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Strong convergence theorems for the approximation of fixed points of demicontinuous pseudocontractive mappings

Chidume C. E, Shehu Y.

Journal of Applied Analysis

2013

Vol. 19, nr 2

213--229

In this paper, we introduce a new explicit iterative scheme for approximation of fixed points of demicontinuous pseudocontractive mappings in uniformly smooth Banach spaces and prove strong convergence of our proposed iterative scheme. Furthermore, we modify our explicit iterative scheme for approximation of zeroes of bounded demicontinuous accretive mappings in uniformly smooth Banach spaces. Our result improves, extends and unifies most of the results that have been proved for this class of mappings.

On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces

Kozlowski W. M.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

113--136

Let C be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space X. We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings Tt: C → C. Each of Tt: is assumed to be pointwise Lipschitzian, that is, there exists a family of functions αt: C → [0, ∞) such that ||Tt(x) — Tt(y)\\ ≤ αt:(x) || - y|| for x,y € C. The paper demonstrates how the weak compactness of C plays an essential role in proving the weak convergence of these processes to common fixed points.