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Approximation of a zero point of accretive operator in Banach spaces. (English) Zbl 1115.47055
The authors consider the composite approximation scheme $\begin{gathered} y_n=\beta_nx_n+(1-\beta_n)J_{r_n}x_n, \\ x_{n+1}=\alpha_nu+(1-\alpha_n)y_n, \end{gathered}$ where $$J_{r_n}$$ is the resolvent of the $$m$$-accretive operator $$A$$ acting in the Banach space $$E$$, $$C\subset E$$ is closed convex, and $$u\in C$$. By assuming that $$E$$ is uniformly convex, $$A^{-1}(0)\neq\varnothing$$ and the sequences $$(\alpha_n)_n$$, $$(\beta_n)_n$$ and $$(r_n)_n$$ satisfy some technical conditions, the authors show that the sequence $$(x_n)_n$$ converges to a zero point of $$A$$. The same conclusion holds true if $$E$$ is reflexive and has a weakly continuous duality map $$J_{\varphi}$$ with gauge $$\varphi$$.

MSC:
 47J25 Iterative procedures involving nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc.
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References:
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