# Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables

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## Abstract

**:**

## 1. Introduction

Let ${G}_{1}$ and ${G}_{2}$ be a group and a metric group with a metric $d(\xb7,\xb7)$, respectively. Given $\epsilon >0$, does there exist a $\delta >0$ such that if a function $h:{G}_{1}\to {G}_{2}$ satisfies the inequality $d\left(\right)open="("\; close=")">h(xy),h(x)h(y)$ for all $x,y\in {G}_{1}$, then there exists a homomorphism $H:{G}_{1}\to {G}_{2}$ with $d\left(\right)open="("\; close=")">h(x),H(x)$ for all $x\in {G}_{1}$?

**Theorem**

**1**

**.**Given $\delta >0$, assume that $f:{E}_{1}\to {E}_{2}$ is a function between Banach spaces such that

## 2. Preliminaries

**Theorem**

**2**

**.**Assume that $f:\mathbb{R}\to \mathbb{R}$ is n times differentiable in a neighborhood N of a point ${t}_{0}$ and ${f}^{(n)}({t}_{0})=0$ and ${f}^{(n)}(t)$ changes sign at ${t}_{0}$. For any $\epsilon >0$, there corresponds a $\delta >0$ such that for every function $g:\mathbb{R}\to \mathbb{R}$ which is n times differentiable in N and satisfies $|f(t)-g(t)|<\delta $ for all $t\in N$, there exists a point ${t}_{1}\in N$ such that ${g}^{(n)}({t}_{1})=0$ and $|{t}_{1}-{t}_{0}|<\epsilon $.

**Remark**

**1.**

- $(i)$
- ε is chosen less than the radius of N;
- $(ii)$
- we choose ${t}_{2}$, ${t}_{3}$ and α such that $|{t}_{2}-{t}_{0}|<\frac{\epsilon}{2}$, $|{t}_{3}-{t}_{0}|<\frac{\epsilon}{2}$, $0<\alpha <\frac{\epsilon}{2n}$, and ${\Delta}_{\alpha}^{n}f({t}_{2}){\Delta}_{\alpha}^{n}f({t}_{3})<0$, where ${\Delta}_{\alpha}^{n}f(t)={\Delta}_{\alpha}\left\{{\Delta}_{\alpha}^{n-1}f(t)\right\}$ and ${\Delta}_{\alpha}f(t)=f(t+\alpha )-f(t)$;
- $(iii)$
- we choose the δ as large as possible with $0<\delta <min\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="\{"\; close="\}">\frac{1}{{2}^{n}}|{\Delta}_{\alpha}^{n}f({t}_{2})|,\frac{1}{{2}^{n}}|{\Delta}_{\alpha}^{n}f({t}_{3})|$.

If a function f has a mean value point η and g is a function quite near to f, does g have a mean value point near η?

**Theorem**

**3**

**.**Assume that $f,h:[a,b]\to \mathbb{R}$ are differentiable functions and η is a Sahoo-Riedel’s point of f in $(a,b)$. If f is twice differentiable at η and

**Theorem**

**4**

**.**Let $a,b,\eta $ be real numbers satisfying $a<\eta <b$. Assume that $f:\mathbb{R}\to \mathbb{R}$ is a twice continuously differentiable function and η is the unique Lagrange’s mean value point of f in an open interval $(a,b)$ and moreover that ${f}^{\u2033}(\eta )\ne 0$. Suppose $g:\mathbb{R}\to \mathbb{R}$ is a differentiable function. Then, for any given $\epsilon >0$, there exists a $\delta >0$ with the property that if $|f(x)-g(x)|<\delta $ for all $x\in [a,b]$, then there is a Lagrange’s mean value point $\xi \in (a,b)$ of g with $|\xi -\eta |<\epsilon $.

**Theorem**

**5**

**.**For every function $f:{\mathbb{R}}^{2}\to \mathbb{R}$ with continuous partial derivatives ${f}_{x}$ and ${f}_{y}$ and for all distinct points $(p,q)$ and $(u,v)$ in ${\mathbb{R}}^{2}$, there exists an intermediate point $(\eta ,\xi )$ on the line segment L joining the points $(p,q)$ and $(u,v)$ such that $f(u,v)-f(p,q)=(u-p){f}_{x}(\eta ,\xi )+(v-q){f}_{y}(\eta ,\xi )$.

## 3. Main Theorem

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

- $(i)$
- by considering $(3)$, we choose $\tilde{\epsilon}$ such that $0<\tilde{\epsilon}<r$;
- $(ii)$
- we choose ${t}_{2}$, ${t}_{3}$ and α such that $|{t}_{2}-{t}_{0}|<\frac{\tilde{\epsilon}}{2}$, $|{t}_{3}-{t}_{0}|<\frac{\tilde{\epsilon}}{2}$, $0<\alpha <\frac{\tilde{\epsilon}}{2}$, and ${\Delta}_{\alpha}{G}_{f}({t}_{2}){\Delta}_{\alpha}{G}_{f}({t}_{3})<0$;
- $(iii)$
- we choose the $\tilde{\delta}$ as large as possible with $0<\tilde{\delta}<min\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="\{"\; close="\}">\frac{1}{2}|{\Delta}_{\alpha}{G}_{f}({t}_{2})|,\phantom{\rule{0.166667em}{0ex}}\frac{1}{2}|{\Delta}_{\alpha}{G}_{f}({t}_{3})|$;
- $(iv)$
- we determine δ by $\delta :=\frac{1}{3}\tilde{\delta}$.

**Corollary**

**1.**

**Corollary**

**2.**

## 4. Example

## 5. Discussions

Assume that L denote the line segment joining two points in the plane and that $f:{\mathbb{R}}^{2}\to \mathbb{R}$ is a twice continuously partial differentiable function. Moreover, suppose $({\eta}_{0},{\xi}_{0})$ is the unique two-dimensional Lagrange’s mean value point of f in L and the condition $(1)$ is fulfilled. Then, for any given $\epsilon >0$, there exists a $\delta >0$ with the property that if a partial differentiable function $g:{\mathbb{R}}^{2}\to \mathbb{R}$ satisfies $|f(x,y)-g(x,y)|<\delta $ for all $(x,y)\in L$, then there exists a two-dimensional Lagrange’s mean value point $({\eta}_{1},{\xi}_{1})$ of g in L with $|({\eta}_{0},{\xi}_{0})-({\eta}_{1},{\xi}_{1})|<\epsilon $.

## Author Contributions

## Funding

## Conflicts of Interest

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Jung, S.-M.; Kim, J.-H.
Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables. *Mathematics* **2018**, *6*, 216.
https://doi.org/10.3390/math6110216

**AMA Style**

Jung S-M, Kim J-H.
Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables. *Mathematics*. 2018; 6(11):216.
https://doi.org/10.3390/math6110216

**Chicago/Turabian Style**

Jung, Soon-Mo, and Ji-Hye Kim.
2018. "Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables" *Mathematics* 6, no. 11: 216.
https://doi.org/10.3390/math6110216