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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 $\langle Au, J(v-u) \rangle \geq 0\;\forall v \in C,$ 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 $$\langle \cdot, \cdot \rangle$$ 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 $x_{n+1}=\alpha_{n}x_n +(1-\alpha_{n})Q_{C}(x_n -\lambda_{n}Ax_n)$ for $$n=1, 2, 3,\dots$$, where $$Q_C$$ is a sunny nonexpansive retraction from $$E$$ onto $$C$$, $$\{\alpha_n\}$$ is a sequence in $$[0,1]$$, and $$\{\lambda_n\}$$ 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 involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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