The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Zhiqun Xue, Yaning Wang and Haiyun Zhou

Abstract and Applied Analysis, 2012, vol. 2012, issue 1

Abstract: Let E be an arbitrary uniformly smooth real Banach space, let D be a nonempty closed convex subset of E, and let T : D → D be a uniformly generalized Lipschitz generalized asymptotically Φ‐strongly pseudocontractive mapping with q ∈ F(T) ≠ ∅. Let {an}, {bn}, {cn}, {dn} be four real sequences in [0,1] and satisfy the conditions: (i) an + cn ≤ 1, bn + dn ≤ 1; (ii) an, bn, dn → 0 as n → ∞ and cn = o(an); (iii) Σn=0∞an=∞. For some x0, z0 ∈ D, let {un}, {vn}, {wn} be any bounded sequences in D, and let {xn}, {zn} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {xn} is equivalent to that of {zn}.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:909187

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