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Publication Details
AFRICAN RESEARCH NEXUS
SHINING A SPOTLIGHT ON AFRICAN RESEARCH
mathematics
Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces
Journal of Mathematical Analysis and Applications, Volume 329, No. 1, Year 2007
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Description
Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C with a fixed point, for some 0 ≤ κ < 1. Given an initial guess x0 ∈ C and given also a real sequence {αn} in (0, 1). The Mann's algorithm generates a sequence {xn} by the formula: xn + 1 = αn xn + (1 - αn) T xn, n ≥ 0. It is proved that if the control sequence {αn} is chosen so that κ < αn < 1 and ∑n = 0∞ (αn - κ) (1 - αn) = ∞, then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions. © 2006 Elsevier Inc. All rights reserved.
Authors & Co-Authors
Marino, Giuseppe
Italy, Rende
Università Della Calabria
Xu, Hongkun
South Africa, Durban
University of Kwazulu-natal
Statistics
Citations: 531
Authors: 2
Affiliations: 2
Identifiers
Doi:
10.1016/j.jmaa.2006.06.055
ISSN:
0022247X
e-ISSN:
10960813