LetEbe an arbitrary uniformly smooth real Banach space, letDbe a nonempty closed convex subset ofE, and letT:D→Dbe a uniformly generalized Lipschitz generalized asymptoticallyΦ-strongly pseudocontractive mapping withq∈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→0asn→∞andcn=o(an); (iii)Σn=0∞an=∞. For somex0,z0∈D, let{un},{vn},{wn}be any bounded sequences inD, 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}.
LetEbe a real uniformly smooth Banach space, andKa nonempty closed convex subset ofE. Assume thatT1+T2:K→Kis a continuous and strongly pseudocontractive mapping, whereT1:K→Kis Lipschitz andT2:K→Khas the bounded range mapping. Then the Ishikawa iterative sequence converges strongly to the unique fixed point ofT1+T2.
The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption
