Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions |
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Authors: | Tae-Hwa Kim Hong-Kun Xu |
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Affiliation: | 1. Division of Mathematical Sciences, Pukyong National University, Busan 608-737, Republic of Korea;2. School of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa |
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Abstract: | Let C be a closed convex subset of a real Hilbert space H and assume that T is an asymptotically 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 modified Mann’s algorithm generates a sequence {xn} via the formula: xn+1=αnxn+(1−αn)Tnxn, n≥0. It is proved that if the control sequence {αn} is chosen so that κ+δ<αn<1−δ for some δ∈(0,1), then {xn} converges weakly to a fixed point of T. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence. |
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Keywords: | primary, 47H09 secondary, 65J15 |
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