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

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

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Title: On the Convergence of Successive Approximations for Quasi-Nonexpansive Mappings Through Abstract Cones
Author: Williams, B. B.; Bolen, J. C.
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.
URI: http://hdl.handle.net/10106/2182
Date: 1975-07

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