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

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 ε F (T) ≠ Ø. Let {a n}, {bn}, {cn}, {dn} be four real sequences in [ 0,1 ] and satisfy the conditions: (i) an + c n ≤ 1, bn + dn ≤ 1; (ii) an, bn, dn → 0 as n → ∞ and cn = o (an); (iii) Σn=0 ∞ a n = ∞. For some x0, z0 ε D, let {un}, {v n}, {wn} be any bounded sequences in D, and let {x n}, {zn} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {xn} is equivalent to that of {zn}. © 2012 Zhiqun Xue et al.