# Ulam Stability for the Composition of Operators

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- (i)
- There exists $C\ge 0$ such that $d(x,N(A\left)\right)\le C\parallel Ax\parallel ,\phantom{\rule{0.166667em}{0ex}}x\in D\left(A\right)$;
- (ii)
- $R\left(A\right)$ is closed.

**Corollary**

**1.**

## 2. Ulam Stability for the Composition of Operators

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Applications

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Example**

**1.**

**Definition**

**2.**

- (i)
- in case $\alpha =\beta =-1$:$${\mathcal{B}}_{n}^{-1,-1}(f;x)=\left\{\begin{array}{cc}f\left(0\right),\hfill & x=0,\hfill \\ {\displaystyle \frac{{\int}_{0}^{1}{t}^{nx-1}{(1-t)}^{n-nx-1}f\left(t\right)dt}{B(nx,n-nx)}},\hfill & 0<x<1,\hfill \\ f\left(1\right),\hfill & x=1;\hfill \end{array}\right.$$
- (ii)
- in case $\alpha =-1,\beta >-1$:$${\mathcal{B}}_{n}^{-1,\beta}(f;x)=\left\{\begin{array}{cc}f\left(0\right),\hfill & x=0,\hfill \\ {\displaystyle \frac{{\int}_{0}^{1}{t}^{nx-1}{(1-t)}^{n-nx+\beta}f\left(t\right)dt}{B(nx,n-nx+\beta +1)}},\hfill & 0<x\le 1;\hfill \end{array}\right.$$
- (iii)
- in case $\alpha >-1$, $\beta =-1$:$${\mathcal{B}}_{n}^{\alpha ,-1}(f;x)=\left\{\begin{array}{cc}{\displaystyle \frac{{\int}_{0}^{1}{t}^{nx+\alpha}{(1-t)}^{n-nx-1}f\left(t\right)dt}{B(nx+\alpha +1,n-nx)}},\hfill & 0\le x<1,\hfill \\ f\left(1\right),\hfill & x=1;\hfill \end{array}\right.$$
- (iv)
- in case $\alpha ,\beta >-1$:$${\mathcal{B}}_{n}^{\alpha ,\beta}(f;x)={\displaystyle \frac{{\int}_{0}^{1}{t}^{nx+\alpha}{(1-t)}^{n-nx+\beta}f\left(t\right)dt}{B(nx+\alpha +1,n-nx+\beta +1)}},\phantom{\rule{0.166667em}{0ex}}0\le x\le 1.$$

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

## 4. An Example and an Open Problem

**Example**

**2.**

**Problem**

**1.**

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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Acu, A.M.; Raşa, I.
Ulam Stability for the Composition of Operators. *Symmetry* **2020**, *12*, 1159.
https://doi.org/10.3390/sym12071159

**AMA Style**

Acu AM, Raşa I.
Ulam Stability for the Composition of Operators. *Symmetry*. 2020; 12(7):1159.
https://doi.org/10.3390/sym12071159

**Chicago/Turabian Style**

Acu, Ana Maria, and Ioan Raşa.
2020. "Ulam Stability for the Composition of Operators" *Symmetry* 12, no. 7: 1159.
https://doi.org/10.3390/sym12071159