# Cascading Operators in CAT(0) Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. Cascading Non-Expansive Mappings

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**1.**

**Definition**

**6.**

**Definition**

**7.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Remark**

**2.**

## 4. Fixed Point Results for Cascading Operators

**Definition**

**8.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Lemma**

**1.**

- 1 .
- If $w\in \mathrm{Fix}\left(T\right)$, then ${lim}_{n\to \infty}d({x}_{n},w)$ exists.
- 2 .
- ${lim}_{n\to \infty}d({x}_{n},T{x}_{n})=0$.

**Proof.**

- By Proposition 1,$$\begin{array}{ccc}\hfill d({x}_{n+1},w)& =& d((1-{\alpha}_{n})T{x}_{n}\oplus {\alpha}_{n}{T}^{2}{x}_{n},w)\le (1-{\alpha}_{n})d(T{x}_{n},w)\hfill \\ & +& {\alpha}_{n}d({T}^{2}{x}_{n},{T}^{2}w)\hfill \\ & \le & (1-{\alpha}_{n}){k}_{n}d({x}_{n},w)+{k}_{n+1}{k}_{n}{\alpha}_{n}d({x}_{n},w)\hfill \\ & =& (1+{\alpha}_{n}{k}_{n}({k}_{n+1}-1))d({x}_{n},w)\hfill \end{array}$$
- Let $r={lim}_{n\to \infty}d({x}_{n},w)$. Since ${x}_{n}\in {C}_{n}$ and $w\in \mathrm{Fix}\left(T\right)$,$$\begin{array}{c}\hfill \underset{n\to \infty}{lim\; sup}d(T{x}_{n},w)=\underset{n\to \infty}{lim\; sup}d(T{x}_{n},Tw)\le \underset{n\to \infty}{lim\; sup}{k}_{n+1}d({x}_{n},w)=r.\end{array}$$Similarly, ${lim\; sup}_{n\to \infty}d(w,{T}^{2}{x}_{n})\le r$, so, by Lemma 4.5 in [17], we have that${lim}_{n\to \infty}d(T{x}_{n},{T}^{2}{x}_{n})=0$.On the other hand,$$\begin{array}{cc}\hfill d({x}_{n+1},T{x}_{n+1})& \le d\left({x}_{n+1},{T}^{2}{x}_{n}\right)+d\left({T}^{2}{x}_{n},T{x}_{n+1}\right)\hfill \\ & \le (1-{\alpha}_{n})d(T{x}_{n},{T}^{2}{x}_{n})+{k}_{n}{\alpha}_{n}d(T{x}_{n},{T}^{2}{x}_{n})\hfill \\ & \le (1-{\alpha}_{n}+{k}_{n}{\alpha}_{n})d\left(T{x}_{n},{T}^{2}{x}_{n}\right)\to 0.\hfill \end{array}$$

**Example**

**4.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Fetter Nathansky, H.; Villada Bedoya, J.
Cascading Operators in CAT(0) Spaces. *Axioms* **2021**, *10*, 20.
https://doi.org/10.3390/axioms10010020

**AMA Style**

Fetter Nathansky H, Villada Bedoya J.
Cascading Operators in CAT(0) Spaces. *Axioms*. 2021; 10(1):20.
https://doi.org/10.3390/axioms10010020

**Chicago/Turabian Style**

Fetter Nathansky, Helga, and Jeimer Villada Bedoya.
2021. "Cascading Operators in CAT(0) Spaces" *Axioms* 10, no. 1: 20.
https://doi.org/10.3390/axioms10010020