ISHIKAWA AND MANN ITERATIVE PROCESSES WITH ERRORS FOR NONLINEAR $\Phi$-STRONGLY QUASI-ACCRETIVE MAPPINGS IN NORMED LINEAR SPACES
Hua‐Cheng Zhou, Y.J. Cho · Journal of the Korean Mathematical Society · 1999
Let X be a real normed linear space. Let T : D(T) ⊂ X \longrightarrow X be a uniformly continuous and ∮-strongly quasi-accretive mapping. Let {n}{{{{ { }`_{n=0 } ^{ } }}}} , {n}{{{{ { }`_{n=0 } ^{ } }}}} be two real sequences in [0, 1] satisfying the following conditions: (ⅰ) n \longrightarrow0, n \longrightarrow0, as n \longrightarrow (ⅱ) {{{{ SUM from { { n}=0} to inf }}}} =. Set Sx=x-Tx for all x D(T). Assume that {u}{{{{ { }`_{n=0 } ^{ } }}}} and {v}{{{{ { }`_{n=0 } ^{ } }}}} are two sequences in D(T) satisfying {{{{ SUM from { { n}=0} to inf }}}}∥un∥ and vn\longrightarrow0 as n\longrightarrow. Suppose that, for any given x0X, the Ishikawa type iteration sequence {xn}{{{{ { }`_{n=0 } ^{ } }}}} with errors defined by (IS)1 xn+1=(1-n)xn+nSyn+un, yn=(1-n)x+nSxn+vn for all n=0, 1, 2 … is well-defined. we prove that {xn}{{{{ { }`_{n=0 } ^{ } }}}} converges strongly to the unique zero of T if and only if {Syn}{{{{ { }`_{n=0 } ^{ } }}}} is bounded. Several related results deal with iterative approximations of fixed points of ∮-hemicontractions by the ishikawa iteration with errors in a normed linear space. Certain conditions on the iterative parameters {n}{{{{ { }`_{n=0 } ^{ } }}}} , {n}{{{{ { }`_{n=0 } ^{ } }}}} and t are also given which guarantee the strong convergence of the iteration processes.