#
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

## References

- Burago, D.; Ferleger, S. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. Math.
**1998**, 147, 695–708. [Google Scholar] [CrossRef] - Bridson, M.; Haefliger, A. Metric Spaces of Non-Positive Curvature; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1999; ISBN 3-540-64324-9. [Google Scholar] [CrossRef]
- Bruhat, F.; Tits, F. Groupes réductifs sur un corps local. I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math.
**1972**, 41, 5–251. [Google Scholar] [CrossRef] - Khamsi, M.A.; Shukri, S. Generalized CAT(0) spaces. Bull. Belg. Math. Soc. Simon Stevin
**2017**, 24, 417–426. [Google Scholar] [CrossRef] - Khamsi, M.A.; Kirk, W.A. An Introduction to Metric Spaces and Fixed Point Theory; John Wiley: New York, NY, USA, 2001; ISBN 0-471-41825-0. [Google Scholar] [CrossRef]
- Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces; Progress in Mathematics; Birkhäuser Boston, Inc.: Boston, MA, USA, 1999; Volume 152, ISBN 0-8176-3898-9. [Google Scholar]
- Kirk, W.A. A fixed point theorem in CAT(0) spaces and $\mathbb{R}$-trees. Fixed Point Theory Appl.
**2004**, 4, 309–316. [Google Scholar] [CrossRef] - Lim, T.C. Fixed point theorem for uniformly Lipschitzian mappings in L
^{p}spaces. Nonlinear Anal.**1983**, 7, 555–563. [Google Scholar] [CrossRef] - Goebel, K.; Kirk, W.A. A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.
**1972**, 35, 171–174. [Google Scholar] [CrossRef] - Kirk, W.A. A fixed point theorem for mappings which do not increase distances. Am. Math. Mon.
**1965**, 72, 1004–1006. [Google Scholar] [CrossRef] - Khamsi, M.A.; Kirk, W.A. On uniformly Lipschitzian multivalued mappings in Banach and metric spaces. Nonlinear Anal. Theory Methods Appl.
**2010**, 72, 2080–2085. [Google Scholar] [CrossRef] - Alfuraidan, M.R.; Khamsi, M.A. A fixed point theorem for monotone asymptotically nonexpansive mappings. Proc. Am. Math. Soc.
**2018**, 146, 2451–2456. [Google Scholar] [CrossRef] - Alfuraidan, M.R.; Khamsi, M.A. Fibonacci-Mann iteration for monotone asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.
**2017**, 96, 307–316. [Google Scholar] [CrossRef] - Calderón, K.; Martínez-Moreno, J.; Rojas, E.M. Hybrid algorithm with perturbations for total asymptotically non-expansive mappings in CAT(0) space. Int. J. Comput. Math.
**2019**. [Google Scholar] [CrossRef] - Pakkaranang, N.; Kumam, P.; Cho, Y.J.; Saipara, P.; Padcharoen, A.; Khaofong, C. Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces. J. Math. Comput. Sci.
**2017**, 2017, 345–354. [Google Scholar] [CrossRef] - Schu, J. Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appli.
**1991**, 158, 407–413. [Google Scholar] [CrossRef] - Lim, T.C. Remarks on some fixed point theorems. Proc. Am. Math. Soc.
**1976**, 60, 179–182. [Google Scholar] [CrossRef] - Opial, Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.
**1967**, 73, 591–597. [Google Scholar] [CrossRef] [Green Version]

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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