×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI