Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ CAT ( 0 ) spaces
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Arabian Journal of Mathematics
Khairul Saleh · Hafiz Fukhar-ud-din
Common fixed point iterations for a class of multi-valued mappings in CAT(0) spaces
Received: 21 July 2019 / Accepted: 20 July 2020 © The Author(s) 2020
Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a C AT (0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of C AT (0) spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces. Mathematics Subject Classification
47H09 · 47H10
1 Introduction The theory of fixed point for multi-valued mappings has become of great interest to many researchers, see, for instance, [3,4,8,11,13,15] and references therein. This concept contributes significantly in convex approximation, fractals, optimal control, digital imaging, and economics. The following statements are quoted from [26]: “Sastry and Babu [21] showed that under certain conditions, Mann and Ishikawa iterative algorithms for a multi-valued nonexpansive mapping with a fixed point x converge to a fixed point y of the mapping. Panyanak [18] extended their results to a uniformly convex Banach space. Song and Wang [23] improved the results of Panyanak [18]. Abbas et al. [1] proposed a one-step iterative scheme to find a common fixed point of two multi-valued nonexpansive mappings in a uniformly convex Banach space. Recently, Uddin et al. [26] discovered a few gaps in the scheme proposed in [1] and came up with a new one-step iterative scheme for fixed points of two multi-valued nonexpansive mappings in C AT (0) space and removed the gaps found in [1]”. Motivated by [1,26], we recommend a new scheme to approximate common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in C AT (0) spaces.
2 Preliminary lemmas We refer to [2] for the following definitions. “Suppose (M, d) is a metric space and a, b ∈ M.” A geodesic path from a to b is a mapping s : [0, d(a, b)] → X , such that s(0) = a, s(d(a, b)) = b and d(s(r ), s(r )) = |r −r | for all r, r ∈ [0, d(a, b)]. A geodesic segment is defined as the image of the geodesic path. If every pair of points in M is joined by a unique geodesic segment, then the space is known as unique geodesic metric K. Saleh (B) Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia E-mail: [email protected] H. Fukhar-ud-din Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
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space. A geodesic triangle is denoted by (a, b, c) in a geodesic metric space (M, d) that consists of three points a, b, c ∈ M (the vertices of the geodesic triangle ) and three geodesic segments among these points. A comparison triangle for this geodesic triangle (a, b, c) in M is a triangle (a, b, c) in R2 , such that dR2 (i, j) = d(i, j) for i, j ∈ {a, b, c}.”
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