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

Abstract EN

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}.