The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption
Zhiqun Xue, Yaning Wang, Haiyun Zhou · Abstract and Applied Analysis · 2012
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) . 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}.