#
Approximations of Fixed Points in the Hadamard Metric Space CAT_{p}(0)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions and Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

## 3. Fixed Point Results in ${\mathbf{CAT}}_{\mathbf{p}}\left(\mathbf{0}\right)$

**Definition**

**3.**

- (1)
- J is asymptotically nonexpansive if there exists $\left\{{\rho}_{n}\right\}$ such that $\underset{n\to \infty}{lim}\phantom{\rule{4pt}{0ex}}{\rho}_{n}=1$ and$$d({J}^{n}\left(x\right),{J}^{n}\left(y\right))\le {\rho}_{n}\phantom{\rule{4pt}{0ex}}d(x,y),$$
- (2)
- J is uniformly Lipschitzian if there exists $\rho \ge 0$ such that$$d({J}^{n}\left(x\right),{J}^{n}\left(y\right))\le \rho \phantom{\rule{4pt}{0ex}}d(x,y),$$
- (3)
- A point $x\in M$ is a fixed point of J if $J\left(x\right)=x$ holds. $Fix\left(J\right)$ will denote the set of fixed points of J.

**Definition**

**4.**

**Lemma**

**2.**

- (1)
- Any minimizing sequence of θ is convergent.
- (2)
- All minimizing sequences of θ converge to the same limit $z\in C$.
- (3)
- z is a minimum point of θ, i.e., $\theta \left(z\right)=inf\left\{\theta \right(x);\phantom{\rule{0.277778em}{0ex}}x\in C\}$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Bachar, M.; Khamsi, M.A. Approximations of Fixed Points in the Hadamard Metric Space *CAT*_{p}(0). *Mathematics* **2019**, *7*, 1088.
https://doi.org/10.3390/math7111088

**AMA Style**

Bachar M, Khamsi MA. Approximations of Fixed Points in the Hadamard Metric Space *CAT*_{p}(0). *Mathematics*. 2019; 7(11):1088.
https://doi.org/10.3390/math7111088

**Chicago/Turabian Style**

Bachar, Mostafa, and Mohamed Amine Khamsi. 2019. "Approximations of Fixed Points in the Hadamard Metric Space *CAT*_{p}(0)" *Mathematics* 7, no. 11: 1088.
https://doi.org/10.3390/math7111088