in Modular Spaces. *Symmetry* **2022**, *14*, 2468.
https://doi.org/10.3390/sym14112468

**AMA Style**

Aboutaib I, Benzarouala C, Brzdęk J, Leśniak Z, Oubbi L.
Ulam Stability of a General Linear Functional Equation in Modular Spaces. *Symmetry*. 2022; 14(11):2468.
https://doi.org/10.3390/sym14112468

**Chicago/Turabian Style**

Aboutaib, Issam, Chaimaa Benzarouala, Janusz Brzdęk, Zbigniew Leśniak, and Lahbib Oubbi.
2022. "Ulam Stability of a General Linear Functional Equation in Modular Spaces" *Symmetry* 14, no. 11: 2468.
https://doi.org/10.3390/sym14112468