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Weak and strong convergence of solutions to accretive operator inclusions and applications. (English) Zbl 0981.47036
Let $$E$$ be a real Banach space and $$A\subset E\times E$$ be an $$m$$-accretive operator. Let $$J_r$$ be the resolvent of $$A$$ for $$r> 0$$. The authors consider the following iterative schemes to solve $$0\in Av$$: \begin{aligned} x_{n+1} & = \alpha_nx+ (1-\alpha_n) J_{r_n} x_n\qquad (n= 0,1,2,\dots),\tag{1}\\ x_{n+1} & = \alpha_nx_n+ (1-\alpha_n) J_{r_n} x_n\qquad (n= 0,1,2,\dots),\tag{2}\end{aligned} where $$x_0 = x\in E$$, $$\{\alpha_n\}$$ is a sequence in $$[0,1]$$ and $$\{r_n\}$$ is a sequence in $$(0,\infty)$$. They prove that the sequence (1) is strongly convergent with the limit $$v\in A^{-1}0$$ and then the convergence of (2). They take into account of computational errors. Next, they apply the iterations (1) and (2) to minimization problems and variational inequalities.

MSC:
 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J25 Iterative procedures involving nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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