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

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  • Zhiqun Xue
  • Yaning Wang
  • Haiyun Zhou

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}.

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Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:909187
DOI: 10.1155/2012/909187
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