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Approximating solutions of maximal monotone operators in Hilbert spaces. (English) Zbl 0992.47022
This article deals with iterates $$x_{n+1}= \alpha_n x+(1- \alpha_n)J_{r_n} x_n\qquad (n= 1,2,\dots)\tag 1$$ and $$x_{n-1}= \alpha_n x_n+ (1- \alpha_n) J_{r_n} x_n\qquad (n= 1,2,\dots),\tag 2$$ where $J_r= (I+ rT)^{-1}$, $\{\alpha_n\}$ is a sequence from $[0,1]$, $\{r_n\}$ a sequence from $(0,\infty)$, $T: H\to 2^H$ a maximal monotone operator in a real Hilbert space. The basic results are (a) a theorem about strong convergence of iterates (1) to $Px$, where $P$ is the metric projection onto $T^{-1}0$; (b) a theorem about weak convergence of iterates (2) to $v\in T^{-1}0= \lim_{n\to\infty} Px_n$, where $P$ is the metric projection onto $T^{-1}0$. In the end of the article the special case when $T=\partial f$ is considered, where $f$ is a proper lower-semicontinuous convex function. The corresponding results is interpreted as theorems of finding a minimizer of $f$.

47H05Monotone operators (with respect to duality) and generalizations
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI
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