Approximating solutions of maximal monotone operators in Hilbert spaces (Q1841213)

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 \(f\).