Research Article Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces Amna Kalsoom, 1 Naeem Saleem , 2 Hüseyin Işık , 3 Tareq M. Al-Shami , 4 Amna Bibi, 1 and Hafsa Khan 1 1 Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan 2 Department of Mathematics, University of Management and Technology, Lahore, Pakistan 3 Department of Engineering Science, Bandırma Onyedi Eylül University, Bandırma 10200, Balıkesir, Turkey 4 Department of Mathematics, Sana’a University, Sana’a, Yemen Correspondence should be addressed to Hüseyin Işık; huseyin.isik@tdtu.edu.vn and Tareq M. Al-Shami; tareqalshami83@gmail.com Received 10 May 2021; Revised 1 July 2021; Accepted 13 July 2021; Published 10 August 2021 Academic Editor: Mustafa Avci Copyright © 2021 Amna Kalsoom et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Fixed points of monotone α-nonexpansive and generalized β-nonexpansive mappings have been approximated in Banach space. Our purpose is to approximate the fixed points for the above mappings in hyperbolic space. We prove the existence and convergence results using some iteration processes. 1. Introduction In 1965, Browder [1], Göhde [2], and Kirk [3] started work- ing in the approximation of fixed point for nonexpansive mappings. Firstly, Browder obtained fixed point theorem for nonexpansive mapping on a subset of a Hilbert space that is closed bounded and convex. Soon after, Browder [1] and Göhde [2] generalized the same result from a Hilbert space to a uniformly convex Banach space. Kirk [3] utilized normal structure property in a reflexive Banach space to sum up the similar results. Recently, Dehici and Najeh [4] and Tan and Cho [5] approximated fixed point result for nonexpansive mappings in Banach space and Hilbert space. Fixed point theory in partially ordered metric spaces has been initiated by Ran and Reurings [6] for finding applica- tion to matrix equation. Nieto and Lopez [7] extended their result for nondecreasing mapping and presented an applica- tion to differential equations. Recently, Song et al. [8] extended the notion of α-nonexpansive mapping to mono- tone α-nonexpansive mapping in order Banach spaces and obtained some existence and convergence theorem for the Mann iteration (see also [9] and the reference therein). Moti- vated by the work of Suzuki [10], Aoyama and Kohsaka [11], Dehaish and Khamsi [9], and Song et al. [8], Pant and Shukla obtained existence results in ordered Banach space for a wider class of nonexpansive mappings [12, 13]. There are many mathematicians who worked on weak and strong con- vergence of nonexpansive mappings and its generalizations by using one step, two step, and multistep iteration process ([8, 14, 15]). We obtain existence results in partial ordered hyperbolic space for monotone generalized α-nonexpansive and monotone generalized β-nonexpansive map. Particu- larly, in Section 3, some auxiliary results and existence theo- rems for monotone α-nonexpansive mappings in ordered hyperbolic spaces are presented. In Section 4, we presented numerical examples and graphical representation. In Section 5, we obtained some existence results for monotone general- ized β-nonexpansive mappings in ordered hyperbolic spaces. 2. Preliminaries In 1976, the concept of Δ -convergence was given by Lim [14]. Lim [14] initiated the idea that in a metric space, Δ -convergence is possible. This concept is adapted for CAT(0) spaces by Kirk and Panyanak [16], and they have indicated that in numerous Banach space, outcomes Hindawi Journal of Function Spaces Volume 2021, Article ID 3243020, 14 pages https://doi.org/10.1155/2021/3243020