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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.

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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