On the Convergence of Successive Approximations for Quasi-Nonexpansive Mappings Through Abstract Cones
Williams, B. B.
Bolen, J. C.
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In a recent paper, Petryshyn and Williamson  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  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 . 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  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.