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The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps. (English) Zbl 1053.47055

The convergence of the modified Mann iteration

${u}_{n+1}=\left(1-{\alpha }_{n}\right){u}_{n}+{\alpha }_{n}{T}^{n}{u}_{n},\phantom{\rule{1.em}{0ex}}n=0,1,2,\cdots ,$

is equivalent to the convergence of the modified Ishikawa iteration

$\begin{array}{cc}& {x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{T}^{n}{y}_{n},\hfill \\ & {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{T}^{n}{x}_{n},\phantom{\rule{1.em}{0ex}}n=0,1,2,\cdots ,\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\}\subset \left(0,1\right),{\alpha }_{n}\to 0,\phantom{\rule{4pt}{0ex}}{\beta }_{n}\to 0,\sum {\alpha }_{n}=+\infty$, when the map $T$ is asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive. Here for a Banach space $X$ and $B$ a nonempty subset of $X$ the map $T\phantom{\rule{0.222222em}{0ex}}B\to B$ is called asymptotically nonexpansive in the intermediate sense (a.n.i.s.) if ${T}^{m}$ is continuous for some $m\in ℕ$ and

$\underset{n\to \infty }{lim sup}\underset{y\in B}{sup}\left(\parallel {T}^{n}x-{T}^{n}y\parallel -\parallel x-y\parallel \right)\le 0;$

and is called strongly successively pseudocontractive (s.s.p.) if there exists $k\in \left(0,1\right)$ and ${n}_{0}\in ℕ$ such that

$\parallel x-y\parallel \le \parallel x-y+t\left[\left(I-{T}^{n}-kI\right)x-\left(I-{T}^{n}-kI\right)y\right]\parallel$

for all $x,y\in B,t>0$ and $n\ge {n}_{0}$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces