On the Convergence of Successive Approximations for Quasi-Nonexpansive Mappings Through Abstract Cones

Abstract

**Please note that the full text is embargoed** ABSTRACT: In a recent paper, Petryshyn and Williamson [4] investigated the convergence of successive approximations of quasi-

nonexpansive mappings in a Banach space. This paper contains an outline, in chronological order, of the main results

concerning the convergence of iteration method and consequently includes a number of references. Perov and Kibenko [3]

employed generalized Banach spaces to extend contraction mapping principal and to show the flexibility of such an approach

in applications. See also Bernfeld and Lakshmikantham [1]. More recently, Eisenfeld and Lakshmikantham [5,6] proved some

fixed point theorems in abstract cones which extend and generalize many known results. In this paper, we extend some main

results of [4] to cone-valued metric spaces. In Sections 2 and 3, we give needed definitions and properties of k-metric

spaces and in Section 4, we prove our main results. For convenience and for future use, we have given more details than

necessary.