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Weak convergence of an iterative sequence for accretive operators in Banach spaces. (English) Zbl 1128.47056

The authors study the generalized variational inequality problem of finding $u\in C$ such that

$〈Au,J\left(v-u\right)〉\ge 0\phantom{\rule{4pt}{0ex}}\forall v\in C,$

where $C$ is a nonempty closed convex subset of a smooth Banach space $E$, $A$ is an accretive operator of $C$ into $E$, $J$ is the duality mapping of $E$ into ${E}^{*}$, and $〈·,·〉$ is the duality paring between $E$ and ${E}^{*}$. To solve this problem, the authors propose the following iterative scheme: ${x}_{1}=x\in C$ and

${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right){Q}_{C}\left({x}_{n}-{\lambda }_{n}A{x}_{n}\right)$

for $n=1,2,3,\cdots$, where ${Q}_{C}$ is a sunny nonexpansive retraction from $E$ onto $C$, $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left[0,1\right]$, and $\left\{{\lambda }_{n}\right\}$ is a sequence of real numbers.

For this iterative scheme, the authors establish a weak convergence result (Theorem 3.1) in a uniformly convex and 2-uniformly smooth Banach space for an $\alpha$-inverse strongly accretive operator. Applications to finding a zero point of an inverse strongly accretive operator and to finding a fixed point of a strictly pseudocontractive mapping are given.

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