Research Article Approximation of Fixed Points and Best Proximity Points of Relatively Nonexpansive Mappings Thabet Abdeljawad , 1,2,3 Kifayat Ullah, 4 Junaid Ahmad , 4 Manuel De La Sen , 5 and Azhar Ulhaq 4 1 Department of Mathematics and General Sciences, Prince Sultan University, P.O.Box 66833, Riyadh 11586, Saudi Arabia 2 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 3 Department of Computer Sciences and Information Engineering, Asia University, Taichung, Taiwan 4 Department of Mathematics, University of Science and Technology, Bannu 28100, Khyber Pakhtunkhwa, Pakistan 5 Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644- Bilbao, Barrio Sarriena, 48940 Leioa, Spain Correspondence should be addressed to Junaid Ahmad; ahmadjunaid436@gmail.com Received 31 August 2020; Revised 9 September 2020; Accepted 16 September 2020; Published 29 October 2020 Academic Editor: Hijaz Ahmad Copyright © 2020 abet Abdeljawad et al. is 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. In this article, we study the Agarwal iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. Using the Von Neumann sequence, we establish the convergence result in a Hilbert space framework. We present a new example of relatively nonexpansive mapping and prove that its Agarwal iterative process is more efficient than the Mann and Ishikawa iterative processes. 1. Introduction Let E be a nonempty subset of a Banach space X. A self-map T of E is said to be nonexpansive mapping if ‖Tu − Tv‖ ≤ ‖u − v‖, forall u, v ∈ E. (1) e class of nonexpansive mappings is important as an application point of view. One of the celebrated result of Kirk [1] states that any self nonexpansive mapping of closed bounded convex subset E of a reflexive Banach space has a fixed point provided that E has normal structure. is result was also independently proved in the same year by Browder [2] and Gohde [3] in uniformly convex Banach space (in short UCBS). After this celebrated result, many generalizations of nonexpansive mappings have been published [4–14]. Among the other things, one of the natural generalization of non- expansive mappings was given by Eldred et al. [15] as follows. Let H and L be two nonempty subsets of a Banach space X.A self-map T of H ∪ L is said to be relatively nonexpansive if ‖Tu − Tv‖ ≤ ‖u − v‖, forall u ∈ H and v ∈ L. (2) Iterative methods played a very important role in var- iational inequalities and many other areas of applied sciences (e.g., see [16–27] and others). One of the earlier iterative scheme is the Picard iteration process, u n+1 � Tu n , which converges very well for Banach contraction mappings. However, this scheme is not suitable for finding fixed points of nonexpansive mappings and hence for the generalized nonexpansive mappings. Let E be a nonempty subset set of a Banach space X. In [30], Eldred and Praveen studied Mann [29] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. In [30], Gopi and Pragadeeswara studied Ishikawa [31] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. Motivated by the above work, we study the Agarwal [32] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. We present a new example of relatively nonexansive mapping and prove Hindawi Journal of Mathematics Volume 2020, Article ID 8821553, 11 pages https://doi.org/10.1155/2020/8821553