chapter  1
42 Pages

Common Fixed Points in Convex Metric Spaces

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

Hafiz Fukhar-ud-din

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia; and Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Ishikawa Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Multistep Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 One-Step Implicit Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

The Banach Contraction Principle (BCP) asserts that a contraction on a complete metric space has a unique fixed point and its proof hinges on “Picard iterations.” This principle is applicable to a variety of subjects such as integral equations, partial differential equations and image processing. This principle breaks down for nonexpansive mappings on metric spaces. This led to the introduction of Mann iterations in a Banach space [33]. Our aim is to study Mann-type iterations for some classes of nonlinear mappings in a metric space. We achieve it through the convex structure introduced by Takahashi [45]. In this chapter, iterative construction of common fixed points of asymptotically (quasi-) nonexpansive mappings [11] by using their explicit and implicit schemes on nonlinear domains such as CAT (0) spaces, hyperbolic spaces, and convex metric spaces[1, 7, 22, 24, 45] will be presented. The new results provide a metric space version of the corresponding known results in Banach spaces and CAT (0) spaces (for example, [12, 25, 32, 26, 47]).